The aim of this paper is to present a study on Ćirić type multi-valued operators. Following the approach given in [16], where the author considered some variants of the multivalued contraction principle given by Nadler [14], respectively a so-called strict multi-valued contraction principle, we will consider here the case of Ćirić type multi-valued operators, see [3].

We also notice that in [24] Reich developed some fixed point theorems for multi-valued generalized contractions. A fully comprehensive study on Reich operators was made in [12] by T. Lazăr et al. Also, qualitative properties, namely data dependence, Ulam-Hyers stability and so on, were studied
for the case of multi-valued $\varphi$-contractions by V.L. Lazăr in [13]. Moreover, C. Chifu and G. Petruşel in [5] studied qualitative properties concerning Hardy-Rogers multi-valued operators (see [7] for the single-valued case) in the framework of b-metric spaces, while T. Lazăr, D. O’Regan et al. [11] studied the case of multi-valued operators of Ćirić type defined on a set endowed with two metrics. Finally, we point out that in [2], M. Boriceanu studied existence and uniqueness of the fixed point and data dependence for multi-valued Ćirić type operators in the context of b-metric spaces. At the same time, Ćirić type multi-valued operators were are studied in [17] and [19].

Regarding terminology and basic concepts for fixed point problems related to multi-valued operators, we will follow the works [1],[9], [18] and [23]. Furthermore, for the approximation of strict fixed points (also called end-points) of multi-valued mappings, we refer to [6], [8] and [22]. Finally, regarding data dependence, multi-valued fractal operators, selections and qualitative properties for the fixed point inclusion and for multi-valued fractals, we will refer to [4], [10] and [20].

Led ( $X,d$ ) be a metric space. Denote by $P(X)$ the family of all nonempty subsets of $X$. Also, $P_{b}(X)$ stands for the family of nonempty, bounded subsets of $X$ and $P_{cl}(X)$ the family of nonempty, closed subsets of $X$. In a similar manner, by $P_{cp}(X)$ we refer to the family of nonempty, compact subsets of $X$. From now on, $\bar{B}\left(x_{0},r\right)$ means the closure in $(X,d)$ of the ball $B\left(x_{0},r\right)$, where $B\left(x_{0},r\right):=\left\{x\in X\mid d\left(x_{0},x\right)<r\right\}$ is the open ball with radius $r>0$ and the center $x_{0}\in X$. By $\widetilde{B}\left(x_{0};r\right):=\left\{x\in X\mid d\left(x_{0},x\right)\leq r\right\}$ we denote the closed ball centered in $x_{0}$ with radius $r$. We recall now some important functionals which will be used through the paper:

• •

  the gap functional $D:P(X)\times P(X)\rightarrow\mathbb{R}_{+},D(A,B):=\inf_{a\in A,b\in B}\{d(a,b)\}$.

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• •

  the generalized Pompeiu-Haussdorf functional $H:P(X)\times P(X)\rightarrow\mathbb{R}_{+}\cup\{+\infty\}$, where $H(A,B)=\max\left\{\sup_{a\in A}D(a,B),\sup_{b\in B}D(b,A)\right\}$.
  Furthermore, if $T:X\rightarrow P(X)$ is a multi-valued operator, then an element $x\in X$ is a fixed point for $T$ if and only if $x\in T(x)$. We denote by $F_{T}$ the set of all fixed points of the operator $T$ and by $(SF)_{T}$ the set of all strict fixed points of $T$, where $x\in X$ is a strict fixed point of $T$ (or an endpoint, or a stationary point) if and only if $\{x\}=Tx$.
  For a multi-valued operator $T:X\rightarrow P(Y)$ we can also define the following useful notions. The graph of the operator $T$, defined by $\operatorname{Graph}(T):=\{(x,y)\in X\times Y\mid y\in T(x)\}$, and the image of the set $Y\in P(X)$ will be denoted by $T(Y):=\bigcup_{x\in Y}T(x)$. A single-valued mapping $t:X\rightarrow Y$ is called a selection of $T$ if for each $x\in X$, we have that $t(x)\in T(x)$.

