Some new results on fixed and best proximity points in preordered metric spaces
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RESEARCH
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Some new results on fixed and best proximity points in preordered metric spaces Alireza Amini-Harandi1,2 , Majid Fakhar2,3 , Hamid Reza Hajisharifi3 and Nawab Hussain4* Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday * Correspondence: [email protected] 4 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article
Abstract In this paper, we first introduce two new classes of (ω, δ )-contractions of the first and second kinds and establish some related new fixed point and best proximity point theorems in preordered metric spaces. Our theorems subsume the corresponding recent results of Samet (J. Optim. Theory Appl. (2013), doi:10.1007/s10957-013-0269-9) and extend and generalize many of the well-known results in the literature. An example is also provided to support our main results. MSC: 47H10; 41A65 Keywords: fixed point; best proximity point; P-property; (ω, δ )-contraction; preordered metric space
1 Introduction and preliminaries Given a metric space (X, d) and a self-mapping T on X, the theory on the existence of a solution to the equation of the form Tx = x has gained impetus because of its applicability to solve many interesting problems that can be formulated as ordinary differential equations, matrix equations etc. For some recent fixed point results, see [–] and references therein. Let A and B be nonempty subsets of X, and let T : A → B be a non-self mapping. The equation Tx = x is unlikely to have a solution, because of the fact that a solution of the preceding equation demands the nonemptiness of A ∩ B. Eventually, it is quite natural to seek an approximate solution x that is optimal in the sense that the distance d(x, Tx) is minimum. The well-known best approximation theorem, due to Fan [], states that if A is a nonempty, compact, and convex subset of a normed linear space X and T is a continuous function from A to X, then there exists a point x in A such that x – Tx = d(Tx, A) = inf{Tx – u : u ∈ A}. Such a point x is called a best approximant point of T in A. Many generalizations and extensions of this theorem appeared in the literature (see [–] and references therein). Best proximity problem for the pairs (A, B) is to find an element x ∈ A such that d(x, Tx) = d(A, B), where d(A, B) = inf{d(x, y) : x ∈ A, y ∈ B}. Since d(A, B) is a lower bound for the function x → d(x, Tx) on A, then the solutions of the best proximity problem are the minimum points of the function x → d(x, Tx) on A. Every solution of the best proximity problem is said to be a best proximity point of T in A. Moreover, if A = B then every best proximity point of T is a fixed point. According to this fact, many authors by motivation of well-known fixed point results obtained sufficient conditions to solving best proximity problems; for more details, see [–] and the references therein. ©2013 Amini-Harandi et al.; licensee Springer. This is an Open Access article distribu
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