Multi-Term Fractional Differential Equations with Generalized Integral Boundary Conditions
Abstract
1. Introduction
2. Preliminary Concepts and Integral Solution
3. Existence and Uniqueness Results
3.1. Existence Result via Kransnoselskii’s Fixed Point Theorem
- are continuous functions.
- There exists a nonnegative constant k such that
- There exist continuous nonnegative functions such that
3.2. Existence Result via Leray-Schauder Nonlinear Alternative
- There exist constants and such that
- There exist a continuous nondecreasing function and a function such that
3.3. Uniqueness Result via Banach Fixed Point Theorem
- There exists a nonnegative constant k such that
- there exists such that
4. Examples
5. Conclusions
- Letting and in the boundary conditions of the problem in Equation (1), we get the ones with the boundary conditions:
- If we choose and for the problem in Equation (1), our results lead to the ones for nonlocal multi-point conditions:
- Fixing and in the results of this paper, we obtain the ones supplemented with parametric type anti-periodic fractional boundary conditions: which are indeed new for multi-term Caputo type fractional differential equations.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Ahmad, B.; Alghanmi, M.; Alsaedi, A.; Ntouyas, S.K. Multi-Term Fractional Differential Equations with Generalized Integral Boundary Conditions. Fractal Fract. 2019, 3, 44. https://doi.org/10.3390/fractalfract3030044
Ahmad B, Alghanmi M, Alsaedi A, Ntouyas SK. Multi-Term Fractional Differential Equations with Generalized Integral Boundary Conditions. Fractal and Fractional. 2019; 3(3):44. https://doi.org/10.3390/fractalfract3030044
Chicago/Turabian StyleAhmad, Bashir, Madeaha Alghanmi, Ahmed Alsaedi, and Sotiris K. Ntouyas. 2019. "Multi-Term Fractional Differential Equations with Generalized Integral Boundary Conditions" Fractal and Fractional 3, no. 3: 44. https://doi.org/10.3390/fractalfract3030044
APA StyleAhmad, B., Alghanmi, M., Alsaedi, A., & Ntouyas, S. K. (2019). Multi-Term Fractional Differential Equations with Generalized Integral Boundary Conditions. Fractal and Fractional, 3(3), 44. https://doi.org/10.3390/fractalfract3030044