Integrable Solutions for Gripenberg-Type Equations with m-Product of Fractional Operators and Applications to Initial Value Problems

Abstract

In this paper, we deal with the existence of integrable solutions of Gripenberg-type equations with m-product of fractional operators on a half-line R+=[0,∞). We prove the existence of solutions in some weighted spaces of integrable functions, i.e., the so-called L1N-solutions. Because such a space is not a Banach algebra with respect to the pointwise product, we cannot follow the idea of the proof for continuous solutions, and we prefer a fixed point approach concerning the measure of noncompactness to obtain our results. Appropriate measures for this space and some of its subspaces are introduced. We also study the problem of uniqueness of solutions. To achieve our goal, we utilize a generalized Hölder inequality on the noted spaces. Finally, to validate our results, we study the solvability problem for some particularly interesting cases and initial value problems.

On Erdélyi–Kober Fractional Operator and Quadratic Integral Equations in Orlicz Spaces

“…Recall that the quadratic integral equations were inspected in Orlicz spaces in [18][19][20] and in L p spaces [9,[21][22][23] by utilizing the measure of noncompactness (MN C) associated with Darbo's fixed-point hypothesis (F P T ) across various sets of assumptions.…”