Some fixed point results linked to α –β rational contractions and modified multivalued Hardy-Rogers operators

Abstract

In this paper two new fixed point results are studied. The first result is a theorem that involves (α −β) type rational singlevalued contractions, in the sense of Geraghty type operators. The second result consists of multivalued modified Hardy Rogers operators, namely the existence of the fixed point, data dependence, local version involving two metrics and homotopy theorems involving two metrics are studied.

Authors

Cristian Daniel Alecsa
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

Keywords

fixed point theorems; complete metric space; rational contractions; multivalued; homotopy; Geraghty;
Hardy-Rogers.

Paper coordinates

C.-D. Alecsa, Some fixed point results linked to α –β rational contractions and modified multivalued Hardy-Rogers operators, J. Fixed Point Theory, 2018, 2018:3.

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J. Fixed Point Theory

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2052-5338

2052-5338

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[3] M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40 (1973), 604-608.

[4] P.S. Kumari, D. Panthi, Connecting various type of cyclic contractions and contractive self-mappings with Hardy-Rogers self-mappings, Fixed Point Theory Appl. 2016 (2016), Article ID 15.22 C.D. ALECSA

[5] T. Lazar, D. O’Regan, A. Petrusel, Fixed points and homotopy results for Ciric-type multivalued operators  on a set with two metrics, Bull. Korean Math. Soc. 45 (2008), no.1, 67-73.

[6] A. Oprea, Fixed point theorems for multivalued generalized contractions of rational type in complete metric spaces, Creat. Math. Inform. 23 (2014), 99-106.

[7] N.S. Papageorgiou, S. Hu, Handbook of Multivalued Analysis (vol. I and II), Kluwer Acad. Publ., Dordrecht (1997 and 1999).

[8] L. Paunovic, P. Kaushik, S. Kumar, Some applications with new admissibility contractions in b-metric spaces,  J. Nonlinear Sci. Appl. 10 (2017), 4162-4174.

[9] A. Petrusel, Multivalued weakly Picard operators and applications, Sci. Math. Jpn. 59, 169-202.

[10] I.A. Rus, Picard operators and applications, Sci. Math. Jpn. 58 (2003), 191-219.

[11] I.A. Rus, A. Petrusel, A. Sîntmarian, Data dependence of the fixed point set of some multivalued weakly Picard operators, Nonlinear Anal. 52 (2003), no.8, 1947-1959.

[12] R.J. Shahkoohi, A. Razani, Some fixed point theorems for rational Geraghty contractive mappings in ordered b-metric spaces, Fixed Point Theory Appl. 2014 (2014), Article ID 373.

[13] W. Sintunavarat, Generalized Ulam-Hyers stability, well-posedness, and limit shadowing of fixed point problems for α −β−contraction mappings in metric spaces, Sci. World J. 2014(2014), article ID 569174, 7 pages.

[14] F. Zabihi, A. Razani, Fixed point theorems for hybrid rational Geraghty contractive mappings in ordered b-metric spaces, J. Appl. Math., 2014(2014), Article ID 929821

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SOME FIXED POINT RESULTS LINKED TO 𝜶𝜷\boldsymbol{\alpha}-\boldsymbol{\beta} RATIONAL CONTRACTIONS AND MODIFIED MULTIVALUED HARDY-ROGERS OPERATORS

CRISTIAN DANIEL ALECSA 1,2,∗
1 Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania
2 Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
Abstract

In this paper two new fixed point results are studied. The first result is a theorem that involves ( αβ\alpha-\beta ) type rational singlevalued contractions, in the sense of Geraghty type operators. The second result consists of multivalued modified Hardy Rogers operators, namely the existence of the fixed point, data dependence, local version involving two metrics and homotopy theorems involving two metrics are studied.

Copyright (C) 2018 C.D. Alecsa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Keywords: fixed point theorems; complete metric space; rational contractions; multivalued; homotopy; Geraghty; Hardy-Rogers.

2010 AMS Subject Classification: 47H10,54H2547\mathrm{H}10,54\mathrm{H}25.

1. Preliminaries for rational Geraghty type mappings

The first idea of the present article is that in the third section we want to prove a theorem based on rational αβ\alpha-\beta-contractions, so we remind the necessary concepts for this type of operators. For more informations, we let the reader follow [13]. We recall the following crucial concepts.

00footnotetext: *Cristian Daniel Alecsa
E-mail address: cristian.alecsa@math.ubbcluj.ro; cristian.alecsa@ictp.acad.ro
Received November 10, 2017

Definition 1.1. Let XX be a nonempty set and α:X×X[0,)\alpha:X\times X\rightarrow[0,\infty) be a mapping. Then f:XXf:X\rightarrow X is called α\alpha-admissible, if it satisfies the following condition for each x,yXx,y\in X, with α(x,y)1α(fx,fy)1\alpha(x,y)\geq 1\Longrightarrow\alpha(fx,fy)\geq 1.

Definition 1.2. The mapping α:X×X[0,)\alpha:X\times X\rightarrow[0,\infty) is called transitive, if for each x,y,zXx,y,z\in X, with α(x,y)1\alpha(x,y)\geq 1 and α(y,z)1\alpha(y,z)\geq 1, we have α(x,z)1\alpha(x,z)\geq 1.

Moreover, let’s denote by YY the set of all functions β:[0,)[0,1)\beta:[0,\infty)\rightarrow[0,1), satisfying limnβ(tn)=1limntn=0\lim_{n\rightarrow\infty}\beta\left(t_{n}\right)=1\Longrightarrow\lim_{n\rightarrow\infty}t_{n}=0.
Also, we recall the definition of αβ\alpha-\beta- contractions, given by Sintunavarat in [13].
Definition 1.3. Let ( X,dX,d ) be a metric space. A mapping f:XXf:X\rightarrow X is called an αβ-\alpha-\beta-contraction, if there exists α:X×X[0,)\alpha:X\times X\rightarrow[0,\infty) and βY\beta\in Y, such that [α(x,y)1+δ]d(fx,fy)δβ(d(x,y))d(x,y)\left[\alpha(x,y)-1+\delta_{*}\right]^{d(fx,fy)}\leq\delta^{\beta(d(x,y))d(x,y)}, for each x,yXx,y\in X, with 1<δδ1<\delta\leq\delta_{*}.

In [13], the author proved a series of theorems such as : existence of a fixed point assuming that the mapping ff is continuous, a theorem in which the continuity is dropped and a theorem for the uniqueness of fixed points. Also, this was based on the result of Geraghty [3] from 1973. Moreover, [8] Paunović et. al. extended the result of W. Sintunavarat in the framework of b-metric spaces. Additionally, they have studied fixed points for a given mapping F:XXF:X\rightarrow X, such that [α(x,y)1+δ]d(Fx,Fy)δλM(x,y)[\alpha(x,y)-1+\delta]^{d(Fx,Fy)}\leq\delta^{\lambda M(x,y)}, for each x,yXx,y\in X, with 1<δ1<\delta, where λ[0,1s]\lambda\in\left[0,\frac{1}{s}\right] and M(x,y)={d(x,y),d(x,Fx),d(y,Fy),d(x,Fy)+d(y,Fx)2s}M(x,y)=\left\{d(x,y),d(x,Fx),d(y,Fy),\frac{d(x,Fy)+d(y,Fx)}{2s}\right\}, where ss was the coefficient of the b-metric space ( X,dX,d ).

Furthermore, Zabihi and Razani [14] considered rational type operators and developed some fixed point results in the framework of complete b -metric spaces. In the context of a metric space ( X,dX,d ), a self mapping ff on XX was considered, satisfying
d(fx,fy)β(d(x,y))M(x,y)+LN(x,y)d(fx,fy)\leq\beta(d(x,y))M(x,y)+L\cdot N(x,y), where L0L\geq 0,
M(x,y)=M(x,y)= max {d(x,y),d(x,fx)d(y,fy)1+d(fx,fy)}\left\{d(x,y),\frac{d(x,fx)d(y,fy)}{1+d(fx,fy)}\right\} and N(x,y)=min{d(x,fx),d(x,fy),d(y,fx),d(y,fy)}N(x,y)=\min\{d(x,fx),d(x,fy),d(y,fx),d(y,fy)\}.

Moreover, since we want to define some other type of αβ\alpha-\beta- contractions, we shall recall that the authors in [12] developed new fixed point theorems involving a new type of rational contractive Geraghty mapping in b-metric spaces. This rational Geraghty mapping is introduced as follows, in the case of metric spaces.

Definition 1.4. Let ( X,dX,d ) be a metric space. A mapping f:XXf:X\rightarrow X is called a rational Geraghty of type II, if d(fx,fy)β(M(x,y))M(x,y)d(fx,fy)\leq\beta(M(x,y))M(x,y), for each x,yXx,y\in X, where βY\beta\in Y and M(x,y)=max{d(x,y),d(x,fx)d(y,fy)1+d(x,y),d(x,fx)d(y,fy)1+d(fx,fy)}M(x,y)=\max\left\{d(x,y),\frac{d(x,fx)d(y,fy)}{1+d(x,y)},\frac{d(x,fx)d(y,fy)}{1+d(fx,fy)}\right\}.

That means that in the third section, we will present a generalized theorem for rational Geraghty mappings of type II.

2. Preliminaries for modified multivalued Hardy-Rogers

In this section, we recall some general notions in the framework of multivalued analysis theory. Also, for the following preliminary notions and lemmas (such as: multivalued weakly Picard operators, data dependence of the fixed point set, Haussdorf metric properties) we refer the reader to [9], [10] and [11].

Let ( X,dX,d ) be a metric space and P(X)P(X) be the family of all nonempty subsets of XX.
We denote by Pcl(X)P_{cl}(X) the family of all nonempty subsets of XX which are closed, by Pb(X)P_{b}(X) the family of all nonempty subsets of XX which are bounded and by Pcp(X)P_{cp}(X) the family of all nonempty subsets of XX which are compact.

Furthermore, we consider the following functionals
D:P(X)×P(X)+,D(A,B)=inf{d(a,b)/aA,bB}D:P(X)\times P(X)\rightarrow\mathbb{R}_{+},D(A,B)=\inf\{d(a,b)/a\in A,b\in B\}
H:Pb(X)×Pb(X)+,H(A,B)=max{supaAD(a,B),supbBD(b,A)}H:P_{b}(X)\times P_{b}(X)\rightarrow\mathbb{R}_{+},H(A,B)=\max\left\{\sup_{a\in A}D(a,B),\sup_{b\in B}D(b,A)\right\}
ρ:Pb(X)×Pb(X)+,ρ(A,B)=sup{D(a,B)/aA}\rho:P_{b}(X)\times P_{b}(X)\rightarrow\mathbb{R}_{+},\rho(A,B)=\sup\{D(a,B)/a\in A\}
We recall some useful results concerning the Haussdorf-Pompeiu generalized functional HH.
Lemma 2.1. Let q>1q>1 and A,BP(X)A,B\in P(X).
Then, for each aAa\in A, there exists bBb\in B such that d(a,b)qH(A,B)d(a,b)\leq qH(A,B).
Lemma 2.2. Let (X,d)(X,d) be a metric space and A,BP(X)A,B\in P(X).
Suppose that there exists η>0\eta>0 such that :
(i) for each aAa\in A, there exists bBb\in B such that d(a,b)ηd(a,b)\leq\eta,
(ii) for each bBb\in B, there exists aAa\in A such that d(a,b)ηd(a,b)\leq\eta.

Then H(A,B)ηH(A,B)\leq\eta.
Moreover, if YY is a nonempty subset of XX and T:YP(X)T:Y\rightarrow P(X) a multivalued operator, then an element xYx\in Y is
(a) a fixed point of TT if and only if xTxx\in Tx;
(b) a strict fixed point of TT if and only if {x}=Tx\{x\}=Tx;

Furthermore, we denote by FTF_{T} the set of all fixed points of TT and by (SF)T(SF)_{T} the set of all strict fixed points of TT.

We also remind the definition of the graphic of a multivalued operator, i.e.
GT:={(x,y)Y×X/yTx}G_{T}:=\{(x,y)\in Y\times X/y\in Tx\}.
Definition 2.3. Let ( X,dX,d ) be a metric space and T:XP(X)T:X\rightarrow P(X) a multivalued operator. We say that TT is a multivalued weakly Picard operator (briefly MWP) if for each xXx\in X and for each yTxy\in Tx, there exists a sequence (xn)X\left(x_{n}\right)\in X, satisfying the following
(i) x0=x,x1=yx_{0}=x,x_{1}=y;
(ii) xn+1Txnx_{n+1}\in Tx_{n}, for each nn\in\mathbb{N};
(iii) the sequence (xn)\left(x_{n}\right) is convergent to a fixed point of TT.

Definition 2.4. Let ( X,dX,d ) be a metric space and T:XP(X)T:X\rightarrow P(X) an MWP.
Then TT is called a cc-weakly Picard operator, with c[0,)c\in[0,\infty), if there exists a selection tt^{\infty} of TT^{\infty}, such that
d(x,t(x,y))cd(x,y)d\left(x,t^{\infty}(x,y)\right)\leq cd(x,y), for each (x,y)GT(x,y)\in G_{T}.
Now we focus our attention to the case of Hardy-Rogers type mappings. In [7], the basic notion of singlevalued Hardy-Rogers contraction appeared.

