On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions

doi:10.1016/j.amc.2015.05.036

Applied Mathematics and Computation

Volume 266, 1 September 2015, Pages 235-243

https://doi.org/10.1016/j.amc.2015.05.036 Get rights and content

Abstract

In this paper, by using the endpoint result for multifunctions, we investigate the existence of solutions for a boundary value problem for fractional differential inclusions with sum and integral boundary conditions. Finally, an example is also given to illustrate the validity of our main result.

Section snippets

Introduction and preliminaries

The topic of fractional differential equations is one of the branches of mathematics which has various important applications in many fields as mathematics, physics, chemistry, biology and many other branches of engineering. Many papers have published about fractional differential equations by researchers which apply the fixed point theory in their existence theorems. For instence, one can find a lot of papers in this field (see [1], [2], [3], [4], [5], [10], [11], [12], [13], [14], [15], [17],

Main result

Now, we are ready to prove our main result. Let $X = {u : u, u^{'},^{c} D_{0}^{p_{i}} u \in C (J, R)}$ endowed with the norm $∥ u ∥ = \sup_{t \in J} | u (t) | + \sup_{t \in J} | u^{'} (t) | + \sum_{i = 1}^{k} \sup_{t \in J} |^{c} D_{0}^{p_{i}} u (t) |$ . Then, (X, ‖ · ‖) is a Banach space [32].

Lemma 2.1

Given y ∈ X, then the unique solution of the problem ${\begin{matrix} ^{c} D_{0}^{α} u (t) = y (t), 0 < t < 1, 2 < α \leq 3, \\ u (0) + \sum_{j = 1}^{m} b_{j} u^{''} (0) = 0, γ_{1} u (η) + γ_{2} \int_{0}^{1} u (τ) d τ = 0, \\ \sum_{j = 1}^{m} b_{j} u^{'} (1) + γ_{3} \int_{0}^{1} u (τ) d τ = 0, \end{matrix}$ is given by $\begin{matrix} u (t) & = & \int_{0}^{t} \frac{{(t - s)}^{α - 1}}{Γ (α)} y (s) d s + \frac{γ_{1}}{Δ} A (t) \int_{0}^{η} \frac{{(η - s)}^{α - 1}}{Γ (α)} y (s) d s \\ + \frac{1}{Δ} (γ_{2} A (t) + γ_{3} B (t)) \int_{0}^{1} \int_{0}^{τ} \frac{{(τ - ω)}^{α - 1}}{Γ (α)} y (ω) d ω d τ \\ + \frac{1}{Δ} B (t) \sum_{j = 1}^{m} b_{j} \int_{0}^{1} \frac{{(1 - s)}^{α - 2}}{Γ (α - 1)} y (s) d s, \end{matrix}$ where $A (t) =$

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On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions

Abstract

Section snippets

Introduction and preliminaries

Main result

J. Appl. Math. Comput.

Bound. Value Prob.

A survey on semilinear differential equations and inclusions invovling Riemann–Liouville fractional derivative

Adv. Diff. Eq.

A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions

Acta Appl. Math.

Existence of solutions for integro-differential equations of fractional order with nonlocal three-point fractional boundary conditions

Adv. Diff. Eq.

Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray–Schauder degree theory

Topol. Methods Nonlinear Anal.

Boundary value problem for fractional differential inclusions with four-point integral boundary conditions

Surv. Math. Appl.

On fractional differential inclusions with with anti-periodic type integral boundary conditions

Bound. Value Probl.

Existence of solutions for fractional differential inclusions with separated boundary conditions in Banach pace

Abstr. Appl. Anal.

Endpoints of set-valued contractions in metric spaces

Nonlinear Anal.

Existence and multiplicity of positive solutions for singular fractional boundary value problems

Comput. Math. Appl.

The existence of solutions for a nonlinear mixed problem of singular fractional differential equations

Adv. Diff. Eq.

Positive solutions of a boundary value problem for nonlinear fractional differential equations

Abstr. Appl. Anal.

Some existence results on nonlinear fractional differential equations

Philos. Trans. Roy. Soc.

Existence of solutions for fractional differential inclusions with separated boundary conditions in Banach space

Adv. Math. Physics