Existence of Solutions for Fractional Integro-Differential Equations with Non-Local Boundary Conditions

Abstract

In this paper, we study the existence of solutions for a new class of boundary value problems of non-linear fractional integro-differential equations. The existence result is obtained with the aid of Schauder type fixed point theorem while the uniqueness of solution is established by means of contraction mapping principle. Then, we present some examples to illustrate our results.

Solvability of Some Types for Multi-fractional Integro-Partial Differential Equation

“…The study of the existence of mild solution for an integro-differential equation description is as follows: Recently, the fractional order differential equation model is more interesting and more descriptive to be applied in various branches such as engineering and science. So, the abstract Cauchy problems with fractional order derivative simulated to some problems of the fractional differential equation have appeared in many articles for many applications in real life (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19). Consider the following fractional order integropartial differential equations: where 1 , 2 , > 0, ( , ) is denoted the deviation x at , 1 < ≤ 2, 1 < 1 + 2 ≤ 2 and 0 < 1 , 2 ≤ 1, ≥ 0, and is a Riemann-Liouville fractional derivative of order > 0, ∈ (X is a Banach space).…”

Holder's Inequality ρ–Mean Continuity for Existence and Uniqueness Solution of Fractional Multi-Integrodifferential Delay System

“…e importance of investigating the solution for fractional order derivatives in integrodifferential equations with a nonlocal initial or boundary condition lies in the fact that they include several classes of fractional order integrodifferential equations, as presented in studies on the existence and uniqueness of nonlocal initial fractional order integrodifferential equations in [1][2][3][4][5] and in some other studies with a nonlocal boundary condition [6], as well as fractional order differential equations involving integral conditions as a boundary condition, as found in multiple papers, including those by [7,8].…”

“…where c D α and I β 1 and I β 2 are the Caputo fractional derivative and fractional integration, respectively, of order 0 < α, β 1 , β 2 < 1, the state x(·) is defined on the Banach space X with the norm ‖·‖, u(·) is the control function in Banach space L 2 (J, V) of admissible control functions, and V is Banach space, where B: V ⟶ X is a linear bounded operator. T(t) { } t≥0 is a strongly continuous semigroup of operators on X generated by A. PC(J, X) � x: [0, b] { ⟶ X, x(t) is continuous at t ≠ t k and left continuous at t � t k and x(t + k )}, the impulsive functions I i : D ⟶ X, i � 1, 2, .…”

Some New Generalization of Darbo’s Fixed Point Theorem and Its Application on Integral Equations

“…In solving functional integral equations, Schauder and Darbo's fixed point theorems play a significant role. We refer (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]) for application of fixed point theorems and measure of noncompactness for solving differential and integral equations.…”