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Fixed points and endpoints of contractive set-valued maps in cone uniform spaces with generalized pseudodistances

Fixed Point Theory and Applications20122012:176

https://doi.org/10.1186/1687-1812-2012-176

• Accepted: 1 October 2012
• Published:

Abstract

We introduce the concept of contractive set-valued maps in cone uniform spaces with generalized pseudodistances and we show how in these spaces our fixed point and endpoint existence theorem of Caristi type yields the fixed point and endpoint existence theorem for these contractive maps.

MSC:47H10, 54C60, 47H09, 54E15, 46A03, 54E50, 46B40.

Keywords

• fixed point
• endpoint
• cone uniform space
• generalized pseudodistance
• set-valued dynamic system
• dynamic process

1 Introduction

Nadler  extended Banach’s fixed point theorem  for set-valued maps in complete metric spaces.

Theorem 1.1 ([, Th. 5])

Let $\left(X,d\right)$ be a complete metric space, let $Cl\left(X\right)$ denote the class of all nonempty closed subsets of X, and let $H:{\left(Cl\left(X\right)\right)}^{2}\to \left[0,\mathrm{\infty }\right]$ be defined by
${\mathrm{\forall }}_{A,B\in Cl\left(X\right)}\left\{H\left(A,B\right)=max\left\{\underset{u\in A}{sup}d\left(u,B\right),\underset{v\in B}{sup}d\left(v,A\right)\right\},$
where, for each $u\in X$ and $V\in Cl\left(X\right)$, $d\left(u,V\right)={inf}_{v\in V}d\left(u,v\right)$. If a set-valued map $T:X\to Cl\left(X\right)$ is H-contractive, i.e., if T satisfies
${\mathrm{\exists }}_{0<\lambda <1}{\mathrm{\forall }}_{x,y\in X}\left\{H\left(T\left(x\right),T\left(y\right)\right)⩽\lambda d\left(x,y\right)\right\},$

then T has a fixed point w in X, i.e., $w\in T\left(w\right)$.

A number of authors introduce the new concepts of set-valued contractions of Nadler type and study the problem concerning the existence of fixed points for such contractions; see, e.g., Aubin and Siegel , de Blasi et al. , Ćirić , Eldred et al. , Feng and Liu , Frigon , Al-Homidan et al. , Jachymski , Kaneko , Klim and Wardowski , Latif and Al-Mezel , Mizoguchi and Takahashi , Pathak and Shahzad , Quantina and Kamran , Reich [17, 18], Reich and Zaslavski [19, 20], Sintunavarat and Kumam , Suzuki , Suzuki and Takahashi , Takahashi  and Zhong et al. . In particular, the significant fixed point existence results of Nadler type were obtained by Suzuki [, Th. 3.7] in metric spaces with τ-distances and by Wardowski  in cone metric spaces.

Recently, Włodarczyk and Plebaniak in  have studied among others the $\mathcal{J}$-families of generalized pseudodistances in cone uniform, uniform and metric spaces which generalize distances of Tataru , w-distances of Kada et al. , τ-distances of Suzuki  and τ-functions of Lin and Du  in metric spaces and distances of Vályi  in uniform spaces.

In the present paper, we introduce the concept of contractive set-valued maps in cone uniform spaces with generalized pseudodistances, and we show how in these spaces our fixed point and endpoint existence theorem of Caristi type [, Th. 4.5] yields the fixed point and endpoint existence theorem for these contractive maps.

It is worth noticing that our fixed point and endpoint existence Theorem 3.1: has a simpler proof; is Nadler type; is new in cone uniform and cone locally convex spaces; is new even in cone metric and metric spaces; and is different from those given in the previous publications on this subject.

This paper is a continuation of [32, 3846].

2 Definitions and notations

We define a real normed space to be a pair $\left(L,\parallel \cdot \parallel \right)$ with the understanding that a vector space L over carries the topology generated by the metric $\left(a,b\right)\to \parallel a-b\parallel$, $a,b\in L$.

A nonempty closed convex set $H\subset L$ is called a cone in L if it satisfies:

(H1) ${\mathrm{\forall }}_{s\in \left(0,\mathrm{\infty }\right)}\left\{sH\subset H\right\}$;

(H2) $H\cap \left(-H\right)=\left\{0\right\}$; and

(H3) $H\ne \left\{0\right\}$.

It is clear that each cone $H\subset L$ defines, by virtue of

an order of L under which L is an ordered normed space with a cone H.

We will write $a{\prec }_{H}b$ to indicate that $a{⪯}_{H}b$, but $a\ne b$. A cone H is said to be solid if $int\left(H\right)\ne \mathrm{\varnothing }$; $int\left(H\right)$ denotes the interior of H. We will write $a{\ll }_{H}b$ to indicate that $b-a\in int\left(H\right)$.

The cone H is normal if a real number $M>0$ such that for each $a,b\in H$, $0{⪯}_{H}a{⪯}_{H}b$ implies $\parallel a\parallel ⩽M\parallel b\parallel$ exists. The number M satisfying above is called the normal constant of H.

Let an element $+\mathrm{\infty }\notin L$ be such that $a{⪯}_{H}+\mathrm{\infty }$ for all $a\in L$.

Let ${2}^{X}$ denote the family of all nonempty subsets of a space X. Recall that a set-valued dynamic system is defined as a pair $\left(X,T\right)$, where X is a certain space and T is a set-valued map $T:X\to {2}^{X}$; in particular, a set-valued dynamic system includes the usual dynamic system where T is a single-valued map. We say that a map $\omega :X\to L\cup \left\{+\mathrm{\infty }\right\}$ is proper if its effective domain, $dom\left(\omega \right)=\left\{x:\omega \left(x\right)\ne +\mathrm{\infty }\right\}$, is nonempty.

Definition 2.1 ([, Def. 2.2])

Let X be a nonempty set, and let L be an ordered normed space with a cone H.
1. (i)

The family $\mathcal{P}=\left\{{p}_{\alpha }:{X}^{2}\to L,\alpha \in \mathcal{A}\right\}$, $\mathcal{A}$-index set, is said to be a $\mathcal{P}$-family of cone pseudometrics on X ($\mathcal{P}$-family for short) if the following three conditions hold:

($\mathcal{P}1$) ${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{x,y\in X}\left\{0{⪯}_{H}{p}_{\alpha }\left(x,y\right)\wedge x=y⇒{p}_{\alpha }\left(x,y\right)=0\right\}$;

($\mathcal{P}2$) ${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{x,y\in X}\left\{{p}_{\alpha }\left(x,y\right)={p}_{\alpha }\left(y,x\right)\right\}$; and

($\mathcal{P}3$) ${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{x,y,z\in X}\left\{{p}_{\alpha }\left(x,z\right){⪯}_{H}{p}_{\alpha }\left(x,y\right)+{p}_{\alpha }\left(y,z\right)\right\}$.

1. (ii)

If $\mathcal{P}$ is a $\mathcal{P}$-family, then the pair $\left(X,\mathcal{P}\right)$ is called a cone uniform space.

2. (iii)

A $\mathcal{P}$-family $\mathcal{P}$ is said to be separating if

($\mathcal{P}4$) ${\mathrm{\forall }}_{x,y\in X}\left\{x\ne y⇒{\mathrm{\exists }}_{\alpha \in \mathcal{A}}\left\{0{\prec }_{H}{p}_{\alpha }\left(x,y\right)\right\}\right\}$.

