# Condensing Mappings and Best Proximity Point Results

Authors: Sarah O. Alshehri; Hamed H. Alsulami; Naseer Shahzad
Journal: Journal of Mathematics
 (2021)

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## Abstract

Best proximity pair results are proved for noncyclic relatively u-continuous condensing mappings. In addition, best proximity points of upper semicontinuous mappings are obtained which are also fixed points of noncyclic relatively u-continuous condensing mappings. It is shown that relative u-continuity ofT is a necessary condition that cannot be omitted. Some examples are given to support our results.


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## Body

## 1. Introduction

The concept of measure of noncompactness was first introduced by Kuratowski [1]. However, the interest in the concept was revived in 1955 when Darbo [2] proved a generalization of Schauder’s fixed point theorem using this concept. Sadovskii [3], in 1967, defined condensing mappings and extended Darbo’s theorem. Since then a lot of work has been done using this concept, and several interesting results have appeared, see, for instance, [4–9].LetW,Z be a nonempty pair in a Banach space (that is, both W and Z are nonempty sets). A mapping T:W∪Z⟶W∪Z is called noncyclic provided TW⊆W and TZ⊆Z. If there exists w,z∈W×Z which satisfies w=Tw, z=Tz, and w−z=distW,Z, then we say that the noncyclic mapping T has a best proximity pair. For a multivalued nonself mapping S:W⟶2Z, a point w∈W is called a fixed point of S if w∈Sw. The necessary condition for the existence of a fixed point for such S is W∩Z≠∅. If W∩Z=∅, then distw,Sw>0 for each w∈W. Best proximity point theorems provide sufficient conditions for the existence of at least one solution for the minimization problem, minw∈Wdistw,Sw. If distw,Sw=distW,Z, the point w is called a best proximity point of S. The existence results of best proximity points for multivalued mappings were obtained in [10–14] and [15]. Best proximity point theorems for relatively nonexpansive and relatively u-continuous were established by Elderd et al. in [16, 17] and by Markin and Shahzad in [18]. In recent years, the topics of best proximity points of single-valued and multivalued mappings have attracted the attention of many researchers, see, for example, the work in [6, 7, 19, 20] and the references cited therein. In this paper, we prove best proximity pair theorems for noncyclic relatively u-continuous condensing mappings. In addition, we obtain best proximity points of upper semicontinuous mappings which are fixed points of noncyclic relatively u-continuous condensing mappings. Also, we give examples to support our results and show by giving an example that relative u-continuity of T is a necessary condition that cannot be omitted. Our results extend and complement results of [6, 7, 11].


