On convergence of Bernstein–Kantorovich operators sequence in variable exponent Lebesgue spaces

Let $E=[0,1]$ and let a function $p(x)\ge1$ be measurable and essentially bounded on $E$. We denote by $L^{p(x)}(E)$ the set of measurable function $f$ on $E$ for which $\int_{E}|f(x)|^{p(x)}dx<\infty$. The convergence of a sequence of operators of Bernstein–Kantorovich $\{K_n(f,x)\}_{n=1}^\infty$ to the function $f$ in Lebesgue spaces with variable exponent $L^{p(x)}(E)$ is studied. The conditions on the variable exponent at which this sequence is uniformly bounded in these spaces are obtained and, as a corollary, it is shown that if $n\to\infty$ then $K_n(f,x)$ converges to function $f$ in the metric of space $L^{p(x)}(E)$ defined by the norm $\|f\|_{p(\cdot)}=\|f\|_{p(\cdot)}(E)=\inf\left\{\alpha>0:\quad\int\limits_E\left|\frac{f(x)}\alpha\right|^{p(x)}dx\le1\right\}$.