1. (iv)

If a $\mathcal{P}$-family $\mathcal{P}$ is separating, then the pair $\left(X,\mathcal{P}\right)$ is called a Hausdorff cone uniform space.

Definition 2.2 ([, Def. 2.3])

Let L be an ordered normed space with a solid cone H, and let $\left(X,\mathcal{P}\right)$ be a Hausdorff cone uniform space with a cone H.
1. (i)
We say that a sequence $\left({w}_{m}:m\in \mathbb{N}\right)$ in X is a $\mathcal{P}$-convergent in X (convergent in X for short) if there exists $w\in X$ such that
${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{{c}_{\alpha }\in L,0\ll {c}_{\alpha }}{\mathrm{\exists }}_{{n}_{0}={n}_{0}\left(\alpha ,{c}_{\alpha }\right)\in \mathbb{N}}{\mathrm{\forall }}_{m\in \mathbb{N};{n}_{0}⩽m}\left\{{p}_{\alpha }\left({w}_{m},w\right){\ll }_{H}{c}_{\alpha }\right\}.$

2. (ii)
We say that a sequence $\left({w}_{m}:m\in \mathbb{N}\right)$ in X is a $\mathcal{P}$-Cauchy sequence in X (Cauchy sequence in X, for short) if
${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{{c}_{\alpha }\in L,0\ll {c}_{\alpha }}{\mathrm{\exists }}_{{n}_{0}={n}_{0}\left(\alpha ,{c}_{\alpha }\right)\in \mathbb{N}}{\mathrm{\forall }}_{m,n\in \mathbb{N};{n}_{0}⩽m

3. (iii)

If every Cauchy sequence in X is convergent in X, then $\left(X,\mathcal{P}\right)$ is called a $\mathcal{P}$-sequentially complete cone uniform space (sequentially complete for short).

Theorem 2.1 ([, Th. 2.3])

Let L be an ordered Banach space with a normal solid cone H, and let $\left(X,\mathcal{P}\right)$ be a Hausdorff cone uniform space with a cone H. The following hold:

(P1) The sequence $\left({w}_{m}:m\in \mathbb{N}\right)$ in X converges to $w\in X$ iff
${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{{\epsilon }_{\alpha }>0}{\mathrm{\exists }}_{{n}_{0}\in \mathbb{N}}{\mathrm{\forall }}_{m\in \mathbb{N};m⩾{n}_{0}}\left\{\parallel {p}_{\alpha }\left({w}_{m},w\right)\parallel <{\epsilon }_{\alpha }\right\}.$
(P2) The sequence $\left({w}_{m}:m\in \mathbb{N}\right)$ in X is a Cauchy sequence in X iff
${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{{\epsilon }_{\alpha }>0}{\mathrm{\exists }}_{{n}_{0}\in \mathbb{N}}{\mathrm{\forall }}_{m,n\in \mathbb{N};m>n⩾{n}_{0}}\left\{\parallel {p}_{\alpha }\left({w}_{m},{w}_{n}\right)\parallel <{\epsilon }_{\alpha }\right\}.$
Definition 2.3 Let L be an ordered Banach space with a cone H.
1. (i)

A subset $D\subset L$ is said to have a minimal (maximal) element if there exists $a\in D$ such that $a{⪯}_{H}b$ ($b{⪯}_{H}a$) for all $b\in D$, and we write then that $a=min\left(D\right)$ ($a=max\left(D\right)$). It is clear that if D has a minimal (maximal) element, then the minimal (maximal) element is unique.

2. (ii)

We say that $a\in L$ is an infimum (supremum) for set $D\subset L$ if ${cl}_{L}\left(D\right)$ has the minimal (maximal) element and $a=min\left({cl}_{L}\left(D\right)\right)$ ($a=max\left({cl}_{L}\left(D\right)\right)$), and we write then that $a=inf\left(D\right)$ ($a=sup\left(D\right)$); here ${cl}_{L}\left(D\right)$ denotes the closure of D in L.

Definition 2.4 Let L be an ordered normed space with a solid cone H. The cone H is called regular if for every increasing (decreasing) sequence $\left({c}_{m}:m\in \mathbb{N}\right)$ in L which is bounded from above (below),

there exists $c\in L$ such that ${lim}_{m\to \mathrm{\infty }}\parallel {c}_{m}-c\parallel =0$. Every regular cone is normal.

Definition 2.5 ([, Def. 2.6])

Let L be an ordered normed space with a normal solid cone H, and let $\left(X,\mathcal{P}\right)$ be a Hausdorff cone uniform space with a cone H.
1. (i)

The family $\mathcal{J}=\left\{{J}_{\alpha }:{X}^{2}\to L,\alpha \in \mathcal{A}\right\}$ is said to be a $\mathcal{J}$-family of cone pseudodistances on X ($\mathcal{J}$-family on X for short) if the following three conditions hold:

($\mathcal{J}1$) ${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{x,y\in X}\left\{0{⪯}_{H}{J}_{\alpha }\left(x,y\right)\right\}$;

($\mathcal{J}2$) ${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{x,y,z\in X}\left\{{J}_{\alpha }\left(x,z\right){⪯}_{H}{J}_{\alpha }\left(x,y\right)+{J}_{\alpha }\left(y,z\right)\right\}$; and

($\mathcal{J}3$) For any sequence $\left({w}_{m}:m\in \mathbb{N}\right)$ in X such that
${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{{\epsilon }_{\alpha }>0}{\mathrm{\exists }}_{{n}_{0}={n}_{0}\left(\alpha ,{\epsilon }_{\alpha }\right)\in \mathbb{N}}{\mathrm{\forall }}_{m,n\in \mathbb{N};{n}_{0}⩽m⩽n}\left\{\parallel {J}_{\alpha }\left({w}_{m},{w}_{n}\right)\parallel <{\epsilon }_{\alpha }\right\},$
if there exists a sequence $\left({v}_{m}:m\in \mathbb{N}\right)$ in X satisfying
${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{{\epsilon }_{\alpha }>0}{\mathrm{\exists }}_{{n}_{0}={n}_{0}\left(\alpha ,{\epsilon }_{\alpha }\right)\in \mathbb{N}}{\mathrm{\forall }}_{m\in \mathbb{N};{n}_{0}⩽m}\left\{\parallel {J}_{\alpha }\left({w}_{m},{v}_{m}\right)\parallel <{\epsilon }_{\alpha }\right\},$
then
${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{{\epsilon }_{\alpha }>0}{\mathrm{\exists }}_{{n}_{0}={n}_{0}\left(\alpha ,{\epsilon }_{\alpha }\right)\in \mathbb{N}}{\mathrm{\forall }}_{m\in \mathbb{N};{n}_{0}⩽m}\left\{\parallel {p}_{\alpha }\left({w}_{m},{v}_{m}\right)\parallel <{\epsilon }_{\alpha }\right\}.$
1. (ii)

Each $\mathcal{P}$-family is a $\mathcal{J}$-family.

