A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator

By J. Sousa, E. Capelas de Oliveira. Differential Equations & Applications, 2017

219 citations recorded by the source corpus.

In this paper, we propose a generalized Gronwall inequality through the fractional integral with respect to another function. The Cauchy-type problem for a nonlinear differential equation involving the $\psi$-Hilfer fractional derivative and the existence and uniqueness of solutions are discussed. Finally, through generalized Gronwall inequality, we prove the continuous dependence of data on the Cauchy-type problem.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/625f9386f89daf41b7463079b51d4164a22cf285

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In Applied Mathematics

Citations in the corpus

Existence uniqueness and stability of mild solutions for semilinear ψ-Caputo fractional evolution equations (2020, background): “11 (Gronwall’s inequality [41, 42]) Let u, v be two integrable functions and h be a continuous function on [a, b].”
On the Ulam–Hyers–Rassias stability for nonlinear fractional differential equations using the $$\psi $$ψ-Hilfer operator (2017, background): “We present the definitions of the ψ-Hilfer fractional derivative and the Riemann-Liouville fractional integral with respect to a function ψ, the spaces in which these fractional operators are defined and Gronwall’s inequality, among other results [26].”
On the existence and stability for noninstantaneous impulsive fractional integrodifferential equation (2018, unknown): “|||||y(t) − Ψ γ (t, 0) x0 − t ∫ 0 N ψ (t, s)f ⎜⎜⎝s, y(s), 1 Γ(α) s ∫ 0 α ψ (s, τ, y(τ))dτ ⎟⎟⎠ ds ||||| ≤ I α,ψ 0+ φ (7)”