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Existence Results for Hybrid Stochastic Differential Equations Involving $\psi$-Hilfer Fractional Derivative

Year 2022, Volume: 14 Issue: 1, 138 - 144, 30.06.2022
https://doi.org/10.47000/tjmcs.987414

Abstract

In this paper, we discuss the existence of solutions for hybrid stochastic differential equations (HSDEs) with the $\psi$-Hilfer fractional derivative. The main tool used in our study is associated with the technique of fixed point theorems due to Dhage.

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References

  • Abbas, S., Benchohra, M., Sivasundaram, S., Dynamics and Ulam stability for Hilfer type fractional differential equations, Nonlinear Stud., 4(2016), 627-637.
  • Abbas, S., Existence of solutions to fractional order ordinary and delay differential equations and applications, Electron. J. Differential Equations, 9(2011), 1-11.
  • Ahmad, B., Ntouyas, S.K., An existence theorem for fractional hybrid differential inclusions of Hadamard type with Dirichlet boundary conditions, Abstr. Appl. Anal., (2014), Article ID 705809, 1-7.
  • Ahmad,. B., Ntouyas, S.K., Initial value problems for hybrid Hadamard fractional differential equations, Electron. J. Differential Equations, 161(2014), 1-8.
  • Ahmed, H.M., Semilinear neutral fractional stochastic integro-differential equations with non local conditions, J. Theor.Probab., 26(4)(2013).
  • Ahmed, H.M., On some fractional stochastic integro-differential equations in Hilbert spaces, Int. J. Math.Math.Sci., 2009(2009), 568-678.
  • Baghani, O., On fractional Langevin equation involving two fractional orders, Commun. Nonlinear. Sci. Numer. Simul., 42(2017), 675-681.
  • El-Borai, M.M., El-Nadi, K, Labib, O., Ahmed, H.M., Semi groups and some fractional stochastic integral equations, Int. J. Pure Appl.Math.Sci., 3(1)(2006), 47-52.
  • Dhage, B.C., On a fixed point theorem in Banach algebras with applications, Appl. Math. Lett., 18(3)(2005), 273-280.
  • Dhage, B.C., Lakshmikantham, V. , Basic results on hybrid differential equations, Nonlinear Anal. Hybrid Syst., 4(3)(2010), 414-424.
  • El-Borai, M.M., El-sayed, W.G., Badr, A.A., Tarek, S.A., Initial value problem for stochastic hybrid Hadamard fractional differential equation, J. Adv. Mat., 16(2019), 1-8.
  • Furati. K.M, Kassim, M.D., Tatar, N.E., Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64(6)(2012), 1616-1626.
  • Furati, K.M., Kassim, M.D., Tatar, N.E., Non-existence of global solutions for a differential equation involving Hilfer fractional derivative, Electron. J. Differential Equations, 235(2013), 1-10.
  • Gu, H., Trujillo, J.J., Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 257(2015), 344-354.
  • Hilfer, R., Application of fractional Calculus in Physics, World Scientific, Singapore, 1999.
  • Hilfer, R., Luchko, Y., Tomovski, Z., Operational method for the solution of fractional differential equations with generalized Riemann-Lioville fractional derivative, Fract. Calc. Appl. Anal., 12(2009), 229-318.
  • Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, in: Mathematics Studies, Vol. 204, Elsevier, 2006.
  • Podlubny, I., Fractional Differential Equations, New York, Academic Press, 1999.
  • Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, 1987, Engl. Trans.from the Russian.
  • Sun, S., Zhao, Y., Han, Z., Li, Y., The existence of solutions for boundary value problem of fractional hybrid differential equations, Commun. Nonlinear Sci. Numer. Simul., 17(2012), 4961-4967.
  • Vanterler da C. Sousa, J., Capelas de Oliveira, E., A Gronwall inequality and the Cauchy-type problem by means of $\psi$-Hilfer operator, Differential Equations and Applications, 11(1)(2019), 87--106.
  • Vanterler da C. Sousa, J., Capelas de Oliveira, E., On the $\psi$-Hilferfractional derivative, Communications in Nonlinear Science and Numerical Simulation, 60(2018), 72-91.
  • Vanterler da C. Sousa, J., Capelas de Oliveira, E., Leibniz type rule: $\psi$-Hilfer fractional operator, Communications in Nonlinear Science and Numerical Simulation, (2019), 1-19.
  • Vivek, D., Kanagarajan, K., Sivasundaram, S., Dynamics and stability of pantograph equations via Hilfer fractional derivative, Nonlinear Stud., 23(4)(2016), 685-698.
  • Vivek, D., Kanagarajan, K., Elsayed, E.M., Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions, Mediterranean Journal of Mathematics, 15(15)(2018), 1-21.
  • Vivek, D., Kanagarajan, K., Elsayed, E.M., Nonlocal initial value problems for implicit differential equations with Hilfer-Hadamard fractional derivative, Nonlinear Analysis: Modelling and Control, 23(3)(2018), 341-360.
  • Wang, J.R., Zhang, Y., Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., 266(2015), 850-859.
  • Zhao, Y., Sun, S., Han, Z., Li, Q., Theory of fractional hybrid differential equations, Comput. Math. Appl., 62(3)(2011), 1312-1324.
Year 2022, Volume: 14 Issue: 1, 138 - 144, 30.06.2022
https://doi.org/10.47000/tjmcs.987414

