Best Proximity Point for Generalized and S -Geraphty Contractions

This paper introduces a new class of mappings called S -Geraphty-contractions and provides su ﬃ cient conditions for the existence and uniqueness of a best proximity point for such mappings. It also presents the best proximity point result for generalized contractions as well. Our results extend and generalize some theorems in the literature.


Introduction and Preliminaries
The center of interest of fixed point theory is the solving of the equation Tx = x where T is a mapping defined on a subset of a metric space, a normed linear space, or a topological vector space. Ever since its appearance, the well-known Banach contraction principle has been extensively studied, and the literature contains numerous interesting extensions and generalizations of the aforementioned result, in particular, Geraphty's generalization of the Banach contraction principle.
Then, T has a unique fixed point. Since S contains the class of constant functions βðtÞ = k ∈ ½0, 1Þ, the previous theorem extends that of Banach.
Another interesting extension of the Banach contraction principle is due to Kirk et al. ([2]). The authors introduced the class of cyclic mappings, i.e., T : A ∪ B ⟶ A ∪ B such that TðAÞ ⊆ B and TðBÞ ⊆ A: And, under a suitable condition on T, proved a fixed point theorem which extends that of Banach. Interestingly, a more important problem than the extension of the Banach principle arose.
Whereas a cyclic mapping does not necessarily have a fixed point, it is desirable to determine an element x which is somehow closest to Tx: More precisely, an element x for which the error dðx, TxÞ assumes the least possible value distðA, BÞ where distðA, BÞ = inf fdðx, yÞ: x ∈ A, y ∈ Bg, such a point is called a best proximity point of the cyclic mapping T: Since 2003, research on best proximity points of cyclic mapping became an important topic in nonlinear analysis and has been studied by many authors [2][3][4][5][6][7][8].
In 2012, Caballero and al, introduced the following contraction.
fixed points and a best proximity point for a nonself mapping T : A ⟶ B is a point x ∈ A such that dðx, TxÞ = distðA, BÞ: Later on, the current authors ( [10]) introduced the notion of tricyclic mappings and the best proximity point thereof. Let A, B, and C be nonempty subsets of a metric space ðX, dÞ: A mapping T : A ∪ B ∪ C ⟶ A ∪ B ∪ C is said to be tricyclic provided that TðAÞ ⊆ B,TðBÞ ⊆ C, and TðCÞ ⊆ A. A best proximity point of T is a point x ∈ A ∪ B ∪ C such that Dðx, Tx, T 2 xÞ = δðA, B, CÞ, where the mapping D : X × X × X ⟶ ½0,+∞Þ is defined by Dðx, y, zÞ = dðx, yÞ + dðy, zÞ + dðz, xÞ, and Some results about the best proximity points of tricyclic mappings can be found in [10][11][12][13][14][15].
In the next section of this paper, taking inspiration from our recent works, we introduce the new class of S − cyclic mappings, which stands somewhere between the two classes of cyclic and tricyclic mappings. We define S − Geraphtycontractions and establish a best proximity point theorem for such mappings. As a special case, we obtain the best proximity point and fixed point theorem for cyclic mapping.
To describe our results, we need some definitions and notations. Given a triad ðA, B, CÞ of nonempty subsets of a metric space ðX, dÞ, then, the proximal pair ðA 0 , B 0 Þ of ðA, BÞ is given by: A pair ðx, yÞ ∈ A × B is said to be proximal in ðA, BÞ if dðx, yÞ = distðA, BÞ: We subsequently use the following notations: Note that ðB × CÞ 0 is included in B 00 × C 00 but the inverse does not always hold, and it is obvious that δðA 00 , B 00 , C 00 Þ = δðA, B, CÞ: Let us illustrate the cases ðB × CÞ 0 ⊂ B 00 × C 00 and ðB × CÞ 0 = B 00 × C 00 with simple examples.
Since ðx n Þ ⊂ A and A is a close subset of the complete metric space ðX, dÞ,x n ⟶ x ′ ∈ A: Since T is continuous, Tx n ⟶ Tx and T 2 x n ⟶ T 2 x: Which implies, Dðx n , Tx n , T 2 x n Þ ⟶ Dðx′, Tx′, T 2 x′Þ: On the other hand, ðDðx n , Tx n , T 2 x n ÞÞ is a constant sequence with the value δðA, B, CÞ: Thus That means x′ is a best proximity point of T. As for the uniqueness, suppose that x 1 and x 2 are two distinct best proximity points of T: That is Taking into account that ðA, B, CÞ has the P-property, we get Which is contradictory with the fact that T is a S -Geraphty-contraction. Indeed, we have And the proof is completed.

