Some weighted modular inequalities in variable Lebesgue spaces

We establish a necessary and sufficient weight condition so that a modular inequality holds for a generalized integral operator $T_{\phi }$ on non-negative non-increasing functions in weighted variable Lebesgue spaces $L_w^{p(x)}\mathrm{.}$ We give a condition under which the mentioned modular inequality leads to a norm inequality. Lastly, examples are given to demonstrate our main result, to obtain modular inequalities for some well known operators.