Abstract fractional integro-differential equations involving nonlocal initial conditions in α-norm

Abstract

In the present paper, we deal with the Cauchy problems of abstract fractional integro-differential equations involving nonlocal initial conditions in a-norm, where the operator A in the linear part is the generator of a compact analytic semigroup. New criterions, ensuring the existence of mild solutions, are established. The results are obtained by using the theory of operator families associated with the function of Wright type and the semigroup generated by A, Krasnoselkii's fixed point theorem and Schauder's fixed point theorem. An application to a fractional partial integrodifferential equation with nonlocal initial condition is also considered. Mathematics subject classification (2000) 26A33, 34G10, 34G20

Existence of mild solutions to semilinear fractional evolution equation using Krasnoselskii fixed point theorem

“…By making uses of the assumption (A2) and the H ölder inequality similarly as (23), the first term φ 1 can be estimated as follows…”

“…Agarwal [21], F. Li, J. Liang and H.K. Xu [38], R. Wang, J. Liu and D. Chen [23] developed this tool to solve fractional evolution equations equipped with non-local initial conditions. N.I.…”

Derivada fracionária 'psi'-Hilfer e estabilidades de Ulam-Hyers

“…As propriedades de estabilidade de todo tipo de equações geram bastante interesse, em particular as estabilidades de Ulam-Hyers e Ulam-Hyers-Rassias, sendo essas investigadas inicialmente para equações diferenciais clássicas [18,53]. A consolidação do cálculo fracionário e seus resultados teóricos e aplicáveis cada vez mais bem posto na comunidade acadêmica, permitiu que a partir de derivadas e integrais fracionárias, pudessem formular problemas fracionários e investigar as estabilidades de Ulam-Hyers de soluções dessas equações fracionárias (diferenciais e integrodiferenciais) [36,59,73,74,75,76].…”