q-Fractional Langevin Differential Equation with q-Fractional Integral Conditions
Abstract
:1. Introduction
2. Preliminary
- 1.
- Definition 4 ⇒ Definition 5;
- 2.
- Definition 6 ⇒ Definition 7.
- 1.
- 2.
- 1.
- 2.
3. Existence of Solution
- .
- are continuous.
- There ∃ and in such way that , and ∀; thus, we have
- There ∃ and in such way that and ∀; thus, we have
- There ∃ and in such a way that and ∀; we thus have
- There ∃, which is bounded and satisfies
- There ∃, which is bounded and satisfies
- There ∃, which is bounded and satisfies
- There ∃ a non-decreasing and fulfilling
- (1)
- and are continuous on J×J,
- (2)
- and ,
- (2)
- and have the following expression:Now taking , we obtainSimilarly,
- Step 1:
- First, we prove that is continuous. Consider a sequence such that . For , we haveHence, by –, we obtainFor each , the sequence as ; thus, by the Lebesgue dominated convergence theorem,This implies that
- Step 2:
- Next, for each . We need to verify that with some . For , we obtainBy and , we haveThus,Similarly,By , we obtainClearly, h and are constants. Using Lemma 3 and (18), we obtain
- Step 3:
- Take in such way that and assume that is bounded. Then, for ,In Step 2, we obtained thatIt is obvious that the right-hand side of (19) tends to zero as . Therefore, from Step 1 to Step 3, the Arzela–Ascoli theorem is a completely continuous mapping.
- Step 4:
- Consider the setWe need to show that is bounded. Let . Then, for some with , we haveThis impliesBy –, we have
4. Stability
5. Examples
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Wang, W.; Khalid, K.H.; Zada, A.; Ben Moussa, S.; Ye, J. q-Fractional Langevin Differential Equation with q-Fractional Integral Conditions. Mathematics 2023, 11, 2132. https://doi.org/10.3390/math11092132
Wang W, Khalid KH, Zada A, Ben Moussa S, Ye J. q-Fractional Langevin Differential Equation with q-Fractional Integral Conditions. Mathematics. 2023; 11(9):2132. https://doi.org/10.3390/math11092132
Chicago/Turabian StyleWang, Wuyang, Khansa Hina Khalid, Akbar Zada, Sana Ben Moussa, and Jun Ye. 2023. "q-Fractional Langevin Differential Equation with q-Fractional Integral Conditions" Mathematics 11, no. 9: 2132. https://doi.org/10.3390/math11092132
APA StyleWang, W., Khalid, K. H., Zada, A., Ben Moussa, S., & Ye, J. (2023). q-Fractional Langevin Differential Equation with q-Fractional Integral Conditions. Mathematics, 11(9), 2132. https://doi.org/10.3390/math11092132