Abstract
In this manuscript, we give some sufficient conditions for existence, uniqueness and various kinds of Ulam stability for a toppled system of fractional order boundary value problems involving the Riemann–Liouville fractional derivative. Applying the Banach contraction principle and the Leray–Schauder result of cone type, uniqueness and existence results are proved for the proposed toppled system. Stability is investigated by using the classical technique of nonlinear functional analysis. The results obtained are well illustrated with the aid of an example.
Highlights
Fractional order differential equations (FODEs) have recently been addressed by many researchers for a variety of problems
1 Introduction FODEs have recently been addressed by many researchers for a variety of problems
The aforesaid equations arise in many engineering and scientific disciplines as the mathematical modeling of processes and systems in the fields of signal and image processing, control theory, physics, blood flow phenomena, polymer rheology, electrodynamics of complex medium, chemistry, aerodynamics, economics, biophysics, etc
Summary
FODEs have recently been addressed by many researchers for a variety of problems. The aforesaid equations arise in many engineering and scientific disciplines as the mathematical modeling of processes and systems in the fields of signal and image processing, control theory, physics, blood flow phenomena, polymer rheology, electrodynamics of complex medium, chemistry, aerodynamics, economics, biophysics, etc. In addition to the aforesaid investigations, many researchers have studied the Ulam stability for differential equations of different orders; see [20, 21, 27, 28, 34, 47, 48] and the references cited therein. Ali et al [8], investigated existence theory and different kinds of stability in the sense of Ulam for the following implicit fractional differential equations:. In view of Lemma 3.1, for t ∈ J, the solution of the proposed system (1.1) is equivalent to the toppled system of integral equations given by. Remark 4.2 Under the hypothesis (H4) and (4.4) and by using Definitions 2.5 and 2.6, one can repeat the process of Lemma 4.1 and Theorem 4.1, system (1.1) will be Ulam–Hyers– Rassias and generalized Ulam–Hyers–Rassias stable
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