# Fixed Point Theorems for Generalized Set-Valued Weakθ-Contractions in Complete Metric Spaces

Authors: Seong-Hoon Cho
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

In this paper, the notion of generalized set-valued weakθ-contractions is introduced and a new fixed point theorem for such contractions is established in the setting metric spaces. The main result is a generalization of fixed point theorems in the literature. An example and an application to generalized differential equation are given to support the validity of the main theorem.


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## Body

## 1. Introduction and Preliminaries

Jleli and Samet [1] introduced the notion of θ-contractions in Branciari metric spaces [2] and obtained a generalization of the Banach contraction principle, where θ:0,∞→1,∞ is a function satisfying the following conditions:(θ1)θ is nondecreasing(θ2)For any sequence tn⊂0,∞,(1)limn→∞θtn=1⇔limn→∞tn=0(θ3)There exist r∈0,1 and l∈0,∞ such that(2)limt→0+θt−1tr=l.Consider the following conditions:(θ4)θ is continuous on 0,∞(θ5)θ is strictly increasing(θ6)For each finite sequence tk>0:k=1,2,⋯,n,(3)θ∑k=1ntk≤∏k=1nθtk(θ7)For any A⊂0,∞ with infA>0,(4)infθA=θinfADenote byΞ the class of all functions θ:0,∞→1,∞ satisfying conditions(5)θ1,θ2,θ3,andθ7.LetΓ be the class of all functions θ:0,∞→1,∞ satisfying conditions(6)θ1,θ2,θ4,andθ6,and Ω the family of all functions θ:0,∞→1,∞ satisfying conditions(7)θ2,θ4,θ5,andθ6,and let Λ be the family of all functions θ:0,∞→1,∞ satisfying conditions(8)θ1,θ2,θ4,θ6,andθ7.Indeed,Λ is also an extension of the family Θ in [3], which satisfies conditions(9)θ1,θ2,θ3,andθ4.Jleli et al. [3] obtained a generalization of the result of [1] with a function θ∈Θ, and Ahmad et al. [4] gave an extention of the result of Jleli and Samet [1] to metric spaces with a simple condition (θ4) instead of (θ3).Since then, Al-Sulami et al. [5] introduced the concept of θR-contractions by using a binary relation and obtained a generalization of the result of [1]. Also, Ahmad et al. [6] gave α-fuzzy fixed point results by introducing the concept of generalized θ-contraction fuzzy mappings in metric spaces.LetX,d be a metric space and CBX be the family of nonempty closed and bounded subsets of X, and let H·,· be the Pompeiu-Hausdorff distance on CBX, i.e., for all A,B∈CBX,(10)HA,B=maxsupa∈Ada,B,supb∈Bdb,A,where da,B=infda,b:b∈B is the distance from the point a to the subset B.ForA,B∈CBX, let DA,B=supx∈Adx,B=supx∈Ainfy∈Bdx,y.Then, we haveDA,B≤HA,B for all A,B∈CBX.Recall that the metric spaceCBX,H is complete if and only if X,d is complete.Nadler [7] extended the Banach contraction principle to set-valued mappings by using Pompeiu-Hausdorff distance.Since then, many researchers (for example, [8–18] and reference therein) have been studying the fixed point results of set-valued maps in not only metric spaces but also various abstract spaces.In particular, Hançer et al. [19] extended the result of [1] to set-valued mappings defined on metric spaces. They proved the following theorem.Theorem 1 [19].
LetX,dbe a complete metric space and letT:X→CBXbe a set-valued map. Assume that there existsk∈0,1such that for allx,y∈XwithHTx,Ty>0,(11)θHTx,Ty≤θdx,yk,where θ∈Ξ.
Then,T has a fixed point.Recently, Durmaz [20] generalized the result of Hançer et al. [19] as follows.Theorem 2 [20].
