Abstract
This research work is dedicated to investigating a class of impulsive fractional order differential equations under the Robin boundary conditions via the application of topological degree theory (TDT). We establish some adequate results for the existence of at most one solution for the consider problem. Further, the whole analysis is illustrated by providing a pertinent example. We keep in mind that the conditions we develop by using TDT are much weaker than using ordinary fixed point theory. Hence TDT can be used as powerful tool for the theoretical analysis of many linear and nonlinear problems.
Highlights
1 Introduction In previous few decades, the area devoted to studying fractional calculus and derivatives and integrals of real or complex order has got proper attention
Keeping in mind the mentioned literature, we investigate the following nonlinear problem of impulsive Fractional ordinary differential equations (FODEs) under Robin boundary conditions (RBCs) for t ∈ [0, T]:
Lemma 3.1 Let Ψ ∈ Υ be the solution of the fractional impulsive problem with η ∈ C([0, 1], R) given by
Summary
The area devoted to studying fractional calculus and derivatives and integrals of real or complex order has got proper attention. Lemma 3.1 Let Ψ ∈ Υ be the solution of the fractional impulsive problem with η ∈ C([0, 1], R) given by. Proof Let Ψ is a solution of (3.1), for η ∈ C([0, 1], R), t ∈ [0, T], and using Lemma 2.2 for the given problem, we have two constants f0, f1, that is,. We show in combined form thta the four operators F1, F2, F3 and F4, satisfy the growth condition and are continuous, we show the operators F1, F2, F3 and F4 are ψ-Lipschitz with constants zero and co-mpact. Proof Let F0, F1, F2, F3 and F4, T : Υ → Υ be the operators defined in the section above
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