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We present now an important concept, which appears naturally by Nadler’s contraction principle. By [21], we recall here the notion of multi-valued weakly Picard operator.

Definition 1.1. Let ( $X,d$ ) be a metric space.
Consider $T:X\rightarrow P(X)$ be a multi-valued operator. By definition, $T$ is a multi-valued weakly Picard operator (briefly MWP operator) if for each $x\in X$ and for each $y\in T(x)$, there exists a sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$, satisfying the following
(i) $x_{0}=x$ and $x_{1}=y$,
(ii) $x_{n+1}\in T\left(x_{n}\right)$, for each $n\in\mathbb{N}$,
(iii) the sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$ is convergent to a fixed point of $T$.

Remark 1.1. A sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$ satisfying conditions $(i)$ and $(ii)$ is called a sequence of successive approximations of $T$ starting from $(x,y)\in\operatorname{Graph}(T)$. If $T:X\rightarrow P(X)$ is a MWP operator, then we define the operator $T^{\infty}:\operatorname{Graph}(T)\rightarrow P\left(F_{T}\right)$, by $T^{\infty}(x,y):=\left\{z\in F_{T}\mid\right.$ there exists a sequence of successive approximations of T starting from ( $\mathrm{x},\mathrm{y}$ ) that converges to z$\}$.

Furthermore, if ( $X,d$ ) is a metric space and $T:X\rightarrow P(X)$ a multi-valued operator, then $T$ is said to be closed if $\operatorname{Graph}(T)$ is a closed set in $X\times X$. By $T^{1}(x):=T(x),\ldots,T^{n}(x):=T\left(T^{n-1}(x)\right)$ we denote the iterates of the multi-valued mapping $T$, while the set $V^{0}(Y;\varepsilon):=\{x\in X\mid D(x,Y)<\varepsilon\}$ is called the (open) $\varepsilon$-neighborhood of $Y\in P(X)$.

From [14], we shall recall some important lemmas that are used throughout the article.

Lemma 1.1. Let $A$ and $B$ from $P(X)$ and $q>1$. Then, for each $a\in A$, there exists $b\in B$, such that $d(a,b)\leq qH(A,B)$.

Lemma 1.2. Let $A$ and $B$ from $P(X)$. Also, consider $\eta>0$, such that
(i) for each $a\in A$, there exists $b\in B$, with $d(a,b)\leq\eta$,
(ii) for each $b\in B$, there exists $a\in A$, with $d(a,b)\leq\eta$.

Then $H(A,B)\leq\eta$.
Now, we recall the basic concepts for the qualitative properties of the fixed point inclusion and of the fixed point iteration. The first two definitions are related to well-posedness of the fixed point problem. For the concept of well-posedness, we let the reader follow [12] and [19].

Definition 1.2. Let ( $X,d$ ) be a metric space and $T:Y\rightarrow P_{cl}(X)$ be a multi-valued operator. Then the fixed point problem is well-posed for $T$ with
respect to the gap functional $D$ if and only if:
(i) $F_{T}=\left\{x^{*}\right\}$;
(ii) if $\left(x_{n}\right)\in X$ has the property that $D\left(x_{n},T\left(x_{n}\right)\right)\rightarrow 0$, then $x_{n}\rightarrow x^{*}$.

Definition 1.3. Let $(X,d)$ be a metric space, $Y\in P(X)$ and $T:Y\rightarrow P_{cl}(X)$ be a multi-valued operator. Then the fixed point problem is well-posed for $T$ with respect to the Pompeiu-Haussdorf functional $H$ if and only if:
(i) $(SF)_{T}=\left\{x^{*}\right\}$;
(ii) if $\left(x_{n}\right)\in X$ is a sequence such that $H\left(x_{n},Tx_{n}\right)\rightarrow 0$, then $x_{n}\rightarrow x^{*}$.

