Fixed point results for non-self operators R₊^{m}-metric spaces

Diana Otrocol
(original), Fixed point theory, html, JCR/ISI, Nonlinear analysis, paper, PDF, works

Abstract

The purpose of this paper is to discuss some problems of the fixed point theory for non-self operators on \({\mathbb R}^m_+\)-metric spaces. The results complement and extend some known results given in the paper: A. Chis-Novac, R. Precup, I.A. Rus, Data dependence of fixed points for non-self generalized contractions, Fixed Point Theory, 10(2009), No. 1, 73–87.

Authors

Veronica Ilea
Babes–Bolyai University, Department of Mathematics, Cluj-Napoca, Romania

Adela Novac
Technical University of Cluj-Napoca, Department of Mathematics, Cluj-Napoca, Romania

Diana Otrocol
Technical University of Cluj-Napoca, Department of Mathematics, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

Diana Otrocol

Keywords

\({\mathbb R}^m_+\)–metric spaces; fixed point; Picard operator; non-self operator; data dependence of the fixed point.

Paper coordinates

V. Ilea, A. Novac, D. Otrocol, Fixed point results for non-self operators R₊m-metric spaces, Fixed Point Theory, 26 (2025) no. 1, 177-188,
https://doi.org/10.24193/fpt-ro.2025.1.10

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About this paper

Journal

Fixed Point Theory

Publisher Name

House of the Book of Science Cluj-Napoca, Romania

DOI

https://doi.org/10.24193/fpt-ro.2025.1.10

Print ISSN

1583-5022

Online ISSN

2066-9208

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[4] M. Frigon, Fixed point and continuation results for contractions and Gauge spaces, Banach Center Publications, 77(2007), 89-114.
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[6] V. Ilea, D. Otrocol, I.A. Rus, M.A. Serban, Applications of fibre contraction principle to some classes of functional integral equations, Fixed Point Theory, 23(2022), no. 1, 279-292.
[7] J.M. Ortega, W.C. Reinboldt, On a class of approximative iterative processes, Arch. Rat. Mech. Anal., 23(1967), 352-365.
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Fixed Point Theory, 26(2025), No.1, 177-188

DOI: 10.24193/fpt-ro.2022.1.XX

http://www.math.ubbcluj.ro/∼nodeacj/sfptcj.html

Fixed point results for non-self operators on ℝ+m\mathbb{R}_{+}^{m}-metric spaces

Veronica Ilea∗, Adela Novac∗∗, Diana Otrocol∗∗

∗Babeş–Bolyai University, Department of Mathematics, 1 M. Kogălniceanu Street, 400084 Cluj-Napoca, Romania
E-mail: veronica.ilea@ubbcluj.ro
∗∗Technical University of Cluj-Napoca, Department of Mathematics, 28 Memorandumului Street, 400114 Cluj-Napoca, Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O.Box. 68-1, 400110 Cluj-Napoca, Romania
E-mail: adela.novac@math.utcluj.ro, diana.otrocol@math.utcluj.ro

Abstract. The purpose of this paper is to discuss some problems of the fixed point theory for non-self operators on ℝ+m\mathbb{R}_{+}^{m}-metric spaces. The results complement and extend some known results given in the paper: A. Chis-Novac, R. Precup, I.A. Rus, Data dependence of fixed points for non-self generalized contractions, Fixed Point Theory, 10(2009), No. 1, 73–87.
Key Words and Phrases: ℝ+m\mathbb{R}_{+}^{m}-metric spaces, fixed point, Picard operator, non-self operator, data dependence of the fixed point.

2020 Mathematics Subject Classification: 47H10, 54H25.

1. Introduction

1.1. Notations

We begin the introduction by some standard notations that will be used throughout the paper.

Let (X,d)(X,d) be a ℝ+m\mathbb{R}_{+}^{m}-metric space, Y⊂XY\subset X a nonempty subset of XX and f:Y→Xf:Y\rightarrow X an operator. In what follow we shall use the following notations:

Ff={x∈Y:f​(x)=x}F_{f}=\{x\in Y:f(x)=x\} - the fixed points set of f.f.

Pc​l​(X)={Y⊂X|Y​ is closed}P_{cl}(X)=\{Y\subset X|Y\text{ is closed}\}

I​(f)={Z⊂Y:f​(Z)⊂Z,Z≠∅}I(f)=\{Z\subset Y:f(Z)\subset Z,Z\neq\varnothing\} - the set of invariant subsets of f.f.

(M​I)f=∪I​(f)(MI)_{f}=\cup I(f)- the maximal invariant subset of f.f.

1.2. Non-self operators on LL-spaces

In what follow we denote an LL-space by (X,→𝐹)(X,\overset{F}{\rightarrow}). Let Y⊂XY\subset X a nonempty subset of XX and f:Y→Xf:Y\rightarrow X an operator. Throughout this paper we consider that Y∈Pc​l​(X)Y\in P_{cl}(X).

(AB)f(x∗)={x∈Y:fn(x)(AB)_{f}(x^{\ast})=\{x\in Y:f^{n}(x) is defined for all n∈ℕn\in\mathbb{N} and fn(x)→x∗∈Ff}f^{n}(x)\rightarrow x^{\ast}\in F_{f}\}-the attraction basin of the fixed point x∗x^{\ast} with respect to f.f.

(B​A)f=∪x∗∈Ff​(A​B)f​(x∗)(BA)_{f}=\underset{x^{\ast}\in F_{f}}{\cup}(AB)_{f}(x^{\ast})- the attraction basin of f.f.

Following [3] we have:

Definition 1.1.

An operator f:Y→Xf:Y\rightarrow X is said to be a Picard operator (PO) if

(i) Ff={xf∗};F_{f}=\{x_{f}^{\ast}\};

(ii) (M​I)f=(B​A)f.(MI)_{f}=(BA)_{f}.

Definition 1.2.

An operator f:Y→Xf:Y\rightarrow X is said to be a weakly Picard operator (WPO) if

(i) Ff≠∅;F_{f}\neq\emptyset;

(ii) (M​I)f=(B​A)f.(MI)_{f}=(BA)_{f}.

Definition 1.3.

For each WPO f:Y→X\,f:Y\rightarrow X we define the operator  f∞:(B​A)f→(B​A)ff^{\infty}:\left(BA\right)_{f}\rightarrow\left(BA\right)_{f}  by f∞​(x)=limn→∞fn​(x).f^{\infty}(x)=\lim\limits_{n\rightarrow\infty}f^{n}(x).

Remark 1.4.

It is clear that f∞​((B​A)f)=Ff,f^{\infty}((BA)_{f})=F_{f}, so f∞f^{\infty} is a set retraction of (B​A)f(BA)_{f} to Ff.F_{f}.

Remark 1.5.