Definition 2.5. Let (X,d)(X,d) be a metric space and T:XXT:X\rightarrow X be an operator such that there exists α,β0\alpha,\beta\geq 0 with α+β<1\alpha+\beta<1, satisfying
d(Tx,Ty)αd(y,Ty)[1+d(x,Tx)]1+d(x,y)+βd(x,y)d(Tx,Ty)\leq\frac{\alpha d(y,Ty)[1+d(x,Tx)]}{1+d(x,y)}+\beta d(x,y), for each x,yXx,y\in X.
In [6], Oprea A. developed a theorem concerning multivalued rational contractions (of Hardy Rogers type).

A multivalued operator T:XP(X)T:X\rightarrow P(X) is called a multivalued rational type contraction, if satisfies the following condition
H(Tx,Ty)αD(y,Ty)[1+D(x,Tx)]1+d(x,y)+βd(x,y)H(Tx,Ty)\leq\frac{\alpha D(y,Ty)[1+D(x,Tx)]}{1+d(x,y)}+\beta d(x,y), for each x,yXx,y\in X.
Oprea has showed that the multivalued rational contractions are MWP-operators and developed theorems for data dependence, fractal theory, Ulam-Hyers stability etc.

In [4], Kumari and Panthi introduced a new type of rational contractions, called modified HardyRogers contractions.

They introduced this as types of cyclic contractions for the case of families of dislocated metric spaces.

We recall the notion of singlevalued contractions in the context of metric spaces, i.e. singlevalued operator that satisfies

d(Tx,Ty)αd(x,y)+βd(x,Ty)+γd(y,Tx)+δd(y,Ty)+ηd(y,Ty)[1+d(x,Tx)]1+d(x,y)+\displaystyle d(Tx,Ty)\leq\alpha d(x,y)+\beta d(x,Ty)+\gamma d(y,Tx)+\delta d(y,Ty)+\eta\frac{d(y,Ty)[1+d(x,Tx)]}{1+d(x,y)}+
λd(y,Ty)+d(y,Tx)1+d(y,Ty)d(y,Tx)+μd(x,Tx)[1+d(y,Tx)]1+d(x,y)+d(y,Ty)\displaystyle\lambda\frac{d(y,Ty)+d(y,Tx)}{1+d(y,Ty)d(y,Tx)}+\mu\frac{d(x,Tx)[1+d(y,Tx)]}{1+d(x,y)+d(y,Ty)}

Also, regarding Hardy Rogers mappings, our purpose to define the concept of modified HardyRogers contractions under the multivalued case shall be presented in the last section, along with some fixed point results.

3. Some theorems regarding rational Geraghty 𝜶𝜷 contractions \boldsymbol{\alpha}\boldsymbol{-}\boldsymbol{\beta}\boldsymbol{-}\boldsymbol{\text{ contractions }}

In this section we present a generalized theorem for rational Geraghty mappings or type II, using the α\alpha-admissibility conditions by Sintunavarat.

Moreover, we will use the same terminology from the first section.
Definition 3.1. Let ( X,dX,d ) be a metric space.
A mapping f:XXf:X\rightarrow X is called an αβ-\alpha-\beta-rational Geraghty mapping of type II if and only if there exists α:X×X[0,)\alpha:X\times X\rightarrow[0,\infty) and βY\beta\in Y, such that [α(x,y)1+δ]d(fx,fy)δβ(M(x,y))M(x,y)\left[\alpha(x,y)-1+\delta_{*}\right]^{d(fx,fy)}\leq\delta^{\beta(M(x,y))M(x,y)}, for each x,yXx,y\in X, with 1<δδ1<\delta\leq\delta_{*}, where M(x,y)=max{d(x,y),d(x,fx)d(y,fy)1+d(x,y),d(x,fx)d(y,fy)1+d(fx,fy)}M(x,y)=\max\left\{d(x,y),\frac{d(x,fx)d(y,fy)}{1+d(x,y)},\frac{d(x,fx)d(y,fy)}{1+d(fx,fy)}\right\}.

Our first main result of this section is the existence theorem for αβ\alpha-\beta-rational Geraghty mappings of type II, using the assumption that ff is continuous. The techniques used in the theorem’s proof follow the same lines as in the theorems from [13].

Theorem 3.2. Let ( X,dX,d ) be a complete metric space and f:XXf:X\rightarrow X an αβ\alpha-\beta rational Geraghty mapping of type I. Also, suppose that the following assumptions hold
(i) ff is α\alpha-admissible,
(ii) α\alpha is transitive,
(iii) there exists x0Xx_{0}\in X, such that α(x0,fx0)1\alpha\left(x_{0},fx_{0}\right)\geq 1,
(iv) ff is continuous.

Then, there exists xXx^{*}\in X, such that x=fxx^{*}=fx^{*}.
Proof. - Let x0Xx_{0}\in X satisfying α(x0,fx0)1\alpha\left(x_{0},fx_{0}\right)\geq 1.
Let’s consider the Picard sequence xn+1=fxnx_{n+1}=fx_{n}, for each nn\in\mathbb{N}.
If there exists nn\in\mathbb{N} such that xn=xn1x_{n}=x_{n-1}, then xn1x_{n-1} is a fixed point and the conclusion holds.
Suppose that for each n,xnxn1n\in\mathbb{N},x_{n}\neq x_{n-1}. So d(xn1,xn)>0d\left(x_{n-1},x_{n}\right)>0, for each nn\in\mathbb{N}.
From condition (i), we know that ff is α\alpha-admissible. Since α(x0,x1)=α(x0,fx0)1\alpha\left(x_{0},x_{1}\right)=\alpha\left(x_{0},fx_{0}\right)\geq 1, then we have that α(x1,x2)=α(x1,fx1)=α(fx0,fx1)1\alpha\left(x_{1},x_{2}\right)=\alpha\left(x_{1},fx_{1}\right)=\alpha\left(fx_{0},fx_{1}\right)\geq 1.

Inductively, one can show that α(xn1,xn)1\alpha\left(x_{n-1},x_{n}\right)\geq 1, for each nn\in\mathbb{N}.
Now, we estimate

δd(xn,xn+1)=δd(fxn1,fxn)δd(fxn1,fxn)\displaystyle\delta^{d\left(x_{n},x_{n+1}\right)}=\delta^{d\left(fx_{n-1},fx_{n}\right)}\leq\delta_{*}^{d\left(fx_{n-1},fx_{n}\right)}\leq
[α(xn1,xn)1+δ]d(fxn1,fxn)δβ(M(xn1,xn))M(xn1,xn),\displaystyle{\left[\alpha\left(x_{n-1},x_{n}\right)-1+\delta_{*}\right]^{d\left(fx_{n-1},fx_{n}\right)}\leq\delta^{\beta\left(M\left(x_{n-1},x_{n}\right)\right)M\left(x_{n-1},x_{n}\right)},}

so d(xn,xn+1)β(M(xn1,xn))M(xn1,xn)d\left(x_{n},x_{n+1}\right)\leq\beta\left(M\left(x_{n-1},x_{n}\right)\right)\cdot M\left(x_{n-1},x_{n}\right).
Moreover, we make the following computations :
M(xn1,xn)=max{d(xn1,xn),d(xn1,fxn1)d(xn,fxn)1+d(xn1,xn),d(xn1,fxn1)d(xn,fxn)1+d(fxn1,fxn)}M\left(x_{n-1},x_{n}\right)=\max\left\{d\left(x_{n-1},x_{n}\right),\frac{d\left(x_{n-1},fx_{n-1}\right)d\left(x_{n},fx_{n}\right)}{1+d\left(x_{n-1},x_{n}\right)},\frac{d\left(x_{n-1},fx_{n-1}\right)d\left(x_{n},fx_{n}\right)}{1+d\left(fx_{n-1},fx_{n}\right)}\right\}
=max{d(xn1,xn),d(xn1,xn)d(xn,xn+1)1+d(xn1,xn),d(xn1,xn)d(xn,xn+1)1+d(xn,xn+1)}=\max\left\{d\left(x_{n-1},x_{n}\right),\frac{d\left(x_{n-1},x_{n}\right)d\left(x_{n},x_{n+1}\right)}{1+d\left(x_{n-1},x_{n}\right)},\frac{d\left(x_{n-1},x_{n}\right)d\left(x_{n},x_{n+1}\right)}{1+d\left(x_{n},x_{n+1}\right)}\right\}
Since d(xn1,xn)d(xn,xn+1)1+d(xn1,xn)=d(xn,xn+1)d(xn1,xn)1+d(xn1,xn)d(xn,xn+1)1=d(xn,xn+1)\frac{d\left(x_{n-1},x_{n}\right)d\left(x_{n},x_{n+1}\right)}{1+d\left(x_{n-1},x_{n}\right)}=d\left(x_{n},x_{n+1}\right)\cdot\frac{d\left(x_{n-1},x_{n}\right)}{1+d\left(x_{n-1},x_{n}\right)}\leq d\left(x_{n},x_{n+1}\right)\cdot 1=d\left(x_{n},x_{n+1}\right) and
d(xn1,xn)d(xn,xn+1)1+d(xn,xn+1)=d(xn1,xn)d(xn,xn+1)1+d(xn,xn+1)d(xn1,xn)1=d(xn1,xn)\frac{d\left(x_{n-1},x_{n}\right)d\left(x_{n},x_{n+1}\right)}{1+d\left(x_{n},x_{n+1}\right)}=d\left(x_{n-1},x_{n}\right)\cdot\frac{d\left(x_{n},x_{n+1}\right)}{1+d\left(x_{n},x_{n+1}\right)}\leq d\left(x_{n-1},x_{n}\right)\cdot 1=d\left(x_{n-1},x_{n}\right),
we get that M(xn1,xn)max{d(xn1,xn),d(xn,xn+1)}M\left(x_{n-1},x_{n}\right)\leq\max\left\{d\left(x_{n-1},x_{n}\right),d\left(x_{n},x_{n+1}\right)\right\}, for each n,n1n\in\mathbb{N},n\geq 1.

Now, we consider two cases.
(I) If max{d(xn1,xn),d(xn,xn+1)}=d(xn,xn+1)\max\left\{d\left(x_{n-1},x_{n}\right),d\left(x_{n},x_{n+1}\right)\right\}=d\left(x_{n},x_{n+1}\right), then we get
d(xn,xn+1)β(M(xn1,xn))d(xn,xn+1)d\left(x_{n},x_{n+1}\right)\leq\beta\left(M\left(x_{n-1},x_{n}\right)\right)\cdot d\left(x_{n},x_{n+1}\right).
Since β(M(xn1,xn))<1\beta\left(M\left(x_{n-1},x_{n}\right)\right)<1, because βY\beta\in Y, we get the contradiction d(xn,xn+1)<d(xn,xn+1)d\left(x_{n},x_{n+1}\right)<d\left(x_{n},x_{n+1}\right).
(II) Then, only the second case is valid, i.e. max{d(xn1,xn),d(xn,xn+1)}=d(xn1,xn)\max\left\{d\left(x_{n-1},x_{n}\right),d\left(x_{n},x_{n+1}\right)\right\}=d\left(x_{n-1},x_{n}\right), that is d(xn,xn+1)β(M(xn1,xn))d(xn1,xn)<d(xn1,xn)d\left(x_{n},x_{n+1}\right)\leq\beta\left(M\left(x_{n-1},x_{n}\right)\right)d\left(x_{n-1},x_{n}\right)<d\left(x_{n-1},x_{n}\right), for each nn\in\mathbb{N}.

So, the sequence (d(xn,xn+1))\left(d\left(x_{n},x_{n+1}\right)\right) is strictly decreasing and nonnegative. It implies that there exists r0r\geq 0, such that d(xn,xn+1)rd\left(x_{n},x_{n+1}\right)\rightarrow r as nn\rightarrow\infty.

Now, we show that r=0r=0.
Let’s suppose that r>0r>0.
We know that d(xn+1,xn+2)β(M(xn,xn+1))d(xn,xn+1)d\left(x_{n+1},x_{n+2}\right)\leq\beta\left(M\left(x_{n},x_{n+1}\right)\right)d\left(x_{n},x_{n+1}\right). Taking the limit as nn\rightarrow\infty, we get that rlimnβ(M(xn,xn+1))rr\leq\lim_{n\rightarrow\infty}\beta\left(M\left(x_{n},x_{n+1}\right)\right)\cdot r.
Because r>0r>0,we get that 1limnβ(M(xn,xn+1))1\leq\lim_{n\rightarrow\infty}\beta\left(M\left(x_{n},x_{n+1}\right)\right).
But β(M(xn,xn+1))<1\beta\left(M\left(x_{n},x_{n+1}\right)\right)<1, so limnβ(M(xn,xn+1))1\lim_{n\rightarrow\infty}\beta\left(M\left(x_{n},x_{n+1}\right)\right)\leq 1. From all this, we find that limnβ(M(xn,xn+1))=1\lim_{n\rightarrow\infty}\beta\left(M\left(x_{n},x_{n+1}\right)\right)=1. This implies that limnM(xn,xn+1)=0\lim_{n\rightarrow\infty}M\left(x_{n},x_{n+1}\right)=0.
Now, because M(xn,xn+1)M\left(x_{n},x_{n+1}\right) is the maximum between three elements, if it’s limit is 0 , so all the elements have the limit 0 . This means that d(xn,xn+1)0d\left(x_{n},x_{n+1}\right)\rightarrow 0. This is a contradiction!

  • We now show that (xn)\left(x_{n}\right) is a Cauchy sequence.

By reductio ad absurdum, let’s suppose that ( xnx_{n} ) is not Cauchy. Then there exists ε>0\varepsilon>0 and there exists nkn_{k} and mkm_{k}, such that nk>mkkn_{k}>m_{k}\geq k, with d(xmk,xnk)d\left(x_{m_{k}},x_{n_{k}}\right) and nkn_{k} being the smallest index satisfying the following

d(xmk,xnk)ε and d(xmk,xnk1)<ε.d\left(x_{m_{k}},x_{n_{k}}\right)\geq\varepsilon\text{ and }d\left(x_{m_{k}},x_{n_{k}-1}\right)<\varepsilon.

