Investigating Asymptotic Stability for Hybrid Cubic Integral Inclusion with Fractal Feedback Control on the Real Half-Axis
Abstract
:1. Introduction and Background
2. Single Valued Problem
- (i)
- Given continuous functions (for ), with as .
- (ii)
- The functions are continuous, non-decreasing, and
- (iii)
- and are continuous functions, and there exists a continuous function such that
- (iv)
- is a Carathéodory function that is measurable in and continuous in there are two integrable functions such that
- (v)
- Let be a Lipschitz function with a Lipschitz constant such thatand In addition, belongs to the space , and we obtain
- (vi)
- The function , is a Carathéodory function that is measurable in and continuous in and there exist measurable and bounded functions , such that
- (vii)
- The following equation has a positive solution r. A positive solution to what follows the equation is r
3. Multi-Valued Problem
- Let satisfy the following assumptions:
- The set is a nonempty, closed, and convex subset for all .
- The set-valued map is continuous and Lipschitzian set-valued map with a nonempty compact convex subset of with a Lipschitz constant , such that
Existence Theorem
4. General Discussion and Examples
- In the case of the presence of a control variable
- (1*)
- Phanograph functional integral inclusion with feedback controlLetting and and , then we have the Phanograph functional-integral inclusion
- (2*)
- Retarded functional integral inclusion with feedback controlLet and Then, we have the functional retarded integral inclusion with feedback control
- (3*)
- For then we obtain the functional retarded integral inclusion with feedback control
- (4*)
- In the case of the absence of control variable we obtain some particular cases which that useful for the theory of qualitative analysis of some functional integral equations and important for some models and real problems.
- (1)
- (2)
- (3)
4.1. Example 1
4.2. Example 2
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Hashem, H.H.G.; El-Sayed, A.M.A.; Al-Issa, S.M. Investigating Asymptotic Stability for Hybrid Cubic Integral Inclusion with Fractal Feedback Control on the Real Half-Axis. Fractal Fract. 2023, 7, 449. https://doi.org/10.3390/fractalfract7060449
Hashem HHG, El-Sayed AMA, Al-Issa SM. Investigating Asymptotic Stability for Hybrid Cubic Integral Inclusion with Fractal Feedback Control on the Real Half-Axis. Fractal and Fractional. 2023; 7(6):449. https://doi.org/10.3390/fractalfract7060449
Chicago/Turabian StyleHashem, Hind H. G., Ahmed M. A. El-Sayed, and Shorouk M. Al-Issa. 2023. "Investigating Asymptotic Stability for Hybrid Cubic Integral Inclusion with Fractal Feedback Control on the Real Half-Axis" Fractal and Fractional 7, no. 6: 449. https://doi.org/10.3390/fractalfract7060449