In terms of weakly Picard self operators the above definitions take the following form:

f:Y→X​ is a PO iff ​f∣(M​I)f:(M​I)f→(M​I)f​ is a PO.f:Y\rightarrow X\,\text{ is a PO \ \ iff\thinspace}\;\;f\mid_{(MI)_{f}}:(MI)_{f}\rightarrow(MI)_{f}\text{ is a PO.}
Remark 1.6.

We have the above notions in each distance structures which induces an LL-space convergence (ℝ+m\mathbb{R}_{+}^{m}-metrics, s​(ℝ+)s(\mathbb{R}_{+}) metrics, KK-metrics, partial metrics, dislocated metrics, …).

For other results on Picard and weakly Picard operators see [1], [2], [4], [8], [9], [11].

1.3. Operators on ℝ+m\mathbb{R}_{+}^{m}-metric spaces

Definition 1.7.

An operator f:X→Xf:X\rightarrow X is an SS-contraction if there exists S∈ℝ+m×mS\in\mathbb{R}_{+}^{m\times m} such that:

  • (i)

    SS is a convergent to zero matrix, i.e. Sn→0S^{n}\rightarrow 0 asn→∞\ n\rightarrow\infty;

  • (ii)

    d​(f​(x),f​(y))≤S​d​(x,y),d(f(x),f(y))\leq Sd(x,y), for all x,y∈Xx,y\in X.

The following results, in ℝ+m\mathbb{R}_{+}^{m}-metric spaces were well known.

Theorem 1.8.

(Saturated Perov-Schröder Theorem) Let (X,d)(X,d) be a complete ℝ+m\mathbb{R}_{+}^{m}-metric space. We suppose that, f:X→Xf:X\rightarrow X and S∈ℝ+m×mS\in\mathbb{R}_{+}^{m\times m} are such that:

  1. (1)

    SS is a matrix convergent to matrix 0;0;

  2. (2)

    d​(f​(x),f​(y))≤S​d​(x,y),∀x,y∈X.d(f(x),f(y))\leq Sd(x,y),\ \forall x,y\in X.

Then:

  1. (i)

    Ff=Ffn={x∗},∀n∈ℕ∗;F_{f}=F_{f^{n}}=\{x^{\ast}\},\ \forall n\in\mathbb{N}^{\ast};

  2. (ii)

    ff is Picard mapping, i.e., fn​(x)→x∗f^{n}(x)\rightarrow x^{\ast} as n→∞,∀x∈X;n\rightarrow\infty,\ \forall x\in X;

  3. (iii)

    d​(x,x∗)≤(I−S)−1​d​(x,f​(x)),∀x∈X.d(x,x^{\ast})\leq(I-S)^{-1}d(x,f(x)),\ \ \forall x\in X.

  4. (iv)

    the fixed point equation, x=f​(x)x=f(x) is Ulam-Hyers stable;

  5. (v)

    if xn∈X,d​(xn,f​(xn))→0x_{n}\in X,\ d(x_{n},f(x_{n}))\rightarrow 0 as n→∞,n\rightarrow\infty, then, xn→x∗x_{n}\rightarrow x^{\ast} as n→∞,n\rightarrow\infty, i.e. the fixed point problem for ff is well posed;

  6. (vi)

    if xn∈X,n∈ℕx_{n}\in X,n\in\mathbb{N} are such that

    d​(xn+1,f​(xn))→0​ as ​n→∞,d(x_{n+1},f(x_{n}))\rightarrow 0\text{ as }n\rightarrow\infty,

    then for all x∈Xx\in X we have

    d​(xn,x∗)→0​ as ​n→∞,d(x_{n},x^{\ast})\rightarrow 0\text{ as }n\rightarrow\infty,

    i.e. ff has the Ostrowski property.

For this result and other results on fixed point theory in a ℝ+m\mathbb{R}_{+}^{m}-metric space see [6], [7], [10], [12].

The aim of this paper is to complement and extend the mentioned results in the case of non-self operators.

2. Metric conditions on non-self operators on ℝ+m\mathbb{R}_{+}^{m}-metrics and fixed points

Let (X,d)(X,d) be a ℝ+m\mathbb{R}_{+}^{m}-metric space, Y⊂XY\subset X a nonempty subset and f:Y→Xf:Y\rightarrow X an operator.

Definition 2.1.

The operators ff is an SS-contraction if S∈ℝ+m×mS\in\mathbb{R}_{+}^{m\times m} and ff is such that

  • (i)

    Sn→0S^{n}\rightarrow 0 asn→∞\ n\rightarrow\infty, i.e. SS is a matrix convergent to 0;

  • (ii)

    d​(f​(x),f​(y))≤S​d​(x,y),d(f(x),f(y))\leq Sd(x,y), for all x,y∈Yx,y\in Y.

Definition 2.2.

The operator ff is a graphic SS-contraction if S∈ℝ+m×mS\in\mathbb{R}_{+}^{m\times m} is convergent to 0 and

d​(f​(x),f2​(x))≤S​d​(x,f​(x)),∀x∈Y.d(f(x),f^{2}(x))\leq Sd(x,f(x)),\forall x\in Y.
Definition 2.3.

The operator ff is a quasi SS-contraction if Ff={x∗},S∈ℝ+m×mF_{f}=\{x^{\ast}\},\ S\in\mathbb{R}_{+}^{m\times m} is convergent to 0 and

d​(f​(x),x∗)≤S​d​(x,x∗),∀x∈Y.d(f(x),x^{\ast})\leq Sd(x,x^{\ast}),\forall x\in Y.

By definition ff satisfies a retraction-displacement condition if Ff={x∗}F_{f}=\{x^{\ast}\} and there exists an increasing function ψ:ℝ+m→ℝ+m,ψ​(0)=0,\psi:\mathbb{R}_{+}^{m}\rightarrow\mathbb{R}_{+}^{m},\psi(0)=0, and is continuous in 0,0, such that

d​(x,x∗)≤ψ​(d​(x,f​(x))),∀x∈Y.d(x,x^{\ast})\leq\psi\left(d(x,f(x))\right),\forall x\in Y.
Definition 2.4.

Let ψ\psi defined as in Definition 2.3. By definition ff is ψ\psi-PO if ff is PO with respect to →𝑑,\overset{d}{\rightarrow}, and

d​(x,x∗)≤ψ​(d​(x,f​(x))),∀x∈(M​I)f.d(x,x^{\ast})\leq\psi\left(d(x,f(x))\right),\forall x\in(MI)_{f}.

In the terms of the above metric conditions we have the following results.

Theorem 2.5.

