Given non-empty subsets A and B of a metric space, let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${S{:}A{\longrightarrow} B}$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${T {:}A{\longrightarrow} B}$$</EquationSource> </InlineEquation> be non-self mappings. Due to the fact that S and T are non-self mappings, the equations Sx=x and Tx=x are likely to have no common solution, known as a common fixed point of the mappings S and T. Consequently, when there is no common solution, it is speculated to determine an element x that is in close proximity to Sx and Tx in the sense that d(x, Sx) and d(x, Tx) are minimum. As a matter of fact, common best proximity point theorems inspect the existence of such optimal approximate solutions, called common best proximity points, to the equations Sx=x and Tx=x in the case that there is no common solution. It is highlighted that the real valued functions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${x{\longrightarrow}d(x, Sx)}$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${x{\longrightarrow}d(x, Tx)}$$</EquationSource> </InlineEquation> assess the degree of the error involved for any common approximate solution of the equations Sx=x and Tx=x. Considering the fact that, given any element x in A, the distance between x and Sx, and the distance between x and Tx are at least d(A, B), a common best proximity point theorem affirms global minimum of both functions <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$${x{\longrightarrow}d(x, Sx)}$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$${x{\longrightarrow}d(x, Tx)}$$</EquationSource> </InlineEquation> by imposing a common approximate solution of the equations Sx=x and Tx=x to satisfy the constraint that d(x, Sx)=d(x, Tx)=d(A, B). The purpose of this article is to derive a common best proximity point theorem for proximally commuting non-self mappings, thereby producing common optimal approximate solutions of certain simultaneous fixed point equations in the event there is no common solution. Copyright Springer Science+Business Media, LLC. 2012
