On Well Posed Generalized Best Approximation Problems

Abstract

Let C be a closed bounded convex subset of X with 0 being an interior point of C and pC be the Minkowski functional with respect to C. Let G be a nonempty closed, boundedly relatively weakly compact subset of a Banach space X. For a point x∈X, we say the minimization problem minC(x, G) is well posed if there exists a unique point z such that pC(z−x)=λC(x, G) and every sequence {zn}⊂G satisfying limn→∞pC(zn−x)=λC(x, G) converges strongly to the point z, where λC(x, G)=infz∈GpC(z−x). Under the assumption that C is both strictly convex and Kadec, we prove that the set Xo(G) of all x∈X such that the problem minC(x, G) is well posed is a residual subset of X extending the results in the case that the modulus of convexity of C is strictly positive due to Blasi and Myjak. In addition, we also prove these conditions are necessary.