Existence of Best Proximity Points: Global Optimal Approximate Solution
DOI:
https://doi.org/10.5644/SJM.09.2.08Keywords:
Optimal approximate solution, common best proximity point, common fixed point, proximally commuting mappings, proximally dominating mappingsAbstract
Given non-empty subsets $A$ and $B$ of a metric space, let $S: A \to B$ and $T: A \to B$ be non-self mappings. Taking into account 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 $x \to d(x, Sx)$ and $x \to d(x, Tx)$ by imposing a common approximate solution of the equations $Sx = x$ and $T x = x$ to satisfy the constraint that $d(x, Sx) = d(x, T x) = d( A, B ).$ In this work we introduce a new notion of proximally dominating type mappings and 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 when there is no common solution. We furnish suitable examples to demonstrate the validity of the hypotheses of our results.
2010 Mathematics Subject Classification. 41A65, 46B20, 47H10
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