Now, the second important concept related to the fixed point problem is limit shadowing or Ostrowski property, which can be found in [12] and [13].

Definition 1.4. Let ( $X,d$ ) be a metric space and $T:X\rightarrow P(X)$ be a multi-valued operator. By definition, the multi-valued operator $T$ has the Ostrowski property, if $F_{T}=\left\{x^{*}\right\}$ and for any sequence $\left(y_{n}\right)_{n\in\mathbb{N}}\subset X$, such that $D\left(y_{n+1},Ty_{n}\right)\rightarrow 0$, we have $\left(y_{n}\right)_{n\in\mathbb{N}}\rightarrow x^{*}$, as $n\rightarrow\infty$.

We introduce now the notions of $\psi$-MWP operator and of generalized Ulam-Hyers stabilites. For the study of generalized Ulam-Hyers stability we refer to [15].

Definition 1.5. Let ( $X,d$ ) be a metric space and $T:X\rightarrow P(X)$ be a MWP operator. Let $\psi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$be continuous in 0 , increasing, such that $\psi(0)=0$. By definition, $T$ is $\psi$-MWP operator, if there exists a selection $t^{\infty}:\operatorname{Graph}(T)\rightarrow F_{T}$ of $T^{\infty}$, such that $d\left(x,t^{\infty}(x,y)\right)\leq\psi(d(x,y))$, for each $(x,y)\in\operatorname{Graph}(T)$.

Definition 1.6. Let ( $X,d$ ) be a metric space and $T:X\rightarrow P(X)$. By definition, the fixed point inclusion

is called generalized Ulam-Hyers stable if and only if there exists an increasing, continuous in 0 function $\psi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$, with $\psi(0)=0$, such that for every $\varepsilon>0$ and for each $y^{*}\in X$ for which $D(y,T(y))\leq\varepsilon$, there exists a solution $x^{*}$ a solution of the fixed point inclusion (1.1), such that $d\left(y^{*},x^{*}\right)\leq\psi(\varepsilon)$.

Definition 1.7. Let ( $X,d$ ) be a metric space and $T:Y\rightarrow P(X)$. By definition, the strict fixed point inclusion

is called generalized Ulam-Hyers stable if and only if there exists an increasing, continuous in 0 function $\psi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$, with $\psi(0)=0$, such that for
every $\varepsilon>0$ and for each $y^{*}\in X$ for which $H(y,T(y))\leq\varepsilon$, there exists a solution $x^{*}$ a solution of the strict fixed point inclusion (1.2), such that $d\left(y^{*},x^{*}\right)\leq\psi(\varepsilon)$.

Finally, following [6], [8] and [22], we recall the last important concepts.
Definition 1.8. Let $X\neq\emptyset$ and $T:X\rightarrow P(X)$ be a multi-valued operator. Then, $T$ has the approximate endpoint property if $\inf_{x\in X}\sup_{y\in Tx}d(x,y)=0$.

The aim of this paper is to extend to the case of Ćirić type multi-valued generalized contractions, the results given in [16], where the author studied extended properties for the fixed point problem related to Nadler’s multivalued contractions through relevant metrical and topological properties. In the present section some variants of the multi-valued Ćirić principle are given. We shall enhance the classical result of Ćirić [3] with additional metrical and topological conclusions with respect to the fixed point problem.
Theorem 2.1 (An extended version of the Ćirić’s multi-valued contraction principle). Let ( $X,d$ ) be a complete metric space and $T:X\rightarrow P_{cl}(X)$ be a multi-valued $\alpha$-Ćirić type operator, i.e., there exists $\alpha\in(0,1)$, such that

where

Then, the following conclusions hold:
(a) there exists $x^{*}\in F_{T}$;
(b) for each $(x,y)\in\operatorname{Graph}(T)$, there exists a sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$ of successive approximations for $T$ starting from $(x,y)$, convergent to a fixed point of $T$;
(c) there exists a selection $t^{\infty}:\operatorname{Graph}(T)\rightarrow F_{T}$ of $T^{\infty}$, such that

