On the measure of noncompactness in $L_p(\mathbb{R}^+)$ and applications to a product of $n$-integral equations

Authors: MOHAMED M. A. METWALI, VISHNU NARAYAN MISHRA

Abstract: In this article, we prove a new compactness criterion in the Lebesgue spaces $L_p({\mathbb{R}}^+), 1 \leq p < \infty$ and use such criteria to construct a measure of noncompactness in the mentioned spaces. The conjunction of that measure with the Hausdroff measure of noncompactness is proved on sets that are compact in finite measure. We apply such measure with a modified version of Darbo fixed point theorem in proving the existence of monotonic integrable solutions for a product of $n$-Hammerstein integral equations $n\geq 2$.

Keywords: Compactness criterion, measure of noncompactness, discontinuous solutions, Hammerstein integral equations, compact in finite measure

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