On measure of noncompactness in variable exponent Lebesgue spaces and applications to integral equations

Abstract

A novel measure of noncompactness is defined in variable exponent Lebesgue spaces $L^{p(\cdot )}$ L p ( ⋅ ) on an unbounded domain $\mathbb{R}^{+}$ R + and its properties are examined. Using the fixed point method, we apply that measure to study the existence theorem for nonlinear integral equations. Our results can be handily applied in studying various types of (differential, integral, functional, and partial differential) equations in $L^{p(\cdot )}$ L p ( ⋅ ) -spaces. The $L^{p(\cdot )} $ L p ( ⋅ ) -spaces are natural extensions of classical constant exponent Lebesgue spaces $L_{p}$ L p , which allows us to bypass several restrictions that were previously discussed in the literature.