By triangular inequality, we have that εd(xmk,xnk)d(xmk,xnk1)+d(xnk1,xnk)<ε+d(xnk1,xnk)\varepsilon\leq d\left(x_{m_{k}},x_{n_{k}}\right)\leq d\left(x_{m_{k}},x_{n_{k}-1}\right)+d\left(x_{n_{k}-1},x_{n_{k}}\right)<\varepsilon+d\left(x_{n_{k}-1},x_{n_{k}}\right).

Since d(xnk1,xnk)0d\left(x_{n_{k}-1},x_{n_{k}}\right)\rightarrow 0, taking kk\rightarrow\infty, it follows that limkd(xmk,xnk)=ε>0\lim_{k\rightarrow\infty}d\left(x_{m_{k}},x_{n_{k}}\right)=\varepsilon>0.
Like in [13], since α\alpha is transitive, we observe that α(xmk,xnk)1,k\alpha\left(x_{m_{k}},x_{n_{k}}\right)\geq 1,k\in\mathbb{N}.

Now, we make the follow estimation

δd(xmk,xnk)\displaystyle\delta^{d\left(x_{m_{k}},x_{n_{k}}\right)} δd(xmk,xmk+1)+d(xmk+1,xnk+1)+d(xnk+1,xnk)\displaystyle\leq\delta^{d\left(x_{m_{k}},x_{m_{k}+1}\right)+d\left(x_{m_{k}+1},x_{n_{k}+1}\right)+d\left(x_{n_{k}+1},x_{n_{k}}\right)}
δd(xmk,xmk+1)+d(xnk,xnk+1)δd(fxmk,fxnk)\displaystyle\leq\delta^{d\left(x_{m_{k}},x_{m_{k}+1}\right)+d\left(x_{n_{k}},x_{n_{k}+1}\right)}\cdot\delta^{d\left(fx_{m_{k}},fx_{n_{k}}\right)}
δd(xmk,xmk+1)+d(xnk,xnk+1)δd(fxmk,fxnk)\displaystyle\leq\delta^{d\left(x_{m_{k}},x_{m_{k}+1}\right)+d\left(x_{n_{k}},x_{n_{k}+1}\right)}\cdot\delta_{*}^{d\left(fx_{m_{k}},fx_{n_{k}}\right)}
δd(xmk,xmk+1)+d(xnk,xnk+1)[α(xmk,xnk)1+δ]d(fxmk,fxnk)\displaystyle\leq\delta^{d\left(x_{m_{k}},x_{m_{k}+1}\right)+d\left(x_{n_{k}},x_{n_{k}+1}\right)}\cdot\left[\alpha\left(x_{m_{k}},x_{n_{k}}\right)-1+\delta_{*}\right]^{d\left(fx_{m_{k}},fx_{n_{k}}\right)}
δd(xmk,xmk+1)+d(xnk,xnk+1)δβ(M(xmk,xnk))M(xmk,xnk)\displaystyle\leq\delta^{d\left(x_{m_{k}},x_{m_{k}+1}\right)+d\left(x_{n_{k}},x_{n_{k}+1}\right)}\cdot\delta^{\beta\left(M\left(x_{m_{k}},x_{n_{k}}\right)\right)M\left(x_{m_{k}},x_{n_{k}}\right)}

So, we get that
d(xmk,xnk)d(xmk,xmk+1)+d(xnk,xnk+1)+β(M(xmk,xnk))M(xmk,xnk)d\left(x_{m_{k}},x_{n_{k}}\right)\leq d\left(x_{m_{k}},x_{m_{k}+1}\right)+d\left(x_{n_{k}},x_{n_{k}+1}\right)+\beta\left(M\left(x_{m_{k}},x_{n_{k}}\right)\right)M\left(x_{m_{k}},x_{n_{k}}\right).
Furthermore

limkM(xmk,xnk)=\displaystyle\lim_{k\rightarrow\infty}M\left(x_{m_{k}},x_{n_{k}}\right)=
limkmax{d(xmk,xnk),d(xmk,xmk+1)d(xnk,xnk+1)1+d(xmk,xnk),d(xmk,xmk+1)d(xnk,xnk+1)1+d(xmk+1,xnk+1)}.\displaystyle\lim_{k\rightarrow\infty}\max\left\{d\left(x_{m_{k}},x_{n_{k}}\right),\frac{d\left(x_{m_{k}},x_{m_{k}+1}\right)d\left(x_{n_{k}},x_{n_{k}+1}\right)}{1+d\left(x_{m_{k}},x_{n_{k}}\right)},\frac{d\left(x_{m_{k}},x_{m_{k}+1}\right)d\left(x_{n_{k}},x_{n_{k}+1}\right)}{1+d\left(x_{m_{k}+1},x_{n_{k}+1}\right)}\right\}.
Since d(xmk,xmk+1)0 and d(xnk,xnk+1)0 as k, we get that\displaystyle\text{ Since }d\left(x_{m_{k}},x_{m_{k}+1}\right)\rightarrow 0\text{ and }d\left(x_{n_{k}},x_{n_{k}+1}\right)\rightarrow 0\text{ as }k\rightarrow\infty\text{, we get that }
limkM(xmk,xnk)limkd(xmk,xnk)=ε>0\displaystyle\lim_{k\rightarrow\infty}M\left(x_{m_{k}},x_{n_{k}}\right)\leq\lim_{k\rightarrow\infty}d\left(x_{m_{k}},x_{n_{k}}\right)=\varepsilon>0

In the above inequality, taking the limit as kk\rightarrow\infty, we have that
εlimkβ(M(xmk,xnk))ε\varepsilon\leq\lim_{k\rightarrow\infty}\beta\left(M\left(x_{m_{k}},x_{n_{k}}\right)\right)\cdot\varepsilon.
Using the fact that ε>0\varepsilon>0, it follows that limkβ(M(xmk,xnk))=1\lim_{k\rightarrow\infty}\beta\left(M\left(x_{m_{k}},x_{n_{k}}\right)\right)=1, i.e. limkM(xmk,xnk)=0\leq\lim_{k\rightarrow\infty}M\left(x_{m_{k}},x_{n_{k}}\right)=0.
Since M(xmk,xnk)M\left(x_{m_{k}},x_{n_{k}}\right) is the maximum of three elements and it has the limit 0 , also because M(xmk,xnk)d(xmk,xnk)M\left(x_{m_{k}},x_{n_{k}}\right)\geq d\left(x_{m_{k}},x_{n_{k}}\right), then all of the elements will have the limit 0 , so d(xmk,xnk)0d\left(x_{m_{k}},x_{n_{k}}\right)\rightarrow 0, which is false; so ( xnx_{n} ) is Cauchy.

  • Since XX is complete with respect to the metric dd, there exists xXx^{*}\in X such that x=limnxnx^{*}=\lim_{n\rightarrow\infty}x_{n}. Because ff is continuous, we infer that
    x=limnxn=limnfxn1=f(limnxn1)=fxx^{*}=\lim_{n\rightarrow\infty}x_{n}=\lim_{n\rightarrow\infty}fx_{n-1}=f\left(\lim_{n\rightarrow\infty}x_{n-1}\right)=fx^{*}, so xx^{*} is a fixed point for the Geraghty-type mapping ff.

Now, also based on [13], we give a theorem where we dropped the continuity of the operator ff.

Theorem 3.3. Let (X,d)(X,d) be a complete metric space and f:XXf:X\rightarrow X an αβ\alpha-\beta rational Geraghty mappings of type I.

Let’s suppose that the following assumptions hold
(i) ff is α\alpha-admissible,
(ii) α\alpha is transitive,
(iii) there exists x0Xx_{0}\in X, such that α(x0,fx0)1\alpha\left(x_{0},fx_{0}\right)\geq 1,
(iv) if ( xnx_{n} ) is a sequence satisfying α(xn,xn+1)1\alpha\left(x_{n},x_{n+1}\right)\geq 1 and xnxx_{n}\rightarrow x implies that α(xn,x)1\alpha\left(x_{n},x\right)\geq 1, for each nn\in\mathbb{N}.

Then, there exists xXx^{*}\in X, such that x=fxx^{*}=fx^{*}.
Proof. In a similar manner like in the previous proof, we can show that (xn)\left(x_{n}\right) is a Cauchy sequence and therefore there exists xXx^{*}\in X, such that xnxx_{n}\rightarrow x^{*} when nn\rightarrow\infty.

From (iv), we have that α(xn,x)1\alpha\left(x_{n},x^{*}\right)\geq 1, for each nn\in\mathbb{N}.
We make the following estimation

δd(x,fx)δd(x,xn+1)+d(xn+1,fx)=\displaystyle\delta^{d\left(x^{*},fx^{*}\right)}\leq\delta^{d\left(x^{*},x_{n+1}\right)+d\left(x_{n+1},fx^{*}\right)}=
δd(x,xn+1)δd(fxn,fx)δd(x,xn+1)δd(fxn,fx)\displaystyle\delta^{d\left(x^{*},x_{n+1}\right)}\cdot\delta^{d\left(fx_{n},fx^{*}\right)}\leq\delta^{d\left(x^{*},x_{n+1}\right)}\cdot\delta_{*}^{d\left(fx_{n},fx^{*}\right)}
δd(x,xn+1)[α(xn,x)1+δ]d(fxn,fx)\displaystyle\leq\delta^{d\left(x^{*},x_{n+1}\right)}\cdot\left[\alpha\left(x_{n},x^{*}\right)-1+\delta_{*}\right]^{d\left(fx_{n},fx^{*}\right)}
δd(x,xn+1)δβ(M(xn,x))M(xn,x)\displaystyle\leq\delta^{d\left(x^{*},x_{n+1}\right)}\cdot\delta^{\beta\left(M\left(x_{n},x^{*}\right)\right)M\left(x_{n},x^{*}\right)}

So, we have that d(x,fx)d(x,xn+1)+β(M(xn,x))M(xn,x)<d(x,xn+1)+M(xn,x)d\left(x^{*},fx^{*}\right)\leq d\left(x^{*},x_{n+1}\right)+\beta\left(M\left(x_{n},x^{*}\right)\right)M\left(x_{n},x^{*}\right)<d\left(x^{*},x_{n+1}\right)+M\left(x_{n},x^{*}\right).
Furthermore, we have that
M(xn,x)=max{d(xn,x),d(xn,xn+1)d(x,fx)1+d(xn,x),d(xn,xn+1)d(x,fx)1+d(xn+1,fx)}M\left(x_{n},x^{*}\right)=\max\left\{d\left(x_{n},x^{*}\right),\frac{d\left(x_{n},x_{n+1}\right)d\left(x^{*},fx^{*}\right)}{1+d\left(x_{n},x^{*}\right)},\frac{d\left(x_{n},x_{n+1}\right)d\left(x^{*},fx^{*}\right)}{1+d\left(x_{n+1},fx^{*}\right)}\right\}.
Taking the limit as nn\rightarrow\infty and using the fact that d(xn,xn+1)0d\left(x_{n},x_{n+1}\right)\rightarrow 0 and that d(xn,x)0d\left(x_{n},x^{*}\right)\rightarrow 0, we get that M(xn1,xn)max{limnd(xn,x),0,0}=limnd(xn,x)=0M\left(x_{n-1},x_{n}\right)\rightarrow\max\left\{\lim_{n\rightarrow\infty}d\left(x_{n},x^{*}\right),0,0\right\}=\lim_{n\rightarrow\infty}d\left(x_{n},x^{*}\right)=0.
Thus d(x,fx)limnd(xn+1,x)+limnd(xn,x)=0d\left(x^{*},fx^{*}\right)\leq\lim_{n\rightarrow\infty}d\left(x_{n+1},x^{*}\right)+\lim_{n\rightarrow\infty}d\left(x_{n},x^{*}\right)=0, so the conclusion holds properly.
Finally, we present the theorem for the uniqueness of the fixed point for the Geraghty type operator.

