# Best Proximity Point for the Sum of Two Non-Self-Operators

Authors: V. Pragadeeswarar; R. Gopi; M. De la Sen
Journal: Journal of Mathematics
 (2020)

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## Abstract

In the present paper, we focus our attention on the existence of the fixed point for the sum of the cyclic contraction and the noncyclic accretive operator. Also, we study the best proximity point for the sum of two non-self-mappings. Moreover, we provide the existence of the best proximity point for the cyclic contraction through the notion of the nonlinearD-set contraction. Finally, we give the existence of the best proximity point for the sum of the nonlinear D-set contraction mapping and partially completely continuous mapping in the setting of the partially ordered complete normed linear space.


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## Body

## 1. Introduction

Fixed point theory plays an important role in the area of nonlinear functional analysis, and it has many applications in the study of nonlinear differential and integral equations. The study of nonlinear equations of the formΓ1η+Γ2η=η, where Γ1,Γ2:ℬ⟶ℬ are mappings on the Banach space ℬ, helps to solve many physical nonlinear real-life problems. For example, Dhage and Otrocol [1] gave the existence and approximation of solutions to the following hybrid differential equation:(1)x′t=ft,xt+gt,maxa≤ξ≤txξ,xa=α0∈ℝ,for all t∈J=a,b and f,g:J×ℝ⟶ℝ are continuous functions. Also, Banaś and Amar [2] obtained the existence of the solution to the nonlinear integral equation of the form(2)xt=at+∫01kt,sfs,xsds+∫01ut,s,xsds,for t∈J=a,b and a∈L1J,f:J×ℝ⟶ℝ,k:J×J⟶ℝ+,andu:J×J×ℝ⟶ℝ. So, the researchers involved in finding the solution of the equation Γ1η+Γ2η=η, which is clearly the problem of finding the sufficient condition for the existence of fixed point for the sum of two mappings.In the sequel, in 1955, Krasnoselskii gave an existence of the solution for the equationΓ1η+Γ2η=η in the Banach space setting, where Γ1 is the contraction and Γ2 is the compact operator. Later, many researchers extended Krasnoselskii’s theorem in different directions (see [3–7] and the references therein). Vijayaraju [7] proved the theorems pointing the existence of the fixed point for a sum of nonexpansive and continuous mappings and also a sum of asymptotically nonexpansive and continuous mappings in the setting of locally convex spaces. O’Regan [5] established the fixed point theorem for the sum of two operators Γ1+Γ2 if Γ1 is compact and Γ2 is nonexpansive. Moreover, the results were used to prove the existence of the solution for second-order boundary value problem. O’Regan and Taoudi [6] proved the fixed point theorems for the sum of two weakly sequentially continuous mappings in the Banach space. Dhage [3] proved the fixed point result by combining two fixed point theorems of Krasnoselskii and Dhage and also derived the existence result for the product of two operators in Banach algebra. Dhage [4] found the local version of fixed point theorems of Krasnoselskii and Nashed et al. By using this result, he provided the application to nonlinear functional integral equations.Agarwal et al. [8] obtained the existence of the fixed point for two mappings, compact mapping and nonexpansive mapping, in the setting of both the weak and the strong topology of a Banach space. Ben Amar and Garcia-Falset [9] proved the existence of fixed point theorems for different kinds of contractions such as nonlinear weakly condensing, 1-set weakly contractive, and pseudo-contractive and nonexpansive operators defined on unbounded domains and provided application to generalized Hammerstein integral equations. Arunchai and Plubtieng [10] improved the Krasnoselskii theorem on fixed points for the sum of operators Γ1+Γ2, where Γ1 is the weakly-strongly continuous mapping and Γ2 is the asymptotically nonexpansive mapping. Banaś and Amar [2] proved some new types of fixed point theorems for the sum of Γ1+Γ2 on an unbounded closed convex subset of a Hausdorff topological vector space, which are used to provide the solution of integral equations in the Lebesgue space. Wang [11] derived fixed point results for the sum of two operators Γ1 and Γ2 if Γ1 is contractive with respect to the measure of weak noncompactness and Γ2 is the ϕ-nonlinear contraction in the setting of the Banach space. By having this result, he obtained the existence of solutions to a nonlinear Hammerstein integral equation in the L1 space. Ben Amar et al. [12] improved Krasnoselskii-type fixed point results for the equation Γ1+Γ2x=x, where the operator Γ1 is ws-compact and Γ2 is ws-compact and asymptotically Φ-nonexpansive operator on an unbounded closed convex subset of a Banach space.Later, the people were interested in finding the fixed point in partially ordered metric spaces which are more general than metric spaces. First, Ran and Reurings [13] initiated the fixed point theorems for the contraction mappings in the partially ordered metric space, which is further improved by Nieto and Rodríguez-López [14] and by Petruşel and Rus [15], and used to give the existence of the solution for boundary value problems of nonlinear first-order ordinary differential equations.Recently, the researchers show interest in finding the optimum solution for real-time modeling problems. In this direction, instead of studying about the fixed point (whenever the fixed point does not exist), the researchers are working on approximating the fixed point in some sense, known as the best proximity point. The existence of best proximity point theorems helps to obtain the optimum solution for different types of modelings. In the literature, there are more number of articles about the existence of the best proximity point for single operators. For example, Basha [16] derived best proximity point theorems for proximal contractions of the first and second kind in the setting of the metric space. Basha [17] proved the existence of the best proximity point for principal cyclic contractive mappings, proximal cyclic contractive mappings, and proximal contractive mappings which are defined on the metric space. In 2000, Eldred and Veeramani [18] proved the best proximity point result for contractive type mappings in the metric space. Al-Thagafi and Shahzad [19] obtained convergence and existence results of the best proximity points for cyclic ϕ-contraction maps in the metric space. For more existence of the best proximity point results, we refer the reader to [20–25].In the light of the above literature survey, we want to find the approximate solution for the fixed point equation of the formΓ1+Γ2x=x, where Γ1,Γ2 are non-self-mappings. So, in this work, we initiate to study the best proximity point for the sum of two non-self-operators, and we provide the existence of the best proximity point for the sum of two operators using best proximity point theorems for the single operator. Additionally, we prove an existence result of the fixed point for the sum of cyclic and noncyclic operators, which involves the concept of accretive operators. Finally, we discuss some notions of the ordered normed linear space, and we find sufficient conditions for the existence of the best proximity point in this space.