Abstract and Applied Analysis
Example 11. Consider X = ℝ 2 with its usual metric. Let A, B, and C be defined by It is easy to see that A 00 = fð0, 0Þg,B 00 = fð1, 0Þg and C 00 = fð2, 0Þg: Hence, A 00 is nonempty and B 00 × C 00 = ðB × CÞ 0 : Let k ∈ ½0, 1Þ and let T k : A ∪ B ⟶ B ∪ C be the mapping defined as Let ð0, Since the constant functions βðtÞ = k where k ∈ ½0, 1Þ, belong to S,T is a S-Geraphty-contraction. Also, ðA, B, CÞ has the P-property. Indeed, if Then x 1 = x 2 = x 3 = ð0, 0Þ, y 1 = y 2 = y 3 = ð1, 0Þ, and z 1 = z 2 = z 3 = ð2, 0Þ: Thus, Dðx 1 , x 2 , x 3 Þ = Dðy 1 , y 2 , y 3 Þ = Dðz 1 , z 2 , z 3 Þ = 0: All things considered, T has a unique best proximity point, clearly ð0, 0Þ: Corollary 12. Let ðA, BÞ be a pair of nonempty closed subsets of a complete metric space ðX, dÞ such that A 0 is nonempty. Let T : A ∪ B ⟶ B ∪ A be a S -Geraphty-contraction satisfying TðA 0 Þ ⊆ B 0 and TðB 0 Þ ⊆ A 0 : Assume the triad ðA, BÞ has the P -property. Then there exists a unique x ′ ∈ A that is both a best proximity point for the cyclic mapping T and a fixed point for the self-mapping T 2 : Proof. The triad ðA, B, AÞ satisfies the condition of the previous theorem. Hence, there exists a unique x′ ∈ A such that Dðx′, Tx′, T 2 x′Þ = 2distðA, BÞ, which implies and that is the wanted result.

S-Min-Max Condition and Generalized Contractions
In this section, using the S-min-max condition, we obtain a best proximity point result for nonself generalized contractions. First, we recall and fix some notions and notations that will subsequently be used. Given mappings R : A ⟶ B, S : B ⟶ C, and T : C ⟶ A, where A, B, and C are nonempty subsets of a metric space ðX, dÞ.
Definition 13 (see [6]). Let A, B be nonempty subsets of a metric space ðX, dÞ: The mapping R : A ⟶ B is called a generalized contraction if, given real numbers a and b with 0 < a ≤ b, there exists a real number αða, bÞ ∈ 0, 1Þ such that for all x 1 , x 2 in A.
Obviously, every generalized contraction is a contractive mapping.
Definition 14 (see [17] where Now, we are at liberty to state the main result of this section.
Theorem 16. Suppose A, B, and C are nonempty closed subsets of a metric space ðX, dÞ and the mappings R, S, and T verify the following conditions: And since the triad ðR, S, TÞ satisfies the S-min-max condition, we obtain Then, we necessarily have And that finishes the proof.
As a special case of our result, we get the following best proximity point theorem, which was proved in [18].
Corollary 17 (see [18]). Let A and B be nonempty, closed subsets of a complete metric space. Let R : A ⟶ B and S : B ⟶ A satisfy the following conditions: R is a generalized contraction: S is a nonexpansive mapping: The pair R, S ð Þsatisfies the min − max condition: Further, for a fixed element y 0 in A, let y 2n+1 = Ry 2n and y 2n = Sy 2n−1: Then, the sequence fy 2n g must converge to a best proximity point x * of R and the sequencefy 2n+1 g must converge to a best proximity point y * of S such that Further, if S has two distinct best proximity points, then dðA, BÞ does not vanish and hence the sets A and B should be disjoint.

Data Availability
No data were used to support this study.