Abstract
This manuscript is devoted to obtain some adequate conditions for existence of at least one solution to fractional pantograph equation (FPE) involving the ψ -fractional derivative. The proposed problem is studied under some boundary conditions. Since stability is an important aspect of the qualitative theory. Therefore, we also discuss the Ulam-Hyers and Ulam-Hyers-Rassias type stabilites for the considered problem. Our results are based on some standard fixed point theorems. For the demonstration of our results, we provide an example.
Highlights
FDEs arise in lots of engineering and clinical disciplines which includes biology, physics, chemistry, economics, signal and image processing, control theory and so on; see the monographs of Hilfer [10], Podlubny [17] and Samko et al [20]
Numbers of applications have been studied by many researchers of these equations in applied sciences including biology, physics, economics, and electrodynamics
One of the important aspects which for nonlinear fractional differential equations (NFDEs) very recently attracted the attentions of researchers is devoted to stability analysis
Summary
A function u is a solution of the ψ-fractional integral equation (2.3) if and only if u is a solution of the ψ-FPE considered by us in (1.1)-(1.2). The problem (2.4) is UHR stable with respect to φ ∈ Φ, if there exists a real number Cf > 0 such that for every ǫ > 0 and for each solution Z ∈ Φ of the inequality (2.6) there exists a solution u ∈ Φ of equation (2.4) with. The equation (2.4) is generalized UHR stable with respect to φ ∈ Φ if there exists a real number Cf,φ > 0 such that for every solution Z ∈ Φ of the inequality (2.7) there exists a solution u ∈ Φ of (2.4) with. (see Theorem 3,( [21])) Let Z, β : I → [0, ∞) be continuous functions where T < ∞.
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