Let (X,d)(X,d) be an ℝ+m\mathbb{R}_{+}^{m}-metric space, Y⊂XY\subset X a nonempty subset and f:Y→Xf:Y\rightarrow X be a SS-contraction with Ff={x∗}F_{f}=\{x^{\ast}\}. Then:

  • (i)

    d​(x,x∗)≤(I−S)−1​d​(x,f​(x)),∀x∈Yd(x,x^{\ast})\leq(I-S)^{-1}d(x,f(x)),\forall x\in Y;

  • (ii)

    if g:Y→Xg:Y\rightarrow X is such that d​(f​(x),g​(x))≤η,∀x∈Y,d(f(x),g(x))\leq\eta,\forall x\in Y, for some η∈(ℝ+∗)m,\eta\in\left(\mathbb{R}_{+}^{\ast}\right)^{m}, then

    d​(y∗,x∗)≤(I−S)−1​η,∀y∗∈Fg;d(y^{\ast},x^{\ast})\leq(I-S)^{-1}\eta,\ \forall y^{\ast}\in F_{g};
  • (iii)

    if y∈Yy\in Y is such that

    d​(y,f​(y))≤ε,d(y,f(y))\leq\varepsilon,

    for some ε∈(ℝ+∗)m,\varepsilon\in\left(\mathbb{R}_{+}^{\ast}\right)^{m}, then

    d​(y,x∗)≤(I−S)−1​ε,d(y,x^{\ast})\leq(I-S)^{-1}\varepsilon,

    i.e., the equation x=f​(x)x=f(x) is Ulam-Hyers stable;

  • (iv)

    xn∈Y,n∈ℕ,d​(xn,f​(xn))→0x_{n}\in Y,\ n\in\mathbb{N},\ d(x_{n},f(x_{n}))\rightarrow 0 as n→∞n\rightarrow\infty implies that xn→x∗x_{n}\rightarrow x^{\ast}as n→∞,n\rightarrow\infty, i.e. the fixed point problem for ff is well posed.

  • (v)

    xn∈Y,n∈ℕ,d​(xn+1,f​(xn))→0x_{n}\in Y,\ n\in\mathbb{N},\ d(x_{n+1},f(x_{n}))\rightarrow 0 as n→∞n\rightarrow\infty implies that xn→x∗x_{n}\rightarrow x^{\ast}as n→∞,n\rightarrow\infty, i.e. ff has the Ostrowski property.

Proof.

(i) We have for x∈Y,x\in Y,

d​(x,x∗)≤d​(x,f​(x))+d​(f​(x),x∗)≤d​(x,f​(x))+S​d​(x,x∗).d(x,x^{\ast})\leq d(x,f(x))+d(f(x),x^{\ast})\leq d(x,f(x))+Sd(x,x^{\ast}).

From this it follows

(I−S)​d​(x,x∗)≤d​(x,f​(x)).(I-S)d(x,x^{\ast})\leq d(x,f(x)).

Since SS is a matrix convergent to 0, there exists (I−S)−1(I-S)^{-1} and (I−S)−1≥0(I-S)^{-1}\geq 0, i.e., the corresponding function, (I−S)−1:ℝ+m×m→ℝ+m,η↦(I−S)−1​η(I-S)^{-1}:\mathbb{R}_{+}^{m\times m}\rightarrow\mathbb{R}_{+}^{m},\ \eta\mapsto(I-S)^{-1}\eta is increasing, with value 0 at 0 and continuous. So we have (i)(i), a retraction-displacement condition.

(ii) From (i),(i),

d​(y∗,x∗)\displaystyle d(y^{\ast},x^{\ast}) ≤(I−S)−1​d​(y∗,f​(y∗))=(I−S)−1​d​(g​(y∗),f​(y∗))\displaystyle\leq(I-S)^{-1}d(y^{\ast},f(y^{\ast}))=(I-S)^{-1}d(g(y^{\ast}),f(y^{\ast}))
≤(I−S)−1​η.\displaystyle\leq(I-S)^{-1}\eta.

(iii) From (i),(i),

d​(y,x∗)≤(I−S)−1​d​(y,f​(y))≤(I−S)−1​ε.d(y,x^{\ast})\leq(I-S)^{-1}d(y,f(y))\leq(I-S)^{-1}\varepsilon.

(iv) The proof follows directly from (i).

(v) Since f:Y→Xf:Y\rightarrow X is an SS-contraction with Ff={x∗}F_{f}=\{x^{\ast}\} it follows that, ff is a quasi SS-contraction. So, we have,

d​(xn+1,x∗)\displaystyle d(x_{n+1},x^{\ast}) ≤d​(xn+1,f​(xn))+d​(f​(xn),x∗)≤\displaystyle\leq d(x_{n+1},f(x_{n}))+d(f(x_{n}),x^{\ast})\leq
≤d​(xn+1,f​(xn))+S​d​(xn,x∗)≤\displaystyle\leq d(x_{n+1},f(x_{n}))+Sd(x_{n},x^{\ast})\leq
≤d​(xn+1,f​(xn))+S​d​(xn,f​(xn−1))+S2​d​(xn−1,x∗)≤…≤\displaystyle\leq d(x_{n+1},f(x_{n}))+Sd(x_{n},f(x_{n-1}))+S^{2}d(x_{n-1},x^{\ast})\leq\ldots\leq
≤d​(xn+1,f​(xn))+S​d​(xn,f​(xn−1))+…+Sn​d​(x1,f​(x0))+Sn+1​d​(x0,x∗).\displaystyle\leq d(x_{n+1},f(x_{n}))+Sd(x_{n},f(x_{n-1}))+\ldots+S^{n}d(x_{1},f(x_{0}))+S^{n+1}d(x_{0},x^{\ast}).

So, d​(xn+1,x∗)→0d(x_{n+1},x^{\ast})\rightarrow 0 as n→∞n\rightarrow\infty, by a Cauchy-Toeplitz lemma (see [13]). ∎

Theorem 2.6.

Let (X,d)(X,d) be an ℝ+m\mathbb{R}_{+}^{m}-metric space, Y⊂XY\subset X a nonempty subset and f:Y→Xf:Y\rightarrow X be an operator such that,

d​(f​(x),f​(y))≤P​d​(x,f​(x))+Q​d​(y,f​(y))+R​d​(x,y),d(f(x),f(y))\leq Pd(x,f(x))+Qd(y,f(y))+Rd(x,y), (2.1)

for all x,y∈Y,x,y\in Y, where P,Q,R∈ℝ+m×mP,Q,R\in\mathbb{R}_{+}^{m\times m}. We suppose that, Ff={x∗}F_{f}=\{x^{\ast}\} and (I−R)−1≥0.(I-R)^{-1}\geq 0. Then we have that:

  • (i)

    d​(x,x∗)≤C​d​(x,f​(x)),∀x∈Yd(x,x^{\ast})\leq Cd(x,f(x)),\ \forall x\in Y, where C:=(I−R)−1​(I+P)C:=(I-R)^{-1}(I+P);

  • (ii)

    if g:Y→Xg:Y\rightarrow X is such that d​(f​(x),g​(x))≤η,∀x∈Y,d(f(x),g(x))\leq\eta,\forall x\in Y, for some η∈(ℝ+∗)m,\eta\in\left(\mathbb{R}_{+}^{\ast}\right)^{m}, then

    d​(y∗,x∗)≤C​η,∀y∗∈Fg;d(y^{\ast},x^{\ast})\leq C\eta,\ \forall y^{\ast}\in F_{g};
  • (iii)

    if y∈Yy\in Y is such that

    d​(y,f​(y))≤ε,d(y,f(y))\leq\varepsilon,

    for some ε∈(ℝ+∗)m,\varepsilon\in\left(\mathbb{R}_{+}^{\ast}\right)^{m}, then

    d​(y,x∗)≤C​ε,d(y,x^{\ast})\leq C\varepsilon,

    i.e., the equation x=f​(x)x=f(x) is Ulam-Hyers stable;

  • (iv)

    xn∈Y,n∈ℕ,d​(xn,f​(xn))→0x_{n}\in Y,\ n\in\mathbb{N},\ d(x_{n},f(x_{n}))\rightarrow 0 as n→∞n\rightarrow\infty implies that xn→x∗x_{n}\rightarrow x^{\ast}as n→∞,n\rightarrow\infty, i.e. the fixed point problem for ff is well posed.