Theorem 3.4. Let’s suppose that all the assumptions from the last theorem are satisfied. Additionally, let’s suppose that one of the following assumptions are valid
(HO) if a,ba,b are two fixed points, then α(a,b)1\alpha(a,b)\geq 1
(H1) for each x,yXx,y\in X, there exists zXz\in X such that α(x,z)1\alpha(x,z)\geq 1 and α(y,z)1\alpha(y,z)\geq 1.
Then, ff admits a unique fixed point.
Proof. Let x,yx^{*},y^{*} two fixed points for the mapping ff.
We consider two cases
(H0) We have that α(x,y)1\alpha\left(x^{*},y^{*}\right)\geq 1.
From the Geraghty condition, it is easy to see that d(x,y)=0d\left(x^{*},y^{*}\right)=0, so the conclusion is true.
(H1) We have that there exists zXz\in X, with α(x,z)1\alpha\left(x^{*},z\right)\geq 1 and α(y,z)1\alpha\left(y^{*},z\right)\geq 1.
Since f is α\alpha-admissible, by induction, we get that α(x,fnz)1\alpha\left(x^{*},f^{n}z\right)\geq 1 and α(y,fnz)1\alpha\left(y^{*},f^{n}z\right)\geq 1.
We make the following estimation

δd(x,fn+1z)=δd(fx,fn+1z)δd(fx,f(fnz))\displaystyle\delta^{d\left(x^{*},f^{n+1}z\right)}=\delta^{d\left(fx^{*},f^{n+1}z\right)}\leq\delta_{*}^{d\left(fx^{*},f\left(f^{n}z\right)\right)}\leq
[α(x,fnz)1+δ]d(fx,f(fnz))\displaystyle{\left[\alpha\left(x^{*},f^{n}z\right)-1+\delta_{*}\right]^{d\left(fx^{*},f\left(f^{n}z\right)\right)}\leq}
δβ(M(x,fnz))M(x,fnz)\displaystyle\delta^{\beta\left(M\left(x^{*},f^{n}z\right)\right)M\left(x^{*},f^{n}z\right)}

So d(x,fn+1z)β(M(x,fnz))M(x,fnz)d\left(x^{*},f^{n+1}z\right)\leq\beta\left(M\left(x^{*},f^{n}z\right)\right)M\left(x^{*},f^{n}z\right), for each nn\in\mathbb{N}.
Now we show that d(x,fnz)0d\left(x^{*},f^{n}z\right)\rightarrow 0 as nn\rightarrow\infty.
By reductio ad absurdum, we suppose that 0<l:=limnd(x,fnz)<0<l:=\lim_{n\rightarrow\infty}d\left(x^{*},f^{n}z\right)<\infty.
We know that
limnM(x,fnz)=limnmax{d(x,fnz),d(x,fx)d(fnz,fn+1z)1+d(x,fnz),d(x,fx)d(fnz,fn+1z)1+d(fx,fn+1z)}\lim_{n\rightarrow\infty}M\left(x^{*},f^{n}z\right)=\lim_{n\rightarrow\infty}\max\left\{d\left(x^{*},f^{n}z\right),\frac{d\left(x^{*},fx^{*}\right)d\left(f^{n}z,f^{n+1}z\right)}{1+d\left(x^{*},f^{n}z\right)},\frac{d\left(x^{*},fx^{*}\right)d\left(f^{n}z,f^{n+1}z\right)}{1+d\left(fx^{*},f^{n+1}z\right)}\right\}.
Since xx^{*} is a fixed point for ff, then limnM(x,fnz)limnd(x,fnz)=l\lim_{n\rightarrow\infty}M\left(x^{*},f^{n}z\right)\leq\lim_{n\rightarrow\infty}d\left(x^{*},f^{n}z\right)=l.
Taking the limit as nn\rightarrow\infty and using the fact that l>0l>0, we get that
llimnβ(M(x,fnz))ll\leq\lim_{n\rightarrow\infty}\beta\left(M\left(x^{*},f^{n}z\right)\right)\cdot l. Now, because βY\beta\in Y, it follows that limnβ(M(x,fnz))1\lim_{n\rightarrow\infty}\beta\left(M\left(x^{*},f^{n}z\right)\right)\leq 1. So, it follows that limnβ(M(x,fnz))=1\lim_{n\rightarrow\infty}\beta\left(M\left(x^{*},f^{n}z\right)\right)=1. This means that limnM(x,fnz)=0\lim_{n\rightarrow\infty}M\left(x^{*},f^{n}z\right)=0.
By the same reasoning as in the last proof, we get that limnd(x,fnz)=0\lim_{n\rightarrow\infty}d\left(x^{*},f^{n}z\right)=0. Furthermore, in a similar way, one can show that fnzyf^{n}z\rightarrow y^{*} as nn\rightarrow\infty, so x=yx^{*}=y^{*}.

Remark 3.5. Taking α(x,y)=1\alpha(x,y)=1, for each x,yXx,y\in X, we get the existence and uniqueness for Geraghty mappings of type II as a corollary.

4. Fixed Point Results for Modified Multivalued Hardy-Rogers contractions

In this section we introduce the concept of modified multivalued Hardy-Rogers contractions and then we present some theorems concerning the existence of a fixed point, data dependence, Ulam-Hyers stability. Also, we present a local version involving two metrics and a homotopy theorem.

The first main result of this section is a fixed point theorem for modified multivalued HardyRogers contractions, regarding the existence of fixed points for these types of self-mappings.

Theorem 4.1. Let ( X,dX,d ) be a complete metric space and T:XPcl(X)T:X\rightarrow P_{cl}(X) be a multivalued modified Hardy Rogers operator, i.e.

H(Tx,Ty)αd(x,y)+βD(x,Ty)+γD(y,Tx)+δD(y,Ty)+\displaystyle H(Tx,Ty)\leq\alpha d(x,y)+\beta D(x,Ty)+\gamma D(y,Tx)+\delta D(y,Ty)+
ηD(y,Ty)[1+D(x,Tx)]1+d(x,y)+λD(y,Ty)+D(y,Tx)1+D(y,Ty)D(y,Tx)+μD(x,Tx)[1+D(y,Tx)]1+d(x,y)+D(y,Ty)\displaystyle\eta\frac{D(y,Ty)[1+D(x,Tx)]}{1+d(x,y)}+\lambda\frac{D(y,Ty)+D(y,Tx)}{1+D(y,Ty)D(y,Tx)}+\mu\frac{D(x,Tx)[1+D(y,Tx)]}{1+d(x,y)+D(y,Ty)}

with all the above coefficients positive.
If α+2β+η+μ+λ+δ<1\alpha+2\beta+\eta+\mu+\lambda+\delta<1, then there exists pXp\in X, such that pFTp\in F_{T}.
Proof. Let’s consider x0Xx_{0}\in X an arbitrary point and q>1q>1.
Let x1Tx0x_{1}\in Tx_{0}.
If H(Tx0,Tx1)=0H\left(Tx_{0},Tx_{1}\right)=0, then Tx0=Tx1Tx_{0}=Tx_{1}, that means x1Tx1x_{1}\in Tx_{1}, i.e. x1FTx_{1}\in F_{T}.
Let’s suppose that H(Tx0,Tx1)0H\left(Tx_{0},Tx_{1}\right)\neq 0.
For x1Tx0x_{1}\in Tx_{0}, we can choose x2Tx1x_{2}\in Tx_{1}, such that d(x1,x2)qH(Tx0,Tx1)d\left(x_{1},x_{2}\right)\leq q\cdot H\left(Tx_{0},Tx_{1}\right).
This means that

d(x1,x2)q[αd(x0,x1)+βD(x0,Tx1)+γD(x1,Tx0)+δD(x1,Tx1)+\displaystyle d\left(x_{1},x_{2}\right)\leq q\left[\alpha d\left(x_{0},x_{1}\right)+\beta D\left(x_{0},Tx_{1}\right)+\gamma D\left(x_{1},Tx_{0}\right)+\delta D\left(x_{1},Tx_{1}\right)+\right.
ηD(x1,Tx1)[1+D(x0,Tx0)]1+d(x0,x1)+λD(x1,Tx1)+D(x1,Tx0)1+D(x1,Tx1)D(x1,Tx0)+μD(x0,Tx0)[1+D(x1,Tx0)]1+d(x0,x1)+D(x1,Tx1)]\displaystyle\left.\eta\frac{D\left(x_{1},Tx_{1}\right)\left[1+D\left(x_{0},Tx_{0}\right)\right]}{1+d\left(x_{0},x_{1}\right)}+\lambda\frac{D\left(x_{1},Tx_{1}\right)+D\left(x_{1},Tx_{0}\right)}{1+D\left(x_{1},Tx_{1}\right)\cdot D\left(x_{1},Tx_{0}\right)}+\mu\frac{D\left(x_{0},Tx_{0}\right)\left[1+D\left(x_{1},Tx_{0}\right)\right]}{1+d\left(x_{0},x_{1}\right)+D\left(x_{1},Tx_{1}\right)}\right]

So, we have that

d(x1,x2)q[αd(x0,x1)+βd(x0,x2)+γd(x1,x1)+δd(x1,x2)+ηd(x1,x2)[1+d(x0,x1)]1+d(x0,x1)+\displaystyle d\left(x_{1},x_{2}\right)\leq q\left[\alpha d\left(x_{0},x_{1}\right)+\beta d\left(x_{0},x_{2}\right)+\gamma d\left(x_{1},x_{1}\right)+\delta d\left(x_{1},x_{2}\right)+\eta\frac{d\left(x_{1},x_{2}\right)\left[1+d\left(x_{0},x_{1}\right)\right]}{1+d\left(x_{0},x_{1}\right)}+\right.
λd(x1,x2)+d(x1,x1)1+D(x1,Tx1)D(x1,Tx0)+μd(x0,x1)[1+d(x1,x1)]1+d(x0,x1)+D(x1,Tx1)]\displaystyle\left.\lambda\frac{d\left(x_{1},x_{2}\right)+d\left(x_{1},x_{1}\right)}{1+D\left(x_{1},Tx_{1}\right)\cdot D\left(x_{1},Tx_{0}\right)}+\mu\frac{d\left(x_{0},x_{1}\right)\left[1+d\left(x_{1},x_{1}\right)\right]}{1+d\left(x_{0},x_{1}\right)+D\left(x_{1},Tx_{1}\right)}\right]

Since
d(x1,x1)=0,d(x1,x2)1+D(x1,Tx1)D(x1,Tx0)d(x1,x2)d\left(x_{1},x_{1}\right)=0,\frac{d\left(x_{1},x_{2}\right)}{1+D\left(x_{1},Tx_{1}\right)\cdot D\left(x_{1},Tx_{0}\right)}\leq d\left(x_{1},x_{2}\right),
that is 1+D(x1,Tx1)D(x1,Tx0)11+D\left(x_{1},Tx_{1}\right)\cdot D\left(x_{1},Tx_{0}\right)\geq 1 and d(x0,x1)1+d(x0,x1)+D(x1,Tx1)d(x0,x1)1+d(x0,x1)d(x0,x1)\frac{d\left(x_{0},x_{1}\right)}{1+d\left(x_{0},x_{1}\right)+D\left(x_{1},Tx_{1}\right)}\leq\frac{d\left(x_{0},x_{1}\right)}{1+d\left(x_{0},x_{1}\right)}\leq d\left(x_{0},x_{1}\right), because 1+d(x0,x1)11+d\left(x_{0},x_{1}\right)\geq 1, we get that
d(x1,x2)q[αd(x0,x1)+βd(x0,x1)+βd(x1,x2)+δd(x1,x2)+ηd(x1,x2)+λd(x1,x2)+μd(x0,x1)]d\left(x_{1},x_{2}\right)\leq q\left[\alpha d\left(x_{0},x_{1}\right)+\beta d\left(x_{0},x_{1}\right)+\beta d\left(x_{1},x_{2}\right)+\delta d\left(x_{1},x_{2}\right)+\eta d\left(x_{1},x_{2}\right)+\lambda d\left(x_{1},x_{2}\right)+\mu d\left(x_{0},x_{1}\right)\right].
This means that
d(x1,x2)qα+β+μ1q(β+δ+η+λ)d(x0,x1)d\left(x_{1},x_{2}\right)\leq q\cdot\frac{\alpha+\beta+\mu}{1-q(\beta+\delta+\eta+\lambda)}\cdot d\left(x_{0},x_{1}\right).
In a similar manner, for x2Tx1x_{2}\in Tx_{1}, there exists x3Tx2x_{3}\in Tx_{2} such that
d(x2,x3)qH(Tx1,Tx2)d\left(x_{2},x_{3}\right)\leq q\cdot H\left(Tx_{1},Tx_{2}\right).
This means that

d(x2,x3)q[αd(x1,x2)+βD(x1,Tx2)+γD(x2,Tx1)+δD(x2,Tx2)+ηD(x2,Tx2)[1+D(x1,Tx1)]1+d(x1,x2)+\displaystyle d\left(x_{2},x_{3}\right)\leq q\left[\alpha d\left(x_{1},x_{2}\right)+\beta D\left(x_{1},Tx_{2}\right)+\gamma D\left(x_{2},Tx_{1}\right)+\delta D\left(x_{2},Tx_{2}\right)+\eta\frac{D\left(x_{2},Tx_{2}\right)\left[1+D\left(x_{1},Tx_{1}\right)\right]}{1+d\left(x_{1},x_{2}\right)}+\right.
λD(x2,Tx2)+D(x2,Tx1)1+D(x2,Tx2)D(x2,Tx1)+μD(x1,Tx1)[1+D(x2,Tx1)]1+d(x1,x2)+D(x2,Tx2)]\displaystyle\left.\lambda\frac{D\left(x_{2},Tx_{2}\right)+D\left(x_{2},Tx_{1}\right)}{1+D\left(x_{2},Tx_{2}\right)\cdot D\left(x_{2},Tx_{1}\right)}+\mu\frac{D\left(x_{1},Tx_{1}\right)\left[1+D\left(x_{2},Tx_{1}\right)\right]}{1+d\left(x_{1},x_{2}\right)+D\left(x_{2},Tx_{2}\right)}\right]

So, we have that

d(x2,x3)q[αd(x1,x2)+βd(x1,x3)+γd(x2,x2)+δd(x2,x3)+ηd(x2,x3)[1+d(x1,x2)]1+d(x1,x2)+\displaystyle d\left(x_{2},x_{3}\right)\leq q\left[\alpha d\left(x_{1},x_{2}\right)+\beta d\left(x_{1},x_{3}\right)+\gamma d\left(x_{2},x_{2}\right)+\delta d\left(x_{2},x_{3}\right)+\eta\frac{d\left(x_{2},x_{3}\right)\left[1+d\left(x_{1},x_{2}\right)\right]}{1+d\left(x_{1},x_{2}\right)}+\right.
λd(x2,x3)+d(x2,x2)1+D(x2,Tx2)D(x2,Tx1)+μd(x1,x2)[1+d(x2,x2)]1+d(x1,x2)D(x2,Tx2)\displaystyle\lambda\frac{d\left(x_{2},x_{3}\right)+d\left(x_{2},x_{2}\right)}{1+D\left(x_{2},Tx_{2}\right)\cdot D\left(x_{2},Tx_{1}\right)}+\mu\frac{d\left(x_{1},x_{2}\right)\left[1+d\left(x_{2},x_{2}\right)\right]}{1+d\left(x_{1},x_{2}\right)_{D}\left(x_{2},Tx_{2}\right)}