## 2. Preliminaries

First, we collect some notions from [26]. Throughout the paper, we denote ℬ,X as the Banach space and metric space, respectively, and let(3)ℬ∗=f:ℬ⟶ℝfis continuous linear transformationbe its dual. For each η∈ℬ, we associate the set(4)Jη=f∈ℬ∗:η,f=η2=f2,where η,f denotes fη. The multivalued operator J:ℬ⟶ℬ∗ is called the duality mapping of ℬ. Suppose Γ is an operator from ℬ to ℬ. Then, the operator −Γ:ℬ⟶ℬ is defined by −Γη=−Γη.Here, we give the definition for weak accretive via the accretive operator in [26].Definition 1.
An operatorΓ:ℬ⟶ℬ is said to be weak accretive if for η1,Γη1,η2,Γη2∈ℬ×ℬ, there exists φ∈Jη1−η2 such that Γη1−Γη2,φ≥0, where J is as in (4).
The following lemma is helpful for our one of the results.Lemma 1.
IfΓ is weak accretive on ℬ, then η1−η2≤η1−η2+λΓη1−Γη2, for all λ≥0, and η1,η2∈ℬ.Proof.
Letη1,η2∈ℬ with η1≠η2. Since Γ is weak accretive, for η1,Γη1,η2,Γη2∈ℬ×ℬ, there exists φ∈Jη1−η2 such that Γη1−Γη2,φ≥0. Now, we have(5)η1−η22=η1−η2,φ≤η1−η2,φ+λΓη1−Γη2,φ=η1−η2+λΓη1−Γη2,φ≤η1−η2+λΓη1−Γη2φ=η1−η2+λΓη1−Γη2η1−η2.Theorem 1 (see [27]).
Letℳ,N be nonempty subsets of X which are complete. Suppose Γ:ℳ∪N⟶ℳ∪N satisfies(1)
Γℳ⊆N and ΓN⊆ℳ.(2)
dΓη,Γξ≤kdη,ξ for η∈ℳ,ξ∈N,where 0<k<1. Then, there exists τ∈ℳ∩N such that Γτ=τ.


