On Some Qualitative Properties of Integrable Solutions for Cauchy-type Problem of Fractional Order

Abstract

The paper discusses the existence of solutions for Cauchytype problem of fractional order in the space of Lebesgue integrable functions on bounded interval. Some qualitative properties of solutions are presented such as monotonicity, uniqueness and continuous dependence on the initial data. The main tools used are measure of weak (strong) noncompactness, Darbo fixed point theorem and fractional calculus.Let R be the field of real numbers, J be the interval [0, 1] and L 1 (J) be the space of Lebesgue integrable functions (equivalence classes of functions) on a measurable subset J of R, with the standard normBy L ∞ (J) we denote the Banach space of essentially bounded measurable functions with the essential supremum norm (denoted by x L∞ ). We will write L 1 and L ∞ instead of L 1 (J) and L ∞ (J) respectively.Definition 2.1.[2] Assume that a function f : J × R → R satisfies the Carathéodory conditions i.e. it is measurable in t for any x ∈ R and continuous in x for almost all t ∈ J. Then to every function x(t) being measurable on J we assign (F x)(t) = f (t, x(t)), t ∈ J.The operator F is called the superposition (Nemytskii) operator.

On solutions of some delay Volterra integral problems on a half-line

“…still being a general form of many previous problems considered, for instance, in [5,7,16,17,25]. It should be stressed that we will investigate these problems on unbounded https://www.journals.vu.lt/nonlinear-analysis interval with integrable solutions as well as locally integrable ones.…”

“…(See[25,31].) If f ∈ L 1 and α ∈ (0, 1), then(a) The operator I α a maps L 1 into itself continuously.…”

Extremal Solutions to a Coupled System of Nonlinear Fractional Differential Equations with ψ-Caputo Fractional Derivatives

“…Recently, more and more attention has been given to the subject of fractional differential and integral equations due to their importance in applications in various branches of applied science and engineering; see, for example, [18,26,27,31,33,34,35]. For basic facts in the fractional calculus, we refer to the books [20,28,30,45].…”