Like before, since
d(x2,x2)=0,d(x2,x3)1+D(x2,Tx2)D(x2,Tx1)d(x2,x3)d\left(x_{2},x_{2}\right)=0,\frac{d\left(x_{2},x_{3}\right)}{1+D\left(x_{2},Tx_{2}\right)\cdot D\left(x_{2},Tx_{1}\right)}\leq d\left(x_{2},x_{3}\right),
that is 1+D(x2,Tx2)D(x2,Tx1)11+D\left(x_{2},Tx_{2}\right)\cdot D\left(x_{2},Tx_{1}\right)\geq 1 and d(x1,x2)1+d(x1,x2)+D(x2,Tx2)d(x1,x2)1+d(x1,x2)d(x1,x2)\frac{d\left(x_{1},x_{2}\right)}{1+d\left(x_{1},x_{2}\right)+D\left(x_{2},Tx_{2}\right)}\leq\frac{d\left(x_{1},x_{2}\right)}{1+d\left(x_{1},x_{2}\right)}\leq d\left(x_{1},x_{2}\right),
because 1+d(x1,x2)11+d\left(x_{1},x_{2}\right)\geq 1, we get that
d(x2,x3)q[αd(x1,x2)+βd(x1,x2)+βd(x2,x3)+δd(x2,x3)+ηd(x2,x3)+λd(x2,x3)+μd(x1,x2)]d\left(x_{2},x_{3}\right)\leq q\left[\alpha d\left(x_{1},x_{2}\right)+\beta d\left(x_{1},x_{2}\right)+\beta d\left(x_{2},x_{3}\right)+\delta d\left(x_{2},x_{3}\right)+\eta d\left(x_{2},x_{3}\right)+\lambda d\left(x_{2},x_{3}\right)+\mu d\left(x_{1},x_{2}\right)\right].
This means that
d(x2,x3)qα+β+μ1q(β+δ+η+λ)d(x1,x2)(qα+β+μ1q(β+δ+η+λ))2d(x0,x1)d\left(x_{2},x_{3}\right)\leq q\frac{\alpha+\beta+\mu}{1-q(\beta+\delta+\eta+\lambda)}\cdot d\left(x_{1},x_{2}\right)\leq\left(q\frac{\alpha+\beta+\mu}{1-q(\beta+\delta+\eta+\lambda)}\right)^{2}\cdot d\left(x_{0},x_{1}\right)
By induction, we infer that d(xn,xn+1)rnd(x0,x1)d\left(x_{n},x_{n+1}\right)\leq r^{n}d\left(x_{0},x_{1}\right), where r:=q(α+β+μ)1q(β+η+λ+δ)r:=\frac{q(\alpha+\beta+\mu)}{1-q(\beta+\eta+\lambda+\delta)}.
Since q>1q>1 is arbitrary taken, we impose the following condition, namely
r<1r<1, which means that q(α+β+μ)<1q(β+η+λ+δ)q(\alpha+\beta+\mu)<1-q(\beta+\eta+\lambda+\delta).
Equivalently, we can take q<1α+2β+μ+η+λ+δq<\frac{1}{\alpha+2\beta+\mu+\eta+\lambda+\delta}.

In this way, we can take q(1,1α+2β+μ+η+λ+δ)q\in\left(1,\frac{1}{\alpha+2\beta+\mu+\eta+\lambda+\delta}\right).
From the hypotheses, we have that α+2β+μ+η+λ+δ<1\alpha+2\beta+\mu+\eta+\lambda+\delta<1, which implies that
1<1α+2β+μ+η+λ+δ1<\frac{1}{\alpha+2\beta+\mu+\eta+\lambda+\delta}, so the definition for qq is correct.
For r0r\geq 0 to take place, we need the relation q<1β+η+λ+μq<\frac{1}{\beta+\eta+\lambda+\mu}.
Now, since q<1α+2β+η+λ+μ+δ1β+η+λ+μq<\frac{1}{\alpha+2\beta+\eta+\lambda+\mu+\delta}\leq\frac{1}{\beta+\eta+\lambda+\mu}, it follows that α+β+δ0\alpha+\beta+\delta\geq 0, which is obviously true.

Now, we show that (xn)n\left(x_{n}\right)_{n\in\mathbb{N}} is a Cauchy sequence, i.e.
d(xn,xn+p)d(xn,xn+1)++d(xn+p1,xn+p)d\left(x_{n},x_{n+p}\right)\leq d\left(x_{n},x_{n+1}\right)+\ldots+d\left(x_{n+p-1},x_{n+p}\right)\leq
(rn++rn+p1)d(x0,x1)=\left(r^{n}+\ldots+r^{n+p-1}\right)\cdot d\left(x_{0},x_{1}\right)=
rn(1++rp1)d(x0,x1)=r^{n}\cdot\left(1+\ldots+r^{p-1}\right)\cdot d\left(x_{0},x_{1}\right)=
rn1rp1rd(x0,x1)rn11rd(x0,x1)r^{n}\cdot\frac{1-r^{p}}{1-r}\cdot d\left(x_{0},x_{1}\right)\leq r^{n}\cdot\frac{1}{1-r}d\left(x_{0},x_{1}\right).
For each p1p\geq 1, letting nn\rightarrow\infty, if follows that ( xnx_{n} ) is a Cauchy sequence.
Because the metric dd is a complete, the sequence ( xnx_{n} ) is convergent.
Then exists pXp\in X, such that xnpx_{n}\rightarrow p.
We now show that pp is a fixed point for the operator TT. We estimate
D(p,Tp)=infyTpd(p,y)d(p,xn+1)+infyTpd(xn+1,y)D(p,Tp)=\inf_{y\in Tp}d(p,y)\leq d\left(p,x_{n+1}\right)+\inf_{y\in Tp}d\left(x_{n+1},y\right)\leq
d(p,xn+1)+D(xn+1,Tp)d(p,xn+1)+H(Txn,Tp)d\left(p,x_{n+1}\right)+D\left(x_{n+1},Tp\right)\leq d\left(p,x_{n+1}\right)+H\left(Tx_{n},Tp\right)\leq
d(p,xn+1)+αd(xn,p)+βD(xn,Tp)+γD(p,Txn)+δD(p,Tp)+ηD(p,Tp)[1+D(xn,Txn)]1+d(xn,p)+d\left(p,x_{n+1}\right)+\alpha d\left(x_{n},p\right)+\beta D\left(x_{n},Tp\right)+\gamma D\left(p,Tx_{n}\right)+\delta D(p,Tp)+\eta\frac{D(p,Tp)\left[1+D\left(x_{n},Tx_{n}\right)\right]}{1+d\left(x_{n},p\right)}+
λD(p,Tp)+D(p,Txn)1+D(p,Tp)D(p,Txn)+μD(xn,Txn)[1+D(p,Txn)]1+d(p,xn)+D(p,Tp)\lambda\frac{D(p,Tp)+D\left(p,Tx_{n}\right)}{1+D(p,Tp)D\left(p,Tx_{n}\right)}+\mu\frac{D\left(x_{n},Tx_{n}\right)\left[1+D\left(p,Tx_{n}\right)\right]}{1+d\left(p,x_{n}\right)+D(p,Tp)}
D(p,Tp)d(p,xn+1)+αd(xn,p)+βd(xn,p)+βD(p,Tp)+γd(p,xn+1)+δD(p,Tp)+\Longrightarrow D(p,Tp)\leq d\left(p,x_{n+1}\right)+\alpha d\left(x_{n},p\right)+\beta d\left(x_{n},p\right)+\beta D(p,Tp)+\gamma d\left(p,x_{n+1}\right)+\delta D(p,Tp)+
ηD(p,Tp)[1+d(xn,xn+1)]1+d(xn,p)+λD(p,Tp)+d(p,xn+1)1+D(p,Tp)D(p,Txn)+μd(xn,xn+1)(1+d(p,xn+1))1+d(xn,p)+D(p,Tp)\eta\frac{D(p,Tp)\left[1+d\left(x_{n},x_{n+1}\right)\right]}{1+d\left(x_{n},p\right)}+\lambda\frac{D(p,Tp)+d\left(p,x_{n+1}\right)}{1+D(p,Tp)D\left(p,Tx_{n}\right)}+\mu\frac{d\left(x_{n},x_{n+1}\right)\left(1+d\left(p,x_{n+1}\right)\right)}{1+d\left(x_{n},p\right)+D(p,Tp)}
Now, we have used the following relations
D(p,Txn)d(p,xn+1)D\left(p,Tx_{n}\right)\leq d\left(p,x_{n+1}\right), for each nn\in\mathbb{N},
D(xn,Tp)=infyTpd(xn,y)d(xn,p)+infyTpd(p,y)=d(xn,p)+D(p,Tp)D\left(x_{n},Tp\right)=\inf_{y\in Tp}d\left(x_{n},y\right)\leq d\left(x_{n},p\right)+\inf_{y\in Tp}d(p,y)=d\left(x_{n},p\right)+D(p,Tp),
D(p,Tp)+D(p,Txn)1+D(p,Tp)D(p,Txn)D(p,Tp)+d(p,xn+1)1+D(p,Tp)D(p,Txn)D(p,Tp)+d(xn+1,p)\frac{D(p,Tp)+D\left(p,Tx_{n}\right)}{1+D(p,Tp)D\left(p,Tx_{n}\right)}\leq\frac{D(p,Tp)+d\left(p,x_{n+1}\right)}{1+D(p,Tp)D\left(p,Tx_{n}\right)}\leq D(p,Tp)+d\left(x_{n+1},p\right),
because D(p,Tp)D(p,Txn)+11D(p,Tp)D\left(p,Tx_{n}\right)+1\geq 1.
Letting nn\rightarrow\infty, we have that
D(p,Tp)βD(p,Tp)+ηD(p,Tp)+λD(p,Tp)+δD(p,Tp)+μ0=(β+η+λ+δ)D(p,Tp)D(p,Tp)\leq\beta D(p,Tp)+\eta D(p,Tp)+\lambda D(p,Tp)+\delta D(p,Tp)+\mu\cdot 0=(\beta+\eta+\lambda+\delta)D(p,Tp).
It follows that [1(β+η+λ+δ)]D(p,Tp)0[1-(\beta+\eta+\lambda+\delta)]D(p,Tp)\leq 0. The inequality 1(β+η+λ+δ)>01-(\beta+\eta+\lambda+\delta)>0, is satisfied since β+η+λ+δα+2β+μ+η+λ+δ<1\beta+\eta+\lambda+\delta\leq\alpha+2\beta+\mu+\eta+\lambda+\delta<1.

We obtain that D(p,Tp)=0D(p,Tp)=0. This means that pTpp\in Tp, i.e. pFTp\in F_{T}.
We have shown that d(xn,xn+p)rn1rp1rd(x0,x1)d\left(x_{n},x_{n+p}\right)\leq r^{n}\frac{1-r^{p}}{1-r}d\left(x_{0},x_{1}\right).
Letting pp\rightarrow\infty, we have d(xn,p)rn1rd(x0,x1)d\left(x_{n},p\right)\leq\frac{r^{n}}{1-r}d\left(x_{0},x_{1}\right).
Letting n=0n=0 in the above inequality, we have d(x0,p)11rd(x0,x1)d\left(x_{0},p\right)\leq\frac{1}{1-r}d\left(x_{0},x_{1}\right), so TT is a MWP operator.

Now we present a theorem concerning the fact that T is a MWP operator.
Theorem 4.2. Let ( X,dX,d ) be a complete metric space and T:XPcl(X)T:X\rightarrow P_{cl}(X) be a multivalued modified Hardy Rogers operator, i.e.