LetX,dbe a complete metric space and letT:X→CBXbe a set-valued map. Assume that there existsk∈0,1such that for allx,y∈XwithHTx,Ty>0,(12)θHTx,Ty≤θmx,yk,where mx,y=maxdx,y,dx,Tx,dy,Ty,1/2dx,Ty+dy,Tx and θ∈Ξ.
Then,T has a fixed point.On the other hand, Khojasteh et al. [21] gave an extension of Banach contraction principle to Θ-metric spaces by introducing the concept of Θ-metrics, where Θ:0,∞×0,∞→0,∞ is a continuous mapping with respect to each variable.The difference between the concept ofΘ-contraction in the sense of Khojasteh et al. and the concept of θ-contraction in the sense of Jleli and Samet is that the concept of Θ-contraction in the sense of Khojasteh is defined in Θ-metric space and the concept of θ-contraction in the sense of Jleli and Samet is defined in metric space.In the paper, we introduce the concept of generalized set-valued weakθ-contraction maps, and we establish a new fixed point theorem for such set-valued contraction maps in the setting of metric spaces.Lemma 3.
LetX,dbe a metric space, and letB∈CBXanda∈X. If0<da,B<c, then there existsb∈Bsuch that(13)0<da,b<c.Proof.
Letε=c−da,B.
Sinceda,B<c, there exists b∈B such that 0<da,b<da,B+ε, by the definition of infimum. Hence, 0<da,b<c.


## 2. Fixed Point Theorems

A functionf:X,d→0,∞, where X,d is metric space, is called lower semicontinuous function if for each x∈X and each sequence xn with limn→∞dx,xn, we have(14)fx≤limn→∞inffxn.Letϕ:1,∞→1,∞ is a strictly increasing and lower semicontinuous function such thatϕ1ϕ−11=1ϕ2For any sequence tn⊂0,∞,(15)∏n=0∞ϕθtn<∞implies∏n=0∞θtn<∞LetX,d be a metric space.A set-valued mappingT:X→CBX is called set-valued weakθ-contraction if there exists a function θ:0,∞→1,∞ such that for all x,y∈X with dx,y>0 and for any u∈Tx, there exists v∈Ty with du,v>0 such that(16)θdu,v≤θdx,yϕθdx,y.A set-valued mappingT:X→CBX is called generalized set-valued weakθ-contraction if there exists a function θ:0,∞→1,∞ such that for all x,y∈X with dx,y>0 and for any u∈Tx, there exists v∈Ty with du,v>0 such that(17)θdu,v≤θmx,yϕθmx,y,where mx,y=maxdx,y,dx,Tx,dy,Ty,1/2dx,Ty+dy,Tx.Remark 4.
Forl>0, ϕθl>1. In fact, ifl>0, thenθl>1. Becauseϕis strictly increasing,ϕθl>ϕ1. It follows from (ϕ1) thatϕθl>1.Now, we prove our main result.Theorem 5.
LetX,dbe a complete metric space, and letT:X→CBXbe a generalized set-valued weakθ-contraction. Ifθ∈Γ, thenThas a fixed point.Proof.
Letx0∈X be a point, and let x1∈Tx0 be such that dx0,x1>0.
Then, it follows from (17) that there exists x2∈Tx1 with dx1,x2>0 such that(18)θdx1,x2≤θmx0,x1ϕθmx0,x1,where mx0,x1=maxdx0,x1,dx0,Tx0,dx1,Tx1,1/2dx0,Tx1+dx1,Tx0.
We deduce thatmx0,x1=maxdx0,x1,dx1,Tx1.
Ifmx0,x1=dx1,Tx1, then it follows from (18) and Remark 4 that(19)θdx1,x2≤θdx1,Tx1ϕθdx1,Tx1<θdx1,Tx1≤θdx1,x2,which is a contradiction.
Thus, we havemx0,x1=dx0,x1, and so,(20)θdx1,x2≤θdx0,x1ϕθdx0,x1.
