On Caputo–Riemann–Liouville Type Fractional Integro-Differential Equations with Multi-Point Sub-Strip Boundary Conditions
Abstract
:1. Introduction
2. Preliminaries
3. Existence and Uniqueness Results
- , and for all , .
- , , for all .
- (i)
- Observe that the continuity of the operator follows from that of f and
- (ii)
- is uniformly bounded on as:
- (iii)
- is equicontinuous.
4. Examples
- (i)
- Let , , and It is easy to see that is satisfied with , andUsing the given data, we have andThen, . Thus, by Theorem 1, the boundary value problem (18) has a unique solution on .
- (ii)
- We choose the following functions in problem (18) for illustrating Theorem 2:Here and as , andFurther,Obviously, and . Moreover, we have
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Alsaedi, A.; Albideewi, A.F.; Ntouyas, S.K.; Ahmad, B. On Caputo–Riemann–Liouville Type Fractional Integro-Differential Equations with Multi-Point Sub-Strip Boundary Conditions. Mathematics 2020, 8, 1899. https://doi.org/10.3390/math8111899
Alsaedi A, Albideewi AF, Ntouyas SK, Ahmad B. On Caputo–Riemann–Liouville Type Fractional Integro-Differential Equations with Multi-Point Sub-Strip Boundary Conditions. Mathematics. 2020; 8(11):1899. https://doi.org/10.3390/math8111899
Chicago/Turabian StyleAlsaedi, Ahmed, Amjad F. Albideewi, Sotiris K. Ntouyas, and Bashir Ahmad. 2020. "On Caputo–Riemann–Liouville Type Fractional Integro-Differential Equations with Multi-Point Sub-Strip Boundary Conditions" Mathematics 8, no. 11: 1899. https://doi.org/10.3390/math8111899