Abstract
In this article, we present some existence of weak solutions for a coupled system of implicit fractional differential equations of Hilfer–Hadamard type. Our approach is based on Mönch’s fixed point theorem associated with the technique of measure of weak noncompactness.
Highlights
1 Introduction In recent years, fractional calculus and fractional differential equations are emerging as a useful tool in modeling the dynamics of many physical systems and electrical phenomena, which has been demonstrated by many researchers in the fields of mathematics, science, and engineering; see [3, 4, 18, 19, 22, 23, 30, 31, 35,36,37,38,39,40]
Considerable attention has been given to the existence of solutions of initial and boundary value problems for fractional differential equations with Hilfer fractional derivative [15, 16, 18, 20, 32, 34] and other problems with Hilfer–Hadamard fractional derivative [28, 29]
We prove the existence of weak solutions for a coupled system of implicit fractional differential equations of Hilfer–Hadamard type
Summary
Fractional calculus and fractional differential equations are emerging as a useful tool in modeling the dynamics of many physical systems and electrical phenomena, which has been demonstrated by many researchers in the fields of mathematics, science, and engineering; see [3, 4, 18, 19, 22, 23, 30, 31, 35,36,37,38,39,40]. Definition 2.11 ([22]) The Hadamard fractional derivative of order q > 0 applied to a function w ∈ ACnδ is defined as. The De Blasi measure of weak noncompactness is the map μ : E → [0, ∞) defined by μ(X) = inf{ε > 0 : there exists a weakly compact set ⊂ E such that X ⊂ εB1 + }. N, are given continuous functions, E is a real (or complex) Banach space with norm · E and dual E∗, such that E is the dual of a weakly compactly generated Banach space X, H I11–γ is the left-sided mixed Hadamard integral of order 1 – γ , and H D1α,β is the Hilfer–Hadamard fractional derivative of order α and type β.
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