# Dynamic Processes, Fixed Points, Endpoints, Asymmetric Structures, and Investigations Related to Caristi, Nadler, and Banach in Uniform Spaces

Authors: Kazimierz Włodarczyk; Robert Plebaniak
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

In uniform spacesX,D with symmetric structures determined by the
D-families of pseudometrics which define uniformity in these spaces, the
new symmetric and asymmetric structures determined by the J-families
of generalized pseudodistances on X are constructed; using these structures the set-valued contractions of two kinds of Nadler type are defined
and the new and general theorems concerning the existence of fixed points
and endpoints for such contractions are proved. Moreover, using these new
structures, the single-valued contractions of two kinds of Banach type are
defined and the new and general versions of the Banach uniqueness and
iterate approximation of fixed point theorem for uniform spaces are established. Contractions defined and studied here are not necessarily continuous. One of the main key ideas in this paper is the application of
our fixed point and endpoint version of Caristi type theorem for dissipative set-valued dynamic systems without lower semicontinuous entropies
in uniform spaces with structures determined by J-families. Results are
new also in locally convex and metric spaces. Examples are provided.


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## Body

## 1. Introduction

The concepts of the symmetric and asymmetric structures became established and investigated in mathematics and in theoretical computer science and are some creative ideas in fixed point theory by which some fascinating results have been achieved. In the proofs of these results, some deep methods based on those symmetric and asymmetric structures do play very important roles. The range of important applications of these results is enormous.Let(X,D) be a uniform space with uniformity defined by a saturated family D={dα:α∈A} of pseudometrics dα:X2→[0;∞), α∈A, uniformly continuous on X2 (D-family, for short); here A is a nonempty index set.It was discovered that theJ-families of generalized pseudodistances defined below generalize: metrics d, distances of Tataru [1], w-distances of Kada et al. [2], τ-distances of Suzuki [3], and τ-functions of Lin and Du [4] in metric spaces (X,d) and also D-families of pseudometrics and distances of Vályi [5] in uniform spaces (X,D).Definition 1 (see [6]).
Let(X,D) be a Hausdorff uniform space.(a)
The familyJ={Jα:α∈A} of maps Jα:X2→[0;∞), α∈A, is said to be a J-family of generalized pseudodistances on X (J-family, for short) if the following two conditions hold:
(J1)
∀α∈A∀x,y,z∈X{Jα(x,z)⩽Jα(x,y)+Jα(y,z)}.(J2)
For any sequences(xm:m∈N) and (ym:m∈N) in X such that ∀α∈A{limn→∞sup⁡m>nJα(xn,xm)=0} and ∀α∈A{limm→∞Jα(xm,ym)=0}, the following holds ∀α∈A{limm→∞dα(xm,ym)=0}.(b)
Define(1)JX,D=J:J=Jα:X2⟶0;∞,α∈AhhhisaJ-familyon⁡X2X.Definition 2.
Let(X,d) be a metric space.(a)
ThenJ={J:X2→[0;∞)} is said to be a J-family on X if J is a generalized pseudodistance on X; that is, the following two conditions hold:
(J1)
∀x,y,z∈X{J(x,z)⩽J(x,y)+J(y,z)}.(J2)
For any sequences(xm:m∈N) and (ym:m∈N) in X such that limn→∞sup⁡m>nJ(xn,xm)=0 and limm→∞J(xm,ym)=0, the following holds limm→∞d(xm,ym)=0.(b)
DefineJ(X,d)={J:J={J:X2→[0;∞)} is a J-family on X}.In the following remark, we list some basic properties ofJ(X,D).Remark 3.
Let(X,D) be a Hausdorff uniform space.(a)
D={dα:α∈A}∈J(X,D) and J(X,D)≠{D}.(b)
([7, Remark 1.1]) Let J={Jα:X2→[0;∞),α∈A}∈J(X,D). If x≠y, x, y∈X, then ∃α∈A{Jα(x,y)>0∨Jα(y,x)>0}.(c)
LetJ={Jα:X2→[0;∞),α∈A}∈J(X,D). If ∀α∈A∀x∈X{Jα(x,x)=0}, then, for each α∈A, Jα is quasipseudometric; examples of J={Jα:X2→[0;∞),α∈A}∈J(X,D) such that the maps Jα, α∈A, are not quasipseudometrics are given in Section 4.Definition 4.
LetX,D be a Hausdorff uniform space. J=Jα:X2→0;∞,α∈A∈JX,D is said to beadmissible if XJ0≠⌀ where
(2)XJ0=x∈X:∀α∈AJαx,x=0.Remark 5.
It is a remarkable fact thatD-family is admissible and XD0=X. Indeed, we have that X=XJ0∪XJ+, where XJ+={x∈X:∃α∈A{Jα(x,x)>0}}. Therefore, by Definition 4, we get the following XD0={x∈X:∀α∈A{dα(x,x)=0}}=X.Let2X denote the family of all nonempty subsets of a space X. Aset-valued dynamic system is defined as a pair (X,T), where X is a certain space and T is a set-valued map T:X→2X; in particular, a set-valued dynamic system includes the usual dynamic system (X,T) where T:X→X is a single-valued map.Let(X,T) be a set-valued dynamic system. By Fix⁡(T) and End(T) we denote the sets of allfixed points andendpoints of T, respectively; that is, Fix⁡(T)={w∈X:w∈T(w)} and End(T)={w∈X:{w}=T(w)}. Adynamic process or atrajectory starting atw0∈X or amotion of the system (X,T) at w0 is a sequence (wm:m∈{0}∪N) defined by wm∈T(wm-1) for m∈N (see, Aubin and Siegel [8], Aubin and Ekeland [9], Aubin and Frankowska [10], and Yuan [11]).Recall that a mapω:X→[0;+∞] isproper if its effective domain, dom(ω)={x:ω(x)≠+∞}, is nonempty.Caristi’s fixed point theorem [12] concerning dissipative single-valued dynamic systems (T,X) in metric spaces (X,d) with lower semicontinuous entropies ω:X→[0;+∞] is equivalent to Ekeland’s variational principle [13–15] providing approximate solutions of nonconvex minimization problems concerning lower semicontinuous maps ω:X→[0;+∞].Theorem 6.
Let(X,d) be a complete metric space.(I)
(Caristi [12]) Let ω:X→[0;+∞] be a map which is proper lower semicontinuous and let (T,X) be a single-valued dynamic system satisfying the condition
(3)∀x∈X{ω(T(x))+d(x,T(x))⩽ω(x)}.
Then Fix⁡(T)≠⌀ (i.e., there exists w∈X such that T(w)=w).(II)
(Ekeland [13–15]) Let ω:X→[0;+∞] be a map which is proper lower semicontinuous. For every ε>0 and for every x0∈dom(ω), there exists u∈X such that ω(u)+εd(x0,u)⩽ω(x0) and ∀x∈X∖{u}{ω(u)<ω(x)+εd(x,u)}.Let(X,D) be a sequentially complete uniform space. We say that a set Y∈2X isclosed in X if Y=cl⁡X(Y), where clX(Y), theclosure of Y in X, denotes the set of all w∈X for which there exists a sequence (wm:m∈N) in Y which converges to w. If a set Y∈2X is closed in X, then (Y,D) is a sequentially complete uniform space.DefineCl(X)={Y∈2X:Y=clX(Y)}; that is, Cl(X) denotes the class of all nonempty closed subsets of X.The following fixed point and endpoint version of Caristi type theorem for dissipative set-valued dynamic systems without lower semicontinuous entropies in uniform spaces with structures determined byJ∈J(X,D) is included in a more general result [6, Theorem 4.5].Theorem 7.
Let(X,D) be a Hausdorff sequentially complete uniform space and let J={Jα:X2→[0;∞),α∈A}∈J(X,D) be admissible.(I)
(Fixed point theorem) Assume the following.
(A1)
The familyΩ={ωα:X→[0;∞],α∈A} satisfies DΩ=⋂α∈A‍dom(ωα)≠⌀.(A2)
Υ={εα,α∈A} is a family of finite positive numbers.(A3)
(X,T) is a set-valued dynamic system.(A4)
For eachx∈X, QJ,Ω,Υ;T(x) is a set defined by
(4)QJ,Ω,Υ;T(x)=y∈Tx∩XJ0:∀α∈Aωαy+εαJαx,y⩽ωαx.(A5)
For eachx∈XJ0, the set QJ,Ω,Υ;T(x) is nonempty.(A6)
For eachx∈XJ0, the set QJ,Ω,Υ;T(x) is a closed subset in X.
Then, there existsw∈DΩ∩XJ0 such that w∈T(w) (i.e., ∀α∈A{Jα(w,w)=0} and w∈Fix⁡(T)).(II)
(Endpoint theorem) Assume, in addition, that
(A7)
for eachx∈XJ0, each dynamic process (wm:m∈{0}∪N) starting at w0=x and satisfying ∀m∈{0}∪N{wm+1∈T(wm)} satisfies ∀m∈{0}∪N{wm+1∈QJ,Ω,Υ;T(wm)}.
Then, there existsw∈DΩ∩XJ0 such that {w}=T(w) (i.e., ∀α∈A{Jα(w,w)=0} and w∈End(T)).It is known that a weaker condition than continuity is lower semicontinuity.Definition 8.
Let(X,D) be a Hausdorff sequentially complete uniform space. Let E⊆X, E≠⌀ and let f:E→[0;∞]. The map f islower semicontinuous onEwith respect toX (written: fis(E,X)-lsc when E≠X and fis lsc when E=X) if the set {y∈E:f(y)⩽c} is a closed subset in X for each c∈[0;∞).The following alternative characterizations of lower semicontinuity hold.Theorem 9.
Let(X,D) be a Hausdorff sequentially complete uniform space. Let E⊆X, E≠⌀ and let f:E→[0;∞]. The following conditions are equivalent.(Z1)
The mapf is lower semicontinuous on E with respect to X.(Z2)
For eachx0∈E,
(5)f(x0)⩽liminf⁡x→x0,x∈Xf(x);
here
(6)liminf⁡x→x0,x∈Xfx=sup⁡inf⁡{f(x):x∈E∩(U∖{x0})}:hhhhhhUopeninX,x0∈U,hhhhhhE∩U∖x0≠⌀.(Z3)
The mapf is sequentially lower semicontinuous on E with respect to X; that is, for each x0∈E,
(7)f(x0)⩽liminf⁡m→∞f(xm)
for any sequence (xm:m∈N) in X such that ∀α∈A{limm→∞dα(xm,x0)=0}; here
(8)liminf⁡m→∞f(xm)=sup⁡{inf⁡{f(xm):m⩾n}:n∈N}.Remark 10 (see [6, Remark 4.6]).
The following hold.(a)
A special case of condition (A6) is a condition (A6′) defined by
(A6′)
for each(x,α)∈XJ0×A, the map ωα(·)+εαJα(x,·):T(x)∩XJ0→[0;∞] is (T(x)∩XJ0,X)-lsc.(b)
IfJ=D, then a special case of condition (A6) is a condition (A6′′) defined by
(A6′′)
for each(x,α)∈X×A, the map ωα(·)+εαdα(x,·):T(x)→[0;∞] is (T(x),X)-lsc.(c)
Theorem7(I) esentially generalizes Theorem 6(I) even in metric spaces.
A classic result of Banach [16], from 1922, is the milestone in the history of fixed point theory and its applications.Theorem 11 (Banach [16]).
Let(X,d) be a complete metric space. Assume that the single-valued dynamic system (X,T) is (d,λ)-contraction; that is,
(9)∃λ∈[0;1)∀x,y∈X{d(T(x),T(y))⩽λd(x,y)}.
Then T has a unique fixed point w in X (i.e., T(w)=w and Fix⁡(T)={w}) and, for each w0∈X, the sequence (wm=T[m](w0):m∈N) satisfies limm→∞d(w,wm)=0.In a slightly different direction is the following elegant result of Nadler on set-valued dynamic systems.Theorem 12 (Nadler [17, Theorem 5]).
Let(X,d)be a complete metric space, let CB(X) denotes the class of all nonempty closed and bounded subsets of X, and let Hd:(CB(X))2→[0;∞) be defined by
(10)∀A,B∈CBXHdA,B=max⁡sup⁡u∈Adu,B,sup⁡v∈Bdv,A,
where ∀u∈X∀V∈CB(X){d(u,V)=inf⁡z∈Vd(u,z)}. Assume that the set-valued dynamic system (X,T) satisfying T:X→CB(X) is (Hd,λ)-contraction; that is,
(11)∃λ∈[0;1)∀x,y∈X{Hd(T(x),T(y))⩽λd(x,y)}.
Then Fix⁡(T)≠⌀ (i.e., there exists w∈X such that w∈T(w)).Remark 13.
Let(X,d) be a complete metric space.(a)
It is well known that Caristi’s fixed point theorem [12] yields Banach’s [16] and Nadler’s [17, Theorem 5] results.(b)
MapsT:(X,d)→(X,d) satisfying (3) are not necessarily continuous.(c)
It is well known that(CB(X),Hd) is a complete metric space and that the continuity of maps T:(X,d)→(X,d) and T:(X,d)→(CB(X),Hd) satisfying conditions (9) and (11) plays an important role in the proofs of Theorems 11 and 12, respectively.Contractions (3) of Caristi, (9) of Banach, (11) of Nadler, and others are among the most important notions in fixed point theory, as well as in its numerous applications. As one will see from the literature, the known results about them have been achieved by employing complicated machineries from various branches of mathematics and the answers for many basic problems about them are still missing. Moreover, examples show that these fundamental results are not optimal even in metric spaces.The several authors have made essential progress in these problems and have solved many cases, and similar methods and ideas have since been applied in greater generality; see for example [1–67] and the references cited therein. However, the complete solutions of some key open problems are still missing.In this paper we show that there are complementary approaches to generalize the Nadler and Banach statements concerning uniform, locally convex, and metric spaces. They involve mixed properties of asymmetric structures and fixed point theory. One of the key ideas in this paper is that in(X,D) the families J={Jα:X2→[0;∞),α∈A}∈J(X,D) construct the symmetric and asymmetric structures on X which generalize the symmetric structure determined by D={dα:α∈A} on X and then, by subtle techniques, we may use stated above Theorem 7.More precisely, let(X,D) be a Hausdorff uniform space. For J={Jα:X2→[0;∞),α∈A}∈J(X,D) and v∈{1,2}, let the distance HvJ on Cl(X) be defined as in Definitions 15 and 23, and let the distance BvJ on X be defined as in Definitions 29 and 33.This paper has two aims.(1)
To determineJ={Jα:X2→[0;∞),α∈A}∈J(X,D), various classes of not necessarily continuous set-valued dynamic systems (X,T) satisfying T:(X,J)→(Cl(X),HvJ), v∈{1,2}, and the conditions guaranteeing that the maps x→inf⁡z∈T(x)Jα(x,z), α∈A, attains its global optimal minimum at a point w (not necessarily unique) satisfying ∀α∈A{Jα(w,T(w))=0} and w∈Fix⁡(T) or w∈End(T).(2)
To determineJ={Jα:X2→[0;∞),α∈A}∈J(X,D), various classes of not necessarily continuous single-valued dynamic systems (X,T) satisfying T:(X,J)→(X,BvJ), v∈{1,2}, and the conditions guaranteeing that the maps x→Jα(x,T(x)), α∈A, attains its unique global optimal approximate minimum at w satisfying Jα(w,T(w))=0, α∈A, T(w)=w and ∀α∈A{limm→∞Jα(w,wm)=limm→∞Jα(wm,w)=limm→∞dα(w,wm)=0}, where (wm=T[m](w0):m∈N) and w0∈X is arbitrary.Remark 14.
(a) The methods of this paper provide a way to compute the fixed point and endpoint theorems in uniform, locally convex and metric spaces with structures determined byJ={Jα:X2→[0;∞),α∈A}∈J(X,D).
(b) Theorems17, 20, 21, 22, 25, 26, 27, 31, 34 and Examples 1–4 and 5–7 shows that our fixed point and endpoint results are new in uniform and locally convex spaces and even in metric spaces, are different from fixed point and endpoint results given in the literature, and their proofs are simpler.


## 2. Fixed Point and Endpoint Theorems for Set-Valued Contractions (of Nadler Type) in Uniform and Metric Spaces

The following definitions will be much used in the sequel.Definition 15.
Let(X,D) be a Hausdorff sequentially complete uniform space, assume that J={Jα:X2→[0;∞),α∈A}∈J(X,D), let
(12)∀α∈A∀u∈X∀V∈Cl(X)Jαu,V=inf⁡Jαu,z:z∈V,
and let v∈{1,2}.(a)
Define onCl(X) thedistanceHvJ, HvJ={Hv;αJ:Cl(X)2→[0;∞],α∈A}, as follows:
(13)∀α∈A∀A,B∈ClXHv;αJ(A,B)=max⁡{sup⁡u∈AJα(u,B),sup⁡z∈BJα(z,A)}ifv=1,∀α∈A∀A,B∈Cl(X){Hv;αJ(A,B)=sup⁡u∈AJα(u,B)}ifv=2.(b)
Let a set-valued dynamic system(X,T) satisfy T:X→Cl(X). If (X,T) satisfies
(14)∀α∈A∃λα∈[0;1)∀x,y∈X{Hv;αJ(T(x),T(y))⩽λαJα(x,y)},
then we say that (X,T) is a (HvJ,Λ)-contraction onXforΛ={λα∈[0;1),α∈A}.Remark 16.
Each(H1J,Λ)-contraction on X is (H2J,Λ)-contraction on X but converse does not hold.One can prove the following characterizations of(HvJ,Λ)-contractions (X,T):Theorem 17.
Let(X,D) be a Hausdorff sequentially complete uniform space, J={Jα:X2→[0;∞),α∈A}∈J(X,D) and v∈{1,2}. Suppose also the following.(I)
A set-valued dynamic system(X,T) satisfies T:X→Cl(X).(II)
There exists a familyΛ={λα∈[0;1),α∈A} such that (X,T) is a (HvJ,Λ)-contraction on X.(III)
The familyΓ={γα∈(0;1),α∈A} satisfies ∀α∈A{λα<γα}.
The following hold.(B1)
∀α∈A∀x∈X{{y∈T(x):Jα(y,T(y))⩽λαJα(x,y)}=T(x)}.(B2)
∀α∈A∀x∈X{UΓ,α(x)≠⌀} where
(15)UΓ,αx=y∈Tx:γαJαx,y⩽Jαx,Tx,α∈A,x∈X.(B3)
∀α∈A∀x∈X{UΓ,α(x)⊂VΓ,α(x)} where
(16)VΓ,αx=y∈T(x):Jα(y,T(y))+(γα-λα)Jα(x,y)hhh⩽Jα(x,T(x)),α∈A,x∈X.(B4)
∀α∈A∀x∈X∀y∈T(x){0⩽Jα(x,T(x))-Jα(y,T(y))⩽(1+λα)Jα(x,y)}.(B5)
∀α∈A∀x∈X∀y∈T(x){Jα(y,T(y))+Jα(x,y)⩾Jα(x,T(x))}.Proof.
Let the familyΩJ={ωαJ:X→[0;∞),α∈A} be defined by
(17)∀α∈A∀x∈X{ωαJ(x)=Jα(x,T(x))}.
Proof of (B1). By assumption (II) and Definitions 15(a) and 15(b),
(18)∀α∈A∀x,y∈Xsup⁡u∈TxJαu,Tysup⁡z∈TyJαz,Txhhhhhhhhhhh⩽max⁡sup⁡u∈TxJαu,Ty,sup⁡z∈TyJαz,Txhhhhhhhhhhhhhhhhhsup⁡z∈TyJαz,Txhhhhhhhhhhhsup⁡z∈TyJαz,Tx⩽λαJα(x,y)ifv=1,∀α∈A∀x,y∈X{sup⁡u∈T(x)Jα(u,T(y))⩽λαJα(x,y)}ifv=2.
Using this, we may thus conclude that
(19)∀α∈A∀x∈X∀y∈TxωαJy⩽sup⁡u∈TxJαu,Tyhhhhhhhhhhhhhhhhi⩽λαJαx,ysup⁡u∈Tx,
and hence
(20)∀α∈A∀x∈X{T(x)⊂{y∈T(x):ωαJ(y)⩽λαJα(x,y)}}.
On the other hand it is clear that
(21)∀α∈A∀x∈X{{y∈T(x):ωαJ(y)⩽λαJα(x,y)}⊂T(x)}.
By applying (20) and (21), we obtain (B1).
Proof of (B2). By (12), we have
(22)∀α∈A∀x∈X{ωαJ(x)=inf⁡y∈T(x)Jα(x,y)}.
Further, by assumption (III), ∀α∈A{γα∈(0;1)}. Hence, for arbitrary and fixed α∈A and x∈X, by (22) and definition of infimum, we obtain that
(23)∃y0∈T(x){γαJα(x,y0)⩽inf⁡y∈T(x)Jα(x,y)=ωαJ(x)}.
Consequently,
(24)∀α∈A∀x∈X∃y0∈T(x){y0∈UΓ,α(x)}.
So we have proved (B2).
Proof of (B3). Let α∈A,  x∈X, and y0∈UΓ,α(x) be arbitrary and fixed. Then, by (B2), we have y0∈T(x) and
(25)γαJα(x,y0)⩽ωαJ(x).
Clearly, by (B1), property y0∈T(x) implies ωαJ(y0)⩽λαJα(x,y0). Thus
(26)-λαJα(x,y0)⩽-ωαJ(y0).
Using (25) and (26) we obtain
(27)(γα-λα)Jα(x,y0)⩽ωαJ(x)-ωαJ(y0).
We proved that
(28)∀α∈A∀x∈XωαJ(x)UΓ,αx⊂ωαJ(x)y∈T(x):(γα-λα)Jα(x,y)hhhhhhhihhhhhhhhhhhh⩽ωαJ(x)-ωαJ(y).
Therefore, (B3) holds.
Proof of (B4). Let α∈A,  x∈X, and y∈T(x) be arbitrary and fixed. Then, by (B1), since y∈T(x), we obtain ωαJ(y)⩽λαJα(x,y). This and (22) imply
(29)-ωαJ(y)⩾-λαJα(x,y)⩾-λαωαJ(x)⩾-ωαJ(x).
Therefore,
(30)∀α∈A∀x∈X∀y∈T(x){0⩽ωαJ(x)-ωαJ(y)}
holds. Next, it follows from (22) and (B1) that
(31)∀α∈A∀x∈X∀y∈T(x)ωαJ(x)-ωαJ(y)⩽ωαJ(x)+ωαJ(y)hhhhhhhhhhhhhhihh⩽(1+λα)Jα(x,y)ωαJ(x).
This shows that (B4) holds.
Proof of (B5). By (30) and (22),
(32)∀α∈A∀x∈X∀y∈T(x){ωαJ(x)-ωαJ(y)⩽ωαJ(x)⩽Jα(x,y)}.
Therefore, (B5) holds.Definition 18.
Let(X,D) be a Hausdorff sequentially complete uniform space and let J={Jα:X2→[0;∞),α∈A}∈J(X,D). We say that the family J iscontinuous in X if, for each x0∈X and for each sequence (xm:m∈N) in X such that
(33)∀α∈A{limm→∞dα(xm,x0)=0},
we have
(34)∀α∈A{liminf⁡m→∞Jα(xm,x0)=liminf⁡m→∞Jα(x0,xm)=0}.Remark 19.
The familyD is continuous in X.
Assertion (B5) says that, for eachx∈X, the set
(35)QJ;Tx=Jαx,Txy∈Tx:∀α∈AJαy,Tyhhhhhhhhhhhhhhh+Jαx,y⩾Jαx,Tx
has the property
(36)⌀≠QJ;T(x)=T(x)∈Cl(X).
Let Υ={εα,α∈A} be a family of positive numbers satisfying ∀α∈A{εα∈(0;1)} and, for each x∈X, let the the set QJ,Υ;T(x) be defined by
(37)QJ,Υ;Tx=y∈T(x)∩XJ0:hhhhhhhhhh∀α∈AXJ0Jα(y,T(y))hhhhhhhhhhhhhhh+εαJα(x,y)⩽Jα(x,TXJ0(x)).
Now, for(HvJ,Λ)-contractions (X,T), we can give the following characterizations of the sets QJ,Υ;T(x), x∈X, defined in (37).Theorem 20.
Let(X,D) be a Hausdorff sequentially complete uniform space, J={Jα:X2→[0;∞),α∈A}∈J(X,D) and v∈{1,2}. Suppose also the following.(I)
J is admissible.(II)
A set-valued dynamic system(X,T) satisfies T:X→Cl(X).(III)
There exists a familyΛ={λα∈[0;1),α∈A} such that (X,T) is a (HvJ,Λ)-contraction on X.(IV)
For each familyΓ={γα∈(0;1),α∈A} satisfying ∀α∈A{λα<γα} and for each x∈X, let the set QJ,Γ-Λ;T(x) be defined by
(38)QJ,Γ-Λ;Tx=y∈T(x)∩XJ0:hhh∀α∈AJαXJ0(y,T(y))hhhhhhhhh+(γα-λα)Jα(x,y)hhhhhhhhh⩽XJ0Jα(x,T(x)).
The following hold.(C1)
If there exists a familyΓ0={γα0∈(0;1),α∈A} satisfying ∀α∈A{λα<γα0} and such that
(39)∀x∈XJ0y∈T(x)∩XJ0:hhhhhhhhh∀α∈Aγα0JαXJ0(x,y)hhhhhhhhhhhhih⩽Jα(x,TXJ0(x))≠⌀,
then ∀x∈XJ0{QJ,Γ0-Λ;T(x)≠⌀}.(C2)
If there exists a familyΓ0={γα0∈(0;1),α∈A} satisfying ∀α∈A{λα<γα0} and such that, for each (x,α)∈XJ0×A, the map
(40)Jα(·,T(·))+(γα0-λα)Jα(x,·):T(x)∩XJ0→[0;∞)
is (T(x)∩XJ0,X)-lsc, then, for each x∈XJ0, QJ,Γ0-Λ;T(x) is a closed subset in X.(C3)
Let the familyJ be continuous in X. Then, for each family Γ={γα∈(0;1),α∈A} satisfying ∀α∈A{λα<γα} and for each x∈XJ0, QJ,Γ-Λ;T(x) is a closed subset in X.(C4)
LetJ=D. If there exists a family Γ0={γα0∈(0;1),α∈A} satisfying ∀α∈A{λα<γα0} and such that
(41)∀x∈Xγα0dαx,y⩽dαx,Txy∈Tx:hhhhhihh∀α∈Aγα0dαx,y⩽dαx,Tx≠⌀,
then ∀x∈X{QD,Γ0-Λ;T(x)≠⌀}.(C5)
LetJ=D. Then, for each family Γ={γα∈(0;1),α∈A} satisfying ∀α∈A{λα<γα} and for each x∈X, QD,Γ-Λ;T(x) is a closed subset in X.Proof.
Let the familyΩJ={ωαJ:X→[0;∞),α∈A} be defined by
(42)∀α∈A∀x∈X{ωαJ(x)=Jα(x,T(x))}.
Proof of (C1). Denote
(43)∀x∈XUΓ0,Ax=⋂α∈A‍UΓ0,αx,∀x∈X{VΓ0,A(x)=⋂α∈A‍VΓ0,α(x)}.
Then, by (B2), (B3), and (IV),
(44)∀x∈XJ0UΓ0,A(x)∩XJ0⊂VΓ0,A(x)∩XJ0hhhhhhhh=QJ,Γ0-Λ;T(x).
Hence, we conclude that, for each x∈XJ0, the set QJ,Γ0-Λ;T(x) is nonempty whenever ∀x∈XJ0{UΓ0,A(x)∩XJ0≠⌀}.
Proof of (C2). The assertion follows immediately from Remark 10(a).
Proof of (C3). The assertion also follows from Remark 10(a). Indeed, let x0∈X be arbitrary and fixed and let a sequence (xm:m∈N) in X be convergent to x0; that is, let ∀α∈A{limm→∞dα(x0,xm)=0}.
Ifm∈N, z∈T(xm) and α∈A are arbitrary and fixed, then, by (J1),
(45)∀u∈T(x0)ωαJ(x0)=Jα(x0,T(x0))⩽Jα(x0,u)hhhhhhhhh⩽Jα(x0,xm)+Jα(xm,z)+Jα(z,u)ωαJ(x0).
This gives
(46)ωαJ(x0)⩽Jα(x0,xm)+Jα(xm,z)+Jα(z,T(x0)).
Hence
(47)ωαJ(x0)⩽Jα(x0,xm)+Jα(xm,z)+sup⁡u∈T(xm)Jα(u,T(x0)).
Furthermore, this holds for each z∈T(xm) and, thus, by (12),
(48)ωαJ(x0)⩽Jα(x0,xm)+ωαJ(xm)+sup⁡u∈T(xm)Jα(u,T(x0)).
However, (X,T) is (HvJ,Λ)-contraction on X. Therefore,
(49)sup⁡u∈TxmJαu,Tx0⩽max⁡sup⁡u∈TxmJαu,Tx0,sup⁡z∈Tx0Jαz,Txm⩽λαJα(xm,x0)ifv=1,sup⁡u∈T(xm)Jα(u,T(x0))⩽λαJα(xm,x0)ifv=2.
Consequently, we obtain that
(50)ωαJx0⩽Jαx0,xm+ωαJxm+sup⁡u∈TxmJαu,Tx0⩽Jαx0,xm+ωαJxm+λαJαxm,x0.
Since the family J is continuous, this implies
(51)ωαJ(x0)⩽liminf⁡m→∞ωαJxm.
Therefore, for each α∈A,  ωαJ(·) is lsc in X.
Moreover, ifm∈N,  x∈X, and α∈A are arbitrary and fixed, then, by (J1),
(52)Jαx,x0⩽Jαx,xm+Jαxm,x0.
Since J is continuous, this gives
(53)Jαx,x0⩽liminf⁡m→∞Jαx,xm.
that is, for each (x,α)∈X×A, the map Jα(x,·) is lsc in X.
Using these two facts, in particular, we have that, for each(x,α)∈XJ0×A, the map
(54)ωαJ·+γα-λαJαx,·:Tx∩XJ0⟶0;∞
is (T(x)∩XJ0,X)-lsc; that is, (A6′) holds.
Proof of (C4). This follows from (C1).
Proof of (C5). This follows from (C3) and Remarks 3(a) and 19.We use notations and auxiliary Theorems17 and 20 above in proving the following basic fixed point and endpoint theorem for set-valued contractions with respect to J∈J(X,D) (of Nadler-type) in uniform spaces (X,D).Theorem 21.
Let(X,D) be a Hausdorff sequentially complete uniform space, J={Jα:X2→[0;∞),α∈A}∈J(X,D) and v∈{1,2}. Suppose also the following.(I)
J is admissible.(II)
A set-valued dynamic system(X,T) satisfies T:X→Cl(X).(III)
There exists a familyΛ={λα∈[0;1),α∈A} such that (X,T) is a (HvJ,Λ)-contraction on X.(IV)
For each familyΓ={γα∈(0;1),α∈A} satisfying ∀α∈A{λα<γα} and for each x∈X, let the set QJ,Γ-Λ;T(x) be defined by
(55)QJ,Γ-Λ;Tx=y∈Tx∩XJ0:∀α∈AJαy,Tyγα-λαJαx,yXJ0⩽Jαx,Txhhhhhhhhhhhhhhhhhhhhhh+γα-λαJαx,yhhhhhhhhhhhhhhhhhhhhhhXJ0:∀α∈A⩽Jαx,Tx.(V)
There exists a familyΓ0={γα0∈(0;1),α∈A} satisfying ∀α∈A{λα<γα0} and such that, for each x∈XJ0,  QJ,Γ0-Λ;T(x) is a nonempty closed subset in X.The following hold,(E1)
(Fixed point theorem)Fix⁡(T)≠⌀ and there exists w∈Fix⁡(T) satisfying ∀α∈A{Jα(w,w)=0}.(E2)
(Endpoint theorem) If, for each x∈XJ0, each dynamic process (wm:m∈{0}∪N) starting at w0=x and satisfying ∀m∈{0}∪N{wm+1∈T(wm)} satisfies ∀m∈{0}∪N{wm+1∈QJ,Γ0-Λ;T(wm)}, then End(T)≠⌀ and ∀w∈End(T)∀α∈A{Jα(w,w)=0}.Proof.
The proof will be broken into five steps.
Step 1. Let the family ΩJ={ωαJ:X→[0;∞),α∈A} be defined by
(56)∀α∈A∀x∈XωαJx=Jαx,Tx.
The family ΩJ satisfies the assumption (A1) of Theorem 7; that is, DΩJ=⋂α∈A‍dom(ωαJ)≠⌀.
Indeed, by (B1),(57)∀x∈XωαJy⩽λαJαx,yy∈Tx:hhhhhhh∀α∈AωαJy⩽λαJαx,y=Tx.
Also, by Definition 1, J={Jα:X2→[0;∞),α∈A} and, by definition of (X,T),  ∀x∈X{⌀≠T(x)}. Hence we conclude that ∀x∈X{⌀≠T(x)⊂DΩJ}.
Step 2. The assumptions (A5) and (A6) of Theorem 7 hold where Υ={εα=γα0-λα,  α∈A} and ΩJ is defined in Step 1.
Indeed, by assumption (V) (i.e., by assumption∀x∈XJ0{⌀≠QJ,Γ0-Λ;T(x)∈Cl(X)}) it follows that
(58)∀x∈XJ0⌀≠QJ,Ω,Υ;T(x)hhhhhhh=y∈Tx∩XJ0:hhhhhhhhhh∀α∈AωαJ(y)+(γα0-λα)Jα(x,y)hhhhhhhhhhhhhhhh⩽ωαJ(x)∈Cl(X).
Step
3. There exists w∈XJ0 such that w∈T(w).
This is a consequence of (I)–(V), Steps 1 and 2, and Theorem7.
Step
4. We now observe that ∀w∈End(T)∀α∈A{Jα(w,w)=0}.
Otherwise,∃w0∈End(T)∃α0∈A{Jα0(w0,w0)>0}. Consequently, for each v∈{1,2},
(59)Jα0w0,w0=Hv;α0J({w0},{w0})=Hv;α0J(T(w0),T(w0))⩽λα0Jα0(w0,w0),
which is absurd.
Step
5. The assertions hold.
This follows from assumptions of Theorem17, Steps 1–4, definition of XJ0, and Theorem 7.As a corollary of the above Theorems17, 20, and 21 we have the following new fixed point and endpoint theorem for set-valued contractions with respect to D-families (of Nadle-type) in uniform spaces (X,D).Theorem 22.
Let(X,D) be a Hausdorff sequentially complete uniform space and let v∈{1,2}. Suppose also the following.(I)
A set-valued dynamic system(X,T) satisfies T:X→Cl(X).(II)
There exists a familyΛ={λα∈[0;1),α∈A} such that (X,T) is a (HvD,Λ)-contraction on X.(III)
For each familyΓ={γα∈(0;1),α∈A} satisfying ∀α∈A{λα<γα} and for each x∈X, let the set QD,Γ-Λ;T(x) be defined by
(60)QD,Γ-Λ;Tx=y∈T(x):hhh∀α∈Adα(y,T(y))+(γα-λα)dα(x,y)hhhhhhhhh⩽dαx,Tx.(IV)
There exists a familyΓ0={γα0∈(0;1),α∈A} satisfying ∀α∈A{λα<γα0} and such that ∀x∈X{QD,Γ0-Λ;T(x)≠⌀}.
The following hold.(F1)
(Closedness property) For each x∈X, QD,Γ0-Λ;T(x) is a closed subset in X.(F2)
(Fixed point theorem)Fix⁡(T)≠⌀.(F3)
(Endpoint theorem) If, for each x∈X, each dynamic process (wm:m∈{0}∪N) starting at w0=x and satisfying ∀m∈{0}∪N{wm+1∈T(wm)} satisfies ∀m∈{0}∪N{wm+1∈QD,Γ0-Λ;T(wm)}, then End(T)≠⌀.We now state consequences of the above in metric spaces.Definition 23.
Let(X,d) be a complete metric space, let J={J:X2→[0;∞)}∈J(X,d), and let v∈{1,2}.(a)
Let(61)∀u∈X∀V∈Cl(X)Ju,V=inf⁡Ju,z:z∈V.
Define HvJ:Cl(X)2→[0;∞] as follows:
(62)∀A,B∈Cl(X)HvJA,B=max⁡sup⁡u∈AJu,B,sup⁡z∈BJz,A,ifv=1,sup⁡u∈AJu,B,ifv=2.(b)
Let a set-valued dynamic system(X,T) satisfy T:X→Cl(X). If (X,T) satisfies
(63)∃λ∈[0;1)∀x,y∈XHvJTx,Ty⩽λJx,y,
then we say that (X,T) is a (HvJ,λ)-contraction onX.(c)
LetXJ0={x∈X:J(x,x)=0}. J is said to beadmissible if XJ0≠⌀.(d)
We say thatJ iscontinuous in X if, for each x0∈X and for each sequence (xm:m∈N) in X such that limm→∞d(xm,x0)=0, we have
(64)liminf⁡m→∞Jxm,x0=liminf⁡m→∞Jx0,xm=0.Remark 24.
LetD={d:X2→[0;∞)}. It is clear that D is J-family; D is admissible; D is continuous; and XD0=X.As corollaries from Theorems17, 20, and 21 and their proofs we get the following three theorems concerning contractions with respect to J∈J(X,d) (of Nadler-type) in metric spaces (X,d).Theorem 25.
Let(X,d) be a complete metric space, J={J:X2→[0;∞)}∈J(X,d) and v∈{1,2}. Suppose also the following.(I)
J is admissible.(II)
A set-valued dynamic system(X,T) satisfies T:X→Cl(X).(III)
There existsλ∈(0;1) such that (X,T) is a (HvJ,λ)-contraction on X.(IV)
For eachγ∈(0;1) satisfying λ<γ and for each x∈X let the set QJ,γ-λ;T(x) be defined by
(65)QJ,γ-λ;Tx=y∈T(x)∩XJ0:Jy,Ty0000+γ-λJx,y⩽XJ0Jx,Tx.The following hold.(G1)
If there existsγ0∈(0;1) satisfying λ<γ0 and such that
(66)∀x∈XJ0y∈Tx∩XJ0:hhhhhhhhγ0Jx,y⩽Jx,Tx≠⌀,
then ∀x∈XJ0{QJ,γ0-λ;T(x)≠⌀}.(G2)
If there existsγ0∈(0;1) satisfying λ<γ0 and such that, for each x∈XJ0, the map
(67)J·,T·+γ0-λJx,·:Tx∩XJ0⟶0;∞
is (T(x)∩XJ0,X)-lsc, then, for each x∈XJ0, QJ,γ0-λ;T(x) is a closed subset in X.(G3)
LetJ be continuous in X. Then, for each γ∈(0,1) satisfying λ<γ and for each x∈XJ0, QJ,γ-λ;T(x) is a closed subset in X.Theorem 26.
Let(X,d) be a complete metric space, J={J:X2→[0;∞)}∈J(X,d) and v∈{1,2}. Suppose also the following.(I)
J is admissible.(II)
A set-valued dynamic system(X,T) satisfies T:X→Cl(X).(III)
There existsλ∈(0;1) such that (X,T) is a (HvJ,λ)-contraction on X.(IV)
For eachγ∈(0;1) satisfying λ<γ and for each x∈X let the set QJ,γ-λ;Tx be defined by
(68)QJ,γ-λ;Tx=y∈T(x)∩XJ0:J(y,T(y))hhh+(γ-λ)J(x,y)⩽XJ0J(x,T(x)).(V)
There existsγ0∈(0;1) satisfying λ<γ0 such that, for each x∈XJ0, QJ,γ0-λ;T(x) is a nonempty closed subset in X.The following hold.(K1)
(Fixed point theorem)Fix⁡(T)≠⌀ and there exists w∈Fix⁡(T) such that J(w,w)=0.(K2)
(Endpoint theorem) If, for each x∈XJ0, each dynamic process (wm:m∈{0}∪N) starting at w0=x and satisfying ∀m∈{0}∪N{wm+1∈T(wm)} satisfies ∀m∈{0}∪N{wm+1∈QJ,γ0-λ;T(wm)}, then End(T)≠⌀ and ∀w∈End(T){J(w,w)=0}.Theorem 27.
Let(X,d) be a complete metric space, D={d:X2→[0;∞)} and v∈{1,2}. Suppose also the following.(I)
A set-valued dynamic system(X,T) satisfies T:X→Cl(X).(II)
There existsλ∈(0;1) such that (X,T) is a (HvD,λ)-contraction on X.(III)
For eachγ∈(0;1) satisfying λ<γ and for each x∈X let the set QD,γ-λ;T(x) be defined by
(69)QD,γ-λ;Tx=y∈T(x):dy,Ty0000+γ-λd(x,y)⩽dx,Tx.
The following hold.(L1)
(Nonemptness and closedness property) For each γ∈(0;1) satisfying λ<γ and for each x∈X,  QD,γ-λ;T(x) is a nonempty closed subset in X.(L2)
(Fixed point theorem)Fix⁡(T)≠⌀.(L3)
(Endpoint theorem) If there exists γ0∈(0;1) satisfying λ<γ0 and such that, for each x∈X, each dynamic process (wm:m∈{0}∪N) starting at w0=x and satisfying ∀m∈{0}∪N{wm+1∈T(wm)} satisfies ∀m∈{0}∪N{wm+1∈QD,γ0-λ;T(wm)}, then End(T)≠⌀.Remark 28.
Theorem27(L2) generalizes Theorem 12 (see Examples 5 and 6).


## 3. Fixed Point Theorems for Single-Valued Contractions (of Banach-Type) in Uniform and Metric Spaces

Definition 29.
Let(X,D) be a Hausdorff sequentially complete uniform space, assume that J={Jα:X2→[0;∞),α∈A}∈J(X,D) and let v∈{1,2}.(a)
Define onX thedistanceBvJ,  BvJ={Bv;αJ:X2→[0;∞],α∈A}, as follows:
(70)∀α∈A∀x,y∈XBv,αJ(x,y)=max⁡Jαx,y,Jαy,x,ifv=1,Jαx,y,ifv=2.(b)
Let(X,T) be a single-valued dynamic system, T:X→X. If (X,T) satisfies
(71)∀α∈A∃λα∈[0;1)∀x,y∈XBv,αJTx,Ty⩽λαJαx,y,
then we say that (X,T) is a (BvJ,Λ)-contraction onXforΛ={λα,α∈A}.Remark 30.
Each(B1J,Λ)-contraction on X is (B2J,Λ)-contraction on X but converse does not hold.We use notations above and Theorem21 in proving the following new fixed point theorem for single-valued contractions with respect to J∈J(X,D) (of Banach-type) in uniform spaces (X,D).Theorem 31.
Let(X,D) be a Hausdorff sequentially complete uniform space, let J={Jα:X2→[0;∞),α∈A}∈J(X,D), and let v∈{1,2}. Suppose also the following.(I)
J is admissible.(II)
(X,T) is a single-valued dynamic system, T:X→X.(III)
There exists a familyΛ={λα∈[0;1),α∈A} such that (X,T) is a (BvJ,Λ)-contraction on X for Λ.(IV)
T(XJ0)⊂XJ0.The following hold.(M1)
T has a unique fixed point w in X; that is, T(w)=w and Fix(T)={w}.(M2)
∀α∈A{Jα(w,w)=0}.(M3)
For eachw0∈X, the sequence (wm=T[m](w0):m∈N) satisfies
(72)∀α∈Alimm→∞Jαw,wm=limm→∞Jαwm,w=0,(73)∀α∈Alimm→∞dαw,wm=0.Proofs of (M1) and (M2).
By Remark30, Definition 29, and the assumptions (I)–(IV) of Theorem 31, we see that
(74)∀x∈XJ0∀α∈AJαTx,T2x⩽λαJαx,Tx,∀x∈XJ0y=Tx∈XJ0.
Let now Δ={δα∈(0;1),α∈A} satisfying ∀α∈A{λα<δα} be arbitrary and fixed. One then immediately finds that
(75)∀x∈XJ0∀α∈AJαTx,T2x⩽λαJαx,Tx+1-δαJαx,Tx
or, equivalently,
(76)∀x∈XJ0∀α∈AJαy,Ty+δα-λαJαx,y0000000000000⩽Jαx,Tx,
where y=T(x)∈XJ0. Consequently, for each x∈XJ0, the singleton set
(77)QJ,Δ-Λ;Tx=y=Tx∈XJ0:0000∀α∈AJαy,Ty0000000000+δα-λαJαx,y0000000000⩽Jαx,Tx={T(x)}
is a nonempty closed subset in X.
From the above and Theorem21 it follows that T has a fixed point w in X (i.e., w=T(w)) and ∀α∈A{Jα(w,w)=0}).
It remains to verify thatFix⁡(T)={w}. Suppose that {u,w}⊂Fix⁡(T). By Definition 29 and assumptions of Theorem 31, we obtain that, if v=1, then
(78)∀α∈A∃λα∈0;1Jα(u,w)⩽max⁡Jαu,w,Jαw,uh=max⁡⁡JαTu,Tw,hhhhhhhJαTw,Tu⩽λαJαu,wh∧Jαw,u⩽max⁡Jαu,w,Jαw,uhhhh=max⁡⁡JαTu,Tw,hhhhhh=max⁡JαTw,Tu⩽λαJαw,u,
and, if v=2, then
(79)∀α∈A∃λα∈0;1Jαu,w=JαTu,Tw⩽λαJαu,whh∧Jαw,u=JαTw,Tuhhhi⩽λαJαw,u.
Hence ∀α∈A{Jα(u,w)=Jα(w,u)=0}. From this information, by Remark 3(b), we deduce that u=w.
Therefore, the assertions (M1) and (M2) hold.
Proof of (M3). Let now w0∈X be arbitrary and fixed and put (wm=T[m](w0):m∈N). By Definition 29, assumptions of Theorem 31 and the fact that T[m](w)=w for m∈N, we obtain that, if v=1, then
(80)∀α∈A∃λα∈0;1∀m∈NJαw,wm+1hhhhhhhhhhhhhhhhh⩽max⁡Jαwm+1,w,Jαw,wm+1hhhhhhhhhhhhhhhhh⩽λαJαw,wmhhhhhhhhhhhhhhhhh∧Jαwm+1,whhhhhhhhhhhhhhhhhhh⩽max⁡Jαwm+1,w,Jαw,wm+1hhhhhhhhhhhhhhhhhhh⩽λαJαwm,w,
and, if v=2, then
(81)∀α∈A∃λα∈[0;1)∀m∈NJαw,wm+1hhhhhhhhhhhhhhhhh⩽λαJαw,wmhhhhhhhhhhhihhhhh∧Jαwm+1,w⩽λαJαwm,w.
Hence
(82)∀α∈A∃λα∈0;1∀m∈NJα(w,wm+1)hhhhhhhhhhhhhhhhh⩽λαmJα(w,w0)hhhhhhhhhihhhhhhh∧Jα(wm+1,w)hhhhhhhhhhhhihhhhh⩽λαmJαw0,w.
This gives the assertion (72), since, by Definition 1,
(83)∀α∈AJαw,w0<+∞,Jαw0,w<+∞.
Finally, letw0∈X be arbitrary and fixed and put (wm=T[m](w0):m∈N), (xm=w:m∈N), and (ym=wm:m∈N). Using assertion (M2), we then have
(84)∀α∈A{limn→∞sup⁡m>nJα(xn,xm)=0},
and, using assertion (72), we get
(85)∀α∈A{limm→∞Jα(xm,ym)=0}.
Hence, using Definition 1  (J2), we find
(86)∀α∈A{limm→∞dα(xm,ym)=limm→∞dα(w,wm)=0}.
Thus (73) holds.Remark 32.
(a) Theorem31 includes Theorem 11 [16] and the result of [52]. Theorem 31 is different from Theorem 11 [16] and the result of [52] even in metric spaces and in uniform spaces, respectively (see Examples 4 and 7).
(b) Letv∈{1,2}. Assumptions (III) and (IV) imply that (XJ0,T) is also a (BvJ,Λ)-contraction on XJ0. However, the dynamic systems (X,T) and (XJ0,T) are not necessarily (BvD,Λ)-contractions on X or XJ0, respectively (see Examples 4 and 7).
(c) Assumptions (II) and (IV) and assertions (M1) and (M2) imply thatw∈XJ0 is a unique fixed point of (X,T) and (XJ0,T). Assertion (M3) implies, in particular, that, for each starting point w0 of the space X, the dynamic process of the system (X,T) converges to w.The above has interesting implications for metric spaces.Definition 33.
Let(X,D) be a complete metric space, assume that J={J:X2→[0;∞)}∈J(X,d) and let v∈{1,2}.(a)
Define onX thedistanceBvJ:X2→[0,∞] as follows:
(87)∀x,y∈XBiJ(x,y)=max⁡Jx,y,Jy,x,ifv=1,Jx,y,ifv=2.(b)
Let(X,T) be a single-valued dynamic system, T:X→X. If (X,T) satisfies
(88)∃λ∈[0;1)∀x,y∈X{BvJ(T(x),T(y))⩽λJ(x,y)}
then we say that (X,T) is a (BvJ,λ)-contraction onXforλ.As a corollary from Theorem31 and its proof we get the following fixed point theorem for single-valued contractions with respect to J∈J(X,d) (of Banach-type) in metric spaces (X,d).Theorem 34.
Let(X,d) be a complete metric space, J={J:X2→[0;∞)}∈J(X,d) and v∈{1,2}. Suppose also the following.(I)
J is admissible.(II)
(X,T) is a single-valued dynamic system, T:X→X.(III)
There existsλ∈[0;1) such that (X,T) is a (BvJ,λ)-contraction on X for λ.(IV)
T(XJ0)⊂XJ0.
The following hold.(S1)
T has a unique fixed point w in X (i.e., T(w)=w and Fix⁡(T)={w}).(S2)
J(w,w)=0.(S3)
For eachw0∈X, the sequence (wm=T[m](w0):m∈N) satisfies limm→∞J(w,wm)=limm→∞J(wm,w)=0 and limm→∞d(w,wm)=0.Remark 35.
Theorem34 generalizes Theorem 11 (see Example 7).


## 4. Examples Illustrating the Results

The following example describes someJ-family in metric spaces.Example 1.
Let(X,d) be a metric space. Let the set E⊂X, containing at least two different points, be arbitrary and fixed and let c>0 satisfy δ(E)<c where δ(E)=sup⁡{d(x,y):x,y∈E}. Let J:X2→[0;∞) be defined by the formulae:
(89)J(x,y)=dx,y,ifE∩x,y=x,y,c,ifE∩{x,y}≠{x,y},x,y∈X. Then J={J:X2→[0;∞)}∈J(X,d) (see [6, Example 6.12]).The following example illustrates the Theorem26(K1) in the case when J={J:X2→[0;∞)}∈J(X,d),  J≠d.Example 2.
LetX=[0;6] be a complete metric space with a metric d:X2→[0;∞), d(x,y)=|x-y|, x,y∈X. Let T1:X→Cl(X) be of the form:
(90)T1(x)=1;2,ifx∈0;6,4;5,ifx=6.
Let E=[0;3)∪(3;6) and let J be of the form:
(91)J(x,y)=d(x,y)ifE∩{x,y}={x,y}8ifE∩{x,y}≠{x,y}.
Clearly, J∈J(X,d) (Example 1).
We observe thatXJ0=[0;3)∪(3;6)≠∅.
Letλ=3/4. We show that (X,T1) is a (H1J,3/4)-contraction on X. Indeed, let x,y∈X be arbitrary and fixed. We consider three cases.
Case  1. If x,y∈[0;6), then we have that T1(x)=T1(y)=[1;2] and H1J(T1(x),T1(y))=0⩽(3/4)J(x,y)=λJ(x,y).
Case  2. If x∈[0;6) and y=6, then y∉E, J(x,y)=8, T1(x)=[1;2] and T1(y)=[4;5]. Hence, we calculate the following.(2.1)
Foru∈T1(x),  J(u,T1(y))=d(u,[4;5])=4-u and, consequently, sup⁡{J(u,T1(y)):u∈T1(x)}=3.(2.2)
Forv∈T1(y),  J(v,T1(x))=d(v,[1;2])=v-2 and, consequently, sup⁡{J(v,T1(x)):v∈T1(y)}=3.(2.3)
By (2.1) and (2.2), forx∈[0;6) and y=6,
(92)H1JT1x,T1y=max⁡⁡sup⁡⁡Ju,T1y:u∈T1x,hhhhhhhhsup⁡Jv,T1x:v∈T1y=3⩽6=λ·8=λJ(x,y).
Case  3. If x=6 and y∈X∖{6}, then also (92) holds.
By Cases  1–3,(X,T1) is a (H1J,λ)-contraction on X.
Now, letγ0=7/8. We prove that, for each x∈XJ0, QJ,γ0-λ;T1(x) is a nonempty closed subset in X. Indeed, for each x∈XJ0=[0;3)∪(3;6), we have T1(x)=[1;2]⊂XJ0, ∀y∈T1(x)=[1;2]{J(y,T1(y))=0} and
(93)QJ,γ0-λ;T1(x)={y∈[1;2]:J(y,T1(y))+18J(x,y)⩽J(x,T1(x))}={y∈[1;2]:18d(x,y)⩽d(x,[1;2])}={y∈[1;2]:18y-x⩽d(x,[1;2])}.
This implies the following.
Case  1. If x∈[0;1], then
(94)QJ,γ0-λ;T1x={y∈[1;2]:y-x⩽8d(x,[1;2])}=y∈1;2:y⩽8-7x.
Case  2. If x∈(1;2], then
(95)QJ,γ0-λ;T1x={y∈[1;2]:18y-x⩽d(x,[1;2])=0}=x.
Case  3. If x∈(2;3)∪(3;6), then
(96)QJ,γ0-λ;T1x={y∈[1;2]:x-y⩽8(x-2)}={y∈[1;2]:y⩾16-7x}.
Assumptions of Theorem26(K1) hold for v=1, Fix⁡(T1)=[1;2], and, for each w∈Fix⁡(T1), J(w,w)=0.The following example illustrates the Theorem26(K2) in the case when J={J},  J≠d.Example 3.
LetX, E, J, λ=3/4, and γ0=7/8 be such as in Example 2 and let T2:X→Cl(X) be of the form:
(97)T2(x)={1}forx∈[0;2)∪{3}∪[5;6){2}forx∈[2;3)∪(3;5)[4;5]forx=6.
Then XJ0=[0;3)∪(3;6)≠∅ and, by analogous considerations as in Example 2, we obtain that (X,T2) is a (H1J,3/4)-contraction on X.
Next, let us observe that, forx∈XJ0,
(98)QJ,γ0-λ;T2(x)=18y∈T2(x):Jy,T2y00000+18J(x,y)⩽Jx,T2x.
Hence we have the following.
Case  1. If x∈[0;2)∪{3}∪[5;6), then T2(x)={1}, J(y,T2(y))=J(1,1)=0 for y∈T2(x) and, consequently,
(99)QJ,γ0-λ;T2x={y∈{1}:J(1,1)+18J(x,1)⩽J(x,1)}=y∈1:18x-1⩽x-1=1.
Case  2. If x∈[2;3)∪(3;5), then T2(x)={2}, J(y,T2(y))=J(2,2)=0 for y∈T2(x) and, consequently,
(100)QJ,γ0-λ;T2x={y∈{2}:J(2,2)+18J(x,2)⩽J(x,2)}=y∈2:18x-2⩽x-2={2}.
Therefore, for each x∈XJ0, each dynamic process (wm:m∈{0}∪N) starting at w0=x and satisfying ∀m∈{0}∪N{wm+1∈T(wm)} satisfies ∀m∈{0}∪N{wm+1∈QJ,γ0-λ;T(wm)}.
Assumptions of Theorem26(K2) hold, End(T2)={1,2} and J(1,1)=J(2,2)=0.The following example illustrates the Theorem34 in the case when J={J:X2→[0;∞)}∈J(X,d),  J≠d.Example 4.
LetX={1,2,3,4}∪[5;6] be a metric space with a metric d:X2→[0;∞),  d(x,y)=|x-y|, x,y∈X. Let E={1,2,4,5} and let J be of the form:
(101)J(x,y)=dx,y,ifE∩{x,y}={x,y},8,ifE∩{x,y}≠{x,y}.
Clearly, J={J:X2→[0;∞)} is a J-family on X (Example 1).
Letλ=3/4 and let T3:X→X be of the form:
(102)T3(x)=2,forx∈{1,2,3}∪(5;6],1,forx∈{4,5}.
ThenXJ0={1,2,4,5}≠∅ and T3:XJ0→XJ0. Thus assumption (IV) of Theorem 34 holds.
We see that(103)∀x,y∈Xmax⁡JT3x,T3y,JT3y,T3xhhhhhhhh=JT3x,T3yhhhhhhhh=dT3x,T3y⩽JT3x,T3y,JT3y,T3xλJx,y;
that is, (X,T3) is a (B1J,λ)-contraction on X. Indeed, we have the following.
Case  1. If x∈{1,2,3}∪(5;6] and y∈{4,5}, then
(104)1=d2,1=J2,1=JT3x,T3y,λJ(x,y)=3λ=94forx=1,y=4orx=2,y=5,2λ=32forx=2,y=4,4λ=3forx=1,y=5,8λ=6forx∈3∪5;6,y∈4,5.
Case  2. If x,y∈{1,2,3}∪(5;6], then
(105)0=d2,2=J2,2=JT3x,T3y.
Case  3. If x,y∈{4,5}, then
(106)0=d(1,1)=J(1,1)=J(T3(x),T3(y)).
Assumptions of Theorem34 hold and the assertions (S1)–(S3) are as follows. Fix⁡(T3)={2},  J(2,2)=0 and, for each w0∈X, the sequence (wm=T[m](w0):m∈N) satisfies
(107)∀w0∈Xlimm→∞J2,wm=limm→∞Jwm,2000000000=limm→∞dwm,2=0.


## 5. Comparisons of Our Results with Nadler’s and Banach’s Results

It is worth noticing that our results in metric spaces include Nadler’s and Banach’s results. Clearly, it is not otherwise. More precisely we have the following.(a)
In Examples5 and 6 below we show that, for each λ∈[0;1), the set-valued dynamic systems (X,T1) and (X,T2) defined in Examples 2 and 3, respectively, are not (Hd,λ)-contractions on X and thus we cannot use Theorem 12.(b)
In Example7 we show that, for each λ∈[0;1), the single-valued dynamic system (X,T3) defined in Example 4 is not (d,λ)-contractions on X and thus we cannot use Theorem 11.Therefore, in our concepts of(HvJ,Λ)-contractive set-valued dynamic systems and (BvJ,Λ)-contractive single-valued dynamic systems, v∈{1,2}, the existence of J-family such that J≠D is essential.Example 5.
Let(X,d) and T1 be such as in Example 2 and let J={d}. We observe that XJ0=X.
Next, we see that, for eachλ∈[0;1), (X,T1) is not a (Hd,λ)-contraction on X. Indeed, suppose that
(108)∃λ∈[0;1)∀x,y∈X{Hd(T1(x),T1(y))⩽λd(x,y)}.
Then, in particular, for x0=3∈X and y0=6∈X, we obtain the following.(1)
T1(x0)=[1;2] and T1(y0)=[4;5].(2)
Foru∈T1(x0)=[1;2],  d(u,T1(y0))=d(u,[4;5])=4-u and, consequently,
(109)sup⁡{d(u,T1(y0)):u∈T1(x0)}=3.(3)
Forz∈T1(y0)=[4;5],  d(z,T1(x0))=d(z,[1;2])=z-2 and, consequently,
(110)sup⁡{d(z,T1(x0)):z∈T1(y0)}=3.(4)
By (2) and (3),(111)Hd(T1(x0),T1(y0))=max⁡{sup⁡u∈T1(x0)d(u,T1(y0)),sup⁡z∈T1(y0)d(z,T1(x0))}=3.Hence, we get(112)∀λ∈[0;1)3=Hd(T1(x0),T1(y0))⩽λd(x0,y0)hhhhhhi<d(x0,y0)=d(3,6)=Hd3,which is absurd.Example 6.
Let(X,d) and T2 be such as in Example 3 and let J={d}. By similar argumentation as in Example 5, we observe that, for each λ∈[0;1), (X,T2) is not a (Hd,λ)-contraction on X.Example 7.
Let(X,d) and T3 be such as in Example 4 and let J=D={d}. Clearly, XD0=X.
We observe that, for eachλ∈[0;1),  (X,T3) is not a (d,λ)-contraction on X. Otherwise, by Definition 29 for J={d} (or by (9)), the following holds:
(113)∃λ∈[0;1)∀x,y∈X{d(T3(x),T3(y))⩽λd(x,y)}.
However, in particular, for x0=3∈X and y0=4∈X, we get d(x0,y0)=1 and then
(114)∀λ∈[0;1)1=d(2,1)=d(T3(x0),T3(y0))hhhhhhhhhi⩽λd(x0,y0)=λ<1,
which is absurd. This gives that the condition (113) does not hold.



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