Proof.

(i) We have for x∈Y,x\in Y,

d​(x,x∗)\displaystyle d(x,x^{\ast}) ≤d​(x,f​(x))+d​(f​(x),x∗)≤d​(x,f​(x))+d​(f​(x),f​(x∗))\displaystyle\leq d(x,f(x))+d(f(x),x^{\ast})\leq d(x,f(x))+d(f(x),f(x^{\ast}))
≤d​(x,f​(x))+P​d​(x,f​(x))+Q​d​(x∗,f​(x∗))+R​d​(x,x∗).\displaystyle\leq d(x,f(x))+Pd(x,f(x))+Qd(x^{\ast},f(x^{\ast}))+Rd(x,x^{\ast}).

From this it follows

(I−R)​d​(x,x∗)≤(I+P)​d​(x,f​(x)),∀x∈Y.(I-R)d(x,x^{\ast})\leq(I+P)d(x,f(x)),\forall x\in Y.

So, d​(x,x∗)≤(I−R)−1​(I+P)​d​(x,f​(x))d(x,x^{\ast})\leq(I-R)^{-1}(I+P)d(x,f(x)).

(ii) By applying (i) to y∗∈Fgy^{\ast}\in F_{g}

d​(y∗,x∗)\displaystyle d(y^{\ast},x^{\ast}) ≤(I−R)−1​(I+P)​d​(y∗,f​(y∗))\displaystyle\leq(I-R)^{-1}(I+P)d(y^{\ast},f(y^{\ast}))
=(I−R)−1​(I+P)​d​(g​(y∗),f​(y∗))\displaystyle=(I-R)^{-1}(I+P)d(g(y^{\ast}),f(y^{\ast}))
≤(I−R)−1​(I+P)​η,∀y∗∈Fg.\displaystyle\leq(I-R)^{-1}(I+P)\eta,\forall y^{\ast}\in F_{g}.

(iii) We apply again (i) and obtain

d​(y,x∗)≤(I−R)−1​(I+P)​d​(y,f​(y))=(I−R)−1​(I+P)​ε,∀y∈Y.d(y,x^{\ast})\leq(I-R)^{-1}(I+P)d(y,f(y))=(I-R)^{-1}(I+P)\varepsilon,\forall y\in Y.

(iv) Let

d​(xn,x∗)\displaystyle d(x_{n},x^{\ast}) ≤d​(xn,f​(xn))+d​(f​(xn),f​(x∗))\displaystyle\leq d(x_{n},f(x_{n}))+d(f(x_{n}),f(x^{\ast}))
≤d​(xn,f​(xn))+P​d​(xn,f​(xn))+Q​d​(x∗,f​(x∗))+R​d​(xn,x∗)\displaystyle\leq d(x_{n},f(x_{n}))+Pd(x_{n},f(x_{n}))+Qd(x^{\ast},f(x^{\ast}))+Rd(x_{n},x^{\ast})

We have

(I−R)​d​(xn,x∗)≤(I+P)​d​(xn,f​(xn)),(I-R)d(x_{n},x^{\ast})\leq(I+P)d(x_{n},f(x_{n})),

so, this implies that xn→x∗x_{n}\rightarrow x^{\ast}as n→∞,n\rightarrow\infty, i.e. the fixed point problem for ff is well posed. ∎

Theorem 2.7.

Let (X,d)(X,d) be an ℝ+m\mathbb{R}_{+}^{m}-metric space, Y⊂XY\subset X a nonempty subset and f:Y→Xf:Y\rightarrow X be an operator such that,

d​(f​(x),f​(y))≤P​d​(x,f​(x))+Q​d​(y,f​(y))+R​d​(x,y),d(f(x),f(y))\leq Pd(x,f(x))+Qd(y,f(y))+Rd(x,y), (2.2)

for all x,y∈Y,x,y\in Y, where P,Q,R∈ℝ+mP,Q,R\in\mathbb{R}_{+}^{m}. We suppose that, (I−P)−1≥0(I-P)^{-1}\geq 0 and the matrix (I−P)−1​(P+R)(I-P)^{-1}(P+R) is convergent to 0. Then we have that:

  • (i)

    d​(f​(x),x∗)≤(I−P)−1​(P+R)​d​(x,x∗),∀x∈Yd(f(x),x^{\ast})\leq(I-P)^{-1}(P+R)d(x,x^{\ast}),\ \forall x\in Y, i.e., ff is a quasi contraction;

  • (ii)

    xn∈Y,n∈ℕ,d​(xn+1,f​(xn))→0x_{n}\in Y,\ n\in\mathbb{N},\ d(x_{n+1},f(x_{n}))\rightarrow 0 as n→∞n\rightarrow\infty implies that xn→x∗x_{n}\rightarrow x^{\ast} as n→∞n\rightarrow\infty.

Proof.
  • (i)
    d​(f​(x),x∗)=d​(f​(x),f​(x∗))\displaystyle d(f(x),x^{\ast})=d(f(x),f(x^{\ast})) ≤P​d​(x,f​(x))+Q​d​(x∗,f​(x∗))+R​d​(x,x∗)\displaystyle\leq Pd(x,f(x))+Qd(x^{\ast},f(x^{\ast}))+Rd(x,x^{\ast})
    ≤P​[d​(x,x∗)+d​(x∗,f​(x))]+R​d​(x,x∗).\displaystyle\leq P[d(x,x^{\ast})+d(x^{\ast},f(x))]+Rd(x,x^{\ast}).
    (I−P)​d​(f​(x),x∗)≤(P+R)​d​(x,x∗).(I-P)d(f(x),x^{\ast})\leq(P+R)d(x,x^{\ast}).

    Thus, follows the conclusion.

