Impulsive Fractional Differential Inclusions and Almost Periodic Waves
Abstract
:1. Introduction
- The important concept of almost periodic waves is introduced to the class of impulsive fractional-order inclusions;
- The effects of considering impulsive perturbations and fractional-order derivatives on the almost periodic properties are investigated and criteria for existence and uniqueness are established;
- The main results are obtained by applying the theory of operators semigroup, Hausdorff measure of noncompactness, fixed point theorems and techniques based on fractional calculus;
- The efficiency of the obtained results is demonstrated on a fractional impulsive GRN model.
2. Preliminaries
- B1
- The set of sequences is uniformly almost periodic;
- B2
- For any there exists a real number such that if the points and belong to the same interval of continuity of, , and satisfy the inequality , then ;
- B3
- For any , there exists a relatively dense set T, such that if , then for all satisfying the condition .
- A1.
- The set of sequences , is uniformly almost periodic, and there exists such that ;
- A2.
- The function is almost periodic in the sense of Weyl piecewise function;
- A3.
- The sequence is almost periodic.
- (a)
- ;
- (b)
- ;
- (c)
- .
- (a)
- for every , and ;
- (b)
- For every , the following equality holds;
- (c)
- For every , the operator is bounded and
- (d)
- For and , we have
- (i)
- a.e. ;
- (ii)
- a.e. .
3. Main Results
4. Application to GRNs
- (a)
- the function is almost periodic in the sense of Weyl;
- (b)
- The sequence , , is almost periodic; then, conditions A2 and A3 are met.
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Stamov, G.; Stamova, I. Impulsive Fractional Differential Inclusions and Almost Periodic Waves. Mathematics 2021, 9, 1413. https://doi.org/10.3390/math9121413
Stamov G, Stamova I. Impulsive Fractional Differential Inclusions and Almost Periodic Waves. Mathematics. 2021; 9(12):1413. https://doi.org/10.3390/math9121413
Chicago/Turabian StyleStamov, Gani, and Ivanka Stamova. 2021. "Impulsive Fractional Differential Inclusions and Almost Periodic Waves" Mathematics 9, no. 12: 1413. https://doi.org/10.3390/math9121413
APA StyleStamov, G., & Stamova, I. (2021). Impulsive Fractional Differential Inclusions and Almost Periodic Waves. Mathematics, 9(12), 1413. https://doi.org/10.3390/math9121413