Sincedx0,x1>0, it follows from Remark 4 that ϕθdx0,x1>1.
Hence,(21)θdx1,x2≤θdx0,x1ϕθdx0,x1<θdx0,x1.
Sinceθ is nondecreasing,(22)dx1,x2<dx0,x1.
Again, there existsx3∈Tx2 with dx2,x3>0 such that(23)θdx2,x3≤θmx1,x2ϕθmx1,x2,where mx1,x2=maxdx1,x2,dx1,Tx1,dx2,Tx2,1/2dx1,Tx2+dx2,Tx1.
We obtain thatmx1,x2=maxdx1,x2,dx2,Tx2.
Assume thatmx1,x2=dx2,Tx2.
Then,(24)θdx2,x3≤θdx2,Tx2ϕθdx2,Tx2<θdx2,Tx2≤θdx2,x3,which is a contradiction.
Thus, we havemx1,x2=dx1,x2, and so,(25)θdx2,x3≤θdx1,x2ϕθdx1,x2,which implies(26)θdx2,x3≤θdx1,x2ϕθdx1,x2<θdx1,x2.
Sinceθ is nondecreasing,(27)dx2,x3<dx1,x2.
Inductively, we have a sequencexn⊂X such that, for all n=1,2,3,⋯,(28)xn∈Txn−1,0<dxn,xn+1<dxn−1,xnθdxn,xn+1≤θdxn−1,xnϕθdxn−1,xn.
Sincedxn−1,xn is a decreasing sequence, there exists r≥0 such that(29)limn→∞dxn−1,xn=r.
We now show thatr=0.
Assume thatr≠0.
By lettingn→∞ in (28), and using lower semi-continuity of ϕ and continuity of θ, we have(30)θr≤θrϕθr.
Sincer>0, it follows from Remark 4 that ϕθr>1.
Hence, we have(31)θr≤θrϕθr<θr,which is a contradiction.
Thus, we have(32)limn→∞dxn−1,xn=0,and so,(33)limn→∞θdxn−1,xn=1.
It follows from (28) that(34)∏k=0nϕθdxk,xk+1θdxn+1,xn+2≤θdx0,x1.
Hence,(35)∏n=0∞ϕθdxn,xn+1≤θdx0,x1<∞,which implies(36)∏n=0∞θdxn,xn+1<∞.
Givenε>0, there exists a positive integer n0 such that for all n≥n0,(37)∏k=n∞θdxk,xk+1<θε.
Hence, we have that for allm>n≥n0,(38)θdxn,xm≤θdxn,xn+1+dxn+1,xn+2+⋯+dxm−1,xm≤θdxn,xn+1×θdxn+1,xn+2×⋯×θdxm−1,xm≤∏k=n∞θdxk,xk+1<θε,which implies(39)∀m>n≥n0,dxn,xm<ε.
Hence,xn is a Cauchy sequence.
SinceX is complete, there exists a point x∗∈X such that(40)limn→∞dx∗,xn=0.
Sincedxn,x∗>0 and xn+1∈Txn∀n=1,2,3,⋯, it follows from (17) that there exists u∈Tx∗ such that(41)θdxn+1,u≤θmxn,x∗ϕθmxn,x∗,where mxn,x∗=maxdxn,x∗,dxn,Txn,dx∗,Tx∗,1/2dxn,Tx∗+dx∗,Txn.
We infer that(42)limn→∞dxn,Txn≤limn→∞dxn,xn+1=0,and so,(43)limn→∞dxn,Txn=0.
Letα=limn→∞1/2dxn,Tx∗+dx∗,Txn.
Then,(44)α≤limn→∞12dxn,Tx∗+dx∗,xn+1≤12dx∗,Tx∗.
Hence,(45)limn→∞mxn,x∗=max0,0,dx∗,Tx∗,α=dx∗,Tx∗.