H(Tx,Ty)αd(x,y)+βD(x,Ty)+γD(y,Tx)+δD(y,Ty)+\displaystyle H(Tx,Ty)\leq\alpha d(x,y)+\beta D(x,Ty)+\gamma D(y,Tx)+\delta D(y,Ty)+
ηD(y,Ty)[1+D(x,Tx)]1+d(x,y)+λD(y,Ty)+D(y,Tx)1+D(y,Ty)D(y,Tx)+μD(x,Tx)[1+D(y,Tx)]1+d(x,y)+D(y,Ty)\displaystyle\eta\frac{D(y,Ty)[1+D(x,Tx)]}{1+d(x,y)}+\lambda\frac{D(y,Ty)+D(y,Tx)}{1+D(y,Ty)D(y,Tx)}+\mu\frac{D(x,Tx)[1+D(y,Tx)]}{1+d(x,y)+D(y,Ty)}

with all the above coefficients positive.
If α+2β+η+μ+λ+δ<1\alpha+2\beta+\eta+\mu+\lambda+\delta<1, then the operator TT is 1(β+η+λ+δ)1(2β+η+λ+δ+α+μ)MWP\frac{1-(\beta+\eta+\lambda+\delta)}{1-(2\beta+\eta+\lambda+\delta+\alpha+\mu)}-MWP.
Proof. From the proof of the previous theorem, we have that d(xn,xn+p)rn1rp1rd(x0,x1)d\left(x_{n},x_{n+p}\right)\leq r^{n}\frac{1-r^{p}}{1-r}d\left(x_{0},x_{1}\right), with rr defined as r=q(α+β+μ)1q(β+η+λ+δ)r=\frac{q(\alpha+\beta+\mu)}{1-q(\beta+\eta+\lambda+\delta)}.
Since the sequence ( xnx_{n} ) was convergent to a fixed point pp of TT, letting pp\rightarrow\infty and then making n=1n=1, we get that d(x1,p)r1rd(x0,x1)d\left(x_{1},p\right)\leq\frac{r}{1-r}d\left(x_{0},x_{1}\right).
Using the triangular inequality, it follows that

d(x0,p)d(x0,x1)+d(x1,p)d(x0,x1)+r1rd(x0,x1)\displaystyle d\left(x_{0},p\right)\leq d\left(x_{0},x_{1}\right)+d\left(x_{1},p\right)\leq d\left(x_{0},x_{1}\right)+\frac{r}{1-r}d\left(x_{0},x_{1}\right)\leq
(1+r1r)d(x0,x1)=11rd(x0,x1)\displaystyle\left(1+\frac{r}{1-r}\right)d\left(x_{0},x_{1}\right)=\frac{1}{1-r}d\left(x_{0},x_{1}\right)

From the definition of r , letting q1q\searrow 1, we obtain

d(x0,p)11α+β+μ1(β+η+λ+δ)d(x0,x1)d\left(x_{0},p\right)\leq\frac{1}{1-\frac{\alpha+\beta+\mu}{1-(\beta+\eta+\lambda+\delta)}}d\left(x_{0},x_{1}\right)

So d(x0,p)1(β+η+λ+δ)1(2β+η+λ+δ+α+μ)d(x0,x1)d\left(x_{0},p\right)\leq\frac{1-(\beta+\eta+\lambda+\delta)}{1-(2\beta+\eta+\lambda+\delta+\alpha+\mu)}d\left(x_{0},x_{1}\right).
But 1(β+η+λ+δ)1(2β+η+λ+δ+α+μ)0\frac{1-(\beta+\eta+\lambda+\delta)}{1-(2\beta+\eta+\lambda+\delta+\alpha+\mu)}\geq 0, which is equivaluent to β+η+λ+δ1\beta+\eta+\lambda+\delta\leq 1 and α+2β+η+μ+λ+δ<1\alpha+2\beta+\eta+\mu+\lambda+\delta<1, which is valid because β+η+λ+δ2β+η+λ+δ+α+μ\beta+\eta+\lambda+\delta\leq 2\beta+\eta+\lambda+\delta+\alpha+\mu, so α+μ0\alpha+\mu\geq 0.

Finally, the conclusion holds properly.
The next two theorems which are presented are related to data dependence and Ulam-Hyers stability. For more information about this notions we remind the articles [6], [10] and [11].

Theorem 4.3. Let ( X,dX,d ) be a complete metric space and T,S:XPcl(X)T,S:X\rightarrow P_{cl}(X) be two multivalued modified Hardy Rogers operators, i.e.

H(Tx,Ty)\displaystyle H(Tx,Ty) αTd(x,y)+βTD(x,Ty)+γTD(y,Tx)+δTD(y,Ty)+ηTD(y,Ty)[1+D(x,Tx)]1+d(x,y)+\displaystyle\leq\alpha_{T}d(x,y)+\beta_{T}D(x,Ty)+\gamma_{T}D(y,Tx)+\delta_{T}D(y,Ty)+\eta_{T}\frac{D(y,Ty)[1+D(x,Tx)]}{1+d(x,y)}+
λTD(y,Ty)+D(y,Tx)1+D(y,Ty)D(y,Tx)+μTD(x,Tx)[1+D(y,Tx)]1+d(x,y)+D(y,Ty) and\displaystyle\lambda_{T}\frac{D(y,Ty)+D(y,Tx)}{1+D(y,Ty)D(y,Tx)}+\mu_{T}\frac{D(x,Tx)[1+D(y,Tx)]}{1+d(x,y)+D(y,Ty)}\text{ and }
H(Sx,Sy)\displaystyle H(Sx,Sy) αSd(x,y)+βSD(x,Sy)+γSD(y,Sx)+δSD(y,Sy)+ηSD(y,Sy)[1+D(x,Sx)]1+d(x,y)+\displaystyle\leq\alpha_{S}d(x,y)+\beta_{S}D(x,Sy)+\gamma_{S}D(y,Sx)+\delta_{S}D(y,Sy)+\eta_{S}\frac{D(y,Sy)[1+D(x,Sx)]}{1+d(x,y)}+
λSD(y,Sy)+D(y,Sx)1+D(y,Sy)D(y,Sx)+μSD(x,Sx)[1+D(y,Sx)]1+d(x,y)+D(y,Sy)\displaystyle\lambda_{S}\frac{D(y,Sy)+D(y,Sx)}{1+D(y,Sy)D(y,Sx)}+\mu_{S}\frac{D(x,Sx)[1+D(y,Sx)]}{1+d(x,y)+D(y,Sy)}

with all the above coefficients positive.
Let’s suppose that αT+2βT+ηT+λT+μT+δT<1\alpha_{T}+2\beta_{T}+\eta_{T}+\lambda_{T}+\mu_{T}+\delta_{T}<1 and αS+2βS+ηS+λS+μS+δS<1\alpha_{S}+2\beta_{S}+\eta_{S}+\lambda_{S}+\mu_{S}+\delta_{S}<1. Also, suppose that there exists τ>0\tau>0, such that H(Sx,Tx)τH(Sx,Tx)\leq\tau, for each xXx\in X.

Then
H(FS,FT)τmax{1(βT+ηT+λT+δT)1(2βT+ηT+λT+δT+μT+αT),1(βS+ηS+λS+δS)1(2βS+ηS+λS+δS+μS+αS)}H\left(F_{S},F_{T}\right)\leq\tau\cdot\max\left\{\frac{1-\left(\beta_{T}+\eta_{T}+\lambda_{T}+\delta_{T}\right)}{1-\left(2\beta_{T}+\eta_{T}+\lambda_{T}+\delta_{T}+\mu_{T}+\alpha_{T}\right)},\frac{1-\left(\beta_{S}+\eta_{S}+\lambda_{S}+\delta_{S}\right)}{1-\left(2\beta_{S}+\eta_{S}+\lambda_{S}+\delta_{S}+\mu_{S}+\alpha_{S}\right)}\right\}.
Proof. Let’s consider x0FSx_{0}\in F_{S}. This means that x0Sx0x_{0}\in Sx_{0}.
Let x:=t(x,y)FTx^{*}:=t^{\infty}(x,y)\in F_{T}, i.e. xTxx^{*}\in Tx^{*}. We denote by x:=x0x:=x_{0}.
From the proofs of the previous theorems, we remind that we have shown d(x,t(x,y))=d(x,x)11rTd(x,y)d\left(x,t^{\infty}(x,y)\right)=d\left(x,x^{*}\right)\leq\frac{1}{1-r_{T}}d(x,y),
where rT:=q(αT+βT+μT)1q(βT+ηT+λT+δT)r_{T}:=\frac{q\left(\alpha_{T}+\beta_{T}+\mu_{T}\right)}{1-q\left(\beta_{T}+\eta_{T}+\lambda_{T}+\delta_{T}\right)}, with qq arbitrary taken as in the previous proofs.
So d(x,t(x,y))11rTd(x,y)=11rTd(x,x1)qτ1rTd\left(x,t^{\infty}(x,y)\right)\leq\frac{1}{1-r_{T}}d(x,y)=\frac{1}{1-r_{T}}d\left(x,x_{1}\right)\leq\frac{q\tau}{1-r_{T}}, with x:=x0x:=x_{0} and y:=x1Tx0y:=x_{1}\in Tx_{0}. This inequality chain is obtained because for x=x0x=x_{0}, there exists y=x1Tx0y=x_{1}\in Tx_{0}, such that d(x0,x1)qH(Sx0,Tx0)qτd\left(x_{0},x_{1}\right)\leq qH\left(Sx_{0},Tx_{0}\right)\leq q\tau.

Analogous, we have that for y0FTy_{0}\in F_{T}, there exists y1Sy0y_{1}\in Sy_{0}, such that
d(y0,t(y0,y1))qτ1rSd\left(y_{0},t^{\infty}\left(y_{0},y_{1}\right)\right)\leq\frac{q\tau}{1-r_{S}},
where rS:=αS+βS+μS1(βS+ηS+λS+δS)r_{S}:=\frac{\alpha_{S}+\beta_{S}+\mu_{S}}{1-\left(\beta_{S}+\eta_{S}+\lambda_{S}+\delta_{S}\right)}.
All the above inequalities implies that H(FS,FT)qτmax{11rT,11rS}H\left(F_{S},F_{T}\right)\leq q\tau\cdot\max\left\{\frac{1}{1-r_{T}},\frac{1}{1-r_{S}}\right\}.
Letting q1q\searrow 1, we get that H(FS,FT)τmax{11rT,11rS}H\left(F_{S},F_{T}\right)\leq\tau\cdot\max\left\{\frac{1}{1-r_{T}},\frac{1}{1-r_{S}}\right\}.
So, the conclusion holds.
Now, the next fixed point theorem involves Ulam-Hyers stability of the fixed point inclusion.
Theorem 4.4. Let T:XPcp(X)T:X\rightarrow P_{cp}(X) be a multivalued modified Hardy Rogers contraction with positive coefficients ( α,β,γ,δ,η,λ,μ\alpha,\beta,\gamma,\delta,\eta,\lambda,\mu ), with α+2β+η+λ+μ+δ<1\alpha+2\beta+\eta+\lambda+\mu+\delta<1.

Let ε>0\varepsilon>0 and xXx\in X, such that Dd(x,Tx)εD_{d}(x,Tx)\leq\varepsilon.
Then, there exists xFTx^{*}\in F_{T} such that d(x,x)ε1(β+η+λ+δ)1(2β+η+α+λ+δ+μ)d\left(x,x^{*}\right)\leq\varepsilon\cdot\frac{1-(\beta+\eta+\lambda+\delta)}{1-(2\beta+\eta+\alpha+\lambda+\delta+\mu)}.
Proof. Let ε>0\varepsilon>0 and xXx\in X, such that Dd(x,Tx)εD_{d}(x,Tx)\leq\varepsilon.
Since TxTx is compact for the above xx, it implies that there exists yTxy\in Tx, such that D(x,Tx)=d(x,y)εD(x,Tx)=d(x,y)\leq\varepsilon.
From the previous proofs, we have that d(x,t(x,y))11rd(x,y)d\left(x,t^{\infty}(x,y)\right)\leq\frac{1}{1-r}d(x,y), with yTxy\in Tx considered above.
Then d(x,x)ε1rd\left(x,x^{*}\right)\leq\frac{\varepsilon}{1-r}, that is d(x,x)ε1(β+η+λ+δ)1(2β+η+α+λ+δ+μ)d\left(x,x^{*}\right)\leq\varepsilon\cdot\frac{1-(\beta+\eta+\lambda+\delta)}{1-(2\beta+\eta+\alpha+\lambda+\delta+\mu)}.
This means that the conclusion is valid under the theorem’s hypotheses.
In the next two theorems and in the last corollary we present local versions involving two metrics and homotopy results with respect to modified multivalued Hardy-Rogers operators. For homotopy-type results we let the reader follow [1] and [2] and [5].

Theorem 4.5. Let ( X,dX,d ) be a complete metric space. Let x0Xx_{0}\in X and r>0r>0.
Let ρ\rho be another metric on XX and let T:B¯ρd(x0,r)P(X)T:\bar{B}_{\rho}^{d}\left(x_{0},r\right)\rightarrow P(X) be a multivalued operator.

Let’s suppose the following assumptions are satisfied
(1) there exists c>0c>0 such that d(x,y)cρ(x,y)d(x,y)\leq c\rho(x,y), for each x,yXx,y\in X
(2) If dρd\neq\rho, then T:B¯ρd(x0,r)P(Xd)T:\bar{B}_{\rho}^{d}\left(x_{0},r\right)\rightarrow P\left(X^{d}\right) is a closed operator,

If d=ρd=\rho, then T:B¯dd(x0,r)Pcl(Xd)T:\bar{B}_{d}^{d}\left(x_{0},r\right)\rightarrow P_{cl}\left(X^{d}\right),
(3) for each x,yB¯ρd(x0,r)x,y\in\bar{B}_{\rho}^{d}\left(x_{0},r\right), we have that TT is a multivalued modified Hardy Rogers contraction with respect to the metric ρ\rho, i.e.