## 3. Main Results

Theorem 2.
Letℳ,N be subsets of ℬ such that ℳ∪N is the subspace. Let Γ1,Γ2:ℳ∪N⟶ℳ∪N, where Γ1 and Γ2 are cyclic and noncyclic mappings which satisfy the following:(1)
−Γ2 is the weak accretive operator, and Γ1 is onto(2)
I−Γ2 is invertible on Γ1ℳ∪N, where I is the identity operator(3)
η=Γ2η+Γ1ξ,ξ∈ℳ⟹η∈N and η=Γ2η+Γ1ξ,ξ∈N⟹η∈ℳ(4)
Γ1η−Γ1ξ≤kη−ξ,k∈0,1, for η∈ℳ and ξ∈N
Then, there existsτ∈ℳ∪N such that Γ1+Γ2τ=τ.Proof.
First, we note that assumption (3) is valid only ifℳ∪N is the subspace. Since Γ1 is onto, Γ1ℳ∪N=ℳ∪N. Since I−Γ2 is invertible on Γ1ℳ∪N, we can define(6)I−Γ2−1Γ1:ℳ∪N⟶ℳ∪N.
Now, we show thatI−Γ2−1Γ1 is the cyclic mapping. Let ξ∈ℳ; then, Γ1ξ∈N. Since I−Γ2 is invertible on Γ1ℳ∪N, there exists unique η such that I−Γ2−1Γ1ξ=η. Then, the equation η=Γ2η+Γ1ξ has a unique solution η. Then, by (3), we obtain I−Γ2−1Γ1ξ∈N. Then, I−Γ2−1Γ1ℳ⊆N. Similarly, we can prove I−Γ2−1Γ1N⊆ℳ. Next, we prove that I−Γ2−1 is nonexpansive on Γ1ℳ∪N. Since I−Γ2 is invertible on Γ1ℳ∪N, it implies that RI−Γ2=Γ1ℳ∪N,whereRI−Γ2is the range ofI−Γ2. Let η,ξ∈Γ1ℳ∪N; then, there exist z1,z2∈Γ1ℳ∪N such that I−Γ2−1η=z1,I−Γ2−1ξ=z2. Using Lemma 1, we get(7)η−ξ=z1−Γ2z1−z2−Γ2z2=z1+−Γ2z1−z2+−Γ2z2=z1−z2+−Γ2z1−−Γ2z2≥z1−z2=I−Γ2−1η−I−Γ2−1ξ.
Therefore, forη∈ℳ,ξ∈N, we obtain(8)I−Γ2−1Γ1η−I−Γ2−1Γ1ξ≤Γ1η−Γ1ξ≤kη−ξ.
Then, the operatorI−Γ2−1Γ1 agrees with the hypothesis of Theorem 1, and then there is τ∈ℳ∩N such that I−Γ2−1Γ1τ=τ, which implies Γ2+Γ1τ=τ.