## 2. Preliminaries

In this section, we present some notions and known results which will be used in the sequel.Definition 1.
LetK be a bounded set in a metric space X. The Kuratowski noncompactness measure αK (or simply, measure of noncompactness) is defined as follows:(1)αK=infη>0:K⊆∪l=1mAl:diamAl≤η,∀1≤l≤m<∞.Theorem 1.
LetX be a metric space. Then, for any nonempty bounded pair C1,C2 in X (that is, both C1 and C2 are nonempty and bounded sets), the following hold:(1)
αC1=0 if and only if C1 is relatively compact(2)
C1⊆C2 implies αC1≤αC2(3)
αC1¯=αC1, where C1¯ denotes the closure of C1(4)
αC1∪C2=maxαC1,αC2(5)
For a normed spaceX:
(i)
αC1+x=αC1(ii)
αC1+C2≤αC1+αC2(iii)
αλC1=λαC1, for any number λ(iv)
αcon¯C1=αconC1=αC1, where conC1 represents the convex hull of C1Theorem 2.
LetFj be a decreasing sequence of nonempty closed subsets of a complete metric space X. If αFj⟶0 as j⟶∞, then ∩j∈ℕFj≠∅.For more details about the measure of noncompactness, see [4].Definition 2.
LetW,Z be a nonempty pair in Banach space X and T:W∪Z⟶W∪Z a mapping. Then, T is said to be noncyclic relatively u-continuous. If T is noncyclic and for each ϵ>0, there is γ>0 such that(2)Tw−Tz<ε+distW,Zwheneverw−z<γ+distW,Z,for each w∈W and z∈Z.Definition 3.
LetW,Z be a nonempty convex pair in Banach space X. A mapping T:W∪Z⟶W∪Z is said to be affine if for each α,β∈0,1 with α+β=1 and x1,x2∈W (respectively, x1,x2∈Z),(3)Tαx1+βx2=αTx1+βTx2.Definition 4.
LetW,Z be a nonempty pair in Banach space X and S:W⟶2Z a multivalued mapping on W, then S is said to be upper semicontinuous if for each closed subset B in Z, S−1B=w∈W:Sw∩B≠∅ is closed in W.Lemma 1.
(see [21]). Let Y be a nonempty, convex, and compact subset of a Banach space X. If f:Y⟶2Y can be written as a finite composition of upper semicontinuous multivalued mappings of nonempty, compact, and convex values, then f has a fixed point.Definition 5.
LetT:W∪Z⟶W∪Z be a noncyclic relatively u-continuous mapping and S:W⟶KCZ be an upper semicontinuous multivalued mapping (here, KCZ denotes the collection of all nonempty, convex, and compact subsets of Z), then by the commutativity of T and S, we mean that TSw⊆STw holds for each w∈W.
GivenW,Z, a nonempty pair in Banach space, its proximal pair W0,Z0 is given by(4)W0=w∈W:w−z∗=distW,Zfor somez∗∈Z,Z0=z∈Z:w∗−z=distW,Zfor somew∗∈W.
Moreover, ifW,Z is a nonempty, convex, and compact pair in X, then W0,Z0 is also a nonempty, convex, and compact pair.Definition 6.
LetX be a normed space. For a nonempty subset C of X, the metric projection operator PC:X⟶2C is given by(5)PCu≔v∈C:u−v=distu,C.
For a nonempty, convex, and compact subsetC of a strictly convex Banach space, PC is a single-valued mapping. Furthermore, for a nonempty, convex, and compact subset C of a Banach space X, PC is upper semicontinuous with nonempty, convex, and compact values.Lemma 2.
(see [11]). Let W,Z be a nonempty, convex, and compact pair in a strictly convex Banach space X. Let T:W∪Z⟶W∪Z be a noncyclic relatively u-continuous and P:W∪Z⟶W∪Z be a mapping given by(6)Pu=PZu,ifu∈W,PWu,ifu∈Z.
Then,TPu=PTu for each u∈W0∪Z0.Theorem 3.
(see [18]). Let W,Z be a nonempty, convex, and compact pair in a strictly convex Banach space X. If T:W∪Z⟶W∪Z is a noncyclic relatively u-continuous mapping. Then, T has best proximity pair.In [6], Gabeleh and Markin introduced the class of noncyclic condensing operators.Recall that a nonempty pairW,Z in a Banach space X is called proximinal if W=W0 and Z=Z0.Definition 7.
LetW,Z be a nonempty convex pair in a strictly convex Banach space X. A mapping T:W∪Z⟶W∪Z is called noncyclic condensing operator provided that, for any nonempty, bounded, closed, convex, proximinal, and T-invariant pair H1,H2⊆W,Z with distH1,H2=distW,Z, there exists k∈0,1 such that(7)αTH1∪TH2≤kαH1∪H2.Lemma 3.
(see [11]). Let W,Z be a nonempty, convex, and compact pair in a strictly convex Banach space X. If T:W∪Z⟶W∪Z is a noncyclic relatively u-continuous mapping, then T is continuous on W0 and Z0.


