Abstract
In this paper, we introduce the notion of an ordered rational proximal contraction in partially ordered b-quasi metric spaces. We shall then prove some best proximity point theorems in partially ordered b-quasi metric spaces.
Highlights
Fixed point theory is one of the most useful techniques in nonlinear functional analysis.The Banach contraction principle [1], which is the simplest statement regarding the fixed points of nonlinear mappings states that every contraction T : X → X on a complete metric space ( X, d) has a unique fixed point
We prove some best proximity point theorems for ordered rational proximal contractions of first and second kind in the setting of partially ordered b-quasi metric spaces
For any fixed x0 ∈ A0 the sequence { xn }, defined by d( xn+1, Txn ) = d( A, B), converges to x. They proved that, if, instead, A is approximatively compact with respect to B, and T is a continuous rational proximal contraction mapping of the second kind, T has a best proximity point
Summary
Fixed point theory is one of the most useful techniques in nonlinear functional analysis. For two given nonempty closed subsets A and B of a complete metric space ( X, d), a non-self contraction T : A → B does not necessarily have a fixed point. In this case, it is quite natural to investigate an element x ∈ X such that d( x, Tx) is in some sense minimum; more precisely, a point x ∈ A for which d( x, Tx) = d( A, B) is called a best proximity point of T. We prove some best proximity point theorems for ordered rational proximal contractions of first and second kind in the setting of partially ordered b-quasi metric spaces
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