Generalized Liouville–Caputo Fractional Differential Equations and Inclusions with Nonlocal Generalized Fractional Integral and Multipoint Boundary Conditions
Abstract
:1. Introduction
2. Preliminaries
- 1.
- if
- 2.
- If
3. Main Results for the Problem (1)
- For a function and a nondecreasing function such that
- there exists a positive constant such that
- and
4. Existence Results for the Problem (2)
4.1. The Carathéodory Case
- is -Carathéodory, where ;
- there exists a continuous nondecreasing function and a function such that
- there exists a constant such that
4.2. The Lipschitz Case
- is such that is measurable for each , where ;
- for almost all and with and for almost all where
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Alsaedi, A.; Alghanmi, M.; Ahmad, B.; Ntouyas, S.K. Generalized Liouville–Caputo Fractional Differential Equations and Inclusions with Nonlocal Generalized Fractional Integral and Multipoint Boundary Conditions. Symmetry 2018, 10, 667. https://doi.org/10.3390/sym10120667
Alsaedi A, Alghanmi M, Ahmad B, Ntouyas SK. Generalized Liouville–Caputo Fractional Differential Equations and Inclusions with Nonlocal Generalized Fractional Integral and Multipoint Boundary Conditions. Symmetry. 2018; 10(12):667. https://doi.org/10.3390/sym10120667
Chicago/Turabian StyleAlsaedi, Ahmed, Madeaha Alghanmi, Bashir Ahmad, and Sotiris K. Ntouyas. 2018. "Generalized Liouville–Caputo Fractional Differential Equations and Inclusions with Nonlocal Generalized Fractional Integral and Multipoint Boundary Conditions" Symmetry 10, no. 12: 667. https://doi.org/10.3390/sym10120667