## 3. Main Results

Throughout this paper, we will assume thatX is a strictly convex Banach space and α is the measure of noncompactness on X.Remark 1.
LetT:W⟶W be condensing in the sense of Definition 7 with k∈0,1. Then, for any bounded subset H of W, T satisfies(8)αTH≤kαH.
To see this, in (7), set W=Z and H1=H2=H. Since H⊆con¯H, then(9)αTH≤αTcon¯H≤kαcon¯H=kαH.Theorem 4.
LetW,Z be a nonempty, convex, and closed pair in X such that W is bounded and W0 is nonempty. Suppose T:W∪Z⟶W∪Z is a noncyclic relatively u-continuous, affine, and condensing mapping. Then, there exists u0,v0∈W×Z such that Tu0=u0, Tv0=v0 and u0−v0=distW,Z. Moreover, if S:W⟶KCZ is an upper semicontinuous multivalued mapping, T and S commute, and for each x∈W0,Sx∩Z0≠∅, there exists w∈W such that Tw=w and distw,Sw=distW,Z.Proof.
We follow [6, 11]. Clearly, W0,Z0 is a nonempty, closed, convex, proximinal, and T-invariant pair. Let w0,z0∈W0×Z0 be such that w0−z0=distW,Z. Suppose ℱ is a family of nonempty, closed, convex, proximinal, and T-invariant pairs C,D⊆W,Z such that w0,z0∈C,D, then ℱ is nonempty. Set F1,F2=∩C,D∈ℱC,D, G1=con¯TF1∪w0, and G2=con¯TF2∪z0. So, w0,z0∈G1×G2 and G1,G2⊆F1,F2. Furthermore, TG1⊆G1 and TG2⊆G2, that is, T is noncyclic on G1∪G2. Also, for x∈G1, x=∑l=1m−1clTwl+cmw0, where for all l∈1,2,…,m−1 with cl≥0 and ∑l=1mcl=1, wl∈F1. Since F1,F2 is proximinal, there is zl∈F2 such that wl−zl=distW,Z, for each l∈1,2,…,m−1. Set y=∑l=1m−1clTzl+cmz0. Then, y∈G2. Moreover,(10)x−y=∑l=1m−1clTwl+cmw0−∑l=1m−1clTzl+cmz0≤∑l=1m−1clTwl−Tzl+cmw0−z0=distW,Z.
So, one can conclude thatG10=G1. Similarly, G20=G2, and hence, G1,G2∈ℱ, that is, G1=F1 and G2=F2. Notice that(11)αG1∪G2=maxαG1,αG2=maxαcon¯TF1∪w0,αcon¯TF2∪z0=maxαTF1,αTF2=αTF1∪TF2=αTG1∪TG2≤kαG1∪G2.
Butk∈0,1, so αG1∪G2=0. We conclude that G1,G2 is a nonempty, compact, and convex pair with distG1,G2=distW,Z. By Theorem 3, there exists u0,v0∈W×Z such that Tu0=u0, Tv0=v0 and u0−v0=distW,Z.
Now, letFixT=x∈W∪Z:Tx=x, FixWT=FixT∩W0, and FixZT=FixT∩Z0. By the above part, the pair FixWT,FixZT is nonempty. Also, it is a convex pair. Indeed, for α,β∈0,1, with α+β=1 and x,y∈FixWT (respectively, FixZT):(12)Tαx+βy=αTx+βTy=αx+βy,and by convexity of W0 (respectively, Z0), we conclude that αx+βy∈FixWT (respectively, FixZT). Furthermore, since T is condensing,(13)αFixWT∪FixZT=αTFixWT∪TFixZT≤kαFixWT∪FixZT,which implies that the pair FixWT,FixZT is compact.