(d) $F_{T}$ is closed in ( $X,d$ );
(e) if $\left(x_{n}\right)_{n\in\mathbb{N}}$ is a sequence of successive approximations for $T$, starting from a pair $(x,y)\in G$ raph $(T)$, which converges to a fixed point $x^{*}(x,y)$ of $T$, then

(f) if $G:X\rightarrow P_{cl}(X)$ is a Ćirić-type multi-valued operator with coefficient $\beta$, and there exists $\eta>0$, such that $H(T(x),G(x))\leq\eta$, for all $x\in X$, then $H\left(F_{T},F_{G}\right)\leq\eta\cdot\max\left\{\frac{1}{1-\alpha},\frac{1}{1-\beta}\right\};$
(g) if $T_{n}:X\rightarrow P_{cl}(X)$ is a sequence of multi-valued $\alpha$-Ćirić-type operators, with $T_{n}(x)\xrightarrow{H}T(x)$ as $n\rightarrow\infty$, uniformly with respect to $x\in X$, then

(h) if there exists $x_{0}\in X$ and $r>0$, such that $D\left(x_{0},T\left(x_{0}\right)\right)<(1-\alpha)r$, then there exists $x^{*}\in F_{T}\cap B\left(x_{0},r\right)$;
(i) if there exists $x_{0}\in X$ and $r>0$ such that $\delta\left(x_{0},T\left(x_{0}\right)\right)<(1-\alpha)r$, then $T:\tilde{B}\left(x_{0},r\right)\rightarrow P\left(\tilde{B}\left(x_{0},\frac{1}{1-\alpha}r\right)\right)$ and there exists $x^{*}\in F_{T}\cap B\left(x_{0},r\right)$;
(j) if $X$ is a Banach space, $U$ an open subset of $X$ and $T:U\rightarrow P_{cl}(X)$ is a Ćirić multi-valued operator, then the associated multi-valued field $G:U\rightarrow P(X),G(x):=x-T(x)$ is open;
(k) there exists a Caristi selection of $T$;
(m) if, additionally, $T:X\rightarrow P_{cp}(X)$, then the fixed point inclusion $x\in T(x)$ is generalized Ulam-Hyers stable;
(n) the multi-valued operator $T$ has the approximate fixed point property;
(o) if the multi-valued operator $T$ is lower semicontinuous, then it has the approximate endpoint property if and only if it has a unique strict fixed point;
(p) if $\alpha<\frac{1}{2}$, then the fixed point set $F_{T}$ is compact.
(q) if $T:X\rightarrow P_{b,cl}(X)$, then for each $p>0$, one has $H\left(F_{p}^{*},F_{T}\right)\leq\frac{p}{1-\alpha}$, where $F_{p}^{*}:=\{x\in X\mid D(x,T(x))<p\}$.

