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

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 nonlinearD-set contraction mapping and partially completely continuous mapping in the setting of the partially ordered complete normed linear space.


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. e study of nonlinear equations of the form Γ 1 η + Γ 2 η � η, where Γ 1 , Γ 2 : 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: x ′ (t) � f(t, x(t)) + g t, max a≤ξ≤t x(ξ) , 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 for t ∈ J � [a, b] and a ∈ L 1 (J), f: J × R ⟶ R, k: J × J ⟶ R + , and u: J × J × R ⟶ R.
So, the researchers involved in finding the solution of the equation Γ 1 η + Γ 2 η � 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 equation Γ 1 η + Γ 2 η � η in the Banach space setting, where Γ 1 is the contraction and Γ 2 is the compact operator. Later, many researchers extended Krasnoselskii's theorem in different directions (see [3][4][5][6][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 Γ 1 + Γ 2 if Γ 1 is compact and Γ 2 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 Γ 1 + Γ 2 , where Γ 1 is the weakly-strongly continuous mapping and Γ 2 is the asymptotically nonexpansive mapping. Banaś and Amar [2] proved some new types of fixed point theorems for the sum of Γ 1 + Γ 2 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 Γ 1 and Γ 2 if Γ 1 is contractive with respect to the measure of weak noncompactness and Γ 2 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 L 1 space. Ben Amar et al. [12] improved Krasnoselskii-type fixed point results for the equation (Γ 1 + Γ 2 )x � x, where the operator Γ 1 is (ws)-compact and Γ 2 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 firstorder 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. e 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-agafi 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][21][22][23][24][25].
In the light of the above literature survey, we want to find the approximate solution for the fixed point equation of the form 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.

Preliminaries
First, we collect some notions from [26]. roughout the paper, we denote B, X as the Banach space and metric space, respectively, and let be its dual. For each η ∈ B, we associate the set where 〈η, f〉 denotes f(η). e multivalued operator J : B ⟶ B * is called the duality mapping of B. Suppose Γ is an operator from B to B. en, the operator Here, we give the definition for weak accretive via the accretive operator in [26].
e following lemma is helpful for our one of the results.

Journal of Mathematics
Theorem 1 (see [27]). Let M, N be nonempty subsets of X which are complete. Suppose Γ:

Main Results
where Γ 1 and Γ 2 are cyclic and noncyclic mappings which satisfy the following: (1) − Γ 2 is the weak accretive operator, and Γ 1 is onto Now, we show that (I − Γ 2 ) − 1 Γ 1 is the cyclic mapping. Let ξ ∈ M; then, en, the equation η � Γ 2 η + Γ 1 ξ has a unique solution η. en, by (3), we obtain ( erefore, for η ∈ M, ξ ∈ N, we obtain en, the operator (I − Γ 2 ) − 1 Γ 1 agrees with the hypothesis of eorem 1, and then there is e pair (M, N) is said to have P-property if for η 1 , η 2 ∈ M and ξ 1 , ξ 2 ∈ N, Definition 2. A function Γ: X ⟶ X is called Lipschitzian if there exists a real constant κ ≥ 0 such that, for all η, ξ ∈ X, Definition 3 (see [20]). Let M, N be nonempty subsets of X. A map Γ: M ⟶ N is said to be a weakly contractive mapping if e following theorem tells that the sum of two non-selfoperators has the best proximity point in Banach space settings. e notion (1/2)N � η: η � (b/2), b ∈ N , where N is the subset of B, is used in the following theorem.  N).

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. e 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. e space (V, ≺ -, ‖ · ‖) is called the partially ordered normed linear space, where ‖ · ‖ is the norm on V.
We denote by P b d,ch (V), P rcp,ch (V) the family of all bounded chains and relatively compact chains of V, respectively.
Definition 6 (see [28]). A mapping μ p : (4) If sequence D n of closed chains in P b d,ch (V) with D n+1 ⊂ D n (n � 1, 2, . . .) and if lim n⟶∞ μ p (D n ) � 0, then the set D ∞ � ∩ ∞ n�1 D n is nonempty. e notion in (1) is known as the kernel of μ p , that is, Clearly, kerμ p ⊂ P rcp,ch (V). And so, D ∞ ∈ kerμ p . Because of μ p (D ∞ ) ≤ μ p (D n ) for all n, we obtain μ p (D ∞ ) � 0. en, D ∞ ∈ kerμ p . e measure μ p is called sublinear if it satisfies And μ p has the maximum property if Finally, μ p is said to be full if (8) kerμ p � P rcp,ch (V).
Definition 8 (see [28]). A nondecreasing mapping Γ: V ⟶ V is said to be partially nonlinear D-set-Lipschitz if there is a D-function χ such that for any bounded chain C in V. If χ(s) < s for s > 0, then Γ is called a partially nonlinear D-set-contraction.
Using the above theorem, here, we provide the result on the best proximity point for the sum of two operators. □ Theorem 6. Let M, N be a nonempty, closed, and partially bounded subset of (2) Γ 1 is partially completely continuous, and where k ∈ (0, (1/2)) and η ∈ M, ξ ∈ N with η and ξ are comparable. (3) Γ 2 is a nonlinear D-set contraction on M, and where k ∈ (0, (1/2)) and η ∈ M, ξ ∈ N with η and ξ are comparable. (4) ere exists an element η 0 ∈ M such that N).

Journal of Mathematics
Let C be a bounded chain in M.

Conclusions
In the nonlinear functional analysis, many mathematical problems can be solved by the existence result of fixed points. e fixed point theorems provide sufficient conditions to ensure the fixed point equation Γx � x, where Γ is the self-mapping, has a solution. In case of nonlinear problems, it is written as Γ 1 x + Γ 2 x � x, where Γ 1 , Γ 2 are selfmappings, and then the fixed point theorems for the sum of two mappings help to obtain the solution for such an equation. Suppose the mappings Γ 1 , Γ 2 are non-self-cases, then the fixed point equation Γ 1 x + Γ 2 x � 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-selfmapping. 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.