Existence Theorems for Mixed Riemann–Liouville and Caputo Fractional Differential Equations and Inclusions with Nonlocal Fractional Integro-Differential Boundary Conditions
Abstract
:1. Introduction
2. Preliminaries
2.1. Fractional Calculus
- (i)
- for
- (ii)
- if then
- (iii)
2.2. Multi-Valued Analysis
- (i)
- is convex (closed) valued if is convex (closed) for all
- (ii)
- is bounded on bounded sets if is bounded in X for all bounded set B of X, i.e.,
- (iii)
- is called upper semi-continuous (u.s.c. for short) on X if for each the set is nonempty, closed subset of X, and for each open set of X containing , there exists an open neighborhood of such that
- (iv)
- is said to be completely continuous if is relatively compact for every bounded subset B of
- (v)
- has a fixed point if there exists such that
2.3. Fixed-Point Theorems
- (i)
- F has a fixed point in , or
- (ii)
- there is a (the boundary of U in C) and with
- (i)
- F has a fixed point in or
- (ii)
- there is a and with
3. Main Results for Single-Valued Problem (5)
- there exist a continuous nondecreasing functions and a function such that
- there exists a constant such that
4. Existence Results for Multi-Valued Problem (6)
4.1. The Upper Semi-Continuous Case
- is -Carathéodory;
- there exists a continuous nondecreasing function and a function such that
- there exists a constant such that
4.2. The Lipschitz Case
- (a)
- Lipschitz if and only if there exists such that
- (b)
- a contraction if and only if it is Lipschitz with .
- (H4)
- is such that is measurable for each
- (H5)
- for almost all and with and for almost all .
5. Examples
5.1. Single-Valued Case
- (i).
- LetPlease note that and thus is satisfied with Since by Theorem 1, the boundary value problem (21), with f given by (22), has a unique solution on .
- (ii).
- With the function f given by (22), we remark that andHence, by Theorem 2, the boundary value problem (21), with f given by (22), has at least one solution on .
- (iii).
- Next considerIt is easy to find that Then by condition with snf we find thatHence, by Theorem 3, the boundary value problem (21), with f given by (23), has at least one solution on .
5.2. Multi-Valued Case
- (I).
- Consider the multi-valued map given byClearly the multi-valued map F satisfies condition and that
- (II).
- Let the multi-valued map be defined by
6. Discussion
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Ntouyas, S.K.; Alsaedi, A.; Ahmad, B. Existence Theorems for Mixed Riemann–Liouville and Caputo Fractional Differential Equations and Inclusions with Nonlocal Fractional Integro-Differential Boundary Conditions. Fractal Fract. 2019, 3, 21. https://doi.org/10.3390/fractalfract3020021
Ntouyas SK, Alsaedi A, Ahmad B. Existence Theorems for Mixed Riemann–Liouville and Caputo Fractional Differential Equations and Inclusions with Nonlocal Fractional Integro-Differential Boundary Conditions. Fractal and Fractional. 2019; 3(2):21. https://doi.org/10.3390/fractalfract3020021
Chicago/Turabian StyleNtouyas, Sotiris K., Ahmed Alsaedi, and Bashir Ahmad. 2019. "Existence Theorems for Mixed Riemann–Liouville and Caputo Fractional Differential Equations and Inclusions with Nonlocal Fractional Integro-Differential Boundary Conditions" Fractal and Fractional 3, no. 2: 21. https://doi.org/10.3390/fractalfract3020021
APA StyleNtouyas, S. K., Alsaedi, A., & Ahmad, B. (2019). Existence Theorems for Mixed Riemann–Liouville and Caputo Fractional Differential Equations and Inclusions with Nonlocal Fractional Integro-Differential Boundary Conditions. Fractal and Fractional, 3(2), 21. https://doi.org/10.3390/fractalfract3020021