2. (iii)
If $\mathcal{J}=\left\{{J}_{\alpha }:{X}^{2}\to L:\alpha \in \mathcal{A}\right\}$ is a $\mathcal{J}$-family, then $X={X}_{\mathcal{J}}^{0}\cup {X}_{\mathcal{J}}^{+}$ where
${X}_{\mathcal{J}}^{0}=\left\{x\in X:{\mathrm{\forall }}_{\alpha \in \mathcal{A}}\left\{0={J}_{\alpha }\left(x,x\right)\right\}\right\}$

and
${X}_{\mathcal{J}}^{+}=\left\{x\in X:{\mathrm{\exists }}_{\alpha \in \mathcal{A}}\left\{0{\prec }_{H}{J}_{\alpha }\left(x,x\right)\right\}\right\}.$

Let $\left(X,\mathcal{P}\right)$ be a sequentially complete cone uniform space. We say that a set $Y\in {2}^{X}$ is closed in X if $Y={cl}_{X}\left(Y\right)$ where ${cl}_{X}\left(Y\right)$, the closure of Y in X, denotes the set of all $w\in X$ for which there exists a sequence $\left({w}_{m}:m\in \mathbb{N}\right)$ in Y which converges to w. If a set $Y\in {2}^{X}$ is closed in X, then $\left(Y,\mathcal{P}\right)$ is a sequentially complete cone uniform space with a cone H. Define $Cl\left(X\right)=\left\{Y\in {2}^{X}:Y={cl}_{X}\left(Y\right)\right\}$; that is, $Cl\left(X\right)$ denotes the class of all nonempty closed subsets of X.

Definition 2.6 Let L be an ordered Banach space with a normal solid cone H, let $\left(X,\mathcal{P}\right)$ be a Hausdorff sequentially complete cone uniform space with a cone H, and let $\mathcal{J}=\left\{{J}_{\alpha }:{X}^{2}\to L,\alpha \in \mathcal{A}\right\}$ be a $\mathcal{J}$-family.
1. (i)

Let $A,B\in Cl\left(X\right)$. We say that a pair $\left(A,B\right)$ is $\mathcal{J}$-admissible if:

2. (a)

For each $\alpha \in \mathcal{A}$, $x\in A$ and $y\in B$, the set ${cl}_{L}\left(\left\{{J}_{\alpha }\left(x,v\right):v\in B\right\}\right)$ has a minimal element, say ${J}_{\alpha }\left(x,B\right)$ (i.e., ${J}_{\alpha }\left(x,B\right)={inf}_{v\in B}{J}_{\alpha }\left(x,v\right)$), and the set ${cl}_{L}\left(\left\{{J}_{\alpha }\left(y,u\right):u\in A\right\}\right)$ has a minimal element, say ${J}_{\alpha }\left(y,A\right)$ (i.e., ${J}_{\alpha }\left(y,A\right)={inf}_{u\in A}{J}_{\alpha }\left(y,u\right)$);

3. (b)
The sets ${cl}_{L}\left(\left\{{J}_{\alpha }\left(u,B\right):u\in A\right\}\right)$ and ${cl}_{L}\left(\left\{{J}_{\alpha }\left(v,A\right):v\in B\right\}\right)$ have maximal elements, say ${J}_{\alpha }\left(A,B\right)$ and ${J}_{\alpha }\left(B,A\right)$, respectively (i.e.,
${J}_{\alpha }\left(A,B\right)=\underset{u\in A}{sup}{J}_{\alpha }\left(u,B\right)=\underset{u\in A}{sup}\underset{v\in B}{inf}{J}_{\alpha }\left(u,v\right)$

and
${J}_{\alpha }\left(B,A\right)=\underset{v\in B}{sup}{J}_{\alpha }\left(v,A\right)=\underset{v\in B}{sup}\underset{u\in A}{inf}{J}_{\alpha }\left(v,u\right),$
respectively); and
1. (c)

For each $\alpha \in \mathcal{A}$, the elements ${J}_{\alpha }\left(A,B\right)$ and ${J}_{\alpha }\left(B,A\right)$ are comparable.

2. (ii)
Let $A,B\in Cl\left(X\right)$, and let a pair $\left(A,B\right)$ be $\mathcal{J}$-admissible. For each $\alpha \in \mathcal{A}$, we define ${\mathcal{H}}^{\mathcal{J}}=\left\{{H}_{\alpha }^{\mathcal{J}}\left(A,B\right),\alpha \in \mathcal{A}\right\}$ where
${\mathrm{\forall }}_{\alpha \in \mathcal{A}}\left\{{H}_{\alpha }^{\mathcal{J}}\left(A,B\right)=max\left\{{J}_{\alpha }\left(A,B\right),{J}_{\alpha }\left(B,A\right)\right\}\right\}.$

Here, for each $\alpha \in \mathcal{A}$, ${H}_{\alpha }^{\mathcal{J}}\left(A,B\right)\in L\cup \left\{+\mathrm{\infty }\right\}$ and by $\left(\mathcal{J}1\right)$ and since H is closed, $0{⪯}_{H}{H}_{\alpha }^{\mathcal{J}}\left(A,B\right)$.
1. (iii)

Let a set-valued dynamic system $\left(X,T\right)$ satisfy $T:X\to Cl\left(X\right)$. We say that $\left(X,T\right)$ is $\mathcal{J}$-admissible if for each $x,y\in X$, a pair $\left(T\left(x\right),T\left(y\right)\right)$ is $\mathcal{J}$-admissible.

2. (iv)
Let $\left(X,T\right)$ satisfy $T:X\to Cl\left(X\right)$, and let $\left(X,T\right)$ be $\mathcal{J}$-admissible. If there exists the family $\mathrm{\Lambda }=\left\{{\lambda }_{\alpha }\in \left(0,1\right),\alpha \in \mathcal{A}\right\}$ such that
${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{x,y\in X}\left\{{H}_{\alpha }^{\mathcal{J}}\left(T\left(x\right),T\left(y\right)\right){⪯}_{H}{\lambda }_{\alpha }{J}_{\alpha }\left(x,y\right)\right\},$

then we say that $\left(X,T\right)$ is ${\mathcal{H}}_{\mathrm{\Lambda }}^{\mathcal{J}}$-contractive.
1. (v)
Let $E\subseteq X$, $E\ne \mathrm{\varnothing }$. The map $F:E\to H\cup \left\{+\mathrm{\infty }\right\}$ is lower semicontinuous on E with respect to X (written: F is $\left(E,X\right)$-lsc when $E\ne X$ and F is lsc when $E=X$) if the set $\left\{y\in E:F\left(y\right){⪯}_{H}c\right\}$ is a closed subset in X for each $c\in H$. Equivalently, for each ${x}_{0}\in E$,
$F\left({x}_{0}\right){⪯}_{H}\underset{x\to {x}_{0},x\in X}{lim inf}F\left(x\right).$

2. (vi)
We say that the family $\mathcal{J}$ is continuous in X if for each ${x}_{0}\in X$ and for each sequence $\left({x}_{m}:m\in \mathbb{N}\right)$ in X converging to ${x}_{0}$, we have
${\mathrm{\forall }}_{\alpha \in \mathcal{A}}\left\{\underset{m\to \mathrm{\infty }}{lim}{J}_{\alpha }\left({x}_{m},{x}_{0}\right)=\underset{m\to \mathrm{\infty }}{lim}{J}_{\alpha }\left({x}_{0},{x}_{m}\right)=0\right\}.$

If $\mathcal{J}=\mathcal{P}$, then $\mathcal{J}$ is continuous in X.

