Nonlocal Boundary Value Problems for Hilfer Generalized Proportional Fractional Differential Equations
Abstract
:1. Introduction
2. Preliminaries
3. An Auxiliary Result
4. Existence and Uniqueness Results in the Scalar Case
- (D1)
- There exists such that:
- (D2)
- is a continuous function such that:
- (D3)
- where is defined by (11).
- (𝔞)
- T admits a fixed–point in ; or
- (𝔞𝔞)
- and with .
- (D4)
- there exist and such that:
- (D5)
- (i)
- If and if , then there exists a constant satisfying .
- (ii)
- If and if then from , there exists a constant K such that:
5. Existence Results in Banach Space
- (1)
- is compact;
- (2)
- ;
- (3)
- (4)
- ;
- (5)
- , ; and
- (6)
- .
- (i)
- is measurable with respect to z for all ;
- (ii)
- is continuous with respect to for .
- (G1)
- The Carathéodory conditions are satisfied by the function ;
- (G2)
- There exist and with φ being nondecreasing such that:
- (G3)
- For each bounded set and for all , we have:
6. Illustrative Examples
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Tariboon, J.; Samadi, A.; Ntouyas, S.K. Nonlocal Boundary Value Problems for Hilfer Generalized Proportional Fractional Differential Equations. Fractal Fract. 2022, 6, 154. https://doi.org/10.3390/fractalfract6030154
Tariboon J, Samadi A, Ntouyas SK. Nonlocal Boundary Value Problems for Hilfer Generalized Proportional Fractional Differential Equations. Fractal and Fractional. 2022; 6(3):154. https://doi.org/10.3390/fractalfract6030154
Chicago/Turabian StyleTariboon, Jessada, Ayub Samadi, and Sotiris K. Ntouyas. 2022. "Nonlocal Boundary Value Problems for Hilfer Generalized Proportional Fractional Differential Equations" Fractal and Fractional 6, no. 3: 154. https://doi.org/10.3390/fractalfract6030154