Forx∈FixWT and u∈Sx, we have(14)Tu∈TSx⊆STx=Sx,that is, Sx is invariant under T. So, by the invariance of Z0 under T, Sx∩Z0≠∅ is invariant under T. So, in view of Remark 3.1, Darbo’s fixed point theorem guarantees the existence of a fixed point for the continuous mapping T: Sx∩Z0⟶Sx∩Z0. Thus, Sx∩FixZT≠∅, for x∈FixWT. Define f:FixWT⟶2FixZT by fx=Sx∩FixZT, for each x∈FixWT. Then, f is an upper semicontinuous multivalued mapping with nonempty, compact, and convex values. Moreover, PW: FixZT⟶FixWT is well-defined. Indeed, for y∈FixZT, there is x∈W such that x−y=distW,Z. So,(15)y=PZxandx=PWy.By relative u-continuity of T, we conclude that Tx−Ty=distW,Z. Thus, Ty=PZTx and Tx=PWTy, by (15), Tx=TPWy=PWTy=PWy. Then, PWy∈FixWT. Consider PW°f:FixWT⟶2FixWT, by Lemma 1, there is w∈FixWT such that w∈PW°fw, that is, Tw=w and w∈PWfw. So, there is z∈fw⊂Sw∩Z0 such that w=PWz. We conclude that z−w=distz,W. But since z∈Z0, there is w∗∈W such that w∗−z=distW,Z. Thus,(16)distW,Z≤distw,Sw≤w−z=distz,W≤z−w∗=distW,Z.
Hence,distw,Sw=distW,Z.Example 1.
Consider the Hilbert spaceX=ℓ2 over ℝ with the basis en:n∈ℕ (the canonical basis) and let(17)W=ζ1e1+ζ2e2:ζ1∈0,4,ζ2=−1andZ=ζ1e1+ζ2e2:ζ1≤0,ζ2=1.
Then,W,Z be a nonempty, convex, and closed pair of X such that W is bounded. Furthermore, distW,Z=2 and(18)W0=−e2andZ0=e2.
Define the mappingT:W∪Z⟶W∪Z by Tζ1e1+ζ2e2=3ζ1/4e1+ζ2e2, for each ζ1e1+ζ2e2∈W∪Z. Then, T is a noncyclic relatively u-continuous, affine, and condensing mapping. Now, define S:W⟶KCZ by Sζe1−e2=−ζe1+e2; then, S is an upper semicontinuous multivalued mapping, T and S commute, and for each x∈W0, Sx∩Z0≠∅. For w=−e2∈W, we have Tw=w and distw,Sw=distW,Z.Example 2.
Consider the Hilbert spaceX=ℓ2 over ℝ with the basis en:n∈ℕ and let(19)W=ζ1e1+ζ2e2:ζ1∈0,4,ζ2∈1,5andZ=ζ1e1+ζ2e2:ζ≥0,ζ2=0.
Then,W,Z be a nonempty, convex, and closed pair of X such that W is bounded with distW,Z=1 and(20)W0=ζ1e1+e2:ζ1∈0,4,Z0=ζe1:ζ∈0,4.
Define the mappingT:W∪Z⟶W∪Z by Tζ1e1+ζ2e2=2ζ1+1/3e1+ζ2e2 for each ζ1e1+ζ2e2∈W∪Z. Then, T is a noncyclic relatively u-continuous, affine, and condensing mapping. Furthermore, for u0,v0=e2,0∈W×Z, we have Tu0=u0, Tv0=v0, and u0−v0=distW,Z. Now, let S:W⟶KCZ given by Sζ1e1+ζ2e2=γe1:γ∈ζ1,4, then S is an upper semicontinuous multivalued mapping, T and S commute, and for each x∈W0, Sx∩Z0≠∅. For w=e1+e2∈W, we have Tw=w and distw,Sw=distW,Z.Remark 2.
The relative u-continuity ofT is necessary in Theorem 4.
To see this, consider the Hilbert spaceX=ℓ2 over ℝ with the basis en:n∈ℕ and let W=x∈X:x≤1, Z=ζe2:ζ∈2,3. Then, W,Z is a nonempty, convex, and closed pair in X such that W is bounded. Obviously, distW,Z=1 and(21)W0=e2,Z0=2e2.
Define the mappingT:W∪Z⟶W∪Z by(22)Tx=∑i=1∞ζi2ei,forx∈W,ζ22+1e2forx∈Z,for x=ζ1,ζ2,ζ3,…∈W∪Z. Then, T is a noncyclic, affine, and condensing mapping. Let S:W⟶KCZ given by Sζ1,ζ2,ζ3,…=2+ζ1e2. Then, S is an upper semicontinuous multivalued mapping, T and S commute, and for each x∈W0, Sx∩Z0≠∅. Here, w=0,0,0,… is the only fixed point of T in W, but distw,Sw>distW,Z. Note that e2−2e2<distW,Z+δ for all δ>0 but Te2−T2e2>distW,Z+1/4.
The following corollary follows immediately from Theorem4.Corollary 1.
