Existence, uniqueness and continuous dependence results of coupled system of Hilfer fractional stochastic pantograph equations with nonlocal integral conditions
DOI:
https://doi.org/10.24193/subbmath.2025.3.02Keywords:
Fractional coupled system, continuous dependence, Hilfer fractional derivative, pantograph equation, fixed point theory, topological degree methodAbstract
This study explores the existence, uniqueness, and continuous dependence of solutions for coupled system of Hilfer fractional stochastic pantograph equations with nonlocal integral conditions. The existence of solutions is demonstrated using topological degree theory for condensing maps. The uniqueness is established via Banach’s contraction principle. To address continuous dependence, the generalized Gronwall inequality is applied. Additionally, a numerical example is provided to illustrate and confirm the theoretical findings.
Mathematics Subject Classification (2010): 26A33, 34A12, 34K50.
Received 12 January 2025; Accepted 18 April 2025.
References
Abbas, M. I., Continuous dependence solutions for Hilfer fractional differential equations with nonlocal conditions, J. Nonlinear Sci. Appl., 12(2019), 573-581.
Abbas, S., A. I. D., Arifi, N. A., Benchohra, M. O. U. F. F. A. K., Henderson, J. O. H. N. Y., Coupled Hilfer and Hadamard random fractional differential systems with finite delay in generalized Banach spaces, Differ. Equ. Appl., 12(2020), 337-353.
Adil, K., Hajra, K., Comparative analysis and definitions of fractional derivatives, J. Differ. Eqns., (2023).
Ahmadova, A., Mahmudov, N. I., Ulam-Hyers stability of Caputo type fractional stochastic neutral differential equations, Statist. Probab. Lett., 168(2021), 108-949.
Alrebdi, R., Al-Jeaid, H. K., Two different analytical approaches for solving the pantograph delay equation with variable coefficient of exponential order, Axioms, (2024).
Arioui, F. Z., Existence results for a coupled system of fractional stochastic differential equations involving Hilfer derivative, Random Oper. Stoch. Equ., 32(2024), 313-327.
Badawi, H., Abu Arqub, O., Shawagfeh, N., Stochastic integrodifferential models of fractional orders and Leffler nonsingular kernels: Well-posedness theoretical results and Legendre Gauss spectral collocation approximations, Chaos Solitons Fractals, 10(2023), 100091.
Balachandran, K., Kiruthika, S., Trujillo, J. J., Existence of solutions of nonlinear fractional pantograph equations, Acta Math. Sin., 33B(2013), 1-9.
Bhairat, S. P., Existence and continuation of solutions of Hilfer fractional differential equations, J. Math. Model., 7(2019).
Blouhi, T., Moghnafi, M., Ahmad, H., Thounthong, P., Existence of coupled systems for impulsive Hilfer fractional stochastic equations with the measure of noncompactness, Filomat, 37(2023), 531-550.
Boufoussi, B., Hajji, S., Continuous dependence on the coefficients for mean-field fractional stochastic delay evolution equations, J. Opt. Soc. Am., 1(2020).
Deimling, K., Nonlinear Functional Analysis, Courier Corporation, (2010).
Dhawan, K., Vats, R. K., Agarwal, R. P., Qualitative analysis of coupled fractional differential equations involving Hilfer derivative, An. Ştiinţ. Ale Univ. Ovidius Constanţa Seria Mat., 30(2022), 191-217.
Diethelm, K., Ford, N. J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265(2002), 229-248.
El-Sayed, A. M. A., Gaafar, F., El-Gendy, M., Continuous dependence of the solution of random fractional-order differential equation with nonlocal conditions, Fract. Differ. Calcul., 7(2017), 135-149.
El-Sayed, A. M. A., Abd-El-Rahman, R. O., El-Gendy, M., Continuous dependence of the solution of a stochastic differential equation with nonlocal conditions, Malaya J. Math., 4(2016), 488-496.
Ferhata, M., Blouhi, T., Topological method for coupled systems of impulsive neutral functional differential inclusions driven by a fractional Brownian motion and Wiener process, Filomat, 38(2024), 6193-6217.
Fredj, F., Hammouche, H., Salim, A., On random fractional differential coupled systems with Hilfer-Katugampola fractional derivative in Banach spaces, J. Math. Sci., (2024).