Hρ(Tx,Ty)αρ(x,y)+βDρ(x,Ty)+γDρ(y,Tx)+δDρ(y,Ty)+\displaystyle H_{\rho}(Tx,Ty)\leq\alpha\rho(x,y)+\beta D_{\rho}(x,Ty)+\gamma D_{\rho}(y,Tx)+\delta D_{\rho}(y,Ty)+
ηDρ(y,Ty)[1+Dρ(x,Tx)]1+ρ(x,y)+λDρ(y,Ty)+Dρ(y,Tx)1+Dρ(y,Ty)Dρ(y,Tx)+μDρ(x,Tx)[1+Dρ(y,Tx)]1+ρ(x,y)+Dρ(y,Ty)\displaystyle\eta\frac{D_{\rho}(y,Ty)\left[1+D_{\rho}(x,Tx)\right]}{1+\rho(x,y)}+\lambda\frac{D_{\rho}(y,Ty)+D_{\rho}(y,Tx)}{1+D_{\rho}(y,Ty)D_{\rho}(y,Tx)}+\mu\frac{D_{\rho}(x,Tx)\left[1+D_{\rho}(y,Tx)\right]}{1+\rho(x,y)+D_{\rho}(y,Ty)}

(4) Dρ(x0,Tx0)<(1θ)rD_{\rho}\left(x_{0},Tx_{0}\right)<(1-\theta)r, with θ:=α+β+μ1(β+δ+η+λ)[0,1)\theta:=\frac{\alpha+\beta+\mu}{1-(\beta+\delta+\eta+\lambda)}\in[0,1), where all the Hardy-Rogers type coefficients are positive.
Then, we have that there exists xB¯ρd(x0,r)x^{*}\in\bar{B}_{\rho}^{d}\left(x_{0},r\right), such that xTxx^{*}\in Tx^{*}.
Proof. From the hypotheses we have that Dρ(x0,Tx0)<(1θ)rD_{\rho}\left(x_{0},Tx_{0}\right)<(1-\theta)r.
Then, for x0x_{0} there exists x1Tx0x_{1}\in Tx_{0} such that
ρ(x0,x1)<(1θ)r\rho\left(x_{0},x_{1}\right)<(1-\theta)r.
This means that x1B¯ρd(x0,r)x_{1}\in\bar{B}_{\rho}^{d}\left(x_{0},r\right). We have that

Hρ(Tx0,Tx1)αρ(x0,x1)+βDρ(x0,Tx1)+γDρ(x1,Tx0)+δDρ(x1,Tx1)+\displaystyle H_{\rho}\left(Tx_{0},Tx_{1}\right)\leq\alpha\rho\left(x_{0},x_{1}\right)+\beta D_{\rho}\left(x_{0},Tx_{1}\right)+\gamma D_{\rho}\left(x_{1},Tx_{0}\right)+\delta D_{\rho}\left(x_{1},Tx_{1}\right)+
ηDρ(x1,Tx1)[1+Dρ(x0,Tx0)]1+ρ(x0,x1)+λDρ(x1,Tx1)+Dρ(x1,Tx0)1+Dρ(x1,Tx1)Dρ(x1,Tx0)+μDρ(x0,Tx0)[1+Dρ(x1,Tx0)]1+ρ(x0,x1)+Dρ(x1,Tx1)\displaystyle\eta\frac{D_{\rho}\left(x_{1},Tx_{1}\right)\left[1+D_{\rho}\left(x_{0},Tx_{0}\right)\right]}{1+\rho\left(x_{0},x_{1}\right)}+\lambda\frac{D_{\rho}\left(x_{1},Tx_{1}\right)+D_{\rho}\left(x_{1},Tx_{0}\right)}{1+D_{\rho}\left(x_{1},Tx_{1}\right)D_{\rho}\left(x_{1},Tx_{0}\right)}+\mu\frac{D_{\rho}\left(x_{0},Tx_{0}\right)\left[1+D_{\rho}\left(x_{1},Tx_{0}\right)\right]}{1+\rho\left(x_{0},x_{1}\right)+D_{\rho}\left(x_{1},Tx_{1}\right)}
αρ(x0,x1)+βρ(x0,x1)+βDρ(x1,Tx1)+δDρ(x1,Tx1)+ηDρ(x1,Tx1)1+ρ(x0,x1)1+ρ(x0,x1)+\displaystyle\leq\alpha\rho\left(x_{0},x_{1}\right)+\beta\rho\left(x_{0},x_{1}\right)+\beta D_{\rho}\left(x_{1},Tx_{1}\right)+\delta D_{\rho}\left(x_{1},Tx_{1}\right)+\eta D_{\rho}\left(x_{1},Tx_{1}\right)\cdot\frac{1+\rho\left(x_{0},x_{1}\right)}{1+\rho\left(x_{0},x_{1}\right)}+
λDρ(x1,Tx1)11+Dρ(x1,Tx1)Dρ(x1,Tx0)+μρ(x0,x1)\displaystyle\lambda D_{\rho}\left(x_{1},Tx_{1}\right)\cdot\frac{1}{1+D_{\rho}\left(x_{1},Tx_{1}\right)D_{\rho}\left(x_{1},Tx_{0}\right)}+\mu\rho\left(x_{0},x_{1}\right)\leq
(α+β+μ)ρ(x0,x1)+(λ+δ+β+η)Dρ(x1,Tx1).\displaystyle(\alpha+\beta+\mu)\rho\left(x_{0},x_{1}\right)+(\lambda+\delta+\beta+\eta)D_{\rho}\left(x_{1},Tx_{1}\right).

Then Hρ(Tx0,Tx1)(α+β+μ)ρ(x0,x1)+(λ+δ+β+η)Dρ(x1,Tx1)H_{\rho}\left(Tx_{0},Tx_{1}\right)\leq(\alpha+\beta+\mu)\rho\left(x_{0},x_{1}\right)+(\lambda+\delta+\beta+\eta)D_{\rho}\left(x_{1},Tx_{1}\right).
Since Dρ(x1,Tx1)Hρ(Tx0,Tx1)D_{\rho}\left(x_{1},Tx_{1}\right)\leq H_{\rho}\left(Tx_{0},Tx_{1}\right), then Dρ(x1,Tx1)α+β+μ1(β+δ+λ+η)ρ(x0,x1)=θρ(x0,x1)<θ(1θ)rD_{\rho}\left(x_{1},Tx_{1}\right)\leq\frac{\alpha+\beta+\mu}{1-(\beta+\delta+\lambda+\eta)}\rho\left(x_{0},x_{1}\right)=\theta\rho\left(x_{0},x_{1}\right)<\theta(1-\theta)r. So there exists x2Tx1x_{2}\in Tx_{1} such that ρ(x1,x2)<θ(1θ)r\rho\left(x_{1},x_{2}\right)<\theta(1-\theta)r.
So ρ(x0,x2)ρ(x1,x2)+ρ(x0,x1)<θ(1θ)r+(1θ)r=(1θ)(1+θ)r=(1θ2)rr\rho\left(x_{0},x_{2}\right)\leq\rho\left(x_{1},x_{2}\right)+\rho\left(x_{0},x_{1}\right)<\theta(1-\theta)r+(1-\theta)r=(1-\theta)(1+\theta)r=\left(1-\theta^{2}\right)r\leq r.
This means that x2B¯ρd(x0,r)x_{2}\in\bar{B}_{\rho}^{d}\left(x_{0},r\right).

In a similar manner, for x1x_{1} and x2x_{2} in B¯ρd(x0,r)\bar{B}_{\rho}^{d}\left(x_{0},r\right) we have that

Hρ(Tx1,Tx2)αρ(x1,x2)+βDρ(x1,Tx2)+γDρ(x2,Tx1)+δDρ(x2,Tx2)+\displaystyle H_{\rho}\left(Tx_{1},Tx_{2}\right)\leq\alpha\rho\left(x_{1},x_{2}\right)+\beta D_{\rho}\left(x_{1},Tx_{2}\right)+\gamma D_{\rho}\left(x_{2},Tx_{1}\right)+\delta D_{\rho}\left(x_{2},Tx_{2}\right)+
ηDρ(x2,Tx2)[1+Dρ(x1,Tx1)]1+ρ(x1,x2)+λDρ(x2,Tx2)+Dρ(x2,Tx1)1+Dρ(x2,Tx2)Dρ(x2,Tx1)+μDρ(x1,Tx1)[1+Dρ(x2,Tx1)]1+ρ(x1,x2)+Dρ(x2,Tx2)\displaystyle\eta\frac{D_{\rho}\left(x_{2},Tx_{2}\right)\left[1+D_{\rho}\left(x_{1},Tx_{1}\right)\right]}{1+\rho\left(x_{1},x_{2}\right)}+\lambda\frac{D_{\rho}\left(x_{2},Tx_{2}\right)+D_{\rho}\left(x_{2},Tx_{1}\right)}{1+D_{\rho}\left(x_{2},Tx_{2}\right)D_{\rho}\left(x_{2},Tx_{1}\right)}+\mu\frac{D_{\rho}\left(x_{1},Tx_{1}\right)\left[1+D_{\rho}\left(x_{2},Tx_{1}\right)\right]}{1+\rho\left(x_{1},x_{2}\right)+D_{\rho}\left(x_{2},Tx_{2}\right)}
αρ(x1,x2)+βρ(x1,x2)+βDρ(x2,Tx2)+δDρ(x2,Tx2)+ηDρ(x2,Tx2)1+ρ(x1,x2)1+ρ(x1,x2)+\displaystyle\leq\alpha\rho\left(x_{1},x_{2}\right)+\beta\rho\left(x_{1},x_{2}\right)+\beta D_{\rho}\left(x_{2},Tx_{2}\right)+\delta D_{\rho}\left(x_{2},Tx_{2}\right)+\eta D_{\rho}\left(x_{2},Tx_{2}\right)\cdot\frac{1+\rho\left(x_{1},x_{2}\right)}{1+\rho\left(x_{1},x_{2}\right)}+
λDρ(x2,Tx2)11+Dρ(x2,Tx2)Dρ(x2,Tx1)+μρ(x1,x2)\displaystyle\lambda D_{\rho}\left(x_{2},Tx_{2}\right)\cdot\frac{1}{1+D_{\rho}\left(x_{2},Tx_{2}\right)D_{\rho}\left(x_{2},Tx_{1}\right)}+\mu\rho\left(x_{1},x_{2}\right)\leq
(α+β+μ)ρ(x1,x2)+(λ+δ+β+η)Dρ(x2,Tx2).\displaystyle(\alpha+\beta+\mu)\rho\left(x_{1},x_{2}\right)+(\lambda+\delta+\beta+\eta)D_{\rho}\left(x_{2},Tx_{2}\right).

Then Hρ(Tx1,Tx2)(α+β+μ)ρ(x1,x2)+(λ+δ+β+η)Dρ(x2,Tx2)H_{\rho}\left(Tx_{1},Tx_{2}\right)\leq(\alpha+\beta+\mu)\rho\left(x_{1},x_{2}\right)+(\lambda+\delta+\beta+\eta)D_{\rho}\left(x_{2},Tx_{2}\right).
Since Dρ(x2,Tx2)Hρ(Tx1,Tx2)D_{\rho}\left(x_{2},Tx_{2}\right)\leq H_{\rho}\left(Tx_{1},Tx_{2}\right), then Dρ(x2,Tx2)α+β+μ1(β+δ+λ+η)ρ(x1,x2)=θρ(x1,x2)<θ2(1θ)rD_{\rho}\left(x_{2},Tx_{2}\right)\leq\frac{\alpha+\beta+\mu}{1-(\beta+\delta+\lambda+\eta)}\rho\left(x_{1},x_{2}\right)=\theta\rho\left(x_{1},x_{2}\right)<\theta^{2}(1-\theta)r. So there exists x3Tx2x_{3}\in Tx_{2} such that ρ(x2,x3)<θ2(1θ)r\rho\left(x_{2},x_{3}\right)<\theta^{2}(1-\theta)r.
So, applying triangular inequality, we obtain
ρ(x0,x3)ρ(x0,x2)+ρ(x2,x3)<(1θ2)r+θ2(1θ)r=(1θ3)rr\rho\left(x_{0},x_{3}\right)\leq\rho\left(x_{0},x_{2}\right)+\rho\left(x_{2},x_{3}\right)<\left(1-\theta^{2}\right)r+\theta^{2}(1-\theta)r=\left(1-\theta^{3}\right)r\leq r.
This means that x3B¯ρd(x0,r)x_{3}\in\bar{B}_{\rho}^{d}\left(x_{0},r\right).

So, we have created a sequence (xn)B¯ρd(x0,r)\left(x_{n}\right)\subset\bar{B}_{\rho}^{d}\left(x_{0},r\right), with the following properties :
(i) xn+1Txnx_{n+1}\in Tx_{n}, for each nn\in\mathbb{N},
(ii) ρ(xn1,xn)θn1(1θ)r\rho\left(x_{n-1},x_{n}\right)\leq\theta^{n-1}(1-\theta)r, for each nn\in\mathbb{N},
(iii) ρ(x0,xn)(1θn)r\rho\left(x_{0},x_{n}\right)\leq\left(1-\theta^{n}\right)r.

It is easy to see that (xn)\left(x_{n}\right) is a Cauchy sequence in ( X,ρX,\rho ).
Using the fact that d(x,y)cρ(x,y)d(x,y)\leq c\rho(x,y), for each x,yXx,y\in X, it implies that (xn)\left(x_{n}\right) is a Cauchy sequence in ( X,dX,d ).
Because ( X,dX,d ) is a complete metric space, there exists xB¯ρd(x0,r)x^{*}\in\bar{B}_{\rho}^{d}\left(x_{0},r\right) such that xn𝑑xx_{n}\xrightarrow{d}x^{*}.