LetW,Z be a nonempty, convex, and closed pair in X such that W is bounded and W0 is nonempty. Suppose T:W⟶W is a continuous, affine, and condensing mapping. If S:W⟶KCW is an upper semicontinuous multivalued mapping, T and S commute, and then there is w∈W which satisfies w∈FixT∩FixS.Theorem 5.
LetW,Z be a nonempty, convex, and closed pair in X such that W is bounded and W0 is nonempty. If T1,T2:W∪Z⟶W∪Z are commuting, noncyclic relatively u-continuous, affine, and condensing mappings, then there exists u0,v0∈W×Z such that T1u0=u0=T2u0, T1v0=v0=T2v0, and u0−v0=distW,Z.Proof.
SinceW0 is nonempty and by relative u-continuity of T1, for w0∈W0, there exists z0∈Z such that w0−z0=distW,Z. Consequently, T1w0−T1z0=distW,Z. That is, W0 is invariant under T1. Thus, Darbo’s fixed point theorem guarantees that there is u∈W0 such that T1u=u. Notice T1FixWT1=FixWT1 and so αFixWT1=αT1FixWT1≤kαFixWT1. Thus, αFixWT1=0, and thus, FixWT1 is compact. Furthermore, T1T2u=T2T1u=T2u. So, T2:FixWT1⟶FixWT1 is a continuous mapping on a compact convex set. By Schauder's fixed point theorem, there is u0∈FixWT1 such that T2u0=u0, that is, u0∈FixWT1∩FixWT2. Let v0 in Z0 be the unique closest point to u0. By relative u-continuity of T1 and T2, we infer that, since u0−v0=distW,Z, T1u0−T1v0=distW,Z and T2u0−T2v0=distW,Z. Hence, T1u0=u0=T2u0, T1v0=v0=T2v0.Lemma 4.
LetW,Z be a nonempty, convex, and closed pair in X such that W is bounded and W0 is nonempty. Let C be the collection of the commuting, noncyclic relatively u-continuous, affine, and condensing mappings on W∪Z. Then, the mappings in C have common fixed points u0∈W0 and v0∈Z0.Proof.
For eachT∈C, consider FixT,FixWT, and FixZT defined previously. Then, FixWT is nonempty, compact, and convex. Let T1,T2,…,Tk be a finite subcollection of C. Assume F=∩1≤i≤kFixWTi≠∅, F1=F∩FixWTk+1=∩1≤i≤k+1FixWTi, and Fn+1=Fn∩FixWTk+n+1=∩1≤i≤k+n+1FixWTi, for n∈ℕ. Then, Fn is a decreasing sequence of compact subsets of X. Furthermore, Fn≠∅ for each n∈ℕ. Indeed, for w∈F and each m∈1,2,…,k, then TmTk+1w=Tk+1Tmw=Tk+1w, and this implies that Tk+1w∈F. Thus, F is invariant under Tk+1. By Schauder's fixed point theorem, we get that F1≠∅. Now, for each n∈ℕ and m∈1,2,…,k+n, pick x∈Fn:(23)TmTk+n+1x=Tk+n+1Tmx=Tk+n+1x,that is, Tk+n+1x∈Fn. So, Tk+n+1:Fn⟶Fn is continuous on Fn, and then there is y∈Fn such that Tk+n+1y=y. Therefore, y∈Fn+1≠∅. By Theorem 2, ∩n∈ℕFn≠∅. Hence, ∩T∈CFixWT≠∅. Similarly, we can show that ∩T∈CFixZT≠∅.Theorem 6.
LetW,Z be a nonempty, convex, and closed pair in X such that W is bounded and W0 is nonempty. Let C be the collection of the commuting, noncyclic, relatively u-continuous, affine, and condensing mappings on W∪Z. Then, there is u0,v0∈W×Z such that, for each T∈C:Tu0=u0, Tv0=v0, and u0−v0=distW,Z.Proof.
