Proximinal Problems In Banach Space Valued Function Spaces

The dissertation is devoted to the Proximinality in Banach space valued Bochner-Lebesgue spaces with variable exponent. The plan is the following:In Chapter 1, there are the literature servey, definitions, notations and the statement of main results.In Chapter 2, We study best simultaneous approximation with respective to Minkowski’norms in Euclidean spaces in Bochner-Lebesgue spaces . Fistly, we give a characterization of distance functions.Then we show that the simultaneous proximinality of Bochner-Lebesgue space whose functions take valyes in a closed separable subspace is equivalent to the simultaneous proximinality of the closed separable subspace.In Chapter 3, The proximinality is proved in Banach space valued grand Bochner-Lebesgue spaces with variable exponent. Firstly, we estimate the distance of f from Lp(·),φ(A,Y) when f∈Lp(·),φ(A, X).Then we obtain the equivalence relation between LP(·),φ(A,Y) is proximinal in Lp(·),φ(A,X) and Y is proximinal in X. Finally, we establish the connection between the proximinality of Lp(·),φ(A,Y) in Lp(·),φ(A,X)and the proximinality of L1(A,Y) in L1(A,X).In Chapter 4, It is studies the simultaneous approximations in Banach space valued Bochner-Lebesgue spaces with variable exponent.