Furthermore, we have two cases to analyze.
I If dρd\neq\rho, since T:B¯ρd(x0,r)P(Xd)T:\bar{B}_{\rho}^{d}\left(x_{0},r\right)\rightarrow P\left(X^{d}\right) is a closed operator, then xTxx^{*}\in Tx^{*}.
II If d=ρd=\rho, we have that Dd(x,Tx)d(x,xn+1)+Dd(xn+1,Tx)d(x,xn+1)+Hd(Txn,Tx)=D_{d}\left(x^{*},Tx^{*}\right)\leq d\left(x^{*},x_{n+1}\right)+D_{d}\left(x_{n+1},Tx^{*}\right)\leq d\left(x^{*},x_{n+1}\right)+H_{d}\left(Tx_{n},Tx^{*}\right)=
d(x,xn+1)+Hρ(Txn,Tx)d\left(x^{*},x_{n+1}\right)+H_{\rho}\left(Tx_{n},Tx^{*}\right). It follows that

Dd(x,Tx)d(x,xn+1)+αd(xn,x)+βDd(xn,Tx)+γDd(x,Txn)+δDd(x,Tx)+\displaystyle D_{d}\left(x^{*},Tx^{*}\right)\leq d\left(x^{*},x_{n+1}\right)+\alpha d\left(x_{n},x^{*}\right)+\beta D_{d}\left(x_{n},Tx^{*}\right)+\gamma D_{d}\left(x^{*},Tx_{n}\right)+\delta D_{d}\left(x^{*},Tx^{*}\right)+
ηDd(x,Tx)[1+Dd(xn,Txn)]1+d(xn,x)+λDd(x,Tx)+Dd(x,Txn)1+Dd(x,Tx)Dd(x,Txn)+μDd(xn,Txn)[1+Dd(x,Txn)]1+d(xn,x)+Dd(x,Tx)\displaystyle\eta\frac{D_{d}\left(x^{*},Tx^{*}\right)\left[1+D_{d}\left(x_{n},Tx_{n}\right)\right]}{1+d\left(x_{n},x^{*}\right)}+\lambda\frac{D_{d}\left(x^{*},Tx^{*}\right)+D_{d}\left(x^{*},Tx_{n}\right)}{1+D_{d}\left(x^{*},Tx^{*}\right)\cdot D_{d}\left(x^{*},Tx_{n}\right)}+\mu\frac{D_{d}\left(x_{n},Tx_{n}\right)\left[1+D_{d}\left(x^{*},Tx_{n}\right)\right]}{1+d\left(x_{n},x^{*}\right)+D_{d}\left(x^{*},Tx^{*}\right)}\leq
d(x,xn+1)+αd(xn,x)+βDd(xn,Tx)+γDd(x,Txn)+δDd(x,Tx)+\displaystyle d\left(x^{*},x_{n+1}\right)+\alpha d\left(x_{n},x^{*}\right)+\beta D_{d}\left(x_{n},Tx^{*}\right)+\gamma D_{d}\left(x^{*},Tx_{n}\right)+\delta D_{d}\left(x^{*},Tx^{*}\right)+
ηDd(x,Tx)[1+d(xn,xn+1)]+λDd(x,Tx)+λd(x,xn+1)+μd(xn,xn+1)[1+d(x,xn)]\displaystyle\eta D_{d}\left(x^{*},Tx^{*}\right)\left[1+d\left(x_{n},x_{n+1}\right)\right]+\lambda D_{d}\left(x^{*},Tx^{*}\right)+\lambda d\left(x^{*},x_{n+1}\right)+\mu d\left(x_{n},x_{n+1}\right)\left[1+d\left(x^{*},x_{n}\right)\right]

Letting nn\rightarrow\infty, we get the following inequality

Dd(x,Tx)[β+δ+λ+η]Dd(x,Tx), i.e. Dd(x,Tx)(β+δ+λ+η)0.D_{d}\left(x^{*},Tx^{*}\right)\leq[\beta+\delta+\lambda+\eta]D_{d}\left(x^{*},Tx^{*}\right),\text{ i.e. }D_{d}\left(x^{*},Tx^{*}\right)(\beta+\delta+\lambda+\eta)\leq 0.

Now, since θ[0,1)\theta\in[0,1), as in the proofs of the previous theorems, it follows that β+δ+η+λα+2β+μ+η+λ+δ<1\beta+\delta+\eta+\lambda\leq\alpha+2\beta+\mu+\eta+\lambda+\delta<1. Moreover, because d=ρd=\rho and TT has closed values, we have that xTxx^{*}\in Tx^{*}.

Now, the last main result of this section involves a theorem regarding the homotopy of a modified multivalued Hardy-Rogers operator.

Theorem 4.6. Let ( X,dX,d ) be a complete metric space and d,ρd,\rho two metrics on XX such that there exists c>0c>0, with d(x,y)cρ(x,y)d(x,y)\leq c\rho(x,y), for each x,yXx,y\in X.

Let U(X,ρ)U\subset(X,\rho) an open subset and V(X,d)V\subset(X,d) a closed subset of XX, such that UVU\subset V.
Let’s consider the multivalued operator G:V×[0,1]P(X)G:V\times[0,1]\rightarrow P(X), which satisfies the following conditions :
(a) xG(x,t)x\notin G(x,t), for each xV\Ux\in V\backslash U and t[0,1]t\in[0,1]
(b) there exists α,β,γ,δ,λ,η,μ\alpha,\beta,\gamma,\delta,\lambda,\eta,\mu positive coefficients with θ[0,1)\theta\in[0,1) as in the previous theorem, such that for each t[0,1]t\in[0,1] and x,yVx,y\in V, we have that
Hρ(G(x,t),G(y,t))Mρ,G(,t)(x,y)H_{\rho}(G(x,t),G(y,t))\leq M_{\rho,G(\cdot,t)}(x,y), where

Mρ,G(,t)(x,y):=αρ(x,y)+βDρ(x,G(y,t))+γDρ(y,G(x,t))+δDρ(y,G(y,t))+\displaystyle M_{\rho,G(\cdot,t)}(x,y)=\alpha\rho(x,y)+\beta D_{\rho}(x,G(y,t))+\gamma D_{\rho}(y,G(x,t))+\delta D_{\rho}(y,G(y,t))+
ηDρ(y,G(y,t))[1+Dρ(x,G(x,t))]1+ρ(x,y)+λDρ(y,G(y,t))+Dρ(y,G(x,t))1+Dρ(y,G(y,t))Dρ(y,G(x,t))+\displaystyle\eta\frac{D_{\rho}(y,G(y,t))\left[1+D_{\rho}(x,G(x,t))\right]}{1+\rho(x,y)}+\lambda\frac{D_{\rho}(y,G(y,t))+D_{\rho}(y,G(x,t))}{1+D_{\rho}(y,G(y,t))D_{\rho}(y,G(x,t))}+
μDρ(x,G(x,t))[1+Dρ(y,G(x,t))]1+ρ(x,y)+Dρ(y,G(y,t))\displaystyle\mu\frac{D_{\rho}(x,G(x,t))\left[1+D_{\rho}(y,G(x,t))\right]}{1+\rho(x,y)+D_{\rho}(y,G(y,t))}

(c) there exists an increasing, continuous function ϕ:[0,1]\phi:[0,1]\rightarrow\mathbb{R}, such that
Hρ(G(x,t),G(x,s))|ϕ(t)ϕ(s)|H_{\rho}(G(x,t),G(x,s))\leq|\phi(t)-\phi(s)|, for each s,t[0,1]s,t\in[0,1] and xVx\in V
(d) G:V×[0,1]P(X,d)G:V\times[0,1]\rightarrow P(X,d) is a closed operator.

Then, we have the following equivalence relation
G(,0)G(\cdot,0) has a fixed point if and only if G(,1)G(\cdot,1) has a fixed point.
Proof. Let’s suppose that G(,0)G(\cdot,0) has a fixed point zz.
From the assumption (a), we get that zUz\in U.
Let’s denote Q:={(t,x)[0,1]×U/xG(x,t)}Q:=\{(t,x)\in[0,1]\times U/x\in G(x,t)\}. Then QQ is nonempty, because (0,z)Q(0,z)\in Q.
On the set QQ, we define a partial order relation, i.e.
(t,x)(s,y)(t,x)\leq(s,y) if and only if tst\leq s and ρ(x,y)21θ|ϕ(t)ϕ(s)|\rho(x,y)\leq\frac{2}{1-\theta}|\phi(t)-\phi(s)|, for each t,s[0,1]t,s\in[0,1] and x,yUx,y\in U.

Let MQM\subset Q, with MM being a totally ordered subset of Q .
Moreover, denote by t:=sup(t,x)Mtt^{*}:=\sup_{(t,x)\in M}t.
Now we define the sequence (tn,xn)M\left(t_{n},x_{n}\right)\in M, such that (tn,xn)(tn+1,xn+1)\left(t_{n},x_{n}\right)\leq\left(t_{n+1},x_{n+1}\right), with tntt_{n}\rightarrow t^{*}.
As in [5], we have that
ρ(xn,xm)21θ|ϕ(tm)ϕ(tn)|\rho\left(x_{n},x_{m}\right)\leq\frac{2}{1-\theta}\left|\phi\left(t_{m}\right)-\phi\left(t_{n}\right)\right|. This implies that ρ(xn,xm)0\rho\left(x_{n},x_{m}\right)\rightarrow 0 and therefore (xn)\left(x_{n}\right) is a Cauchy sequence with respect to the metric ρ\rho.

Using the fact that (X,d)(X,d) is a complete metric space and that there exists c>0c>0, such that for each x,yX,d(x,y)cρ(x,y)x,y\in X,d(x,y)\leq c\rho(x,y), we obtain that xnxx_{n}\rightarrow x^{*}, with x(X,d)x^{*}\in(X,d).

Since xnG(xn,tn)x_{n}\in G\left(x_{n},t_{n}\right) and GG is a dd-closed operator, we have that xG(x,t)x^{*}\in G\left(x^{*},t^{*}\right).
From assumption (a),xU(a),x^{*}\in U and therefore (t,x)Q\left(t^{*},x^{*}\right)\in Q.
Since MM is a totally ordered subset of Q , it follows that (t,x)(t,x)(t,x)\leq\left(t^{*},x^{*}\right), for each (t,x)M(t,x)\in M, so (t,x)\left(t^{*},x^{*}\right) is an upper bound for MM.

Using the well known Zorn’s Lemma, QQ admits a maximal element, i.e. (t0,x0)Q\left(t_{0},x_{0}\right)\in Q.
Now we show that t0=1t_{0}=1.
Let’s suppose the contrary, i.e. that t0<1t_{0}<1. We choose r>0r>0 and t(t0,1)t\in\left(t_{0},1\right) such that Bρ(x0,r)UB_{\rho}\left(x_{0},r\right)\subset U, with r:=21θ|ϕ(t)ϕ(t0)|r:=\frac{2}{1-\theta}\left|\phi(t)-\phi\left(t_{0}\right)\right|.
Then Dρ(x0,G(x0,t))Dρ(x0,G(x0,t0))+Hρ(G(x0,t0),G(x0,t))D_{\rho}\left(x_{0},G\left(x_{0},t\right)\right)\leq D_{\rho}\left(x_{0},G\left(x_{0},t_{0}\right)\right)+H_{\rho}\left(G\left(x_{0},t_{0}\right),G\left(x_{0},t\right)\right)\leq
|ϕ(t)ϕ(t0)|+Dρ(x0,G(x0,t0))\left|\phi(t)-\phi\left(t_{0}\right)\right|+D_{\rho}\left(x_{0},G\left(x_{0},t_{0}\right)\right).

Since (t0,x0)Q\left(t_{0},x_{0}\right)\in Q, it implies that x0G(x0,t0)x_{0}\in G\left(x_{0},t_{0}\right), therefore Dρ(x0,G(x0,t0))=0D_{\rho}\left(x_{0},G\left(x_{0},t_{0}\right)\right)=0.
So Dρ(x0,G(x0,t))|ϕ(t)ϕ(t0)|(1θ)r2<(1θ)rD_{\rho}\left(x_{0},G\left(x_{0},t\right)\right)\leq\left|\phi(t)-\phi\left(t_{0}\right)\right|\leq\frac{(1-\theta)r}{2}<(1-\theta)r.
But we know that B¯ρd(x0,r)V\bar{B}_{\rho}^{d}\left(x_{0},r\right)\subset V, so G:B¯ρd(x0,r)Pcl(X)G:\bar{B}_{\rho}^{d}\left(x_{0},r\right)\rightarrow P_{cl}(X) satisfies the assumptions of the previous theorem, therefore for each t[0,1]t\in[0,1], there exists xB¯ρd(x0,r)x\in\bar{B}_{\rho}^{d}\left(x_{0},r\right) satisfying the property that GG has a fixed point, that is xG(x,t)x\in G(x,t), which implies that (t,x)Q(t,x)\in Q.
But ρ(x0,x)21θ|ϕ(t)ϕ(t0)|=r\rho\left(x_{0},x\right)\leq\frac{2}{1-\theta}\left|\phi(t)-\phi\left(t_{0}\right)\right|=r. This means that (t0,x0)<(t,x)\left(t_{0},x_{0}\right)<(t,x), which is a contraction. So t0=1t_{0}=1.

For the other implication, we show that G(,1)G(\cdot,1) has a fixed point by swapping tt with 1t1-t in the first part of the proof. So, we get the desired result.

When the metric functionals dd and ρ\rho are identical, we have the following corollary.
Corollary 4.7. Let UVXU\subset V\subset X, with ( X,dX,d ) a complete metric space, UU open and VV closed.
Let G:V×[0,1]P(X)G:V\times[0,1]\rightarrow P(X) a closed operator, satisfying the following assumptions
(a) xG(x,t)x\notin G(x,t), for each xV\Ux\in V\backslash U and t[0,1]t\in[0,1]
(b) G(,t)G(\cdot,t) be a Hardy Rogers modified multivalued contraction with respect to dd, for each t[0,1]t\in[0,1]
(c) Hd(G(x,t),G(x,s))|ϕ(t)ϕ(s)|H_{d}(G(x,t),G(x,s))\leq|\phi(t)-\phi(s)|, for each t,s[0,1]t,s\in[0,1] and for each xVx\in V, with ϕ:[0,1]\phi:[0,1]\rightarrow\mathbb{R} increasing and continuous.

Then
G(,0)G(\cdot,0) has a fixed point if and only if G(,1)G(\cdot,1) has a fixed point.

Conflict of Interests

The authors declare that there is no conflict of interests.

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