3 Statement of result

Let $\left(X,T\right)$ be a set-valued dynamic system. By $Fix\left(T\right)$ and $End\left(T\right)$ we denote the sets of all fixed points and endpoints of T, respectively, i.e., $Fix\left(T\right)=\left\{w\in X:w\in T\left(w\right)\right\}$ and $End\left(T\right)=\left\{w\in X:\left\{w\right\}=T\left(w\right)\right\}$. A dynamic process or a trajectory starting at ${w}_{0}\in X$ or a motion of the system $\left(X,T\right)$ at ${w}_{0}$ is a sequence $\left({w}_{m}:m\in \left\{0\right\}\cup \mathbb{N}\right)$ defined by ${w}_{m}\in T\left({w}_{m-1}\right)$ for $m\in \mathbb{N}$ (see, Aubin-Siegel  and Yuan ).

The aim of this paper is to prove the following fixed point and endpoint existence general result of Nadler type.

Theorem 3.1 (i) Assume that:

(A1) L is an ordered Banach space with a regular solid cone H;

(A2) $\left(X,\mathcal{P}\right)$ is a Hausdorff sequentially complete cone uniform space with a cone H;

(A3) $\mathcal{J}=\left\{{J}_{\alpha }:{X}^{2}\to L,\alpha \in \mathcal{A}\right\}$ is a $\mathcal{J}$-family on X such that ${X}_{\mathcal{J}}^{0}\ne \mathrm{\varnothing }$;

(A4) The set-valued dynamic system $\left(X,T\right)$ satisfies $T:X\to Cl\left(X\right)$ and is $\mathcal{J}$-admissible;

(A5) There exists the family $\mathrm{\Lambda }=\left\{{\lambda }_{\alpha }\in \left(0,1\right),\alpha \in \mathcal{A}\right\}$ such that $\left(X,T\right)$ is ${\mathcal{H}}_{\mathrm{\Lambda }}^{\mathcal{J}}$-contractive;

(A6) For each $x\in X$, the set ${Q}_{\mathcal{J};T}\left(x\right)$ is of the form:
$\begin{array}{rcl}{Q}_{\mathcal{J};T}\left(x\right)& =& \left\{y\in T\left(x\right)\cap {X}_{\mathcal{J}}^{0}:{\mathrm{\forall }}_{\alpha \in \mathcal{A}}\left\{{J}_{\alpha }\left(y,T\left(y\right)\right)\\ +\left({\gamma }_{\alpha }-{\lambda }_{\alpha }\right){J}_{\alpha }\left(x,y\right){⪯}_{H}{J}_{\alpha }\left(x,T\left(x\right)\right)\right\}\right\},\end{array}$

where the family $\mathrm{\Gamma }=\left\{{\gamma }_{\alpha }\in \left(0,1\right),\alpha \in \mathcal{A}\right\}$ satisfies ${\mathrm{\forall }}_{\alpha \in \mathcal{A}}\left\{{\lambda }_{\alpha }<{\gamma }_{\alpha }\right\}$;

(A7) For each $x\in {X}_{\mathcal{J}}^{0}$, the set ${Q}_{\mathcal{J};T}\left(x\right)$ is a nonempty subset in X; and

(A8) For each $x\in {X}_{\mathcal{J}}^{0}$, the set ${Q}_{\mathcal{J};T}\left(x\right)$ is a closed subset in X.

Then the following hold:

(${\mathrm{a}}_{1}$) $Fix\left(T\right)\ne \mathrm{\varnothing }$; and

(${\mathrm{a}}_{2}$) For each $w\in Fix\left(T\right)$, ${\mathrm{\forall }}_{\alpha \in \mathcal{A}}\left\{{J}_{\alpha }\left(w,w\right)=0\right\}$.

1. (ii)

(A9) For each $x\in {X}_{\mathcal{J}}^{0}$, each dynamic process $\left({w}_{m}:m\in \left\{0\right\}\cup \mathbb{N}\right)$ starting at ${w}_{0}=x$ and satisfying ${\mathrm{\forall }}_{m\in \left\{0\right\}\cup \mathbb{N}}\left\{{w}_{m+1}\in T\left({w}_{m}\right)\right\}$ satisfies ${\mathrm{\forall }}_{m\in \left\{0\right\}\cup \mathbb{N}}\left\{{w}_{m+1}\in {Q}_{\mathcal{J};T}\left({w}_{m}\right)\right\}$.

Then the assertions $\left({\mathrm{a}}_{1}\right)$ and $\left({\mathrm{a}}_{2}\right)$ are of the forms:

(${\mathrm{a}}_{1}^{\mathrm{\prime }}$) $End\left(T\right)\ne \mathrm{\varnothing }$; and

(${\mathrm{a}}_{2}^{\mathrm{\prime }}$) For each $w\in End\left(T\right)$, ${\mathrm{\forall }}_{\alpha \in \mathcal{A}}\left\{{J}_{\alpha }\left(w,w\right)=0\right\}$.

Remark 3.1 (i) Assume that:

(A10) ${\mathrm{\forall }}_{x\in {X}_{\mathcal{J}}^{0}}\left\{\left\{y\in T\left(x\right)\cap {X}_{\mathcal{J}}^{0}:{\mathrm{\forall }}_{\alpha \in \mathcal{A}}\left\{{\gamma }_{\alpha }{J}_{\alpha }\left(x,y\right){⪯}_{H}{J}_{\alpha }\left(x,T\left(x\right)\right)\right\}\right\}\ne \mathrm{\varnothing }\right\}$.

Then (A7) holds.

1. (ii)

Assume that one of the following conditions holds:

(A11) For each $\left(x,\alpha \right)\in {X}_{\mathcal{J}}^{0}×\mathcal{A}$, the map
${J}_{\alpha }\left(\cdot ,T\left(\cdot \right)\right)+\left({\gamma }_{\alpha }-{\lambda }_{\alpha }\right){J}_{\alpha }\left(x,\cdot \right):T\left(x\right)\cap {X}_{\mathcal{J}}^{0}\to H\cup \left\{+\mathrm{\infty }\right\}$

is $\left(T\left(x\right)\cap {X}_{\mathcal{J}}^{0},X\right)$-lsc;

(A12) The family $\mathcal{J}$ is continuous in X.

Then (A8) holds.

4 Proof of Theorem 3.1

We will use the following fixed point and endpoint existence general result of Caristi type.

Theorem 4.1 ([, Th. 4.5 ])
1. (i)

Assume that:

(C1) L is an ordered Banach space with a regular solid cone H;

(C2) $\left(X,\mathcal{P}\right)$ is a Hausdorff sequentially complete cone uniform space with a cone H;

(C3) The family $\mathcal{J}=\left\{{J}_{\alpha }:{X}^{2}\to L,\alpha \in \mathcal{A}\right\}$ is a $\mathcal{J}$-family on X such that ${X}_{\mathcal{J}}^{0}\ne \mathrm{\varnothing }$;

(C4) The family $\mathrm{\Omega }=\left\{{\omega }_{\alpha }:X\to H\cup \left\{+\mathrm{\infty }\right\},\alpha \in \mathcal{A}\right\}$ satisfies ${D}_{\mathrm{\Omega }}={\bigcap }_{\alpha \in \mathcal{A}}dom\left({\omega }_{\alpha }\right)$;

(C5) $\left(X,T\right)$ is a set-valued dynamic system;

(C6) $\left\{{\epsilon }_{\alpha },\alpha \in \mathcal{A}\right\}$ is a family of finite positive numbers;

(C7) For each $x\in X$, the set ${Q}_{\mathcal{J},\mathrm{\Omega };T}\left(x\right)$ is of the form:
${Q}_{\mathcal{J},\mathrm{\Omega };T}\left(x\right)=\left\{y\in T\left(x\right)\cap {X}_{\mathcal{J}}^{0}:{\mathrm{\forall }}_{\alpha \in \mathcal{A}}\left\{{\omega }_{\alpha }\left(y\right)+{\epsilon }_{\alpha }{J}_{\alpha }\left(x,y\right){⪯}_{H}{\omega }_{\alpha }\left(x\right)\right\}\right\};$

(C8) For each $x\in {X}_{\mathcal{J}}^{0}$, the set ${Q}_{\mathcal{J},\mathrm{\Omega };T}\left(x\right)$ is a nonempty subset of X; and

(C9) For each $x\in {X}_{\mathcal{J}}^{0}$, the set ${Q}_{\mathcal{J},\mathrm{\Omega };T}\left(x\right)$ is a closed subset in X.

