New Results on Controllability for a Class of Fractional Integrodifferential Dynamical Systems with Delay in Banach Spaces †
Abstract
:1. Introduction
2. Preliminaries
- (I)
- is strongly continuous on and ;
- (II)
- , for all and every ;
- (III)
- (1)
- Let be bounded sets of E and Then
- (i)
- if, and only if, is relatively compact;
- (ii)
- if ;
- (iii)
- ;
- (iv)
- ;
- (v)
- ;
- (2)
- Let be bounded. Then, is bounded in E and
- (3)
- Let be bounded and equicontinuous. Then, is continuous on V, and
- (4)
- Let be countable. If there exists such that then is integrable on V, and
3. Main Results
- Step I. From (H3), we have
- Step II. is equicontinuous.
- Step III. is continuous on .
- Step IV. Mönch’s condition holds.
4. An Example
5. Conclusions and Future Work
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Zhao, D. New Results on Controllability for a Class of Fractional Integrodifferential Dynamical Systems with Delay in Banach Spaces. Fractal Fract. 2021, 5, 89. https://doi.org/10.3390/fractalfract5030089
Zhao D. New Results on Controllability for a Class of Fractional Integrodifferential Dynamical Systems with Delay in Banach Spaces. Fractal and Fractional. 2021; 5(3):89. https://doi.org/10.3390/fractalfract5030089
Chicago/Turabian StyleZhao, Daliang. 2021. "New Results on Controllability for a Class of Fractional Integrodifferential Dynamical Systems with Delay in Banach Spaces" Fractal and Fractional 5, no. 3: 89. https://doi.org/10.3390/fractalfract5030089
APA StyleZhao, D. (2021). New Results on Controllability for a Class of Fractional Integrodifferential Dynamical Systems with Delay in Banach Spaces. Fractal and Fractional, 5(3), 89. https://doi.org/10.3390/fractalfract5030089