ABSTRACT
Many problems in mathematics such as equilibrium and variational inequalities, mathematical economics, game theory and optimization can be formulated as equations of the form T x = x , $ T x = x, $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351243377/a2fb70c4-ea85-4ffd-9347-9273d5aafc73/content/inline-math6_1.tif"/> where T is a self mapping in some suitable framework, because of its ability to solve ordinary differential equations, integral equations, matrix equations and others. However, given nonempty subsets A and B of X. A mapping T : A → B $ T : A \rightarrow B $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351243377/a2fb70c4-ea85-4ffd-9347-9273d5aafc73/content/inline-math6_2.tif"/> (or T : A ∪ B → A ∪ B $ T : A \cup B \rightarrow A \cup B $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351243377/a2fb70c4-ea85-4ffd-9347-9273d5aafc73/content/inline-math6_3.tif"/> ) using the equation T x = x $ Tx =x $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351243377/a2fb70c4-ea85-4ffd-9347-9273d5aafc73/content/inline-math6_4.tif"/> does not necessarily have a solution. It is worthwhile to find an approximate solution x under mapping T so that the error d(x, Tx) is minimal. This is the basis of the optimal approximation theory that includes a generalized fixed point. Whenever A coincides with B, the optimization problem known as a best proximity point of the mapping T reduces to a fixed point problem. Fan [7] introduced a classical best approximation theorem: if A is a nonempty compact convex subset of a Hausdorff locally convex topological vector space B and T : A → B $ T : A \rightarrow B $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351243377/a2fb70c4-ea85-4ffd-9347-9273d5aafc73/content/inline-math6_5.tif"/> is a continuous mapping, there exists an element x ∈ A $ x\in A $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351243377/a2fb70c4-ea85-4ffd-9347-9273d5aafc73/content/inline-math6_6.tif"/> such that d ( T x , x ) = d ( T x , A ) : = inf { d ( T x , y ) : y ∈ A } . $ d(Tx,x) = d(Tx, A):=\inf \{d(Tx,y) : y \in A\}. $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351243377/a2fb70c4-ea85-4ffd-9347-9273d5aafc73/content/inline-math6_7.tif"/> Several authors, including Prolla [15], Reich [16], Sehgal [20,21], derived extensions of Fan’s theorem in many directions.
