Cone Metric Version of Existence and Convergence for Best Proximity Points

In 2011, Gabeleh and Akhar [3] introduced semi-cyclic-contraction and considered the existence and convergence results of best proximity points in Banach spaces. In this paper, the author introduces a cone semicyclic φ-contraction pair in cone metric spaces and considers best proximity points for the pair in cone metric spaces. His results generalize the corresponding results in [1–5]..


Introduction and Preliminaries
The existence and the convergence of best proximity points may be very applicable in nonlinear analysis including optimization problems by considering the strong applications of fixed point theory.
A cone metric space is a generalization of a metric space by replacing the real numbers with Banach spaces ordered by the given cone [4,6].Generally, it is not possible to find the exact solution to the equation T x = x for a non-self mapping T : A → B defined on a cone metric space (X, d), where A and B are nonempty subsets of (X, d), Hence to focus the study on the problem of finding an element x which is the closest proximity to T x is considerable in optimization senses.To consider a global minimization problem for a mapping G : A → (E, P ), a Banach space ordered by a given cone P , defined by G(x) = d(x, T x) due to the fact that d(A, B) ≤ d(x, T x) for all x ∈ A, is very reasonable and interesting.
In 2007, Al-Thagafi and Shahzad [1] introduced a class of cyclic φ-contractions in metric spaces which contains a class of cyclic contractions as a subclass, introduced by Eldred and Veeramani [2].They obtained convergence and existence results of best proximity points for a cyclic φ-contractions in metric spaces and proved the existence of a best proximity point for a cyclic contraction in a reflexive Banach spaces which provide a positive answer to Eldred and Veeramani's question [2].
In 2011, Gabeleh and Akhar [3] introduced semi-cyclic-contraction and considered the existence and convergence results of best proximity points in Banach spaces.
In this paper, we introduce a cone semi-cyclic φ-contraction pair and obtain the existence and convergence results for the pair in cone metric spaces.Our results generalize the corresponding results in [1][2][3][4][5].
A nonempty subset P of a real Banach space E is called a (pointed) cone if and only if (P1) P is closed, P ̸ = {0}, (P2) for a, b ∈ R with a, b ≥ 0, x, y ∈ P implies ax + by ∈ P and (P3) x ∈ P and − x ∈ P implies x = 0.
We define a partial ordering '≼' with respect to P as follows; for x, y ∈ E, we say that x ≼ y if and only if y − x ∈ P, x ≪ y if and only if y − x ∈ intP , where intP denotes the interior of P, x ≺ y if and only if x ≼ y and x ̸ = y.A cone P is said to be normal if there is a number K > 0 such that for all x, y ∈ E, 0 ≤ x ≤ y implies ∥x∥ ≤ K∥y∥.
Let M be a nonempty set and (E, P ) a Banach space with a given cone P .A mapping d : M × M → (E, P ) satisfying the conditions (d1) 0 ≼ d(x, y) for all x, y ∈ M and d(x, y) = 0 if and only if x = y, (d2) d(x, y) = d(y, x) for all x, y ∈ M and (d3) d(x, y) ≼ d(x, z) + d(y, z) for all x, y, z ∈ M is called a cone metric on M and (M, d) is called a cone metric space.Definition 1.1.[5].Let (M, d) be a cone metric space.A subset A of M is said to be bounded above if there exists c ∈ intP such that c − d(x, y) ∈ P for all x, y ∈ A, and is said to be bounded if δ(A) = sup{d(x, y) : x, y, ∈ A} exists in E.
Let {x n } be a sequence in (M, d) and x ∈ (M, d).If for every c ∈ intP , there is a natural nember N such that for all n > N, c − d(x n , x) ∈ intP, then we say that {x n } converges to x with respect to P and denote as lim n→∞ x n = x.Lemma 1.1.[4] Let P be a normal cone.Let {x n } and {y n } be sequences in (M, d).
(i) {x n } converges to x with respect to P if and only if d(x n , x) → 0 as n → ∞; (iii) If x n → x and y n → y as n → ∞ with respect to P and y n − x n ∈ P for all n ∈ N, then y − x ∈ P .

Properties of Constructed Sequences
Let A and B be nonempty subsets of a cone metric space (M, d) and S, T :
Remark 2.1.(i) A cone semi-cyclic k-contraction pair is a cone semi-cyclic φ-contraction pair with φ(x) = (1 − k)x for x ∈ E. In this case the pair (S, T ) satisfies for some k ∈ (0, 1) (ii) When S = T, T is called a cone cyclic k-contraction.

Cone Metric Version of Existence and Convergence for Best Proximity Points
Proof.For a strictly increasing mapping φ : Thus we have Hence the sequence {d( Consequently, from (2.2) and (2.3), we have d(A, B) − d(x 0 , y 0 ) ∈ P \ intP ⊂ P, which is a contradiction.Hence we have lim The same method also shows that {d(y n+1 , T y n )} is decreasing and lim Proof.Now Inductively, we have Letting n → ∞, we have lim We have the corresponding result to Theorem 2.1 as a corollary in metric spaces as follows; Corollary 2.4.[1].Let A and B be nonempty subsets of a metric space X and T : A ∪ B → A ∪ B be a cyclic φ-contraction mapping.For x 0 ∈ A, define x n+1 = T x n for each n ≥ 0, then

1 ) 2 . 1 .
For a cone semi-cyclic φ-contraction pair (S, T ), letting x 0 ∈ A, y 0 = Sx 0 ∈ B, x 1 = T y 0 ∈ A and y 1 = Sx 1 ∈ B inductively, we have two sequences {x n } in A and {y n } in B such thatx n+1 = T y n and y n = Sx n for n ∈ N ∪ {0}.(2.TheoremLet A and B be nonempty subsets of a cone metric space (X, d) and (S, T ) a cone semi-cyclic φ-contraction pair.Then for the sequences {x n } and {y n } generated as (2.1), two new sequences {d(x n , Sx n )} and {d(y n+1 , T y n )} are decreasing and converge to d(A, B) in E.

Corollary 2 . 2 .Corollary 2 . 3 .
n→∞ d(y n+1 , T y n ) = d(A, B).For a sequence {x n } constructed as x n+1 = T x n for x 0 ∈ A (n ∈ N∪{0}), where T : A∪B → A∪B is a cone cyclic φ-contraction, we have the same result as d(x n , x n+1 ) ↓ d(A, B).Let A and B be nonempty subsets of a cone metric space (M, d).Let T : A ∪ B → A ∪ B be a cone cyclic k-contraction.Then d(x n , x n+1 ) converges d(A, B), where x 0 is a given point of A and x n+1 = T x n (n ∈ N ∪ {0}).
d(x n , x n+1 ) → d(A, B) as n → ∞.Now we show that the generated sequences {x n } and {y n } in (2.1) are bounded.
Theorem 2.5.Let A and B be nonempty subsets of a normal cone metric space (X, d) and (S, T ) be a cone semicyclic φ-contraction pair.For given point x 0 ∈ A, the sequences {x n } and {y n } generated as (2.1) are bounded.Proof.We show that{x n } is bounded.Since d(x n , 0) ≼ d(x n , Sx n ) + d(Sx n , 0) and {d(x n , Sx n )} is bounded, it is enough to show that {Sx n } is bounded.For the unbounded mapping φ, take M ∈ E such that φ(M ) ≻ d(x 0 , x 1 ) + φ ( d(A, B) ) .If {Sx n } is not bounded, thenthere exists a natural number N ∈ N such that d(x 1 , Sx N ) ≻ M and d(x 1 , Sx N −1 ) ≼ M.