Then there exists $w\in {D}_{\mathrm{\Omega }}\cap {X}_{\mathcal{J}}^{0}$ such that
1. (c)

$w\in T\left(w\right)$.

1. (ii)

(C10) For each $x\in {X}_{\mathcal{J}}^{0}$, each dynamic process $\left({w}_{m}:m\in \left\{0\right\}\cup \mathbb{N}\right)$ starting at ${w}_{0}=x$ and satisfying ${\mathrm{\forall }}_{m\in \left\{0\right\}\cup \mathbb{N}}\left\{{w}_{m+1}\in T\left({w}_{m}\right)\right\}$ satisfies ${\mathrm{\forall }}_{m\in \left\{0\right\}\cup \mathbb{N}}\left\{{w}_{m+1}\in {Q}_{\mathcal{J},\mathrm{\Omega };T}\left({w}_{m}\right)\right\}$.

Then assertion $\left(\mathrm{c}\right)$ is of the form:

(${\mathrm{c}}^{\mathrm{\prime }}$) $\left\{w\right\}=T\left(w\right)$.

Remark 4.1 ([, Remark 4.6 ])
1. (i)

A special case of condition (C9) is a condition ($\mathrm{C}{9}^{\mathrm{\prime }}$) defined by:

($\mathrm{C}{9}^{\mathrm{\prime }}$) For each $\left(x,\alpha \right)\in {X}_{\mathcal{J}}^{0}×\mathcal{A}$, the map
${\omega }_{\alpha }\left(\cdot \right)+{\epsilon }_{\alpha }{J}_{\alpha }\left(x,\cdot \right):T\left(x\right)\cap {X}_{\mathcal{J}}^{0}\to H\cup \left\{+\mathrm{\infty }\right\}$

is $\left(T\left(x\right)\cap {X}_{\mathcal{J}}^{0},X\right)$-lsc.

1. (ii)

If $\mathcal{J}=\mathcal{P}$, then a special case of condition (C9) is a condition ($\mathrm{C}{9}^{\mathrm{\prime }\mathrm{\prime }}$) defined by:

($\mathrm{C}{9}^{\mathrm{\prime }\mathrm{\prime }}$) For each $\left(x,\alpha \right)\in X×\mathcal{A}$, the map
${\omega }_{\alpha }\left(\cdot \right)+{\epsilon }_{\alpha }{p}_{\alpha }\left(x,\cdot \right):T\left(x\right)\to H\cup \left\{+\mathrm{\infty }\right\}$

is $\left(T\left(x\right),X\right)$-lsc.

The proof will be broken into seven steps.

Step 1. Let $\mathrm{\Omega }=\left\{{\omega }_{\alpha }:X\to L,\alpha \in \mathcal{A}\right\}$ where
${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{x\in X}\left\{{\omega }_{\alpha }\left(x\right)={J}_{\alpha }\left(x,T\left(x\right)\right)\right\}.$
Indeed, by (A4), (A5) and (iii) and (iv) of Definition 2.6, we obtain
${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{x,y\in X}\left\{\underset{u\in T\left(x\right)}{sup}{J}_{\alpha }\left(u,T\left(y\right)\right){⪯}_{H}{H}_{\alpha }^{\mathcal{J}}\left(T\left(x\right),T\left(y\right)\right){⪯}_{H}{\lambda }_{\alpha }{J}_{\alpha }\left(x,y\right)\right\}.$
Hence, in particular, for $u=y$, we get
${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{x\in X}{\mathrm{\forall }}_{y\in T\left(x\right)}\left\{{\omega }_{\alpha }\left(y\right){⪯}_{H}{\lambda }_{\alpha }{J}_{\alpha }\left(x,y\right)\right\}.$

This implies (4.1).

Note that
${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{x\in X}\left\{{J}_{\alpha }\left(x,T\left(x\right)\right)=\underset{y\in T\left(x\right)}{inf}{J}_{\alpha }\left(x,y\right)\right\}.$
(4.5)

This, by (A6) (recall that ${\mathrm{\forall }}_{\alpha \in \mathcal{A}}\left\{{\gamma }_{\alpha }\in \left(0,1\right)\right\}$), implies (4.2).

i.e., (4.3) holds.

By (4.3) and ($\mathcal{J}$1),
${\mathrm{\forall }}_{\alpha \in \mathcal{A}}{\mathrm{\forall }}_{x\in X}{\mathrm{\forall }}_{y\in T\left(x\right)}\left\{0{⪯}_{H}{\omega }_{\alpha }\left(x\right)-{\omega }_{\alpha }\left(y\right)\right\}$

Consequently, (4.4) holds.

Step 2. The family Ω defined in Step 1 satisfies (C4).

Indeed, by (4.1),
${\mathrm{\forall }}_{x\in X}\left\{\left\{y\in T\left(x\right):{\mathrm{\forall }}_{\alpha \in \mathcal{A}}\left\{{\omega }_{\alpha }\left(y\right){⪯}_{H}{\lambda }_{\alpha }{J}_{\alpha }\left(x,y\right)\right\}\right\}=T\left(x\right)\right\}.$

Also, by Definition 2.5, $\mathcal{J}=\left\{{J}_{\alpha }:{X}^{2}\to L,\alpha \in \mathcal{A}\right\}$, ($\mathcal{J}$1) holds and, by (A4), ${\mathrm{\forall }}_{x\in X}\left\{\mathrm{\varnothing }\ne T\left(x\right)\right\}$. Hence, we conclude that ${\mathrm{\forall }}_{x\in X}\left\{\mathrm{\varnothing }\ne T\left(x\right)\subset {D}_{\mathrm{\Omega }}\right\}$.

Step 3. Assumptions (C8) and (C9) hold where ${\mathrm{\forall }}_{\alpha \in \mathcal{A}}\left\{{\epsilon }_{\alpha }={\gamma }_{\alpha }-{\lambda }_{\alpha }\right\}$ and Ω is defined in Step 1.

