Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems.

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. RL 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. [ABSTRACT FROM AUTHOR]