Proof. (a), (b), (c) and (e) (In fact (a) and (b) means that $T$ is a MWP operator, while (a), (b) and (c) can be concise represented by saying that $T$ is a $\psi$-MWP operator, with $\psi(t)=\frac{1}{1-\alpha}t$ ).
Let $x_{0}\in X$ and $x_{1}\in T\left(x_{0}\right)$ be arbitrary elements. Then $H\left(T\left(x_{0}\right),T\left(x_{1}\right)\right)\leq\alpha M\left(x_{0},x_{1}\right)$. Furthermore, consider $q\in\left(1,\frac{1}{\alpha}\right)$.
Now, for $x_{1}$, there exists $x_{2}\in T\left(x_{1}\right)$, such that $d\left(x_{1},x_{2}\right)\leq qH\left(T\left(x_{0}\right),T\left(x_{1}\right)\right)$, so $d\left(x_{1},x_{2}\right)\leq q\alpha M\left(x_{0},x_{1}\right)$. We consider the following cases:
If $M\left(x_{0},x_{1}\right)=d\left(x_{0},x_{1}\right)$, then $d\left(x_{1},x_{2}\right)\leq(q\alpha)d\left(x_{0},x_{1}\right)$.
If $M\left(x_{0},x_{1}\right)=D\left(x_{0},T\left(x_{0}\right)\right)\leq d\left(x_{0}\right),x_{1}$, then $d\left(x_{1},x_{2}\right)\leq(q\alpha)d\left(x_{0},x_{1}\right)$.
If $M\left(x_{0},x_{1}\right)=D\left(x_{1},T\left(x_{1}\right)\right)\leq d\left(x_{1},x_{2}\right)$, then $d\left(x_{1},x_{2}\right)\leq(q\alpha)d\left(x_{1},x_{2}\right)$, which is a contradiction, So $M\left(x_{0},x_{1}\right)$ can not be $d\left(x_{1},x_{2}\right)$.

Finally, if $M\left(x_{0},x_{1}\right)=\frac{1}{2}\left[D\left(x_{1},T\left(x_{0}\right)\right)+D\left(x_{0},T\left(x_{1}\right)\right)\right]$, then by using the fact that $D\left(x_{1},T\left(x_{0}\right)\right)\leq d\left(x_{1},x_{1}\right)=0$ and the fact that $D\left(x_{0},T\left(x_{1}\right)\right)\leq d\left(x_{0},x_{2}\right)\leq d\left(x_{0},x_{1}\right)+d\left(x_{1},x_{2}\right)$, it follows that $M\left(x_{0},x_{1}\right)\leq\frac{1}{2}d\left(x_{0},x_{1}\right)+\frac{1}{2}d\left(x_{1},x_{2}\right)$. So $d\left(x_{1},x_{2}\right)\leq\frac{q\alpha}{2}d\left(x_{0},x_{1}\right)+\frac{q\alpha}{2}d\left(x_{1},x_{2}\right)$. Then $d\left(x_{1},x_{2}\right)\leq\frac{q\alpha}{2-q\alpha}d\left(x_{0},x_{1}\right)$.
Since $q\in\left(1,\frac{1}{\alpha}\right)$, we get that $d\left(x_{1},x_{2}\right)\leq(q\alpha)d\left(x_{0},x_{1}\right)$.
Let us denote by $\lambda:=q\alpha$. Then, by all the cases $d\left(x_{1},x_{2}\right)\leq\lambda d\left(x_{0},x_{1}\right)$.
Also, denote by $\rho_{n}:=d\left(x_{n},x_{n+1}\right)$, for each $n\in\mathbb{N}$. By induction, we can construct a sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$, such that for $x_{n}\in T\left(x_{n-1}\right)$, there exists $x_{n+1}\in T\left(x_{n}\right)$, for which $\rho_{n}\leq\lambda\rho_{n-1}$, for each $n\in\mathbb{N}$. Then $\rho_{n}\leq\lambda^{n}\rho_{0}$, so by triangle inequality $d\left(x_{n},x_{n+p}\right)\leq\lambda^{n}\frac{1-\lambda^{p}}{1-\lambda}\rho_{0}$. Taking $n\rightarrow\infty$, it follows up that the sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$ is Cauchy, so there exists $x^{*}\in X$, such that $x_{n}\rightarrow x^{*}$. Furthermore, in the estimate $d\left(x_{n},x_{n+p}\right)\leq\lambda^{n}\frac{1-\lambda^{p}}{1-\lambda}\rho_{0}$. taking $p\rightarrow\infty$, it follows that $d\left(x_{n},x^{*}\right)\leq\frac{\lambda^{n}}{1-\lambda}d\left(x_{0},x_{1}\right)$. Taking $n=0$ and making $q\searrow 1$, it follows the estimate $d\left(x,x^{*}\right)=d\left(x,t^{\infty}(x,y)\right)\leq\frac{1}{1-\alpha}d(x,y)$, with $y\in T(x)$. Here, we denoted by $x:=x_{0}$ and $y:=x_{1}\in T\left(x_{0}\right)$.
The final step is to show that $x^{*}\in F_{T}$, i.e., to prove that $D\left(x^{*},T\left(x^{*}\right)\right)=0$. We have the following estimation:

