Consider the problem of computing a common solution to the simultaneous equations <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$Tx=x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$Fx=x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> in the framework of metric spaces. Nevertheless, if T is not a self-mapping and F is a self-mapping in particular, then it may be the case that the equation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$Tx=x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> has no solution and the equation <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Fx=x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> has a solution, in which case the system comprising the equations <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Tx=x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$Fx=x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> is inconsistent. Eventually, it is of paramount interest and fundamental significance to identify a point in the space that serves as an approximate solution of the first equation <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$Tx=x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation>, with the least possible error, and serves as an exact solution of the second equation <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$Fx=x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation>. In view of the fact that for an approximate solution <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$x^{}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>x</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation> of the equation <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$Tx=x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation>, the quantum <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$d(x^{}, Tx^{*})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mmultiscripts> <mi>x</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo>,</mo> <mi>T</mi> <mmultiscripts> <mi>x</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> scales the error due to approximation, one is conclusively interested in the constrained global minimization of the real valued error function <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$x \longmapsto d(x, Tx)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>⟼</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>T</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> subject to the constraint <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$Fx=x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation>. The purpose of this paper is to resolve the preceding constrained global minimization problem in some special interesting cases, thereby generalizing some best proximity point theorems and fixed point theorems. It is remarked that unlike the preceding endeavor, the common best proximity point theorems accomplish unconstrained global minimization. Further, the results presented in this article generalize the most celebrated contraction principle due to Banach.

Constrained best proximity point theorems: constrained global minimization with one or more constraints

期刊论文
发表日期：

2025年2月23日

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S. Sadiq Basha

摘要

Consider the problem of computing a common solution to the simultaneous equations $Tx=x$ $T x = x$ and $Fx=x$ $F x = x$ in the framework of metric spaces. Nevertheless, if T is not a self-mapping and F is a self-mapping in particular, then it may be the case that the equation $Tx=x$ $T x = x$ has no solution and the equation $Fx=x$ $F x = x$ has a solution, in which case the system comprising the equations $Tx=x$ $T x = x$ and $Fx=x$ $F x = x$ is inconsistent. Eventually, it is of paramount interest and fundamental significance to identify a point in the space that serves as an approximate solution of the first equation $Tx=x$ $T x = x$ , with the least possible error, and serves as an exact solution of the second equation $Fx=x$ $F x = x$ . In view of the fact that for an approximate solution $x^{*}$ $x^{*}$ of the equation $Tx=x$ $T x = x$ , the quantum $d(x^{*}, Tx^{*})$ $d (x^{*}, T x^{*})$ scales the error due to approximation, one is conclusively interested in the constrained global minimization of the real valued error function $x \longmapsto d(x, Tx)$ $x ⟼ d (x, T x)$ subject to the constraint $Fx=x$ $F x = x$ . The purpose of this paper is to resolve the preceding constrained global minimization problem in some special interesting cases, thereby generalizing some best proximity point theorems and fixed point theorems. It is remarked that unlike the preceding endeavor, the common best proximity point theorems accomplish unconstrained global minimization. Further, the results presented in this article generalize the most celebrated contraction principle due to Banach.