Abstract
In this paper, the existence of multiplicity distinct weak solutions is proved for differentiable functionals for perturbed systems of impulsive nonlinear fractional differential equations. Further, examples are given to show the feasibility and efficacy of the key findings. This work is an extension of the previous works to Banach space.
Highlights
This paper explores the perturbed impulsive fractional differential system
In [36], we investigated the existence of solutions of the periodic boundary value problem for a nonlinear impulsive fractional differential equation with periodic boundary conditions: D2αuðtÞ = f ðt, u, DuÞ, ð7Þ
In this work, we extend the last work [38] to Banach space, where we show that there are at least three weak solutions for the system (1), which involves two parameters λ and μ
Summary
The left and right Riemann–Liouville fractional derivatives of order αi for the function u are defined in the following forms, respectively, 0Dαt i uðtÞ Θp3 − θp2 Fðt, Θ3Þdt and every nonnegative function G : 1⁄20, T × Rn ⟶ R satisfying Gθ ≥ 0, there exists δλ,G > 0 given by (39) such that, for each μ ∈ 1⁄20, δλ,G1⁄2, the system (1) has at least three solutions u1, u2, and u3 such that maxt∈1⁄20,T ju1ðtÞj < θ1, maxt∈1⁄20,T ju2ðtÞj < θ2, and maxt∈1⁄20,Tju3ðtÞj < θ3.
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