On a class of retarded integro-differential equations with nonlocal initial conditions

Abstract

a b s t r a c tOf concern is the following Cauchy problem for fractional integro-differential equations with time delay and nonlocal initial conditiongenerator of a solution operator on a complex Banach space X , the convolution integral in the equation is known as the Riemann-Liouville fractional integral, κ(t) : [0, +∞) → [−τ , +∞) representing the delay property, is a function, and H t is an operator defined from [−τ , 0] × C([−τ , 0], X ) into X for some T > 0 which constitutes a nonlocal condition. The local existence and uniqueness of mild solutions for the Cauchy problem, under various criteria, are proved. Moreover, we present an existence result of the global solution. Also an example is given to illustrate the applications of the abstract results.

On fractional integro-differential inclusions with state-dependent delay in Banach spaces

“…Wang and Chen [20] considered a Cauchy problem for fractional integro-differential equations with time delay and nonlocal initial conditions. Uniqueness and existence results of mild solutions on a semi-infinite interval have been established by Benchohra and Litimein [21].…”

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

“…Note also that motivated by mathematical models in such areas as fluids dynamics and geophysics, a number of theoretical results were also obtained for the ultraparabolic and second order evolution equations [131][132][133][134]. Other classes of motivational models for NICs have traditionally been integro-differential and dynamic inclusions on time scales [135][136][137][138], along with delay differential equations and reaction-diffusion systems [139]. While we were already mentioning fractional diffusion problems as one of the key motivations in this field, it is also worthwhile to point out that fractional mathematical models with NICs have been a subject of interest [140] where one of the tools for the analysis of their well-posedness relies on Mittag-Leffler functions, traditionally useful in nonlocal models [141,142].…”

Stability for a class of fractional partial integro-differential equations

“…Let us take note that, in these works, the models under investigation have neither nonlocal nor impulsive condition. We are also concerned with the existence results in [36], in which the model considered involved a nonlocal condition. It should be noted that, in the above-mentioned works, no attempt has been made to consider stability problems.…”