Monotonicity and the Dominated Farthest Points Problem in Banach Lattice

H R Khademzadeh,H Mazaheri

doi:10.1155/2014/616989

Abstract

We introduce the dominated farthest points problem in Banach lattices. We prove that for two equivalent norms such thatXbecomes an STM and LLUM space the dominated farthest points problem has the same solution. We give some conditions such that under these conditions the Fréchet differentiability of the farthest point map is equivalent to the continuity of metric antiprojection in the dominated farthest points problem. Also we prove that these conditions are equivalent to strong solvability of the dominated farthest points problem. We prove these results in STM, reflexive STM, and UM spaces. Moreover, we give some applications of the stated results in Musielak-Orlicz spacesL ϕ(μ)andE ϕ(μ)over nonatomic measure spaces in terms of the functionϕ. We will prove that the Fréchet differentiability of the farthest point map and the conditionsϕ∈Δ2andϕ>0in reflexive Musielak-Orlicz function spaces are equivalent.

Highlights

The problem of farthest points in Banach spaces is studied with many authors
Musielak-Orlicz spaces under the natural ordering, when endowed with each of the following norms, become a Banach lattice (e.g., [12, 13])
We prove that if (X, ‖ ⋅ ‖1) and (X, ‖ ⋅ ‖2) are two Banach lattices with the same order such that ‖ ⋅ ‖1 and ‖ ⋅ ‖2 are equivalent norms and (X, ‖ ⋅ ‖1) is an STM space and (X, ‖ ⋅ ‖2) is an LLUM space, the problem of the dominated farthest points has the same solution in two spaces

Summary

Introduction

The problem of farthest points in Banach spaces is studied with many authors (see [1,2,3]). Fitzpatrick in [2] gives some conditions such that farthest point map is Frechet differentiable and the metric antiprojection is continuous; he showed that these conditions are equivalent. Balashov and Ivanov in [1] proved that in Hilbert spaces, the set of conditions for the existence, uniqueness, and Lipschitz dependence (on x) of the metric antiprojection of x on the set A for points x that are sufficiently far from the set A is equivalent to the strong convexity of the set A. Equivalency of the Frechet differentiability of the farthest point map and the conditions φ ∈ Δ 2 and φ > 0 in reflexive Musielak-Orlicz function spaces is the final result which will be proved

Preliminaries

Dominated Farthest Points Problem in Banach Lattices