Best Proximity Results with Applications to Nonlinear Dynamical Systems
Hamed H Al-Sulami, Nawab Hussain, Jamshaid Ahmad
Mathematics · 2019
doi:10.3390/math7100900 (opens in a new tab)
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Abstract
Best proximity point theorem furnishes sufficient conditions for the existence and computation of an approximate solution <inline-formula> <math display="inline"> <semantics> <mi>ω</mi> </semantics> </math> </inline-formula> that is optimal in the sense that the error <inline-formula> <math display="inline"> <semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi>ω</mi> <mo>,</mo> <mi mathvariant="script">J</mi> <mi>ω</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> assumes the global minimum value <inline-formula> <math display="inline"> <semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϑ</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. The aim of this paper is to define the notion of Suzuki <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-<inline-formula> <math display="inline"> <semantics> <mo>Θ</mo> </semantics> </math> </inline-formula>-proximal multivalued contraction and prove the existence of best proximity points <inline-formula> <math display="inline"> <semantics> <mi>ω</mi> </semantics> </math> </inline-formula> satisfying <inline-formula> <math display="inline"> <semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi>ω</mi> <mo>,</mo> <mi mathvariant="script">J</mi> <mi>ω</mi> <mo>)</mo> <mo>=</mo> <mi>σ</mi> <mo>(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϑ</mi> <mo>)</mo> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">J</mi> </semantics> </math> </inline-formula> is assumed to be continuous or the space <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">M</mi> </semantics> </math> </inline-formula> is regular. We derive some best proximity results on a metric space with graphs and ordered metric spaces as consequences. We also provide a non trivial example to support our main results. As applications of our main results, we discuss some variational inequality problems and dynamical programming problems.
Abstract licence: CC BY
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