## 4. Best Proximity Point Theorems

The following notions are used in this article: letℳ,N be nonempty subsets of X.(9)dℳ,N=infdη,ξ:η∈ℳ,ξ∈N,ℳ0=η∈ℳ:dη,ξ′=dℳ,Nfor someξ′∈N,N0=ξ∈N:dη′,ξ=dℳ,Nfor someη′∈ℳ.The pairℳ,N is said to have P-property if for η1,η2∈ℳ and ξ1,ξ2∈N,(10)dη1,ξ1=dℳ,N,dη2,ξ2=dℳ,N,⟹dη1,η2=dξ1,ξ2.Definition 2.
A functionΓ:X⟶X is called Lipschitzian if there exists a real constant κ≥0 such that, for all η,ξ∈X,(11)dΓη,Γξ≤κdη,ξ.Definition 3 (see [20]).
Letℳ,N be nonempty subsets of X. A map Γ:ℳ⟶N is said to be a weakly contractive mapping if(12)dΓη,Γξ≤dη,ξ−χdη,ξ,for all η,ξ∈ℳ, where χ:0,∞⟶0,∞ is a nondecreasing and continuous function such that χ is positive on 0,∞,χ0=0 and limt⟶∞χt=∞.Theorem 3 (see [20]).
LetX be complete and ℳ,N⊆X be nonempty, closed sets such that ℳ0 is nonempty. Let Γ:ℳ⟶N be a weakly contractive mapping such that Γℳ0⊆N0. Assume the pair ℳ,N has the P-property. Then, there exists unique η∗ in ℳ such that dη∗,Γη∗=dℳ,N.The following theorem tells that the sum of two non-self-operators has the best proximity point in Banach space settings. The notion1/2N=η:η=b/2,b∈N, where N is the subset of ℬ, is used in the following theorem.Theorem 4.
Letℳ,N be two nonempty convex, closed subsets of ℬ. Assume ℳ0 is nonempty. Let Γ1,Γ2:ℳ⟶N be two mappings which satisfy(1)
Ifη∈ℳ, then Γ1η+Γ2η∈N(2)
Γ1 is k/2-Lipschitzian, where k∈0,1(3)
Γ2η−Γ2ξ≤k/2η−ξ−χη−ξ, for all η,ξ∈ℳ, where χ is as in Definition 3 and k∈0,1(4)
Γ1ℳ0⊆1/2N0 and Γ2ℳ0⊆1/2N0
If the pairℳ,N has the P-property, then there exists unique η∗ in ℳ such that η∗−Γ1+Γ2η∗=dℳ,N.Proof.
Because of (1), we defineΓ1+Γ2:ℳ⟶N by Γ1+Γ2η=Γ1η+Γ2η. First, we show that Γ1+Γ2 satisfies Γ1+Γ2ℳ0⊆N0. For η∈ℳ0, we have Γ1η∈1/2N0 and Γ2η∈1/2N0. Therefore, there exist ξ1,ξ2∈N0 such that Γ1η=ξ1/2,Γ2η=ξ2/2. Then, Γ1+Γ2η=Γ1η+Γ2η=ξ1/2+ξ2/2. Since N0 is convex, Γ1+Γ2η∈N0. This shows Γ1+Γ2ℳ0⊆N0. Now we prove the mapping Γ1+Γ2 is weakly contractive. For η,ξ∈ℳ0, we have(13)Γ1+Γ2η−Γ1+Γ2ξ=Γ1η−Γ1ξ+Γ2η−Γ2ξ≤Γ1η−Γ1ξ+Γ2η−Γ2ξ≤kη−ξ2+kη−ξ2−χη−ξ=kη−ξ−χη−ξ≤η−ξ−χη−ξ.
This givesΓ1+Γ2 is a weakly contractive mapping from ℳ0 to N0. Clearly, the pair ℳ0,N0 has the P-property, and dℳ0,N0=dℳ,N. Then, the map Γ1+Γ2 satisfies the requirements of Theorem 3, and there exists η∗∈ℳ0 such that η∗−Γ1+Γ2η∗=dℳ,N.