Lettingn→∞ in (41), and using continuity of θ and lower semicontinuity of ϕ, we have(46)θdx∗,u≤θdx∗,Tx∗ϕθdx∗,Tx∗.
Assume thatx∗∉Tx∗.
Then,dx∗,Tx∗>0, and so, ϕθdx∗,Tx∗>1 by Remark 4. Thus, we have(47)θdx∗,Tx∗≤θdx∗,u≤θdx∗,Tx∗ϕθdx∗,Tx∗<θdx∗,Tx∗,which is a contradiction.
Hence, we havex∗∈Tx∗.Corollary 6.
LetX,dbe a complete metric space, and letT:X→CBXbe a set-valued map such that(48)forallx,y∈Xwithinfa∈Txda,Ty>0,θDTx,Ty<θmx,yϕθmx,y.
If eitherθ∈Ω or θ∈Λ, then T has a fixed point.Proof.
Suppose thatθ∈Ω.
Sinceθ is strictly increasing, there exists the inverse function θ−1:1,∞→1,∞ such that(49)DTx,Ty<θ−1θmx,yϕθmx,y.
Letu∈Tx.
Then, it follows from the conditioninfa∈Txda,Ty>0,∀x,y∈X that(50)0<du,Ty<θ−1θmx,yϕθmx,y.
By Lemma3, there exists v∈Ty such that(51)0<du,v<θ−1θmx,yϕθmx,y.
Hence,(52)θdu,v<θmx,yϕθmx,y.
By Theorem5, T has a fixed point.
Assume thatθ∈Λ.
Letu∈Tx.
It follows from (θ7) that(53)infz∈Tyθdu,z=θinfz∈Tydu,z=θdu,Ty.
Becauseθ is nondecreasing,(54)infz∈Tyθdu,z≤θDTx,Ty,and so,(55)infz∈Tyθdu,z<θmx,yϕθmx,y.
Let(56)ε=θmx,yϕθmx,y−infz∈Tyθdu,z.
By the definition of infimum, there existsv∈Ty such that(57)θdu,v<infz∈Tyθdu,z+ε.
Thus,(58)θdu,v<θmx,yϕθmx,y.
By Theorem5, T has a fixed point.Corollary 7.
LetX,dbe a complete metric space, and letT:X→CBXbe a set-valued map such that(59)forallx,y∈Xwithinfa∈Txda,Ty>0,θHTx,Ty<θmx,yϕθmx,y.
If eitherθ∈Ω or θ∈Λ, then T has a fixed point.Corollary 8.
LetX,dbe a complete metric space, and letT:X→CBXbe a set-valued map satisfying the following condition: there exists θ∈Γ such that for all x,y∈X with dx,y>0 and for any u∈Tx, there exists v∈Ty with du,v>0 such that(60)θdu,v≤θmx,yk,where k∈0,1.
Then,T has a fixed point.Proof.
We define a functionϕ:1,∞→1,∞ by(61)ϕt=t1−k.
Then,ϕ is nondecreasing and lower semicontinuous such that (ϕ1) and (ϕ2) hold. It follows from (60) that for all x,y∈X with dx,y>0 and for any u∈Tx, there exists v∈Ty with du,v>0 with(62)θdu,v≤θmx,yk=θmx,yθmx,y1−k=θmx,yϕθmx,y.
By Theorem5, T has a fixed point.Corollary 9.
LetX,dbe a complete metric space, and letT:X→CBXbe a set-valued map such that(63)forallx,y∈Xwithinfa∈Txda,Ty>0,θDTx,Ty<θmx,yk,where k∈0,1.
If eitherθ∈Ω or θ∈Λ, then T has a fixed point.Corollary 10.
LetX,dbe a complete metric space, and letT:X→CBXbe a set-valued map such that(64)forallx,y∈Xwithinfa∈Txda,Ty>0,θHTx,Ty<θmx,yk,where k∈0,1.
If eitherθ∈Ω or θ∈Λ, then T has a fixed point.Corollary 11.
