Simultaneous Approximations in Banach Space-Valued Bochner–Lebesgue Spaces with Variable Exponent

Abstract

Let $(A,𝒜,\mu )$ be a σ-finite positive measure space and let X be a Banach space. Denote $L^{p(·)}(A,𝒜,X)$ the X-valued Bochner–Lebesgue spaces with variable exponent $p(·),$ where $p(·)$ be a μ measurable function on A and take values in $[1,\infty ].$ We establish some N-simultaneous proximinality results of $L^{p(·)}(A,{𝒜}_0,Y)$ in $L^{p(·)}(A,𝒜,X)$ if $1<\text{ess}\underset{y\in A}{\inf }p(y)$ and $\text{ess}\underset{y\in A}{\sup }p(y)<\infty ,$ where ${𝒜}_0$ be a sub-algebra of $𝒜$ and Y be a nonempty locally weakly compact convex subset of X.

Keywords:

• Bochner–Lebesgue space
• simultaneous approximation
• variable exponent

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 41A28
• 41A65
• 46G10

Additional information

Funding

The second author is partially supported by the National Natural Science Foundation of China (Grant No. 11761026 and 11761027) and Hainan Province Natural Science Foundation of China (2018CXTD338).