By (4.2) and (4.3), in particular,
${\mathrm{\forall }}_{x\in {X}_{\mathcal{J}}^{0}}{\mathrm{\forall }}_{\alpha \in \mathcal{A}}\left\{\mathrm{\varnothing }\ne \left\{y\in T\left(x\right):{\omega }_{\alpha }\left(y\right)+\left({\gamma }_{\alpha }-{\lambda }_{\alpha }\right){J}_{\alpha }\left(x,y\right){⪯}_{H}{\omega }_{\alpha }\left(x\right)\right\}\right\}$
and, by (A7) and (A8), the following property concerning intersection of these sets holds: for each $x\in {X}_{\mathcal{J}}^{0}$,
$\left\{y\in T\left(x\right)\cap {X}_{\mathcal{J}}^{0}:{\mathrm{\forall }}_{\alpha \in \mathcal{A}}\left\{{\omega }_{\alpha }\left(y\right)+\left({\gamma }_{\alpha }-{\lambda }_{\alpha }\right){J}_{\alpha }\left(x,y\right){⪯}_{H}{\omega }_{\alpha }\left(x\right)\right\}\right\}={Q}_{\mathcal{J},\mathrm{\Omega };T}\left(x\right)$

is a nonempty closed subset in X.

Step 4. The assertions of Theorem 3.1 hold.

This follows from Assumptions (A1)-(A9), Steps 1-3, definition of ${X}_{\mathcal{J}}^{0}$ and Theorem 4.1.

Step 5. Assumption (A10) implies (A7).

Indeed, denote
${\mathrm{\forall }}_{x\in X}\left\{{U}_{\mathcal{J}}\left(x\right)=\bigcap _{\alpha \in \mathcal{A}}{U}_{\alpha }\left(x\right)\right\}$
and
${\mathrm{\forall }}_{x\in X}\left\{{V}_{\mathcal{J}}\left(x\right)=\bigcap _{\alpha \in \mathcal{A}}{V}_{\alpha }\left(x\right)\right\}.$
By (4.2) and (4.3),
${\mathrm{\forall }}_{x\in {X}_{\mathcal{J}}^{0}}\left\{{U}_{\mathcal{J}}\left(x\right)\cap {X}_{\mathcal{J}}^{0}\subset {V}_{\mathcal{J}}\left(x\right)\cap {X}_{\mathcal{J}}^{0}\subset {Q}_{\mathcal{J};T}\left(x\right)\right\}.$

Hence, we conclude that for each $x\in {X}_{\mathcal{J}}^{0}$, the set ${Q}_{\mathcal{J};T}\left(x\right)$ is nonempty whenever ${\mathrm{\forall }}_{x\in {X}_{\mathcal{J}}^{0}}\left\{{U}_{\mathcal{J}}\left(x\right)\cap {X}_{\mathcal{J}}^{0}\ne \mathrm{\varnothing }\right\}$.

Step 6. Assumption (A11) implies (A8).

This follows from Remark 4.1(i)

Step 7. Assumption (A12) implies (A8).

Let ${x}_{0}$ be arbitrary and fixed, and let a sequence $\left({x}_{m}:m\in \mathbb{N}\right)$ in X be convergent to ${x}_{0}$, i.e., let ${\mathrm{\forall }}_{\alpha \in \mathcal{A}}\left\{{lim}_{m\to \mathrm{\infty }}{p}_{\alpha }\left({x}_{0},{x}_{m}\right)=0\right\}$ (see Definition 2.2 and Theorem 2.1). If $m\in \mathbb{N}$, $v\in T\left({x}_{m}\right)$ and $\alpha \in \mathcal{A}$ are arbitrary and fixed, then by ($\mathcal{J}$1),
${\omega }_{\alpha }\left({x}_{0}\right)={J}_{\alpha }\left({x}_{0},T\left({x}_{0}\right)\right){⪯}_{H}{J}_{\alpha }\left({x}_{0},{x}_{m}\right)+{J}_{\alpha }\left({x}_{m},v\right)+{J}_{\alpha }\left(v,T\left({x}_{0}\right)\right).$
Since $v\in T\left({x}_{m}\right)$ and T satisfy (A5), this implies
$\begin{array}{rcl}{\omega }_{\alpha }\left({x}_{0}\right)& {⪯}_{H}& {J}_{\alpha }\left({x}_{0},{x}_{m}\right)+{J}_{\alpha }\left({x}_{m},T\left({x}_{m}\right)\right)+\underset{v\in T\left({x}_{m}\right)}{sup}{J}_{\alpha }\left(v,T\left({x}_{0}\right)\right)\\ {⪯}_{H}& {J}_{\alpha }\left({x}_{0},{x}_{m}\right)+{J}_{\alpha }\left({x}_{m},T\left({x}_{m}\right)\right)+{H}_{\alpha }^{\mathcal{J}}\left(T\left({x}_{m}\right),T\left({x}_{0}\right)\right)\\ {⪯}_{H}& {J}_{\alpha }\left({x}_{0},{x}_{m}\right)+{\omega }_{\alpha }\left({x}_{m}\right)+{\lambda }_{\alpha }{J}_{\alpha }\left({x}_{m},{x}_{0}\right),\end{array}$
that is,
$0{⪯}_{H}{J}_{\alpha }\left({x}_{0},{x}_{m}\right)+{\omega }_{\alpha }\left({x}_{m}\right)+{\lambda }_{\alpha }{J}_{\alpha }\left({x}_{m},{x}_{0}\right)-{\omega }_{\alpha }\left({x}_{0}\right).$
Hence, by (A4), since H is closed, using the fact that $\mathcal{J}$ is continuous and taking the limit as $m\to \mathrm{\infty }$, we get
$0{⪯}_{H}\underset{m\to \mathrm{\infty }}{lim inf}{\omega }_{\alpha }\left({x}_{m}\right)-{\omega }_{\alpha }\left({x}_{0}\right).$
Therefore, for each $\alpha \in \mathcal{A}$,
${\omega }_{\alpha }\left({x}_{0}\right){⪯}_{H}\underset{m\to \mathrm{\infty }}{lim inf}{\omega }_{\alpha }\left({x}_{m}\right),$
i.e., ${\omega }_{\alpha }$ is lsc in X. Moreover, if $m\in \mathbb{N}$, $x\in X$ and $\alpha \in \mathcal{A}$ are arbitrary and fixed, then by ($\mathcal{J}$1),
${J}_{\alpha }\left(x,{x}_{0}\right){⪯}_{H}{J}_{\alpha }\left(x,{x}_{m}\right)+{J}_{\alpha }\left({x}_{m},{x}_{0}\right),$
that is,
$0{⪯}_{H}{J}_{\alpha }\left(x,{x}_{m}\right)+{J}_{\alpha }\left({x}_{m},{x}_{0}\right)-{J}_{\alpha }\left(x,{x}_{0}\right).$
Since H is closed and $\mathcal{J}$ is continuous, this implies
$0{⪯}_{H}\underset{m\to \mathrm{\infty }}{lim inf}{J}_{\alpha }\left(x,{x}_{m}\right)-{J}_{\alpha }\left(x,{x}_{0}\right),$
that is, for each $\left(x,\alpha \right)\in X×\mathcal{A}$, the map ${J}_{\alpha }\left(x,\cdot \right)$ is lsc in X. Hence, in particular, we conclude that for each $\left(x,\alpha \right)\in {X}_{\mathcal{J}}^{0}×\mathcal{A}$, the map
${\omega }_{\alpha }\left(\cdot \right)+\left({\gamma }_{\alpha }-{\lambda }_{\alpha }\right){J}_{\alpha }\left(x,\cdot \right):T\left(x\right)\cap {X}_{\mathcal{J}}^{0}\to H\cup \left\{+\mathrm{\infty }\right\}$

is $\left(T\left(x\right)\cap {X}_{\mathcal{J}}^{0},X\right)$-lsc, that is, ($\mathrm{C}{9}^{\mathrm{\prime }}$) holds.

