Proximinality in Banach space valued grand Bochner-Lebesgue spaces with variable exponent

Let $(A,\mathscr{A},\mu)$ be a $\sigma$-finite complete measure space and $p(\cdot)$ be a $\mu$-measurable function on $A$ which takes values in $(1,\infty).$ Let $Y$ be a subspace of a Banach space $X.$ Denote $\widetilde{L}^{p(\cdot),\varphi}(A, Y)$ and $\widetilde{L}^{p(\cdot),\varphi}(A, X)$ by grand Bochner-Lebesgue spaces with variable exponent $p(\cdot)$ whose functions take values in $Y$ and $X$ respectively. Firstly, we estimate the distance of $f$ from $\widetilde{L}^{p(\cdot),\varphi}(A, Y)$ when $f\in \widetilde{L}^{p(\cdot),\varphi}(A, X).$ Then we obtain that $\widetilde{L}^{p(\cdot),\varphi}(A, Y)$ is proximinal in $\widetilde{L}^{p(\cdot),\varphi}(A, X)$ if $Y$ is weakly $\mathcal{K}$-analytic and proximinal in $X.$ Finally, we establish the connection between the proximinality of $\widetilde{L}^{p(\cdot),\varphi}(A, Y)$ in $\widetilde{L}^{p(\cdot),\varphi}(A, X)$ and the proximinality of $L^1(A, Y)$ in $L^1(A, X).$