Moreover, since

by letting $n\rightarrow\infty$, we obtain that $\lim_{n\rightarrow\infty}M\left(x_{n},x^{*}\right)\leq D\left(x^{*},T\left(x^{*}\right)\right)$. This means that

and the conclusion follows.
(d) We know that $F_{T}\in P(X)$. We shall show that $F_{T}$ is closed in $(X,d)$. For this, let $x_{n}\in F_{T}$, such that $x_{n}\rightarrow x^{*}$. So, for each $n\in\mathbb{N},D\left(x_{n},T\left(x_{n}\right)\right)=0$.

We shall show that $x^{*}\in F_{T}$, i.e. $x^{*}\in T\left(x^{*}\right)$. Also, since the operator $T$ has closed values, then it is enough to show that $D\left(x^{*},T\left(x^{*}\right)\right)=0$. We have the following inequalities :

We have the following cases:
If $M\left(x^{*},x_{n}\right)=d\left(x^{*},x_{n}\right)$, then $D\left(x^{*},T\left(x^{*}\right)\right)\leq(1+\alpha)d\left(x_{n},x^{*}\right)\rightarrow 0$.
Furthermore, if $M\left(x^{*},x_{n}\right)=D\left(x_{n},T\left(x_{n}\right)\right)=0$, then $D\left(x^{*},T\left(x^{*}\right)\right)\leq d\left(x^{*},x_{n}\right)\rightarrow 0$. Moreover, if $M\left(x^{*},x_{n}\right)=D\left(x^{*},T\left(x^{*}\right)\right)$, then we obtain that

Finally, if $M\left(x^{*},x_{n}\right)=\frac{1}{2}\left[D\left(x_{n},T\left(x^{*}\right)\right)+D\left(x^{*},T\left(x_{n}\right)\right)\right]\leq\frac{1}{2}D\left(x_{n},T\left(x^{*}\right)\right)+\frac{1}{2}d\left(x^{*},x_{n}\right)$.
Also, $D\left(x_{n},T\left(x^{*}\right)\right)\leq H\left(T\left(x_{n}\right),T\left(x^{*}\right)\right)\leq\alpha M\left(x_{n},x^{*}\right)\leq\frac{\alpha}{2}D\left(x_{n},T\left(x^{*}\right)\right)+\frac{\alpha}{2}d\left(x^{*},x_{n}\right)$, so $D\left(x_{n},T\left(x^{*}\right)\right)\leq\frac{\alpha}{2-\alpha}d\left(x_{n},x^{*}\right)$. This implies that