  • (ii)

    Let

    d​(xn+1,x∗)\displaystyle d(x_{n+1},x^{\ast}) ≤d​(xn+1,f​(xn))+d​(f​(xn),x∗)\displaystyle\leq d(x_{n+1},f(x_{n}))+d(f(x_{n}),x^{\ast})
    ≤d​(xn+1,f​(xn))+(I−P)−1​(P+R)​d​(xn,x∗)\displaystyle\leq d(x_{n+1},f(x_{n}))+(I-P)^{-1}(P+R)d(x_{n},x^{\ast})

    We have

    d​(xn+1,x∗)\displaystyle d(x_{n+1},x^{\ast}) ≤(I−P)−1​(P+R)​d​(xn,x∗)\displaystyle\leq(I-P)^{-1}(P+R)d(x_{n},x^{\ast})
    ≤(I−P)−1​(P+R)​[d​(xn,f​(xn−1))+d​(f​(xn−1,x∗))]\displaystyle\leq(I-P)^{-1}(P+R)\left[d(x_{n},f(x_{n-1}))+d(f(x_{n-1},x^{\ast}))\right]
    ≤[(I−P)−1​(P+R)]2​d​(xn−1,x∗)\displaystyle\leq\left[(I-P)^{-1}(P+R)\right]^{2}d(x_{n-1},x^{\ast})
    ≤…\displaystyle\leq\ldots
    ≤[(I−P)−1​(P+R)]n+1​d​(x0,x∗),\displaystyle\leq\left[(I-P)^{-1}(P+R)\right]^{n+1}d(x_{0},x^{\ast}),

    so, this implies that xn+1→x∗x_{n+1}\rightarrow x^{\ast} as n→∞n\rightarrow\infty.

∎

By a similar proofs as above we have the following results:

Theorem 2.8.

Let (X,d)(X,d) be an ℝ+m\mathbb{R}_{+}^{m}-metric space and Y⊂XY\subset X a nonempty subset. We suppose that f:Y→Xf:Y\rightarrow X is a ψ\psi-PO with Ff={x∗}.F_{f}=\{x^{\ast}\}. Then we have that:

  • (i)

    if g:Y→Xg:Y\rightarrow X is such that d​(f​(x),g​(x))≤η,∀x∈Y,d(f(x),g(x))\leq\eta,\forall x\in Y, for some η∈(ℝ+∗)m,\eta\in\left(\mathbb{R}_{+}^{\ast}\right)^{m}, then

    d​(y∗,x∗)≤ψ​(η),∀y∗∈Fg∩(M​I)f;d(y^{\ast},x^{\ast})\leq\psi(\eta),\ \forall y^{\ast}\in F_{g}\cap(MI)_{f};
  • (ii)

    if y∈Y∩(M​I)fy\in Y\cap(MI)_{f} is such that

    d​(y,f​(y))≤ε,d(y,f(y))\leq\varepsilon,

    for some ε∈(ℝ+∗)m,\varepsilon\in\left(\mathbb{R}_{+}^{\ast}\right)^{m}, then

    d​(y,x∗)≤ψ​(ε),d(y,x^{\ast})\leq\psi(\varepsilon),

    i.e., the equation x=f​(x)x=f(x) is Ulam-Hyers stable;

  • (iii)

    xn∈(M​I)f,n∈ℕ,d​(xn,f​(xn))→0x_{n}\in(MI)_{f},\ n\in\mathbb{N},\ d(x_{n},f(x_{n}))\rightarrow 0 as n→∞n\rightarrow\infty implies that xn→x∗x_{n}\rightarrow x^{\ast}as n→∞,n\rightarrow\infty, i.e. the fixed point problem for ff is well posed.

Proof.
  • (i)
    d​(y∗,x∗)\displaystyle d(y^{\ast},x^{\ast}) ≤ψ​(d​(y∗,f​(y∗)))=ψ​(d​(f​(y∗),g​(y∗)))\displaystyle\leq\psi(d(y^{\ast},f(y^{\ast})))=\psi(d(f(y^{\ast}),g(y^{\ast})))
    ≤ψ​(η).\displaystyle\leq\psi(\eta).
  • (ii)
    d​(y,x∗)≤ψ​(d​(y,f​(y)))≤ψ​(ε)d(y,x^{\ast})\leq\psi(d(y,f(y)))\leq\psi(\varepsilon)
  • (iii)
    d​(xn,x∗)≤ψ​(d​(xn,f​(xn)))​→n→∞​ψ​(0)=0d(x_{n},x^{\ast})\leq\psi(d(x_{n},f(x_{n})))\underset{n\rightarrow\infty}{\rightarrow}\psi(0)=0

    So xn→x∗x_{n}\rightarrow x^{\ast}as n→∞.n\rightarrow\infty.

∎

Theorem 2.9.

Let (X,d)(X,d) be an ℝ+m\mathbb{R}_{+}^{m}-metric space and Y⊂XY\subset X a nonempty subset. We suppose that f:Y→Xf:Y\rightarrow X is a quasi SS-contraction. Then the following implication holds:

xn∈Y,n∈ℕ,d​(xn+1,f​(xn))→0​as ​n→∞​ implies that ​xn→x∗​as ​n→∞.x_{n}\in Y,\ n\in\mathbb{N},\ d(x_{n+1},f(x_{n}))\rightarrow 0\ \text{as\ }n\rightarrow\infty\text{\ implies that }x_{n}\rightarrow x^{\ast}\ \text{as\ }n\rightarrow\infty.
Proof.

The conclusion follows from Theorem 2.7. ∎

Remark 2.10.

The Theorem 2.6 and Theorem 2.7, in the case P:=α∈ℝ+,Q:=β∈ℝ+,R:=γ∈ℝ+P:=\alpha\in\mathbb{R}_{+},\ Q:=\beta\in\mathbb{R}_{+},R:=\gamma\in\mathbb{R}_{+}, take the following form:

Theorem 2.11.

Let (X,d)(X,d) be an ℝ+\mathbb{R}_{+}-metric space, Y⊂XY\subset X a nonempty subset and f:Y→Xf:Y\rightarrow X be an operator such that,

d​(f​(x),f​(y))≤α​d​(x,f​(x))+β​d​(y,f​(y))+γ​d​(x,y),d(f(x),f(y))\leq\alpha d(x,f(x))+\beta d(y,f(y))+\gamma d(x,y), (2.3)

for all x,y∈Y,x,y\in Y, where α,β,γ∈ℝ+\alpha,\beta,\gamma\in\mathbb{R}_{+}. We suppose that, Ff={xf∗}F_{f}=\{x_{f}^{\ast}\} and γ<1.\gamma<1. Then we have that:

  • (i)

    d​(x,x∗)≤C​d​(x,f​(x)),∀x∈Yd(x,x^{\ast})\leq Cd(x,f(x)),\ \forall x\in Y, where C:=1+α1−γC:=\dfrac{1+\alpha}{1-\gamma};

  • (ii)

    if g:Y→Xg:Y\rightarrow X is such that d​(f​(x),g​(x))≤η,∀x∈Y,d(f(x),g(x))\leq\eta,\forall x\in Y, for some η∈ℝ+∗,\eta\in\mathbb{R}_{+}^{\ast}, then

    d​(y∗,x∗)≤C​η,∀y∗∈Fg;d(y^{\ast},x^{\ast})\leq C\eta,\ \forall y^{\ast}\in F_{g};
  • (iii)

    if y∈Yy\in Y is such that

    d​(y,f​(y))≤ε,d(y,f(y))\leq\varepsilon,

    for some ε∈ℝ+∗,\varepsilon\in\mathbb{R}_{+}^{\ast}, then

    d​(y,x∗)≤C​ε,d(y,x^{\ast})\leq C\varepsilon,

    i.e., the equation x=f​(x)x=f(x) is Ulam-Hyers stable;

  • (iv)

    xn∈Y,n∈ℕ,d​(xn,f​(xn))→0x_{n}\in Y,\ n\in\mathbb{N},\ d(x_{n},f(x_{n}))\rightarrow 0 as n→∞n\rightarrow\infty implies that xn→x∗x_{n}\rightarrow x^{\ast}as n→∞,n\rightarrow\infty, i.e. the fixed point problem for ff is well posed.

