μ-pseudo almost automorph mild solutions to the fractional integro-differential equation with uniform continuity

Abstract

Abstract Our aim in the article is to study the existence of μ-pseudo almost automorph mild solutions to the following fractional integro-differential equation: Dαu(t)=Au(t)+∫−∞ta(t−s)Au(s)ds+f(t,u(t)),t∈R, $$ D^{\alpha}u(t)=Au(t)+ \int_{-\infty}^{t}a(t-s)Au(s)\,ds+f\bigl(t,u(t)\bigr), \quad t\in\mathbb{R}, $$ where for α>0 $\alpha>0$, the fractional derivative Dα $D^{\alpha}$ is understood in the sense of Weyl, and A is a closed linear operator defined on Banach space X $\mathbb{X}$, a∈Lloc1(R+) $a\in L_{\mathrm{loc}}^{1}(\mathbb {R_{+}})$ is a scalar-valued kernel. The novelty of this work is a study of this equation with a μ- Sp $S^{p}$-pseudo almost automorph nonlinear term satisfying the condition of “uniform continuity” instead of some “Lipschitz” type conditions supposed in the literature. We utilize Schauder’s fixed point theorem. An example is provided to explain our abstract results.

Keywords

• Mild solutions
• μ- S p $S^{p}$ -pseudo almost automorphy
• Fixed point theorem
• Fractional integro-differential equation