Guida, K., Hilal, K., Ibnelazyz, L., Existence results for a class of coupled Hilfer fractional pantograph differential equations with nonlocal integral boundary value conditions, Adv. Math. Phys., 2020(2020), 1-8.
Hilfer, R., Applications of fractional calculus in physics, World Scientific, Singapore, 1999.
Isaia, F., On a nonlinear integral equation without compactness, Acta Math. Univ. Comen., 75(2006), 233-240.
Jin, Y., He, W., Wang, L., Mu, J., Existence of mild solutions to delay diffusion equations with Hilfer fractional derivative, Fractals and Fractional, 8(2024), 367.
Kandouci, A., Existence of mild solution result for fractional neutral stochastic integro-differential equations with nonlocal conditions and infinite delay, Malaya J. Matem., 3(2015), 1-13.
Kumar, A., Mohan, M. T., Well-posedness of a class of stochastic partial differential equations with fully monotone coefficients perturbed by Lévy noise, Anal. Math. Phys., 41(2024), 44.
Levakov, A. A., Vas'kovskii, M. M., Properties of solutions of stochastic differential equations with standard and fractional Brownian motions, Differ. Equ., 52(2016), 972-980.
Luo, D., Zada, A., Shaleena, S., Ahmad, M., Analysis of a coupled system of fractional differential equations with non-separated boundary conditions, Adv. Differ. Equ., 2020(2020), 1-24.
Ma, Y. K., Raja, M. M., Vijayakumar, V., Shukla, A., Albalawi, W., Nisar, K. S., Existence and continuous dependence results for fractional evolution integrodifferential equations of order r ∈ (1, 2), Alexandria Eng. J., 61(2022), 9929-9939.
Mchiri, L., Exponential stability in mean square of neutral stochastic pantograph integro-differential equations, Filomat, 36(2022), 6457-6472.
Nisar, K. S., Efficient results on Hilfer pantograph model with nonlocal integral condition, Alex. Eng. J., 80(2023), 342-347.
Nisar, K. S., Jothimani, K., Ravichandran, C., Optimal and total controllability approach of non-instantaneous Hilfer fractional derivative with integral boundary condition, PLoS One, 19(2024), e0297478.
Podlubny, I., Fractional Differential equation, Academic Press, San Diego, 1999.
Radhakrishnan, B., Sathya, T., Alqudah, M. A., et al., Existence results for nonlinear Hilfer pantograph fractional integro-differential equations, Qual. Theory Dyn. Syst., 2024(2024), 237.
Salamooni, A. Y., Pawar, D. D., Continuous dependence of a solution for fractional order Cauchy-type problem, Partial Differ. Equ. Appl. Math., 4(2021), 100110.
Sugumaran, H., Vivek, D., Elsayed, E., On the study of pantograph differential equations with proportional fractional derivative, Math. Sci. Appl. E-Notes, 11(2023), 97-103.
Sweis, H. A. A., Arqub, O. A., Shawagfeh, N., Hilfer fractional delay differential equations: Existence and uniqueness computational results and pointwise approximation utilizing the Shifted-Legendre Galerkin algorithm, Alexandria Eng. J., (2023).
Xiao, G., Wang, J., O’Regan, D., Existence, uniqueness and continuous dependence of solutions to conformable stochastic differential equations, Chaos Solitons Fractals, 139(2020), 110269.
Xiao, G., Wang, J., O’Regan, D., Existence and stability of solutions to neutral conformable stochastic functional differential equations, Qual. Theory Dyn. Syst., 21(2022), 1-22.
Yang, H., Yang, Z., Wang, P., Han, D., Mean-square stability analysis for nonlinear stochastic pantograph equations by transformation approach, J. Math. Anal. Appl., 479(2019), 977-986.
Zentar, O., Ziane, M., and Khelifa, S., Coupled fractional differential systems with random effects in Banach spaces, Random Oper. Stoch. Equ., 29(2021), 251-263.
Zhang, L., Liu, X., Some existence results of coupled Hilfer fractional differential system and differential inclusion on the circular graph, Qual. Theory Dyn. Syst., 23 (2024), (Suppl 1), 259.
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