Thus, by all cases $x^{*}\in F_{T}$, so $F_{T}$ is closed.
(f) By (a),(b),(c) and (e), we have that $d\left(x,x^{*}\right)\leq\frac{1}{1-\alpha}d(x,y)$, where $x$ is an arbitrary element of $X$ and $y\in T(x)$, where $x^{*}\in F_{T}$. Taking $x=y^{*}\in F_{G}$, then we obtain that $d\left(x^{*},y^{*}\right)\leq\frac{1}{1-\alpha}d\left(y,y^{*}\right)$, where $y\in T\left(y^{*}\right)$. Furthermore, since $y\in T\left(y^{*}\right)$ is arbitrary, we can make the following assertion: for $y^{*}\in F_{G}$, there exists $y\in T\left(y^{*}\right)$, such that $d\left(y,y^{*}\right)\leq H\left(G\left(y^{*}\right),T\left(y^{*}\right)\right)\leq\eta$, so $d\left(x^{*},y^{*}\right)\leq\frac{\eta}{1-\alpha}$.
Now, also from the global principle of the existence of the fixed point of $G$, we get that $d\left(x,x^{*}\right)\leq\frac{1}{1-\beta}d(x,y)$, with $x^{*}\in F_{G},x$ is an arbitrary element of $X$ and $y\in Gx$.
Taking $x=y^{*}\in F_{T}$, then we obtain that $d\left(x^{*},y^{*}\right)\leq\frac{1}{1-\beta}d\left(y,y^{*}\right)$, where $y\in G\left(y^{*}\right)$.
As in the first case, since $y\in G\left(y^{*}\right)$ is arbitrary, then for $y^{*}\in F_{T}$, there exists

Vol. LVII (2019) Some variants of Ćirić’s multi-valued contraction principle 31
$y\in G\left(y^{*}\right)$, such that $d\left(y^{*},y\right)\leq H\left(T\left(y^{*}\right),G\left(y^{*}\right)\right)\leq\eta$. So $d\left(x^{*},y^{*}\right)\leq\frac{\eta}{1-\beta}$. From the first case we get that for $y^{*}\in F_{G}$, there exists $x^{*}\in F_{T}$, such that

while from the second case we infer that for $y^{*}\in F_{T}$, there exists $x^{*}\in F_{G}$, such that

By Lemma 1.2, we get the conclusion $H\left(F_{T},F_{G}\right)\leq\eta\cdot\max\left\{\frac{1}{1-\alpha},\frac{1}{1-\beta}\right\}$.
$(\mathrm{g})$ Let $\varepsilon>0$ be an arbitrary fixed element. Since $T_{n}(x)\xrightarrow{H}T(x)$ as $n\rightarrow\infty$, uniformly for each $x\in X$, then for all $x\in X$, we have that $\lim_{n\rightarrow\infty}H\left(T_{n}x,Tx\right)=0$.
This means that for $\varepsilon>0$, there exists $N(\varepsilon)\in\mathbb{N}$, such that for each $n\geq N(\varepsilon)$, we have that $\sup_{x\in X}H\left(T_{n}(x),T(x)\right)<\varepsilon$. From the conclusion (f) of data dependence, we have that for $\varepsilon>0$, there exists $N(\varepsilon)\in\mathbb{N}$, such that for all $n\geq N(\varepsilon)$, one has $H\left(F_{T_{n}},F_{T}\right)<\frac{1}{1-\alpha}\cdot\varepsilon$. So, the conclusion is valid.
(h) Let $s\in(0,r)$, such that $\tilde{B}\left(x_{0},s\right)\subset B\left(x_{0},r\right)$, with $D\left(x_{0},T\left(x_{0}\right)\right)<(1-\alpha)s<(1-\alpha)r$. Since $D\left(x_{0},T\left(x_{0}\right)\right)<(1-\alpha)s$, then there exists $x_{1}\in T\left(x_{0}\right)$, such that $d\left(x_{0},x_{1}\right)<(1-\alpha)s<s$, so $x_{1}\in B\left(x_{0},s\right)\subset\tilde{B}\left(x_{0},s\right)$. From the hypothesis, we have that $H\left(T\left(x_{0}\right),T\left(x_{1}\right)\right)\leq\alpha M\left(x_{0},x_{1}\right)$, where:

