Non-local boundary value problems for impulsive fractional integro-differential equations in Banach spaces

Abstract

In this study, we establish some conditions for existence and uniqueness of the solutions to semilinear fractional impulsive integro-differential evolution equations with non-local conditions by using Schauder's fixed point theorem and the contraction mapping principle. MSC: 26A33; 34A37

Evolution fractional differential problems with impulses and nonlocal conditions

“…The other one is based on the fact that the Caputo fractional derivative depends significantly on the initial point, leading to a change of the equation on each interval (t k , t k+1 ). Thus with this approach the lower limit of the Caputo fractional derivative is given by the impulse time t k (see, for example, [1,2,8,7,12,24,28,29]).…”

“…We apply a similar approach in the framework of fractional differential inclusion in [4,5] and in comparison with the literature on the subject, this is the main novelty of the paper. For instance, in [2,8,12,29] the existence, uniqueness and controllability ( [29]) of the solution of a problem similar to (2) via fixed point theorems is proved under Lipschitz regularity assumptions on the nonlinear part, the nonlocal condition and the impulse functions; applying the monotone iterative technique in the presence of upper and lower solutions, in [24] the existence of extremal solutions is obtained under monotonicity and compactness like assumptions on the nonlinear term and on the nonlocal condition and under monotonicity assumptions on the impulse functions; in [1] the compactness of the α-resolvent family generated by the linear part is assumed; in [7] and in [28] the Lipschitz regularity of the nonlinear term, the nonlocal condition and the impulse functions, or alternatively the compactness of the α−resolvent family generated by the linear part, of the nonlinear term, of the nonlocal condition and of the impulse function are taken as main hypotheses.…”

“…Notice that the last two terms on the right hand side of equation 12 and applying the Laplace transform to (12), recalling that the Laplace transform of the convolution product of two functions is the product of the corresponding Laplace transforms, we obtain…”

Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

“…Several works on boundary value problems for an impulsive differential equation with anti-periodic boundary conditions were conducted, and results on the existence of solutions for mixed-type fractional integro-differential equation were established (see, e.g., [5,19,30,42,51,52,54,55]). More recently, the theory of existence, uniqueness, and stability analysis for impulsive fractional differential equations with different kinds of fractional operators and initial/boundary conditions has attracted the attention of many researchers; for an overview of the literature, we refer the reader to [8-11, 49, 50].…”

Impulsive fractional differential inclusions with flux boundary conditions

“…That is why, it is owing to the fact that each of fractional calculus and impulsive theory serves very practical instruments for mathematical modeling of many concepts in different branches of science and engineering [1][2][3][4][5][6][7]. See [8][9][10][11][12][13][14][15][16][17][18][19][20][21] for some recent works on fractional differential equations and inclusions, and see [22][23][24][25][26][27][28][29][30][31] for the ones on impulsive fractional differential equations and inclusions.…”