Theorem 2.12.

Let (X,d)(X,d) be an ℝ+\mathbb{R}_{+}-metric space, Y⊂XY\subset X a nonempty subset and f:Y→Xf:Y\rightarrow X be an operator such that,

d​(f​(x),f​(y))≤α​d​(x,f​(x))+β​d​(y,f​(y))+γ​d​(x,y),d(f(x),f(y))\leq\alpha d(x,f(x))+\beta d(y,f(y))+\gamma d(x,y), (2.4)

for all x,y∈Y,x,y\in Y, where α,β,γ∈ℝ+\alpha,\beta,\gamma\in\mathbb{R}_{+}. We suppose that, α<1\alpha<1. Then we have that:

  • (i)

    d​(f​(x),x∗)≤α+γ1−α​d​(x,x∗),∀x∈Yd(f(x),x^{\ast})\leq\dfrac{\alpha+\gamma}{1-\alpha}d(x,x^{\ast}),\ \forall x\in Y, i.e., ff is a quasicontraction;

  • (ii)

    xn∈Y,n∈ℕ,d​(xn+1,f​(xn))→0x_{n}\in Y,\ n\in\mathbb{N},\ d(x_{n+1},f(x_{n}))\rightarrow 0 as n→∞n\rightarrow\infty implies that xn→x∗x_{n}\rightarrow x^{\ast}as n→∞n\rightarrow\infty.

3. Fibre non-self contraction principle

In this section we obtain the fibre contraction principle for non-self operators in ℝ+m\mathbb{R}_{+}^{m}-metric space.

Theorem 3.1.

Let (X,d)(X,d) be an ℝ+m\mathbb{R}_{+}^{m}-metric space, Y⊂XY\subset X a nonempty set and (Y1,d)(Y_{1},d) a complete metric space. Let g:Y→X,g:Y\rightarrow X, h​(x,⋅):Y1→Y1h(x,\cdot):Y_{1}\rightarrow Y_{1} and f:Y×Y1→X×Y1,f:Y\times Y_{1}\rightarrow X\times Y_{1},\, f​(x,y)=(g​(x),h​(x,y)).f(x,y)=(g(x),h(x,y)). We suppose that:

(i) gg is a PO;

(ii) there exists SS a convergent to zero matrix such that

d​(h​(x,y),h​(x,z))≤S​d​(y,z), d(h(x,y),h(x,z))\leq Sd(y,z),\text{ }

for all x∈(A​B)gx\in(AB)_{g} and y,z∈Y1;y,z\in Y_{1};

(iii) ff is continuous.

Then ff is a PO.

Proof.

First of all we remark that (M​I)f=(M​I)g×Y1(MI)_{f}=(MI)_{g}\times Y_{1} and (M​I)g=(A​B)g.(MI)_{g}=(AB)_{g}. Let x0∈(A​B)gx_{0}\in(AB)_{g} and y0∈Y1y_{0}\in Y_{1}. Define xn+1=g​(xn),x_{n+1}=g(x_{n}), yn+1=h​(xn,yn)y_{n+1}=h(x_{n},y_{n}) for n∈ℕ.n\in\mathbb{N}. It is clear that xn→x∗∈Fgx_{n}\rightarrow x^{\ast}\in F_{g} as n→∞.n\rightarrow\infty. Since h​(x∗,⋅)h(x^{\ast},\cdot) is an SS-contraction, Fh​(x∗,⋅)={y∗}.F_{h(x^{\ast},\cdot)}=\{y^{\ast}\}. Let us prove that yn→y∗.y_{n}\rightarrow y^{\ast}. We have

d​(yn+1,y∗)\displaystyle d(y_{n+1},y^{\ast}) =d​(h​(xn,yn),y∗)\displaystyle=d(h(x_{n},y_{n}),y^{\ast})
≤d​(h​(xn,yn),h​(xn,y∗))+d​(h​(xn,y∗),y∗)\displaystyle\leq d(h(x_{n},y_{n}),h(x_{n},y^{\ast}))+d(h(x_{n},y^{\ast}),y^{\ast})
≤S​d​(yn,y∗)+d​(h​(xn,y∗),y∗)\displaystyle\leq Sd(y_{n},y^{\ast})+d(h(x_{n},y^{\ast}),y^{\ast})
…\displaystyle...
≤Sn+1​d​(y0,y∗)+Sn​d​(h​(x0,y∗),y∗)+\displaystyle\leq S^{n+1}d(y_{0},y^{\ast})+S^{n}d(h(x_{0},y^{\ast}),y^{\ast})+
…+S​d​(h​(xn−1,y∗),y∗)+d​(h​(xn,y∗),y∗).\displaystyle...+Sd(h(x_{n-1},y^{\ast}),y^{\ast})+d(h(x_{n},y^{\ast}),y^{\ast}).

Then d​(yn+1,y∗)→0,d(y_{n+1},y^{\ast})\rightarrow 0, by a Cauchy-Toeplitz lemma (see [13]), so ff is a PO. ∎

The above result is very useful to study of the differentiability of solutions of operator equations with respect to a parameter. For example, let us consider the following equation

x​(t,λ)=F​(t,x​(t,λ),λ), ​t∈[a,b], ​λ∈J⊂ℝx(t,\lambda)=F(t,x(t,\lambda),\lambda),\text{\ \thinspace}t\in[a,b],\text{ }\lambda\in J\subset\mathbb{R} (3.1)

and F:[a,b]×ℝ+m×J→ℝ+m.F:[a,b]\times\mathbb{R}_{+}^{m}\times J\rightarrow\mathbb{R}_{+}^{m}. We suppose that:

(H1) J⊂ℝJ\subset\mathbb{R} is a compact interval;

(H2) F∈C​([a,b]×ℝ+m×J,ℝ+m);F\in C([a,b]\times\mathbb{R}_{+}^{m}\times J,\mathbb{R}_{+}^{m});

(H3) F​(t,⋅,⋅)∈C1​(ℝ+m×J)F(t,\cdot,\cdot)\in C^{1}(\mathbb{R}_{+}^{m}\times J)\; for every t∈[a,b];t\in[a,b];

(H4) (|∂Fj∂ui​(t,u,λ)|)i,j=1m≤S,\left(\left|\dfrac{\partial F_{j}}{\partial u_{i}}(t,u,\lambda)\right|\right)_{i,j=1}^{m}\leq S, SS convergent to zero, for every t∈[a,b],t\in[a,b], u∈ℝ+m,u\in\mathbb{R}_{+}^{m}, αi∈ℝ,λ∈J,i=1,m¯.\alpha_{i}\in\mathbb{R},\ \lambda\in J,\ i=\overline{1,m}.