## 5. Best Proximity Point Theorems in the Ordered Normed Linear Space

In this section, first we extract some notions from [28] to obtain best proximity point results. Let V be the real vector space. The pair V,≺– is called the partially ordered linear space, where ≺– is the partial order. Two elements η,ξ∈V are called comparable if either η≺–ξ or ξ≺–η holds. A nonempty subset C of V is said to be a chain or totally ordered if any two elements of C are comparable. The space V,≺–,· is called the partially ordered normed linear space, where · is the norm on V.Definition 4 (see [28]).
A mappingΓ:V⟶V is called monotone nondecreasing if η≺–ξ implies Γη≺–Γξ for all η,ξ∈V. Similarly, Γ is called monotone nonincreasing if η≺–ξ implies Γξ≺–Γη for all η,ξ∈V.Definition 5 (see [28]).
A subsetC of V,≺–,· is called partially bounded if every chain in C is bounded.
We denote byPbd,chV,Prcp,chV the family of all bounded chains and relatively compact chains of V, respectively.Definition 6 (see [28]).
A mappingμp:Pbd,chV⟶ℝ+=0,∞ is called a partial measure of noncompactness in V if it satisfies(1)
∅≠μp−10⊂Prcp,chV.(2)
μpD¯=μpD.(3)
μp is nondecreasing, i.e., if D1⊂D2⟹μpD1≤μpD2.(4)
If sequenceDn of closed chains in Pbd,chV with Dn+1⊂Dnn=1,2,… and if limn⟶∞μpDn=0, then the set D¯∞=∩n=1∞Dn is nonempty.The notion in (1) is known as the kernel ofμp, that is,(14)kerμp=D∈Pbd,chV:μpD=0.Clearly,kerμp⊂Prcp,chV. And so, D∞∈kerμp. Because of μpD∞≤μpDn for all n, we obtain μpD∞=0. Then, D∞∈kerμp. The measure μp is called sublinear if it satisfies(5)
μpD1+D2≤μpD1+μpD2 for all D1,D2∈Pbd,chV.(6)
μpλD=λμpD for λ∈ℝ.Andμp has the maximum property if(7)
μpD1+D2=maxμpD1,μpD2.Finally,μp is said to be full if(8)
kerμp=Prcp,chV.Definition 7 (see [28]).
A mappingχ:0,∞⟶0,∞ is called a D-function if it is monotone nondecreasing and upper semicontinuous with χ0=0.Definition 8 (see [28]).
A nondecreasing mappingΓ:V⟶V is said to be partially nonlinear D-set-Lipschitz if there is a D-function χ such that(15)μpΓC≤χμpC,for any bounded chain C in V. If χs<s for s>0, then Γ is called a partially nonlinear D-set-contraction.Lemma 2 (see [28]).
Ifχ is a D-function with χs<s for s>0, then limn⟶∞χns=0 for all s∈0,∞ and vice-versa.Definition 9 (see [27]).
Letℳ,N be nonempty subsets of X. A map Γ:ℳ∪N⟶ℳ∪N is a partially cyclic contraction map if it satisfies(1)
Γℳ⊆N and ΓN⊆ℳ(2)
dΓη,Γξ≤kdη,ξ+1−kdℳ,N, for some k∈0,1 and for η∈ℳ,ξ∈N, with η and ξ are comparable
From (2), one can easily verify thatdΓη,Γξ≤dη,ξ holds.Lemma 3 (see [18]).
Letℳ,N be nonempty subsets of X. Suppose Γ:ℳ∪N⟶ℳ∪N is a cyclic contraction map. Then, for any η0 in ℳ∪N, we have dηn,Γηn⟶dℳ,N, where ηn+1=Γηn,n=0,1,2,3,….First, we show the result for a single operator.Theorem 5.
Letℳ,N be two nonempty closed and partially bounded subsets of V,≺–,·. Let Γ:ℳ∪N⟶ℳ∪N be nondecreasing, partially cyclic contraction, and partially nonlinear D-set contraction in ℳ∪N. If there exists an element η0∈ℳ such that η0≺–Γη0, then Γ has a best proximity point.Proof.
SinceΓ is cyclic, for η0∈ℳ, then Γη0∈N. Define η1=Γη0. Now, again Γη1∈ℳ, and define η2=Γη1. In the same way, we construct ηn+1=Γηn such that η2n⊆ℳ and η2n+1⊆N, for n=0,1,2,…. Since η0≺–Γη0 and Γ is nondecreasing, it implies that η1≺–η2. Again using Γ is nondecreasing, we obtain η0≺–η1≺–η2≺–⋯. Denote(16)D2n=η2n,η2n+2,η2n+4,…,for n=0,1,2,…. Here, each D2n is a bounded chain in ℳ, and D2n=Γ2D2n−2,n=1,2,…. From the construction of D2n, we obtain(17)D0⊇D2⊇D4⊇⋯.
Consequently,(18)D¯0⊇D¯2⊇D¯4⊇⋯.
Now,(19)μpD2n=μpΓ2D2n−2≤χμpΓD2n−2≤χ2μpD2n−2≤χ4μpD2n−4⋮≤χ2nμpD0.
