Mild Solutions for <i>w</i>-Weighted, Φ-Hilfer, Non-Instantaneous, Impulsive, <i>w</i>-Weighted, Fractional, Semilinear Differential Inclusions of Order <i>μ</i> ∈ (1, 2) in Banach Spaces

Abstract

The aim of this work is to obtain novel and interesting results for mild solutions to a semilinear differential inclusion involving a w-weighted, Φ-Hilfer, fractional derivative of order μ∈(1,2) with non-instantaneous impulses in Banach spaces with infinite dimensions when the linear term is the infinitesimal generator of a strongly continuous cosine family and the nonlinear term is a multi-valued function. First, we determine the formula of the mild solution function for the considered semilinear differential inclusion. Then, we give sufficient conditions to ensure that the mild solution set is not empty or compact. The desired results are achieved by using the properties of both the w-weighted Φ-Laplace transform, w-weighted ψ-convolution and the measure of non-compactness. Since the operator, the w-weighted Φ-Hilfer, includes well-known types of fractional differential operators, our results generalize several recent results in the literature. Moreover, our results are novel because no one has previously studied these types of semilinear differential inclusions. Finally, we give an illustrative example that supports our theoretical results.

Keywords

• fractional differential inclusions
• w-weighted Φ-Hilfer fractional derivative
• infinitesimal generator of cosine family
• w-weighted Φ-Laplace transform
• measure of non-compactness
• non-instantaneous impulses