(H5) equation (3.1) has at least one solution.

Then we have:

Theorem 3.2.

Under the conditions (H1)-(H5) the equation (3.1) has in C​([a,b]×J,ℝ+m)C([a,b]\times J,\mathbb{R}_{+}^{m}) a unique solution x∗x^{\ast} and x∗​(t,⋅)∈C1​(J)x^{\ast}(t,\cdot)\in C^{1}(J) for every t∈[a,b].t\in[a,b].

Proof.

Let X=C​([a,b]×J,ℝ+m)X=C([a,b]\times J,\mathbb{R}_{+}^{m}) with norm ∥.∥C\left\|.\right\|_{C} and let B:C​([a,b]×J,ℝ+m)→C​([a,b]×J,ℝ+m)B:C([a,b]\times J,\mathbb{R}_{+}^{m})\rightarrow C([a,b]\times J,\mathbb{R}_{+}^{m}) be defined by B​(x)​(t,λ)=F​(t,x​(t,λ),λ).B(x)(t,\lambda)=F(t,x(t,\lambda),\lambda).

From conditions (H4) and (H5) it follows that FB={x∗}.F_{B}=\{x^{\ast}\}. Let Y={x∈C([a,b]×J,ℝ+m):Y=\{x\in C([a,b]\times J,\mathbb{R}_{+}^{m}): B​(x)​(t,λ)∈ℝ+m,B(x)(t,\lambda)\in\mathbb{R}_{+}^{m}, ∀t∈[a,b],\forall t\in[a,b], λ∈J}.\lambda\in J\}. It is clear that x∗∈Y,B​(Y)⊂Yx^{\ast}\in Y,\,B(Y)\subset Y and B:Y→YB:Y\rightarrow Y is a PO. Let x0∈Yx^{0}\in Y be such that there exists ∂xi0∂λ\dfrac{\partial x_{i}^{0}}{\partial\lambda} and ∂xi0∂λ∈C​([a,b]×J).\dfrac{\partial x_{i}^{0}}{\partial\lambda}\in C([a,b]\times J). Let us suppose that there exists ∂xi∗∂λ.\dfrac{\partial x_{i}^{\ast}}{\partial\lambda}. Then we have that

∂xi∗​(t,λ)∂λ=∂Fi​(t,x∗​(t,λ),λ)∂xi⋅∂xi∗​(t,λ)∂λ+∂Fi​(t,x∗​(t,λ),λ)∂λ,i=1,m¯.\dfrac{\partial x_{i}^{\ast}(t,\lambda)}{\partial\lambda}=\dfrac{\partial F_{i}(t,x^{\ast}(t,\lambda),\lambda)}{\partial x_{i}}\cdot\dfrac{\partial x_{i}^{\ast}(t,\lambda)}{\partial\lambda}+\dfrac{\partial F_{i}(t,x^{\ast}(t,\lambda),\lambda)}{\partial\lambda},\ i=\overline{1,m}.

This relation suggests us to consider the following operators:

Ci:Y×C​([a,b]×J)→C​([a,b]×J)C_{i}:Y\times C([a,b]\times J)\rightarrow C([a,b]\times J)

defined by

Ci​(x,y)​(t,λ)=∂Fi​(t,x​(t,λ),λ)∂xi⋅y​(t,λ)+∂Fi​(t,x​(t,λ),λ)∂λC_{i}(x,y)(t,\lambda)=\dfrac{\partial F_{i}(t,x(t,\lambda),\lambda)}{\partial x_{i}}\cdot y(t,\lambda)+\dfrac{\partial F_{i}(t,x(t,\lambda),\lambda)}{\partial\lambda}

and

A:Y×C​([a,b]×J)→Y×C​([a,b]×J)A:Y\times C([a,b]\times J)\rightarrow Y\times C([a,b]\times J)

with

A​(x,y)=(B​(x),C​(x,y)).A(x,y)=(B(x),C(x,y)).

From Theorem 3.1 we have that AA is a PO. This implies that the sequences xn+1=B​(xn),x_{n+1}=B(x_{n}), yn+1=C​(xn,yn)y_{n+1}=C(x_{n},y_{n}) are convergent, xn→x∗,x_{n}\rightarrow x^{\ast}, yn→y∗y_{n}\rightarrow y^{\ast} and x∗=B​(x∗),x^{\ast}=B(x^{\ast}), y∗=C​(x∗,y∗).y^{\ast}=C(x^{\ast},y^{\ast}).

Let us take yi0=∂xi0∂λ.y_{i}^{0}=\dfrac{\partial x_{i}^{0}}{\partial\lambda}. Then yi,n=∂xi,n∂λ.y_{i,n}=\dfrac{\partial x_{i,n}}{\partial\lambda}. So

xn→x∗ as n→∞, with respect to the norm ∥⋅∥Cx_{n}\rightarrow x^{\ast}\text{ \ as }n\rightarrow\infty,\text{ with respect to the norm }\left\|\cdot\right\|_{C}

and

∂xi,n∂λ→yi∗​ as ​n→∞.\dfrac{\partial x_{i,n}}{\partial\lambda}\rightarrow y_{i}^{\ast}\text{ \ as }n\rightarrow\infty.

These imply that y∗∈C1​([a,b]×J,ℝ+m)y^{\ast}\in C^{1}([a,b]\times J,\mathbb{R}_{+}^{m}) and yi∗=∂xi∗∂λ,i=1,m¯.y_{i}^{\ast}=\dfrac{\partial x_{i}^{\ast}}{\partial\lambda},\ i=\overline{1,m}. ∎

For other results regarding fibre contractions, see [5], [14], [15].

4. Data dependence in terms of ψ\psi-PO

Let (X,d)(X,d) be a ℝ+m\mathbb{R}_{+}^{m}-metric space, Y⊂XY\subset X a nonempty subset of XX and f,g:Y→Xf,g:Y\rightarrow X  two operators.

Theorem 4.1.

Assume that the following conditions are satisfied:

(i) ff is ψ\psi-PO with Ff={x∗}F_{f}=\{x^{\ast}\};

(ii) Fg⊂(B​A)fF_{g}\subset(BA)_{f} ;

(iii) there exists η∈ℝ+m\eta\in\mathbb{R}_{+}^{m} such that

d​(f​(x),g​(x))≤η​ for all ​x∈Y.d(f(x),g(x))\leq\eta\text{ \ \ \ for all\thinspace\ }x\in Y.

