Abstract
By using variational methods and critical point theory, the authors establish the existence of infinitely many weak solutions for impulsive differential inclusions involving two parameters and the p-Laplacian and having Dirichlet boundary conditions.
Highlights
Impulsive boundary value problems for differential equations and inclusions have been intensively studied in recent years
Such problems appear in mathematical models with sudden changes in their states such as in population dynamics, pharmacology, optimal control, etc
Impulsive problems for fractional equations are often treated by topological methods such as in [3, 7, 8, 23]
Summary
Because of its wide applicability in the modeling of many phenomena in various fields of physics, chemistry, biology, engineering and economics, the theory of fractional differential equations has recently been attracting increasing interest; see for instance the monographs of Hilfer [19], Kilbas et al [24], Miller and Ross [29], Podlubny [31], Samko et al [33], the papers [1, 2, 4, 5, 6, 9, 25, 27, 35, 37] and references therein. Impulsive boundary value problems for differential equations and inclusions have been intensively studied in recent years. The left and right Riemann-Liouville fractional derivatives of order α > 0 of u are defined by aDtαu(t). (i) If γ ∈ (n − 1, n) and u ∈ ACn([a, b], R), the left and right Caputo fractional derivatives of order γ for the function u exist almost everywhere on [a, b], and are given by caDtγ u(t). Dtn−1u(t) u(n−1)(t) and ct Dbn−1u(t) = (−1)(n−1)u(n−1)(t) for every t ∈ [a, b] With these definitions, we have the following formulas for fractional integration by parts and the composition of the Riemann-Liouville fractional integration operator with the Caputo fractional differentiation operator as proved in [24] and [33].
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