On best proximity point theorems without ordering. (English) Zbl 1429.90055

Farajzadeh, A. P.; Plubtieng, S.; Ungchittrakool, K.

Abstr. Appl. Anal. 2014, Article ID 130439, 5 p. (2014).

Summary: Recently, S. S. Basha [Optim. Lett. 7, No. 5, 1035–1043 (2013; Zbl 1267.90104)] addressed a problem that amalgamates approximation and optimization in the setting of a partially ordered set that is endowed with a metric. He assumed that if \(A\) and \(B\) are nonvoid subsets of a partially ordered set that is equipped with a metric and \(S\) is a non-self-mapping from \(A\) to \(B\), then the mapping \(S\) has an optimal approximate solution, called a best proximity point of the mapping \(S\), to the operator equation \(S x = x\), when \(S\) is a continuous, proximally monotone, ordered proximal contraction. In this note, we are going to obtain his results by omitting ordering, proximal monotonicity, and ordered proximal contraction on \(S\).

MSC:

90C26	Nonconvex programming, global optimization
90C30	Nonlinear programming
41A65	Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B20	Geometry and structure of normed linear spaces
47H10	Fixed-point theorems
54H25	Fixed-point and coincidence theorems (topological aspects)

Citations:

Zbl 1267.90104

References:

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