Then

d​(x∗,y∗)≤ψ​(η),∀y∗∈Fg.d(x^{\ast},y^{\ast})\leq\psi(\eta),\ \forall y^{\ast}\in F_{g}.
Proof.

Let y∗∈Fg,y∗∈(B​A)​(x∗).y^{\ast}\in F_{g},\ y^{\ast}\in(BA)(x^{\ast}). Then

d​(y∗,x∗)\displaystyle d(y^{\ast},x^{\ast}) ≤ψ​(d​(y∗,f​(y∗)))=ψ​(d​(g​(y∗),f​(y∗)))\displaystyle\leq\psi(d(y^{\ast},f(y^{\ast})))=\psi(d(g(y^{\ast}),f(y^{\ast})))
≤ψ​(η).\displaystyle\leq\psi(\eta).

∎

Another result in the case of strict φ\varphi-contractions is the following.

Theorem 4.2.

Assume that the following conditions are satisfied:

(i) ff is a strict φ\varphi-contraction with Ff={xf∗};F_{f}=\{x_{f}^{\ast}\};

(ii) Fg≠∅;F_{g}\neq\emptyset;

(iii) there exists η∈ℝ+m\eta\in\mathbb{R}_{+}^{m} such that

d​(f​(x),g​(x))≤η, for all ​x∈Y.d(f(x),g(x))\leq\eta,\text{ \ for all\ }x\in Y.

Then

d​(xg∗,xf∗)≤ψφ​(η), for all ​xg∗∈Fg.d(x_{g}^{\ast},x_{f}^{\ast})\leq\psi_{\varphi}(\eta),\text{ \ for all \thinspace}x_{g}^{\ast}\in F_{g}.
Proof.

Let xg∗∈Fg.x_{g}^{\ast}\in F_{g}. We have

d​(xg∗,xf∗)\displaystyle d(x_{g}^{\ast},x_{f}^{\ast}) ≤d​(xg∗,f​(xg∗))+d​(f​(xg∗),xf∗)\displaystyle\leq d(x_{g}^{\ast},f(x_{g}^{\ast}))+d(f(x_{g}^{\ast}),x_{f}^{\ast})
=d​(g​(xg∗),f​(xg∗))+d​(f​(xg∗),f​(xf∗))\displaystyle=d(g(x_{g}^{\ast}),f(x_{g}^{\ast}))+d(f(x_{g}^{\ast}),f(x_{f}^{\ast}))
≤η+φ​(d​(xg∗,xf∗)).\displaystyle\leq\eta+\varphi(d(x_{g}^{\ast},x_{f}^{\ast})).

Hence

d​(xg∗,xf∗)−φ​(d​(xg∗,xf∗))≤η.d(x_{g}^{\ast},x_{f}^{\ast})-\varphi(d(x_{g}^{\ast},x_{f}^{\ast}))\leq\eta.

Then

d​(xg∗,xf∗)≤ψφ​(η).d(x_{g}^{\ast},x_{f}^{\ast})\leq\psi_{\varphi}(\eta).

∎

We also have the following result:

Theorem 4.3.

Assume that the following conditions are satisfied:

(i) there exist P,Q∈ℝ+m×m,P,Q\in\mathbb{R}_{+}^{m\times m}, PP convergent to 0 matrix, such that

d​(f​(x),f​(y))≤P​d​(x,y)+Q​[d​(x,f​(x))+d​(y,f​(y))]d(f(x),f(y))\leq Pd(x,y)+Q[d(x,f(x))+d(y,f(y))]

for all x,y∈X,x,y\in X, and let Ff={xf∗};F_{f}=\{x_{f}^{\ast}\};

(ii) Fg≠∅;F_{g}\neq\emptyset;

(iii) there exists η∈ℝ+m\eta\in\mathbb{R}_{+}^{m} such that

d​(f​(x),g​(x))≤η, for all ​x∈Y.d(f(x),g(x))\leq\eta,\;\text{ for all\thinspace\ }x\in Y.

Then

d​(xg∗,xf∗)≤(I−P)−1​(I+Q)​η, for all ​xg∗∈Fg.d(x_{g}^{\ast},x_{f}^{\ast})\leq(I-P)^{-1}(I+Q)\eta,\text{ \ for all\thinspace\ }x_{g}^{\ast}\in F_{g}. (4.1)
Proof.

Let xg∗∈Fg.x_{g}^{\ast}\in F_{g}. We have

d​(xg∗,xf∗)\displaystyle d(x_{g}^{\ast},x_{f}^{\ast}) ≤d​(xg∗,f​(xg∗))+d​(f​(xg∗),xf∗)\displaystyle\leq d(x_{g}^{\ast},f(x_{g}^{\ast}))+d(f(x_{g}^{\ast}),x_{f}^{\ast})
=d​(g​(xg∗),f​(xg∗))+d​(f​(xg∗),f​(xf∗))\displaystyle=d(g(x_{g}^{\ast}),f(x_{g}^{\ast}))+d(f(x_{g}^{\ast}),f(x_{f}^{\ast}))
≤η+P​d​(xg∗,xf∗)+Q​[d​(xg∗,f​(xg∗))+d​(xf∗,f​(xf∗))]\displaystyle\leq\eta+Pd(x_{g}^{\ast},x_{f}^{\ast})+Q\left[d(x_{g}^{\ast},f(x_{g}^{\ast}))+d(x_{f}^{\ast},f(x_{f}^{\ast}))\right]
=η+P​d​(xg∗,xf∗)+Q​d​(xg∗,f​(xg∗))\displaystyle=\eta+Pd(x_{g}^{\ast},x_{f}^{\ast})+Qd(x_{g}^{\ast},f(x_{g}^{\ast}))
=η+P​d​(xg∗,xf∗)+Q​d​(g​(xg∗),f​(xg∗))\displaystyle=\eta+Pd(x_{g}^{\ast},x_{f}^{\ast})+Qd(g(x_{g}^{\ast}),f(x_{g}^{\ast}))
≤η+Q​η+P​d​(xg∗,xf∗).\displaystyle\leq\eta+Q\eta+Pd(x_{g}^{\ast},x_{f}^{\ast}).

Then

(I−P)​d​(xg∗,xf∗)≤(I+Q)​η\left(I-P\right)d(x_{g}^{\ast},x_{f}^{\ast})\leq\left(I+Q\right)\eta

so

d​(xg∗,xf∗)≤(I−P)−1​(I+Q)​η, for all ​xg∗∈Fg.d(x_{g}^{\ast},x_{f}^{\ast})\leq\left(I-P\right)^{-1}\left(I+Q\right)\eta,\text{ for all\thinspace\ }x_{g}^{\ast}\in F_{g}.

∎

References

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Received: ; Accepted:

2025

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