LetX,dbe a complete metric space, and letT:X→CBXbe a set-valued weakθ-contraction. Ifθ∈Γ, thenThas a fixed point.Corollary 12.
LetX,dbe a complete metric space, and letT:X→CBXbe a set-valued map such that(65)forallx,y∈Xwithinfa∈Txda,Ty>0,θDTx,Ty<θdx,yϕθdx,y.
If eitherθ∈Ω or θ∈Λ, then T has a fixed point.Corollary 13.
LetX,dbe a complete metric space, and letT:X→CBXbe a set-valued map such that(66)forallx,y∈Xwithinfa∈Txda,Ty>0,θHTx,Ty<θdx,yϕθdx,y.
If eitherθ∈Ω or θ∈Λ, then T has a fixed point.Corollary 14.
LetX,dbe a complete metric space, and letT:X→CBXbe a set-valued map satisfying the following condition: there exist θ∈Γ such that for all x,y∈X with dx,y>0and for any u∈Tx, there exists v∈Ty with du,v>0 such that(67)θdu,v≤θdx,yk,where k∈0,1.
Then,T has a fixed point.Corollary 15.
LetX,dbe a complete metric space, and letT:X→CBXbe a set-valued map such that for allx,y∈Xwithmx,y>0and for anyu∈Tx, there existsv∈Tysuch that(68)du,v≤mx,y−φmx,y,where φ:0,∞→0,∞ is nondecreasing and lower semicontinuous such that φ−10=0. Then, T has a fixed point.Proof.
Letθt=et,∀t>0.
From (68), we have that for all x,y∈X with mx,y>0 and for any u∈Tx, there exists v∈Ty such that(69)θdu,v=edu,v≤emx,y−φmx,y=emx,yeφmx,y.
Letφt=lnϕθt,∀t≥0, where ϕ:1,∞→1,∞ is nondecreasing and lower semicontinuous such that ϕ−11=1.
Then,φ is nondecreasing and lower semicontinuous, and(70)φt=0⇔ϕθt=1⇔θt=et=1⇔t=0.
It follows from (69) that for all x,y∈X with mx,y>0 and for any u∈Tx, there exists v∈Ty such that(71)θdu,v≤θmx,yelnϕθmx,y=θmx,yϕθmx,y.
By Theorem5, T has a fixed point.Remark 16.
Corollary15is an extention of Theorem 1 of [22] to set-valued maps and is a modification of Theorem 18of [23].By takingθt=2−2/πarctan1/tα, where α∈0,1,t>0 in Theorem 5, we obtain the following result.Corollary 17.
LetX,dbe a complete metric space, and letT:X→CBXbe a set-valued map. If for allx,y∈Xwithdx,y>0and for anyu∈Tx, there existsv∈Tywithdu,v>0such that(72)2−2πarctan1du,vα≤2−2/πarctan1/mx,yαϕ2−2/πarctan1/mx,yα,where α∈0,1, then T has a fixed point.We give an example to illustrate Theorem5. This example shows that the main theorem is a genuine generalization of Nadler’s contraction principle, Theorem 1 and Theorem 2.LetX=1/2n:n=1,2,3,⋯∪0, and let dx,y=∣x−y∣ for all x,y∈X.Define a mapT:X→CBX by(73)Tx=1,12n+1,x=12n,n=1,2,3,⋯,0,x=0,and a function θ:0,∞→1,∞ by(74)θt=et.Letϕt=t1/2,∀t>1.Then,ϕ is strictly increasing and lower semicontinuous satisfying conditions (ϕ1) and (ϕ2).We now show thatT is generalized set-valued weak θ-contraction.Letx=0,y=1/2n.We have that(75)mx,y=maxd0,12n,d0,T0,d12n,T12n,12d0,T12n+d12n,T0=12n=dx,y.Then, foru=0∈T0, there