5 Remarks, examples and comparisons

Remark 5.1 Examples 5.1 and 5.2 illustrate a fixed point version and an endpoint version of Theorem 3.1, respectively, in cone metric spaces with $\mathcal{J}$-family where $\mathcal{J}=\left\{J\right\}$ and $J\ne p$.

Example 5.1 If
$X=\left\{\mathbf{N}=\left(n,n\right):n\in \left\{1,2,3,4,5,6\right\}\right\}=\left\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4},\mathbf{5},\mathbf{6}\right\},$
$L={\mathbb{R}}^{2}$, $H=\left\{\left(x,y\right)\in L:x,y⩾0\right\}\subset {\mathbb{R}}^{2}$ and, for each $\beta >0$, $p:{X}^{2}\to L$ is defined by the formula
$p\left(\mathbf{N},\mathbf{M}\right)=\left(|n-m|,\beta |n-m|\right),\phantom{\rule{1em}{0ex}}\mathbf{N}=\left(n,n\right),\mathbf{M}=\left(m,m\right)\in X,$

then $\left(X,\mathcal{P}\right)$, $\mathcal{P}=\left\{p\right\}$ is a cone metric space; let in the sequel $\beta =2$.

Let $T:X\to Cl\left(X\right)$ be of the form:
Let $W=\left\{\mathbf{1},\mathbf{2},\mathbf{4},\mathbf{5}\right\}$, and let $J:{X}^{2}\to L$ be of the form:

$\mathbf{N},\mathbf{M}\in X$. Clearly, $\mathcal{J}=\left\{J\right\}$ is a $\mathcal{J}$-family on X (see [, Ex. 5.1]).

We observe that ${X}_{J}^{0}=\left\{\mathbf{1},\mathbf{2},\mathbf{4},\mathbf{5}\right\}\ne \mathrm{\varnothing }$.

We show that $\left(X,T\right)$ is $\mathcal{J}$-admissible and ${\mathcal{H}}_{3/4}^{J}$-contractive on X where
${\mathrm{\forall }}_{A,B\in Cl\left(X\right)}\left\{{H}^{J}\left(A,B\right)=max\left\{\underset{\mathbf{N}\in A}{sup}J\left(\mathbf{N},B\right),\underset{\mathbf{M}\in B}{sup}J\left(\mathbf{M},A\right)\right\}\right\}.$

Indeed, let $\lambda =3/4$, and let $\mathbf{N},\mathbf{M}\in X$ be arbitrary and fixed.

We consider three cases:

Case 1. If $\mathbf{N},\mathbf{M}\in X\setminus \left\{\mathbf{6}\right\}$, then by definition of T, we have that $T\left(\mathbf{N}\right)=T\left(\mathbf{M}\right)=\left\{\mathbf{1},\mathbf{2}\right\}$ and
${H}^{J}\left(T\left(\mathbf{N}\right),T\left(\mathbf{M}\right)\right)=\left(0,0\right)=\mathbf{0}{⪯}_{H}\left(3/4\right)J\left(\mathbf{N},\mathbf{M}\right)=\lambda J\left(\mathbf{N},\mathbf{M}\right).$
Case 2. If $\mathbf{N}\in X\setminus \left\{\mathbf{6}\right\}$ and $\mathbf{M}=\mathbf{6}$, then by definition of T, $T\left(\mathbf{N}\right)=\left\{\mathbf{1},\mathbf{2}\right\}$ and $T\left(\mathbf{M}\right)=\left\{\mathbf{4},\mathbf{5}\right\}$. Hence, by definition of J, we calculate:
1. (i)
$J\left(\mathbf{1},T\left(\mathbf{M}\right)\right)=p\left(\mathbf{1},\left\{\mathbf{4},\mathbf{5}\right\}\right)=\left(3,6\right)$, $J\left(\mathbf{2},T\left(\mathbf{M}\right)\right)=p\left(\mathbf{2},\left\{\mathbf{4},\mathbf{5}\right\}\right)=\left(2,4\right)$ and
$sup\left\{J\left(\mathbf{U},T\left(\mathbf{M}\right)\right):\mathbf{U}\in T\left(\mathbf{N}\right)\right\}=\left(3,6\right);$

2. (ii)
$J\left(\mathbf{4},T\left(\mathbf{N}\right)\right)=p\left(\mathbf{4},\left\{\mathbf{1},\mathbf{2}\right\}\right)=\left(2,4\right)$, $J\left(\mathbf{5},T\left(\mathbf{N}\right)\right)=p\left(\mathbf{5},\left\{\mathbf{1},\mathbf{2}\right\}\right)=\left(3,6\right)$ and
$sup\left\{J\left(\mathbf{V},T\left(\mathbf{N}\right)\right):\mathbf{V}\in T\left(\mathbf{M}\right)\right\}=\left(3,6\right);$

3. (iii)
By (i) and (ii),
$\begin{array}{rcl}{H}^{J}\left(T\left(\mathbf{N}\right),T\left(\mathbf{M}\right)\right)& =& max\left\{sup\left\{J\left(\mathbf{U},T\left(\mathbf{M}\right)\right):\mathbf{U}\in T\left(\mathbf{N}\right)\right\},\\ sup\left\{J\left(\mathbf{V},T\left(\mathbf{N}\right)\right):\mathbf{V}\in T\left(\mathbf{M}\right)\right\}\right\}=\left(3,6\right).\end{array}$

Consequently,
${H}^{J}\left(T\left(\mathbf{N}\right),T\left(\mathbf{M}\right)\right)=\left(3,6\right){⪯}_{H}\mathbf{6}=\left(3/4\right)\cdot \mathbf{8}=\lambda J\left(\mathbf{N},\mathbf{M}\right)$

for $\mathbf{N}\in X\setminus \left\{\mathbf{6}\right\}$ and $\mathbf{M}=\mathbf{6}$.

Case 3. If $\mathbf{N}=\mathbf{6}$ and $\mathbf{M}\in X\setminus \left\{\mathbf{6}\right\}$, then by analogous considerations as in Case 2, we get
${H}^{J}\left(T\left(\mathbf{N}\right),T\left(\mathbf{M}\right)\right)=\left(3,6\right){⪯}_{H}\mathbf{6}=\left(3/4\right)\cdot \mathbf{8}=\lambda J\left(\mathbf{N},\mathbf{M}\right).$

Thus, T is $\mathcal{J}$-admissible and ${\mathcal{H}}_{3/4}^{J}$-contractive on X.

