On positive solutions of a system of equations generated by Hadamard fractional operators

Abstract

This paper is devoted to studying some systems of quadratic differential and integral equations with Hadamard-type fractional order integral operators. We concentrate on general growth conditions for functions generating right-hand side of considered systems, which leads to the study of Hadamard-type fractional operators on Orlicz spaces. Thus we need to prove some properties of such type of operators. In contrast to the case of Caputo or Riemann-Liouville type of fractional operators, it is not a convolution-type operator, so we need to study some of their new properties. Some more general problems than systems of quadratic integral equations are also studied, and the results are new even in the context of a single integral equation with the Hadamard fractional operator. The paper concludes with illustrative examples.

On the Generalization of Tempered-Hilfer Fractional Calculus in the Space of Pettis-Integrable Functions

“…} and, by our assumptions, the last set is bounded, for any ε > 0 there exists (sufficiently small) τ such that t t−τ G(t, s)V(s) ds < ε. (39) From this it follows that we can cover the set t t−τ G(t, s)v(s) ds : s ∈ [t − τ, t], v ∈ V by balls with radius less than ε and then by definition of the De Blasi measure of weak noncompactness µ we obtain In view of Ambrosetti's Lemma 2, v is a continuous function. Note, that from our assumption it follows that s → G(t, s)v(s) is continuous on [a, t − τ], so is uniformly continuous.…”

“…, where α > 0, µ ∈ R + is a generalized version of the classical Hadamard model of fractional calculus. Obviously, in the particular choice µ = 0, we cover the standard version of the Hadamard fractional integral, discussed, among others, by Cicho ń and Salem in [29,39,40] (cf. also [22,[41][42][43][44][45]), for the existence of solutions to the fractional Cauchy problem.…”

Study on the Existence of Solutions for a Class of Nonlinear Neutral Hadamard-Type Fractional Integro-Differential Equation with Infinite Delay

“…The Mellin's transformation of Hadamard fractional calculus was investigated in [17]. Some papers on Hadamard fractional differential boundary value problems [5,[18][19][20][21][22][23], integrodifferential equations [5,18,24], neutral equations [24] and coupled systems [5,25] have been published. Compared with the research results of Riemann-Liouville and Caputo fractional differential equations, the study on Hadamard fractional differential equation are relatively rare.…”