The relationship between directional derivatives of generalized distance functions and the existence of generalized nearest points in Banach spaces is investigated. Let G be any nonempty closed subset in a compact locally uniformly convex Banach space. It is proved that if the one-sided directional derivative of the generalized distance function associated to G at x equals to 1 or − 1, then the generalized nearest points to x from G exist. We also give a partial answer (Theorem 3.5) to the open problem put forward by S. Fitzpatrick (1989, Bull. Austral. Math. Soc. 39, 233-238).

Let Z be a closed, boundedly relatively weakly compact, nonempty subset of a Banach space X, and J : Z → R a lower semicontinuous function bounded from below. If X 0 is a convex subset in X and X 0 has approximatively Z-property (K), then the set of all points x in X 0 \ Z for which there exists z 0 ∈ Z such that J (z 0 ) + x − z 0 = ϕ(x) and every sequence {z n } ⊂ Z satisfying lim n→∞ [J (z n ) + x − z n ] = ϕ(x) for x contains a subsequence strongly convergent to an element of Z is a dense G δ -subset of X 0 \ Z. Moreover, under the assumption that X 0 is approximatively Z-strictly convex, we show more, namely that the set of all points x in X 0 \ Z for which there exists a unique point z 0 ∈ Z such that J (z 0 ) + x − z 0 = ϕ(x) and every sequence {z n } ⊂ Z satisfying lim n→∞ [J (z n ) + x − z n = ϕ(x) for x converges strongly to z 0 is a dense G δ -subset of X 0 \ Z. Here ϕ(x) = inf{J (z) + x − z ; z ∈ Z}. These extend S. Cobzas's result [

The relation between directional derivatives of generalized distance functions and the existence of generalized nearest points in Banach spaces is investigated. We show that if the generalized function generated by a closed set has a one-sided directional derivative equal to 1 or -1, then the existence of generalized nearest points follows. We also give a partial answer to an open problem proposed by S. Fitzpatrick.

In this paper, we investigate common solutions to a family of mixed equilibrium problems with a relaxed η-α-monotone mapping and a nonlinear operator equation