Optimal Approximate Solutions of Fixed Point Equations

and Applied Analysis 3 Proof. As T and S form a K-Cyclic map, d x1, x2 d Tx0, Sx1 ≤ k d x0, Tx0 d x1, Sx1 1 − 2k d A,B k d x0, x1 d x1, x2 1 − 2k d A,B . 3.1 So, it follows that d x1, x2 ≤ k/ 1 − k d x0, x1 1 − k/ 1 − k d A,B . Similarly, it can be seen that d x2, x3 ≤ ( k 1 − k )2 d x0, x1 [ 1 − ( k 1 − k )2] d A,B . 3.2 Hence, it follows by induction that d xn, xn 1 ≤ ( k 1 − k )n d x0, x1 [ 1 − ( k 1 − k )n] d A,B . 3.3 Therefore, d xn, xn 1 → d A,B because of the fact that k < 1/2. Lemma 3.2. LetA and B be non-empty closed subsets of a metric space. Let the mappings T : A → B and S : B → A form a K-Cyclic map between A and B. For a fixed element x0 in A, let x2n 1 Tx2n and x2n Sx2n−1. Then, the sequence {xn} is bounded. Proof. It follows from Lemma 3.1 that d x2n−1, x2n is convergent and hence it is bounded. Further, since S and T form a K-cyclic mapping, it follows that d x2n, Tx0 ≤ k d x2n−1, x2n d x0, Tx0 1 − 2k d A,B . 3.4 Therefore, the subsequence {x2n} is bounded. Similarly, it can be shown that {x2n 1} is also bounded. Lemma 3.3. LetA and B be non-empty closed subsets of a metric space. Let the mappings T : A → B and S : B → A form a K-Cyclic map between A and B. For a fixed element x0 in A, let x2n 1 Tx2n and x2n Sx2n−1. Suppose that the sequence {x2n} has a subsequence converging to some element x in A. Then, x is a best proximity point of T . Proof. Suppose that a subsequence {x2nk} converges to x inA. It follows from Lemma 3.1 that d x2nk−1, x2nk converges to d A,B . As S and T form a K-cyclic mapping, it follows that d A,B ≤ d x2nk , Tx ≤ k d x2nk−1, x2nk d x, Tx 1 − 2k d A,B . 3.5 Therefore, d x, Tx d A,B . The preceding two lemmas yield the following best proximity point theorem for Kcyclic mappings in the setting of metric spaces. 4 Abstract and Applied Analysis Corollary 3.4. Let A and B be two non-empty and closed subsets of a metric space. Let the mappings T : A → B and S : B → A form a K-Cyclic map between A and B. If A is boundedly compact, then T has a best proximity point. The following lemma, due to Eldred and Veeramani 10 , will be required subsequently to establish the next best proximity point theorem of this section. Lemma 3.5. LetA be a non-empty, closed, and convex subset and B be a non-empty and closed subset of a uniformly convex Banach space. Suppose that {xn} and {yn} are sequences in A and {zn} is a sequence in B satisfying the following conditions: a ‖yn − zn‖ → d A,B , b for every > 0, ‖xm − zn‖ ≤ d A,B , for sufficiently large values of m and n. Then, for every > 0, ‖xm − yn‖ ≤ for sufficiently large values of m and n. The following best proximity point theorem is for K-cyclic mappings in the setting of uniformly convex Banach spaces. Theorem 3.6. Let A and B be non-empty, closed, and convex subsets of a uniformly convex Banach space. If the mappings T : A → B and S : B → A form a K-Cyclic map betweenA and B, then there exist a unique element x ∈ A and a unique element y ∈ B such that