Some important results for the conformable fractional stochastic pantograph differential equations in the \(\mathbf{L}^{\mathrm{p}}\) space
Authors
M. I. Liaqat
- Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New MuslimTown, Lahore 54600, Pakistan.
F. Ud Din
- Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New MuslimTown, Lahore 54600, Pakistan.
A. Akgul
- Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon.
- Siirt University, Art and Science Faculty, Department of Mathematics, 56100 Siirt, Turkey.
M. B. Riaz
- IT4Innovations, VSB-Technical University of Ostrava, Ostrava, Czech Republic.
- Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon.
Abstract
Important mathematical topics include existence, uniqueness, continuous dependency, regularity, and the averaging principle. In this research work, we establish these results for the conformable fractional stochastic pantograph differential equations (CFSPDEs) in \(\mathbf{L}^{\mathrm{p}}\) space. The situation of \(\mathrm{p}=2\) is generalized by the obtained findings. First, we establish the existence and uniqueness results by applying the contraction mapping principle under a suitably weighted norm and demonstrating the continuous dependency of solutions on both the initial values and fractional exponent \(\phi\). The second section is devoted to examining the regularity of time. As a result, we find that, for each \(\Phi\in(0,\phi-\frac{1}{2})\), the solution to the considered problem has a \(\Phi\)-Hölder continuous version. Next, we study the averaging principle by using Jensen's, Grönwall-Bellman's, Hölder's, and Burkholder-Davis-Gundy's inequalities. To help with the understanding of the theoretical results, we provide three applied examples at the end.
Share and Cite
ISRP Style
M. I. Liaqat, F. Ud Din, A. Akgul, M. B. Riaz, Some important results for the conformable fractional stochastic pantograph differential equations in the \(\mathbf{L}^{\mathrm{p}}\) space, Journal of Mathematics and Computer Science, 37 (2025), no. 1, 106--131
AMA Style
Liaqat M. I. , Ud Din F., Akgul A., Riaz M. B., Some important results for the conformable fractional stochastic pantograph differential equations in the \(\mathbf{L}^{\mathrm{p}}\) space. J Math Comput SCI-JM. (2025); 37(1):106--131
Chicago/Turabian Style
Liaqat, M. I. , Ud Din, F., Akgul, A., Riaz, M. B.. "Some important results for the conformable fractional stochastic pantograph differential equations in the \(\mathbf{L}^{\mathrm{p}}\) space." Journal of Mathematics and Computer Science, 37, no. 1 (2025): 106--131
Keywords
- Pantograph problem
- existence and uniqueness
- continuous dependency
- regularity
- averaging principle
- conformable fractional derivative
MSC
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