Letting limit superior asn⟶∞ and using Lemma 2, we obtain(20)μpD¯2n=limn⟶∞μpD2n≤limn⟶∞supχ2nμpD0=limn⟶∞χ2nμpD0=0.
Therefore, by condition (4) ofμp, we get(21)D¯∞=∩n=1∞D2n≠∅,D∞∈Prcp,ch.
Sincelimn⟶∞μpD2n=0, then for ε>0, there exists n0∈ℕ such that μpD2n<ε, for n≥n0. This implies that D¯2n0 and D¯0 are compact chains in ℳ. Hence, η2n has a convergent subsequence, that is, η2nk⟶η∈ℳ. Now,(22)dℳ,N≤η−η2nk−1≤η−η2nk+η2nk−η2nk−1.
Ask⟶∞ and by Lemma 3, we obtain(23)dℳ,N≤limk⟶∞η−η2nk−1≤dℳ,N.
Therefore,limk⟶∞η−η2nk−1=dℳ,N. Since Γ is a partially cyclic contraction, we have(24)dℳ,N≤η2nk−Γη≤η2nk−1−η.
Ask⟶∞, we get(25)dℳ,N≤limk⟶∞η2nk−Γη≤dℳ,N,which implies that η−Γη=dℳ,N.
Using the above theorem, here, we provide the result on the best proximity point for the sum of two operators.Theorem 6.
Letℳ,N be a nonempty, closed, and partially bounded subset of V,≺−,·. Let Γ1,Γ2:ℳ∪N⟶ℳ∪N be nondecreasing cyclic mappings which satisfy(1)
η∈ℳ⟹Γ1η+Γ2η∈N and η∈N⟹Γ1η+Γ2η∈ℳ.(2)
Γ1 is partially completely continuous, and(26)Γ1η−Γ1ξ≤k2η−ξ+12dℳ,N,wherek∈0,1/2 and η∈ℳ,ξ∈N with η and ξ are comparable.(3)
Γ2 is a nonlinear D-set contraction on ℳ, and(27)Γ2η−Γ2ξ≤k2η−ξ,wherek∈0,1/2 and η∈ℳ,ξ∈N with η and ξ are comparable.(4)
There exists an elementη0∈ℳ such that η0≺−Γ1η0+Γ2η0.Then, there existsη∗∈ℳ such that η∗−Γ1+Γ2η∗=dℳ,N.Proof.
DefineΓ1+Γ2:ℳ∪N⟶ℳ∪N by Γ1+Γ2η=Γ1η+Γ2η. Clearly, because of (1), Γ1+Γ2 is a cyclic mapping. And one can easily prove Γ1+Γ2 is nondecreasing on ℳ∪N. Since k<1/2 implies 1/2<1−k, now, we show that Γ1+Γ2 is a partial contraction. For η∈ℳ and ξ∈N with η and ξ are comparable, we have(28)Γ1+Γ2η−Γ1+Γ2ξ=Γ1η+Γ2η−Γ1ξ−Γ2ξ≤Γ1η−Γ1ξ+Γ2η−Γ2ξ≤kη−ξ2+dℳ,N2+kη−ξ2≤kη−ξ+1−kdℳ,N,which show that Γ1+Γ2 is a partial contraction. Next, we prove that Γ1+Γ2 is a partially nonlinear D-set contraction. Let C be a bounded chain in ℳ. Therefore, Γ1+Γ2C⊆Γ1C+Γ2C. By conditions sublinearity and full of μp, we get(29)μpΓ1+Γ2C≤μpΓ1C+Γ2C≤μpΓ1C+μpΓ2C≤χμpC.
Therefore, the mappingΓ1+Γ2 follows Theorem 5, and there exists η∗∈ℳ such that η∗−Γ1+Γ2η∗=dℳ,N.


## 6. Conclusions

In the nonlinear functional analysis, many mathematical problems can be solved by the existence result of fixed points. The fixed point theorems provide sufficient conditions to ensure the fixed point equationΓx=x, where Γ is the self-mapping, has a solution. In case of nonlinear problems, it is written as Γ1x+Γ2x=x, where Γ1,Γ2 are self-mappings, and then the fixed point theorems for the sum of two mappings help to obtain the solution for such an equation. Suppose the mappings Γ1,Γ2 are non-self-cases, then the fixed point equation Γ1x+Γ2x=x does not possess a solution. In the literature, there are many research papers which deal with the existence of the best proximity point for the equation of the form Γx=x, where Γ is the non-self-mapping. However, there is no single research work which gives the existence result of the best proximity point for the sum of mappings. So, we want to obtain an approximate solution via finding the best proximity point for such an equation in some sense. So, in this research article, we study the existence of the best proximity point for the sum of two non-self-mappings using best proximity point theorems for a single operator. Moreover, using the notion accretive operators, we prove an existence result of the fixed point for the sum of cyclic and noncyclic operators. Also, we study some notions of the ordered normed linear space, and we provide sufficient conditions for the existence of the best proximity point.



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