Nonlocal and multiple-point fractional boundary value problem in the frame of a generalized Hilfer derivative

• Main Authors: Wasfi Shatanawi, Abdellatif Boutiara, Mohammed S. Abdo, Mdi B. Jeelani, Kamaleldin Abodayeh
• Format: Article
• Language: English
• Published: SpringerOpen 2021-06-01
• Series: Advances in Difference Equations
• Subjects: Nonlocal fractional differential equations; Generalized Hilfer fractional derivative; Existence and Ulam stability; Fixed point theorem
• Online Access: https://doi.org/10.1186/s13662-021-03450-5

Abstract The aim of this manuscript is to handle the nonlocal boundary value problem for a specific kind of nonlinear fractional differential equations involving a ξ-Hilfer derivative. The used fractional operator is generated by the kernel of the kind k ( ϑ , s ) = ξ ( ϑ ) − ξ ( s ) $k(\vartheta,s)=\xi (\vartheta )-\xi (s)$ and the operator of differentiation D ξ = ( 1 ξ ′ ( ϑ ) d d ϑ ) ${ D}_{\xi } = ( \frac{1}{\xi ^{\prime }(\vartheta )}\frac{d}{d\vartheta } ) $ . The existence and uniqueness of solutions are established for the considered system. Our perspective relies on the properties of the generalized Hilfer derivative and the implementation of Krasnoselskii’s fixed point approach and Banach’s contraction principle with respect to the Bielecki norm to obtain the uniqueness of solution on a bounded domain in a Banach space. Besides, we discuss the Ulam–Hyers stability criteria for the main fractional system. Finally, some examples are given to illustrate the viability of the main theories.