exists v=1/2n+1∈T1/2n such that(76)θdu,v=θd0,12n+1=θ12n+1=e1/2n−1/2n+1=e1/2ne1/2n1/2=θ1/2nϕθ1/2n≤θmx,yϕθmx,y.Ifx=1/2n and y=1/2mm>n, then we have(77)mx,y=maxd12n,12n,d12n,T12n,d12m,T12m,12d12m,T12n+d12n,T12m=12n−12m.Foru=1/2n+1∈T1/2n, there exists v=1/2m+1∈T1/2m such that(78)θdu,v=θ12n+1,12m+1=e1/2n+1−1/2m+1=e1/2n−1/2m−1/2n+1+1/2m+1=e1/2n−1/2me1/2n−1/2m1/2=θ1/2n−1/2mϕθ1/2n−1/2m≤θmx,yϕθmx,y.Hence,T is a generalized set-valued weak θ-contraction.Thus, all hypotheses of Theorem5 are satisfied, and T has a fixed point x∗=0.Note that Nadler’s contraction principle is not satisfied. In fact, ifx=0,y=1/n, then(79)HT0,T12n≤kd0,12n,k∈0,1,which implies(80)1≤k12n.Hence,(81)k≥2n,∀n=1,2,3,⋯,and hence,(82)k>1,which is a contradiction. Thus, Nadler’s contraction principle does not hold.Also, note that Theorem1 and Theorem 2 cannot be used here. In fact, if(83)θHT0,T12n≤θm0,12nk,where k∈0,1andθt=et,∀t>0, then(84)e1≤e1/2nk=ek1/2n,and so, k>1. Hence, condition (12) is not satisfied. Since m0,1/2n=d0,1/2n, condition (11) does not hold. Thus, Theorem 1 and Theorem 2 are not applicable here.


## 3. Application to Initial Value Problem

LetB⊂ℝ be an origin-centered closed ball, and let CCB the family of nonempty closed convex subsets of B with Hausdorff metric D generated by the norm ∥·∥ of ℝn.LetR:0,a×B→CCB be a continuous map.LetX=C0,a,ℝn be the space of all continuous functions from 0,a into ℝn, and let(85)dx,y=∥x−y∥=maxt∈0,a∣xt−yt∣for all x,y∈X.Then,X,d is a complete metric space.Note that every continuous function from a compact space into a metric space is bounded.Consider the following initial value problem:(86)ẋ∈Rt,xt,x0=x0.The existence of solutions for generalized differential equation (86) under certain conditions on R is proved by Casting [24], Filippove [25], and Herrmes [26].We give an application of our fixed point result to the problem on the existence of solutions for generalized differential equation (86).Theorem 18.
Suppose that for eachx,y∈Xand(87)ut=x0+∫0tzsds,zx∈Rs,xs,there exists ws∈Rs,ys with(88)vt=x0+∫0twsds,and ∣u−v∣>0 such that(89)θ∣u−v∣ϕθ∣x−y∣≤θ∣x−y∣.
Then, the initial value problem (86) has a solution.Proof.
Define a mappingT:X→CBX by(90)Tx=u∈X:ut=x0+∫0tzsds,zx∈Rs,xs.
Letx,y∈X and u∈Tx be given.
By assumption, there existsv∈Ty with du,v≥∣u−v∣>0 such that(91)θ∣u−v∣ϕ∣x−y∣≤θ∣x−y∣.
Hence,(92)θ∣u−v∣ϕ∣x−y∣≤θ∣x−y∣≤θ∥x−y∥.
By taking maximum in0,a,(93)θ∥u−v∥ϕ∥x−y∥≤θ∥x−y∥,and so,(94)θ∥u−v∥≤ϕ∥x−y∥θ∥x−y∥.
Thus, we have(95)θdu,v≤ϕdx,yθdx,y.
By Corollary11, T has a fixed point, say x∗ which is a solution of (86). That is, ẋ∗∈Rt,x∗t.



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