Abstract
In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP): c D 0 + q u ( t ) = f ( t , u ( t ) ) , t ∈ [ 0 , T ] , u ( k ) ( 0 ) = ξ k , u ( T ) = ∑ i = 1 m β i R L I 0 + p i u ( η i ) , where n - 1 < q < n , n ≥ 2 , m , n ∈ N , ξ k , β i ∈ R , k = 0 , 1 , ⋯ , n - 2 , i = 1 , 2 , ⋯ , m , and c D 0 + q is the Caputo fractional derivatives, f : [ 0 , T ] × C ( [ 0 , T ] , E ) → E , where E is the Banach space. The space E is chosen as an arbitrary Banach space; it can also be R (with the absolute value) or C ( [ 0 , T ] , R ) with the supremum-norm. R L I 0 + p i is the Riemann–Liouville fractional integral of order p i > 0 , η i ∈ ( 0 , T ) , and ∑ i = 1 m β i η i p i + n - 1 Γ ( n ) Γ ( n + p i ) ≠ T n - 1 . Via the fixed point theorems of Krasnoselskii and Darbo, the authors study the existence of solutions to this problem. An example is included to illustrate the applicability of their results.
Highlights
In this paper, we study and investigate an interesting Caputo fractional derivative q and Riemann–Liouville integral boundary value problem (BVP): c D0+ u(t) = f (t, u(t)), t ∈ [0, T ], u ( k ) (0) = ξ k, u ( T ) =
The fractional differential equations have an important role in numerous fields of study carried out by mathematicians, physicists, and engineers
We present the existence of solutions for Problem (6) via the fixed point theorems of Krasnoselskii and Darbo
Summary
The fractional differential equations have an important role in numerous fields of study carried out by mathematicians, physicists, and engineers. Fractional integro-differential equations involving the Caputo–Fabrizio derivative have been studied by many researchers from different points of view (see, for example, [5,6,7,8], and the references therein). In [19], the authors investigated the existence and uniqueness of solutions of the nonlocal fractional integral condition. Inspired by the above papers in [15,16,17,18,19], the objective of this paper is to derive the existence solution of the fractional differential equations and nonlocal fractional integral conditions:.
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