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Choukri Derbazi

Arabian Journal of Mathematics

· Hadda Hammouche

Boundary value problems for Caputo fractional differential equations with nonlocal and fractional integral boundary conditions

Received: 12 January 2019 / Accepted: 11 July 2020 © The Author(s) 2020

Abstract In this paper, we study the existence and uniqueness of solutions for fractional differential equations with nonlocal and fractional integral boundary conditions. New existence and uniqueness results are established using the Banach contraction principle. Other existence results are obtained using O’Regan fixed point theorem and Burton and Kirk fixed point. In addition, an example is given to demonstrate the application of our main results. Mathematics Subject Classification

34A08 · 26A33

1 Introduction Fractional differential equations have gained much importance and attention due to the fact that they have been proved to be valuable tools in the modeling of many phenomena in engineering and sciences such as physics, mechanics, economics, biology, etc. (see, for example, [16,19,23,24] and references therein). Actually, the concepts of fractional derivatives are not the only generalization of the ordinary derivatives, but also it has been found that they can efficiently and properly describe the behavior of many physical systems (real-life phenomena) more accurately than integer-order derivatives. Different kind of fixed point theorems are widely used as fundamental tools to prove the existence and uniqueness of solutions for various classes of fractional differential equations; for example, we refer the reader to [1–4,9,12,14,15,17,18,21,25,27] and the references cited therein. Moreover, some mathematicians considered Caputo fractional differential equations with a nonlinear term depending on the Caputo derivative (see Benchohra and Souid [5], Benchohra and Lazreg [6], Benchohra et al. [7], El-Sayed and Bin-Taher [10,11], Guezane-Lakoud and Khaldi [12], Guezane-Lakoud and Bensebaa [13], Houas and Benbachir [14], Nieto et al. [20], and Yan et al. [27]). Motivated by the above papers, in this paper, we establish various existence and uniqueness results of solutions for a boundary value problem of nonlinear fractional differential equations of order α ∈ (2, 3] with nonlocal and fractional integral boundary conditions given by: ⎧ cD α u(t) = f (t, u(t), cD β u(t)), t ∈ J := [0, 1] ⎪ ⎪ 0+ 0+ ⎪ ⎨u(0) = g(u), (1) ⎪ u  (0) = a I0σ+1 u(η1 ), 0 < η1 < 1 ⎪ ⎪ ⎩c β1 D 0+ u(1) = bI0σ+2 u(η2 ), 0 < η2 < 1, C. Derbazi (B) , H. Hammouche Laboratory of Mathematics and Applied Sciences, University of Ghardaia, 47000 Metlili, Algeria E-mail: [email protected] E-mail: [email protected]

123

Arab. J. Math.

where cD κ0+ is the Caputo fractional derivative of order κ ∈ {α, β, β1 }, such that 2 < α ≤ 3, and 0 < β, β1 ≤ 1, and f : [0, 1] × R2 −→ R and g : C([0, 1], R) −→ R are continuous functions, I0σ+i is the Riemann–Liouville fractional integral of order σi > 0, i = 1, 2, η1 , η2 , a, b are suitably chosen real constants, such that: 

  aη1σ1 +1 b