Fractional Langevin Equations with Nonlocal Integral Boundary Conditions
Abstract
:1. Introduction
2. Preliminaries and Relevant Lemmas
- 1.
- 2.
- 1.
- 2.
- is compact and continuous on Ω
- 3.
- is a contraction mapping on Ω.
- (i)
- has a fixed point in , or
- (ii)
- there is (the boundary of U in C) and such that .
3. Main Results
- ()
- The function is a jointly continuous.
- ()
- The function f satisfies
- ()
- There exists a nonnegative function such that
- ()
- There exist two nonnegative functions such that
4. An Example
5. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
References
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Salem, A.; Alzahrani, F.; Almaghamsi, L. Fractional Langevin Equations with Nonlocal Integral Boundary Conditions. Mathematics 2019, 7, 402. https://doi.org/10.3390/math7050402
Salem A, Alzahrani F, Almaghamsi L. Fractional Langevin Equations with Nonlocal Integral Boundary Conditions. Mathematics. 2019; 7(5):402. https://doi.org/10.3390/math7050402
Chicago/Turabian StyleSalem, Ahmed, Faris Alzahrani, and Lamya Almaghamsi. 2019. "Fractional Langevin Equations with Nonlocal Integral Boundary Conditions" Mathematics 7, no. 5: 402. https://doi.org/10.3390/math7050402
APA StyleSalem, A., Alzahrani, F., & Almaghamsi, L. (2019). Fractional Langevin Equations with Nonlocal Integral Boundary Conditions. Mathematics, 7(5), 402. https://doi.org/10.3390/math7050402