Based on the previous lemma, the mappings inC have a fixed point in common u0∈W, that is, Tu0=u0, for each T∈C. Let v0∈Z be the unique closest point to u0. By relative u-continuity of T, since u0−v0=distW,Z,(24)u0−Tv0=Tu0−Tv0=distW,Z,for eachT∈C.
Hence,Tv0=v0.Theorem 7.
LetW,Z be a nonempty, convex, and closed pair in X such that W is bounded and W0 is nonempty. Let C be the collection of the commuting, noncyclic relatively u-continuous, affine, and condensing mappings on W∪Z. If S:W⟶KCZ is an upper semicontinuous multivalued mapping such that, for each x∈W0: Sx∩Z0≠∅. If C commutes with S, then there exists w∈W such that(25)Tw=wand distw,Sw=distW,Z.Proof.
By Lemma4, ∩T∈CFixWT,∩T∈CFixZT is a nonempty compact convex pair. Also, in view to the proof of Theorem 4, for T∈C and for each x∈FixWT, we have Sx and Z0 are invariant under T. So, Sx∩∩T∈CFixZT≠∅.
Definef:∩T∈CFixWT⟶2∩T∈CFixZT by fx=Sx∩∩T∈CFixZT, for x∈∩T∈CFixWT. Then, f is an upper semicontinuous multivalued mapping with nonempty, compact, and convex values. Moreover, PW: ∩T∈CFixZT⟶∩T∈CFixWT is well-defined. Indeed, for y∈∩T∈CFixZT, there exists x∈W such that x−y=distW,Z. So,(26)y=PZxandx=PWy,By relative u-continuity of T, one can conclude that Tx−Ty=distW,Z. Thus, Ty=PZTx and Tx=PWTy, and by (26), Tx=TPWy=PWTy=PWy. Thus, PWy∈∩T∈CFixWT. Note that PW°f:∩T∈CFixWT⟶2FixWT, and by Lemma 1, there is w∈∩T∈CFixWT such that w∈PW°fw, that is, for T∈C, we have Tw=w and w∈PWfw. So, there is z∈fw=Sw∩∩T∈CFixZT such that w=PWz. We infer that z−w=distz,W. But z∈Z0, then there is w∗∈W such that w∗−z=distW,Z. Then,(27)distW,Z≤distw,Sw≤w−z≤z−w∗=distW,Z.
Hence,distw,Sw=distW,Z.Example 3.
LetX=ℓ2 over ℝ with the basis en:n∈ℕ and let(28)W=ζ1e1+ζ2e2:ζ1∈−3,−1,ζ2∈−8,8andZ=ζ1e1+ζ2e2:ζ1∈0,3,ζ2∈ℝ.
Then,W,Z be a nonempty, convex, and closed pair in X such that W is bounded. Furthermore, distW,Z=1 and(29)W0=−e1+ζ2e2:ζ2∈−8,8andZ0=ζ2e2:ζ2∈−8,8.
ConsiderT1,T2:W∪Z⟶W∪Z given by(30)T1ζ1e1+ζ2e2=ζ1e1+ζ22e2andT2ζ1e1+ζ2e2=ζ1e1+ζ24e2,for each ζ1e1+ζ2e2∈W∪Z. Then, T1andT2 are noncyclic, affine, and condensing mappings. Furthermore, T1 and T2 commute.
DefineS:W⟶KCZ by Sζ1e1+ζ2e2=γe1+ζ2e2:γ∈0,−ζ1, then S is an upper semicontinuous multivalued mapping that commutes with T1 and T2 and satisfies that, for each x∈W0: Sx∩Z0≠∅. For w=−e1 and z=0,T1w=T2w=w and T1z=T2z=z. Furthermore, w−z=distW,Z and distw,Sw=distW,Z.


## 4. Conclusion

We have proved some best proximity pair theorems for noncyclic relatively u-continuous and condensing mappings. We have also obtained best proximity points of upper semicontinuous mappings which are fixed points of noncyclic relatively u-continuous condensing mappings. Moreover, we have given some examples to support our results. It has been shown that relative u-continuity ofT is a necessary condition that cannot be omitted. We have extended recent results of [6, 11].



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