We consider the following cases:
If the maximum is $d\left(x_{0},x_{1}\right)$, then $H\left(T\left(x_{0}\right),T\left(x_{1}\right)\right)\leq\alpha d\left(x_{0},x_{1}\right)$.
If the maximum is $H\left(T\left(x_{0}\right),T\left(x_{1}\right)\right)$, then, since $\alpha<1$, we obtain a contradiction.
From the above cases, it follows that $H\left(T\left(x_{0}\right),T\left(x_{1}\right)\right)\leq\alpha d\left(x_{0},x_{1}\right)$. Since $D\left(x_{1},T\left(x_{1}\right)\right)\leq H\left(T\left(x_{0}\right),T\left(x_{1}\right)\right)\leq\alpha d\left(x_{0},x_{1}\right)<\alpha(1-\alpha)s$, then there exists $x_{2}\in T\left(x_{1}\right)$ for which $d\left(x_{1},x_{2}\right)<\alpha(1-\alpha)s$.

Furthermore, by triangle inequality one can obtain $d\left(x_{0},x_{2}\right)\leq d\left(x_{0},x_{1}\right)+d\left(x_{1},x_{2}\right)<(1-\alpha)s+\alpha(1-\alpha)s=\left(1-\alpha^{2}\right)s<s$, so $x_{2}\in T\left(x_{1}\right)\cap\tilde{B}\left(x_{0},s\right)$.
By induction, we can construct a sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$, with $x_{n}$ from $T\left(x_{n-1}\right)\cap\tilde{B}\left(x_{0},s\right)$, such that :
$\left\{\begin{array}[]{l}x_{n+1}\in T\left(x_{n}\right),\text{ for each }n\in\mathbb{N},\\ d\left(x_{n-1},x_{n}\right)\leq\alpha^{n-1}(1-\alpha)s,\text{ for each }n\in\mathbb{N}^{*},\\ d\left(x_{0},x_{n}\right)\leq\left(1-\alpha^{n}\right)s,\text{ for all }n\in\mathbb{N}.\end{array}\right.$
It follows that the sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$ is Cauchy, so there exists $x^{*}\in\tilde{B}\left(x_{0},s\right)$, such that $x_{n}\rightarrow x^{*}$.
As in the proof of (a),(b),(c) and (e), one can show that $x^{*}\in T\left(x^{*}\right)$. Moreover, since $x_{n}\in\tilde{B}\left(x_{0},s\right)$ and $\tilde{B}\left(x_{0},s\right)$ is closed in $X$, then $x^{*}\in\tilde{B}\left(x_{0},s\right)\subset B\left(x_{0},r\right)$.
(i) Let $u\in T\left(x_{0}\right)$. Then, by applying the triangle inequality, we get that $d\left(z,x_{0}\right)\leq d(z,u)+d\left(u,x_{0}\right)$, so $d\left(z,x_{0}\right)\leq d(z,u)+\delta\left(x_{0},T\left(x_{0}\right)\right)$. Now, taking $\inf_{u\in T\left(x_{0}\right)}$, it follows that $d\left(z,x_{0}\right)\leq D\left(z,T\left(x_{0}\right)\right)+\delta\left(x_{0},T\left(x_{0}\right)\right)$.
We first show that $T\left(\tilde{B}\left(x_{0},r\right)\right)\subset\tilde{B}\left(x_{0},\frac{1}{1-\alpha}r\right)$.
Let $y\in\tilde{B}\left(x_{0},r\right)$. We will show that $T(y)\subset\tilde{B}\left(x_{0},\frac{1}{1-\alpha}r\right)$.
So, take $z\in T(y)$. The aim is to show that $z\in\tilde{B}\left(x_{0},\frac{1}{1-\alpha}r\right)$, i.e., $d\left(z,x_{0}\right)\leq\frac{1}{1-\alpha}r$.
Then $d\left(z,x_{0}\right)\leq D\left(z,T\left(x_{0}\right)\right)+\delta\left(x_{0},T\left(x_{0}\right)\right)<H\left(T(y),T\left(x_{0}\right)\right)+(1-\alpha)r$. So $d\left(z,x_{0}\right)<\alpha M\left(y,x_{0}\right)+(1-\alpha)r$.
We know that: