# Fixed Point Theorems for Multivalued NonselfG-Almost Contractions in Banach Spaces Endowed with Graphs

Authors: J. Tiammee; P. Charoensawan; S. Suantai
Journal: Journal of Function Spaces
 (2017)

---

## Abstract

In this paper, we prove some fixed point theorems for multivalued nonselfG-almost contractions in Banach spaces with a directed graph and give some examples to illustrate our main results. The main results in this paper extend and generalize many known results in the literature therein.


---

## Body

## 1. Introduction

Fixed point theory plays a very important role in nonlinear analysis, economics, and applications. The important tool for solving existence problems in many branches of mathematics and applied sciences is Banach contraction principle [1]. It has been generalized in many directions by many authors ([2–15]).On the other hand, the fixed point theory for nonself mappings has been studied by many authors ([3–5]). In 2013, Berinde and Păcurar [5] introduced the new type of contraction for nonself single valued mappings.Definition 1.
LetK be a nonempty subset of a normed space X. A mapping T:K→X is said to be almost contractive if there exist δ∈[0,1) and L≥0 such that (1)Tx-Ty≤δx-y+Ly-Txfor all x,y∈KBy the above definition, they proved some fixed point theorem for almost contractive mappings in Banach spaces.Later, Alghamdi et al. [4] extended the almost contractive mappings to a multivalued nonself almost contractions.Definition 2.
Let(X,d) be a complete convex metric space and K a nonempty subset of X. A mapping T:K→CB(X) is said to be amultivalued almost contraction if there exist δ∈[0,1) and L≥0 such that (2)HTx,Ty≤δdx,y+Ldy,Txfor all x,y∈K, where H(·,·) is the Pompeiu-Hausdorff metric induced by d.They proved a fixed point theorem for multivalued nonself almost contraction on convex metric spaces.LetG=(V(G),E(G)) be a directed graph, where V(G) is a set of vertices of graph and E(G) is a set of its edges. Assume that G has no parallel edges. We denote by G-1 the directed graph obtained from G by reversing the direction of edges: that is, (3)EG-1=x,y:y,x∈EG.Letx and y be two vertices in G; apath in G from x to y of length n∈N∪{0} is a sequence {xi}i=0n of n+1 vertices such that x0=x, xn=y, (xi-1,xi)∈E(G) for each i=1,2,…,n. A graph G is said to beconnected if there exists a (directed) path between any two vertices of G. We denote(4)xGN=y∈X:thereisapathinGoflengthNfromxtoy.On the other hand, in 2008, Jachymski [6] extended Banach contraction principle by introducing the notion of G-contraction and proved a fixed point theorem for G-contractions in a metric space endowed with a directed graph. These results have been generalized by some authors in several ways (see [7–14]).Motivated by the previous results, we introduced the concept of multivalued nonselfG-almost contractions in normed spaces and establish some fixed point theorems for this this type of mappings in normed spaces endowed with a directed graph. We give some examples to illustrate our main results.


## 2. Preliminaries

In this section, we give some basic and useful definitions and known results that are useful for the main results in this paper.LetX be a normed space. We denote by CB(X) the class of all nonempty closed and bounded subsets of X. For any A,B∈CB(X), define the function H:CB(X)×CB(X)→R+ by (5)HA,B=max⁡δA,B,δB,A,where(6)δA,B=sup⁡da,B:a∈A,δB,A=sup⁡db,A:b∈B,da,C=inf⁡a-x:x∈C.Note thatH is called thePompeiu-Hausdorff metric induced by metric d.The following lemma is very useful for our main results.Lemma 3 (see [2]).
Let(X,d) be a metric space. If A,B∈CB(X) and a∈A, then, for each ϵ>0, there exists b∈B such that d(a,b)≤H(A,B)+ϵ.LetX be a normed space, K a nonempty closed subset of X, and T:K→CB(X) a nonself mapping. If x∈K with Tx⊂K, then we can always choose an y∈∂K such that y=(1-λ)x+λz, where 0<λ<1 and z∈Tx∖K, which expresses the fact that (7)x-z=x-y+y-z.We denote(8)PTx=y∈∂K:x-z=x-y+y-zwherez∈Tx.The following property was introduced by Jachymski [6]. It is very useful for our main results.Property A (see [7]).
With any sequence(xn) in X, if xn→x and (xn,xn+1)∈E(G) for all n∈N, then (xn,x)∈E(G) for any n∈N.In order to obtain our main results, we need the following definition of domination in graphs [16, 17].LetG=(V(G),E(G)) be a directed graph. A set X⊆V(G) is called a dominating set if, for all v∈V(G)∖X, there exists x∈X such that (x,v)∈E(G) and we say that x dominates v or v is dominated by x. For any v∈V, a set X⊆V isdominated by v if (v,x)∈E(G) for any x∈X and we say that Xdominatesv if (x,v)∈E(G) for all x∈X.


## 3. Main Results

In this section, we prove some fixed point theorems for multivalued nonselfG-almost contractions in a Banach space endowed with a directed graph. First, we recall some definitions which are useful for our main results.Definition 4.
LetK be a nonempty subset of a normed space X and G=(V(G),E(G)) be a directed graph such that V(G)=K. A mapping T:K→CB(X) is said to be G-almost contraction(1)
if there existδ∈(0,1) and L≥0 with δ(1+L)<1 such that, for any x,y∈K, (9)HTx,Ty≤δx-y+Ly-Txwhenever (x,y)∈E(G),(2)
ifu∈Tx and v∈Ty are such that d(u,v)≤d(x,y); then (u,v)∈E(G).First, we prove a fixed point theorem for aG-almost contraction in a Banach space endowed with a directed graph.Theorem 5.
LetK be a nonempty closed subset of a Banach space X and G=(V(G),E(G)) be a directed graph such that V(G)=K. Let T:K→CB(X) be a G-almost contraction. Suppose that(1)
T satisfies Rothe’s boundary condition: that is, x∈∂K⇒Tx⊂K,(2)
K has Property A,(3)
there isx0∈X such that (x0,y)∈E(G) for some y∈Tx0(4)
PT(x) is dominated by x for all x∈C with Tx∉K.Then T has a fixed point in K.Proof.
Letx0∈K be such that (x0,y1)∈E(G), where y1∈Tx0. If y1∈K, we denote x1=y1: that is, (x0,x1)∈E(G). If y1∉K, there exists x1∈∂K such that (10)x0-x1+x1+y1=x0-y1.Since PT(x0) is dominated by x0, we obtain (x0,x1)∈E(G). If x0=x1 then x0∈Tx0. Suppose x0≠x1; we can choose k0∈N such that (11)δk0<1-δx0-x1.By Lemma 3, we can choose y2∈Tx1 such that (12)y1-y2≤HTx0,Tx1+δk0≤δx0-x1+Ldx1,Tx0+δk0=δx0-x1+δk0<x0-x1.Hence, (y1,y2)∈E(G). If y2∈K, we denote x2=y2. If y2∉K, then there exists x2∈∂K such that (13)x1-x2+x2-y2=x1-y2.If x1=x2 then x1∈Tx1. Suppose x1≠x2; we can choose k1∈N with k1>k0 such that (14)δk1<1-δx1-x2.By Lemma 3, we can choose y3∈Tx2 such that (15)y2-y3≤HTx1,Tx2+δk1≤δx1-x2+Ldx2,Tx2+δk1=δx1-x2+δk1<x1-x2.Hence, (y1,y2)∈E(G). By induction, we obtain two sequences {xn} and {yn} such that (i)
yn+1∈Txn;(ii)
yn-yn+1≤H(Txn-1,Txn)+δkn, where
(a)
yn=xn if yn∈K,(b)
yn≠xn whenever yn∉K, and then xn∈∂K is such that (16)xn-1-xn+xn-yn=xn-1-yn,(c)
{kn} is a sequence of positive integers with δkn<(1-δ)xn-xn+1.Suppose that (17)P=xk∈xn:xk=yk,Q=xk∈xn:xk≠yk.Note that {yn}⊂K and (yn,yn+1)∈E(G) for all n∈N. Moreover, if xk∈Q, then both xk-1 and xk belong to the set P. Since xk-1 and xk are not in the set Q for all k, we cannot have two consecutive terms of {xk} in the set Q.
Now, we claim that{xn} is a Cauchy sequence. We conclude that there are three possibilities.
Case  1. If xn,xn+1∈P, then xn∈Txn-1 and xn+1∈Txn. Since T is a G-almost contraction and (xn,xn+1)∈E(G), we have (18)xn-xn+1=HTxn-1-Txn+δkn-1≤δxn-1-xn+Ldxn,Txn-1+δkn-1=δxn-1-xn+δkn-1.
Case  2. If xn∈P and xn+1∈Q, then xn∈Txn-1 and yn+1≠xn+1: that is, (19)xn-xn+1+xn+1-Txn=xn-Txn.Since T is a G-almost contraction and (xn,xn+1)∈E(G), we have (20)xn-xn+1≤xn-xn+1+xn+1-yn+1=xn-yn+1=yn-yn+1≤HTxn-1-Txn+δkn-1≤δxn-1-xn+Ldxn,Txn-1+δkn-1=δxn-1-xn+δkn-1.
Case  3. If xn∈Q and xn+1∈P, then xn-1∈P and yn≠xn such that (21)xn-1-xn+xn-Txn-1=xn-1-Txn-1.Moreover, yn-1=xn-1 and yn∈Txn-1. Since T is a G-almost contraction, we have (22)xn-xn+1=xn-xn+1≤xn-yn+yn,yn+1≤xn-yn+HTxn-1-Txn+δkn-1≤xn-yn+δxn-1-xn+Ldxn,Txn-1+δkn-1≤xn-yn+xn-1-xn+Ldxn,Txn-1+δkn-1=xn-1-yn+Ldxn,Txn-1+δkn-1≤xn-1-yn+Lxn,yn+δkn-1=xn-1-yn+Lxn-1,yn-Lxn-1,xn+δkn-1=1+Lxn-1-yn-Lxn-1,xn+δkn-1≤1+Lyn-1-yn+δkn-1≤1+LHxn-2,xn-1+1+Lδkn-2+δkn-1≤1+Lδxn-2-xn-1+1+LLdxn-1,Txn-2+1+Lδkn-2+δkn-1=1+Lδxn-2-xn-1+1+Lδkn-2+δkn-1.We denote h=(1+L)δ<1, so (23)xn-xn+1≤hyn-2-yn-1+hδkn-3+δkn-1.Therefore, combining Cases 1, 2, and 3, we obtain (24)xn-xn+1≤hmax⁡xn-1-xn,xn-2-xn-1+hδkn-3+δkn-1≤hmax⁡xn-1-xn,xn-2-xn-1+hkn-3+1+hkn-1≤hmax⁡xn-1-xn,xn-2-xn-1+hn-3+hn-1for all n≥2. By inductivity, it follows that, for any n>1, (25)xn-xn+1≤hn/2max⁡x0-x1,x1-x2+hn-1/2n.Now, for any n>m, we obtain (26)xm-xn≤xm-xm+1+xm+1-xm+2+⋯+xn-1-xn≤hm-1/2+hm/2+⋯+hn-2/2max⁡x0-x1,x1-x2+hm-1/2m+hm/2m+1+⋯++hn-1/2n.This implies that {xn} is a Cauchy sequence. Since C is closed and {xn}⊂C, there exists x∈C such that limn→∞xn=x. Let {xnk}⊂P be an infinite subsequence of {xn}. Since X has Property A, (xnk,x)∈E(G) for all k∈N. Then we have (27)dx,Tx≤x-xnk+1+dxnk+1,Tx=x-xnk+1+HTxnk,Tx≤x-xnk+1+δxnk-x+Ldx,Txnk=1+Lx-xnk+1+δxnk-x.Letting k→∞, we obtain (28)dx,Tx=0and hence x∈Tx: that is, x is a fixed point of T. This completes the proof.As a consequence of Theorem5, by putting E(G)=C×C, we obtain the following corollary.Corollary 6.
LetK be a nonempty closed subset of a Banach space X and T:K→CB(X) be an almost contractive mapping with Rothe’s boundary condition. Then T has a fixed point.Remark 7.
(i) In Theorem5, if we take E(G)=X×X, we obtain Theorem  9 of Alghmdi et al. [4] in Banach spaces.
(ii) In Theorem5, if we take E(G)=X×X and T:K→X, we obtain the existence result which is similar to Berinde and Păcurar’s result [5].
(iii) In Theorem5, if we take T:X→X and L=0, we obtain the existence result which is similar to Jachymski’s result [5] in Banach spaces.
(iv) In Theorem5, if we take E(G)=X×X and L=0, we obtain the existence result which is similar to Nadler’s theorem [5] in Banach spaces.Next, we give an example to illustrate Theorem5.Example 8.
LetX=R and K=[0,∞) with the usual norm ∥x-y∥=|x-y| and G=(V(G),E(G)) be such that V(G)=K, (29)EG=12n,122n+1,12n,0,122n+1,122n+2:n≥2∪12,0,0,0.Notice that K has Property A. Define a mapping T:K→CB(X) by (30)Tx=0ifx=0,-18ifx=12,122k,122k+1ifx=122k+1,k∈N∪0122k+2,122k+3ifx=122k,k∈N∪0∖12.We see that the only x∈K with Tx∉K is x=1/2, and we see that PT(1/2)={0} and (1/2,0)∈E(G); hence, PT(x) is dominated by x. By direct computation, we can show that T is G-almost contraction with δ=1/4 and L=2. Moreover, we see that (31)T∂K=T0,1=0,1⟹T0andT1aresubsetofK,which implies T satisfies Rothe’s boundary condition. Choose x0=0∈∂C, we see that (x0,Tx0)=(0,0)∈E(G). Therefore, all conditions of Theorem 5 are satisfied and we see that F(T)={0}.



---

*Source: 7053849-2017-01-31.xml*