Abstract
In this paper, the concept of upper and lower solutions method combined with the fixed point theorem is used to investigate the existence of oscillatory and nonoscillatory solutions for a class of initial value problem for Caputo–Hadamard impulsive fractional differential inclusions.
Highlights
Fractional differential equations and integrals are valuable tools in the modeling of many phenomena in various fields of science and engineering
This paper deals with the existence of oscillatory and nonoscillatory solutions for the following class of initial value problems for the Caputo–Hadamard impulsive fractional differential inclusion: HcDαtk y(t) ∈ F t, y(t), a.e. t ∈ J =, (1)
The following concept of lower and upper solutions was introduced by Benchohra and Boucherif [8, 9] for initial initial value problems for impulsive differential inclusions of first order
Summary
Fractional differential equations and integrals are valuable tools in the modeling of many phenomena in various fields of science and engineering. This paper deals with the existence of oscillatory and nonoscillatory solutions for the following class of initial value problems for the Caputo–Hadamard impulsive fractional differential inclusion: HcDαtk y(t) ∈ F t, y(t) , a.e. t ∈ J = (tk, tk+1),. Let C(J, R) be the space of all continuous functions from J into R. y ∞ = sup y(t). If the multivalued map G is completely continuous with nonempty compact values, G is u.s.c. if and only if G has a closed graph (i.e. xn → x∗, yn → y∗, yn ∈ G(xn) imply y∗ ∈ G(x∗)). Lemma 2.1 ([17]) Let G be a completely continuous multivalued map with nonempty compact values, G is u.s.c. if and only if G has a closed graph. Let us recall some definitions and properties of Hadamard fractional integration and differentiation
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