Let now $\gamma =7/8$. Then for each $\mathbf{N}\in {X}_{J}^{0}=\left\{\mathbf{1},\mathbf{2},\mathbf{4},\mathbf{5}\right\}$, we have $T\left(\mathbf{N}\right)=\left\{\mathbf{1},\mathbf{2}\right\}$ and using the fact that $T\left({X}_{J}^{0}\right)\subset {X}_{J}^{0}$, we obtain
$\begin{array}{rcl}{Q}_{J;T}\left(\mathbf{N}\right)& =& \left\{\mathbf{M}\in \left\{\mathbf{1},\mathbf{2}\right\}:J\left(\mathbf{M},T\left(\mathbf{M}\right)\right)+\left(1/8\right)J\left(\mathbf{N},\mathbf{M}\right){⪯}_{H}J\left(\mathbf{N},T\left(\mathbf{N}\right)\right)\right\}\\ =& \left\{\mathbf{M}\in \left\{\mathbf{1},\mathbf{2}\right\}:\left(1/8\right)p\left(\mathbf{N},\mathbf{M}\right){⪯}_{H}p\left(\mathbf{N},\left\{\mathbf{1},\mathbf{2}\right\}\right)\right\}.\end{array}$
and
${Q}_{J;T}\left(\mathbf{N}\right)=\left\{\mathbf{M}\in \left\{\mathbf{1},\mathbf{2}\right\}:\left(1/8\right)p\left(\mathbf{N},\mathbf{M}\right){⪯}_{H}\left(n-2,2n-4\right)\right\}=\left\{\mathbf{1},\mathbf{2}\right\}$

for $\mathbf{N}=\left(n,n\right)\in \left\{\mathbf{4},\mathbf{5}\right\}$.

Assumptions (A1)-(A8) of Theorem 3.1 hold, $Fix\left(T\right)=\left\{\left(1,1\right),\left(2,2\right)\right\}$ and $J\left(\left(1,1\right),\left(1,1\right)\right)=J\left(\left(2,2\right),\left(2,2\right)\right)=0$.

Example 5.2 Let X, W, J, λ and γ be such as in Example 5.1, and let $T:X\to Cl\left(X\right)$ be of the form:
Then ${X}_{J}^{0}=\left\{\mathbf{1},\mathbf{2},\mathbf{4},\mathbf{5}\right\}$ and
${Q}_{J;T}\left(\mathbf{N}\right)=\left\{\mathbf{M}\in T\left(\mathbf{N}\right):J\left(\mathbf{M},T\left(\mathbf{M}\right)\right)+\left(1/8\right)J\left(\mathbf{N},\mathbf{M}\right){⪯}_{H}J\left(\mathbf{N},T\left(\mathbf{N}\right)\right)\right\}$
for $\mathbf{N}\in {X}_{J}^{0}$ since $T\left({X}_{J}^{0}\right)\subset {X}_{J}^{0}$. Hence:
and
${Q}_{J;T}\left(\mathbf{5}\right)=\left\{\mathbf{M}\in \left\{\mathbf{1}\right\}:J\left(\mathbf{1},\mathbf{1}\right)+\left(1/8\right)J\left(\mathbf{5},\mathbf{1}\right){⪯}_{H}J\left(\mathbf{5},\mathbf{1}\right)\right\}=\left\{\mathbf{1}\right\}.$

Assumptions (A1)-(A9) of Theorem 3.1 hold, $End\left(T\right)=\left\{\left(1,1\right),\left(2,2\right)\right\}$ and $J\left(\left(1,1\right),\left(1,1\right)\right)=J\left(\left(2,2\right),\left(2,2\right)\right)=0$.

Remark 5.2 In Example 5.3, we show that in our concept of ${\mathcal{H}}_{\mathrm{\Lambda }}^{\mathcal{J}}$-contractive set-valued dynamic systems, the existence of $\mathcal{J}$-family such that $\mathcal{J}\ne \mathcal{D}$ is essential; from Example 5.3, it follows that for maps defined in Examples 5.1 and 5.2, we cannot use Theorem 3.1 when $\mathcal{J}=\left\{p\right\}$.

Example 5.3 (a) Let X and T be such as in Example 5.1. We observe that for each $\lambda \in \left(0,1\right)$, T is not ${\mathcal{H}}_{\lambda }^{p}$-contractive on X.

Otherwise, $J=p$, ${X}_{p}^{0}=X$ and
${\mathrm{\exists }}_{\lambda \in \left(0,1\right)}{\mathrm{\forall }}_{\mathbf{N},\mathbf{M}\in X}\left\{{H}^{p}\left(T\left(\mathbf{N}\right),T\left(\mathbf{M}\right)\right)⩽\lambda p\left(\mathbf{N},\mathbf{M}\right)\right\}.$
However, for ${\mathbf{N}}_{0}=\mathbf{3}$ and ${\mathbf{M}}_{0}=\mathbf{6}$ from X, we obtain:
1. (i)

$T\left({\mathbf{N}}_{0}\right)=\left\{\mathbf{1},\mathbf{2}\right\}$ and $T\left({\mathbf{M}}_{0}\right)=\left\{\mathbf{4},\mathbf{5}\right\}$;

2. (ii)
$p\left(\mathbf{1},T\left({\mathbf{M}}_{0}\right)\right)=p\left(\mathbf{1},\left\{\mathbf{4},\mathbf{5}\right\}\right)=\left(3,6\right)$, $p\left(\mathbf{2},T\left({\mathbf{M}}_{0}\right)\right)=p\left(\mathbf{2},\left\{\mathbf{4},\mathbf{5}\right\}\right)=\left(2,4\right)$ and
$sup\left\{p\left(\mathbf{U},T\left({\mathbf{M}}_{0}\right)\right):\mathbf{U}\in T\left({\mathbf{N}}_{0}\right)\right\}=\left(3,6\right);$

3. (iii)
$p\left(\mathbf{4},T\left({\mathbf{N}}_{0}\right)\right)=p\left(\mathbf{4},\left\{\mathbf{1},\mathbf{2}\right\}\right)=\left(2,4\right)$, $p\left(\mathbf{5},T\left({\mathbf{N}}_{0}\right)\right)=p\left(\mathbf{5},\left\{\mathbf{1},\mathbf{2}\right\}\right)=\left(3,6\right)$ and
$sup\left\{p\left(\mathbf{V},T\left({\mathbf{N}}_{0}\right)\right):\mathbf{V}\in T\left({\mathbf{M}}_{0}\right)\right\}=\left(3,6\right);$

4. (iv)
By (i)-(iii),
$\begin{array}{rcl}{H}^{p}\left(T\left({\mathbf{N}}_{0}\right),T\left({\mathbf{M}}_{0}\right)\right)& =& max\left\{sup\left\{p\left(\mathbf{U},T\left({\mathbf{M}}_{0}\right)\right):\mathbf{U}\in T\left({\mathbf{N}}_{0}\right)\right\},\\ sup\left\{p\left(\mathbf{V},T\left({\mathbf{N}}_{0}\right)\right):\mathbf{V}\in T\left({\mathbf{M}}_{0}\right)\right\}\right\}=\left(3,6\right).\end{array}$

Consequently, for each $\lambda \in \left(0,1\right)$,
$\begin{array}{rcl}\left(3,6\right)& =& {H}^{p}\left(T\left({\mathbf{N}}_{0}\right),T\left({\mathbf{M}}_{0}\right)\right){⪯}_{H}\lambda p\left({\mathbf{N}}_{0},{\mathbf{M}}_{0}\right){\prec }_{H}p\left({\mathbf{N}}_{0},{\mathbf{M}}_{0}\right)\\ =& p\left(\mathbf{3},\mathbf{6}\right)=\left(3,6\right).\end{array}$
It is absurd.
1. (b)

Let X and T be such as in Example 5.2. By similar argumentation as in $\left(\mathrm{a}\right)$, we observe that for each $\lambda \in \left(0,1\right)$, T is not ${\mathcal{H}}_{\lambda }^{p}$-contractive on X.

Authors’ Affiliations

(1)
Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of Łódź, Banacha 22, Łódź, 90-238, Poland

References 