Abstract

Project Number

-

References

  • Abbas, S., Benchohra, M., Sivasundaram, S., Dynamics and Ulam stability for Hilfer type fractional differential equations, Nonlinear Stud., 4(2016), 627-637.
  • Abbas, S., Existence of solutions to fractional order ordinary and delay differential equations and applications, Electron. J. Differential Equations, 9(2011), 1-11.
  • Ahmad, B., Ntouyas, S.K., An existence theorem for fractional hybrid differential inclusions of Hadamard type with Dirichlet boundary conditions, Abstr. Appl. Anal., (2014), Article ID 705809, 1-7.
  • Ahmad,. B., Ntouyas, S.K., Initial value problems for hybrid Hadamard fractional differential equations, Electron. J. Differential Equations, 161(2014), 1-8.
  • Ahmed, H.M., Semilinear neutral fractional stochastic integro-differential equations with non local conditions, J. Theor.Probab., 26(4)(2013).
  • Ahmed, H.M., On some fractional stochastic integro-differential equations in Hilbert spaces, Int. J. Math.Math.Sci., 2009(2009), 568-678.
  • Baghani, O., On fractional Langevin equation involving two fractional orders, Commun. Nonlinear. Sci. Numer. Simul., 42(2017), 675-681.
  • El-Borai, M.M., El-Nadi, K, Labib, O., Ahmed, H.M., Semi groups and some fractional stochastic integral equations, Int. J. Pure Appl.Math.Sci., 3(1)(2006), 47-52.
  • Dhage, B.C., On a fixed point theorem in Banach algebras with applications, Appl. Math. Lett., 18(3)(2005), 273-280.
  • Dhage, B.C., Lakshmikantham, V. , Basic results on hybrid differential equations, Nonlinear Anal. Hybrid Syst., 4(3)(2010), 414-424.
  • El-Borai, M.M., El-sayed, W.G., Badr, A.A., Tarek, S.A., Initial value problem for stochastic hybrid Hadamard fractional differential equation, J. Adv. Mat., 16(2019), 1-8.
  • Furati. K.M, Kassim, M.D., Tatar, N.E., Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64(6)(2012), 1616-1626.
  • Furati, K.M., Kassim, M.D., Tatar, N.E., Non-existence of global solutions for a differential equation involving Hilfer fractional derivative, Electron. J. Differential Equations, 235(2013), 1-10.
  • Gu, H., Trujillo, J.J., Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 257(2015), 344-354.
  • Hilfer, R., Application of fractional Calculus in Physics, World Scientific, Singapore, 1999.
  • Hilfer, R., Luchko, Y., Tomovski, Z., Operational method for the solution of fractional differential equations with generalized Riemann-Lioville fractional derivative, Fract. Calc. Appl. Anal., 12(2009), 229-318.
  • Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, in: Mathematics Studies, Vol. 204, Elsevier, 2006.
  • Podlubny, I., Fractional Differential Equations, New York, Academic Press, 1999.
  • Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, 1987, Engl. Trans.from the Russian.
  • Sun, S., Zhao, Y., Han, Z., Li, Y., The existence of solutions for boundary value problem of fractional hybrid differential equations, Commun. Nonlinear Sci. Numer. Simul., 17(2012), 4961-4967.
  • Vanterler da C. Sousa, J., Capelas de Oliveira, E., A Gronwall inequality and the Cauchy-type problem by means of $\psi$-Hilfer operator, Differential Equations and Applications, 11(1)(2019), 87--106.
  • Vanterler da C. Sousa, J., Capelas de Oliveira, E., On the $\psi$-Hilferfractional derivative, Communications in Nonlinear Science and Numerical Simulation, 60(2018), 72-91.
  • Vanterler da C. Sousa, J., Capelas de Oliveira, E., Leibniz type rule: $\psi$-Hilfer fractional operator, Communications in Nonlinear Science and Numerical Simulation, (2019), 1-19.
  • Vivek, D., Kanagarajan, K., Sivasundaram, S., Dynamics and stability of pantograph equations via Hilfer fractional derivative, Nonlinear Stud., 23(4)(2016), 685-698.
  • Vivek, D., Kanagarajan, K., Elsayed, E.M., Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions, Mediterranean Journal of Mathematics, 15(15)(2018), 1-21.
  • Vivek, D., Kanagarajan, K., Elsayed, E.M., Nonlocal initial value problems for implicit differential equations with Hilfer-Hadamard fractional derivative, Nonlinear Analysis: Modelling and Control, 23(3)(2018), 341-360.
  • Wang, J.R., Zhang, Y., Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., 266(2015), 850-859.
  • Zhao, Y., Sun, S., Han, Z., Li, Q., Theory of fractional hybrid differential equations, Comput. Math. Appl., 62(3)(2011), 1312-1324.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Dvivek Vivek 0000-0003-0951-8060

Elsayed Elsayed 0000-0003-0894-8472

Kangarajan K. 0000-0001-5556-2658

Project Number -
Publication Date June 30, 2022
Published in Issue Year 2022 Volume: 14 Issue: 1

Cite

APA Vivek, D., Elsayed, E., & K., K. (2022). Existence Results for Hybrid Stochastic Differential Equations Involving $\psi$-Hilfer Fractional Derivative. Turkish Journal of Mathematics and Computer Science, 14(1), 138-144. https://doi.org/10.47000/tjmcs.987414
AMA Vivek D, Elsayed E, K. K. Existence Results for Hybrid Stochastic Differential Equations Involving $\psi$-Hilfer Fractional Derivative. TJMCS. June 2022;14(1):138-144. doi:10.47000/tjmcs.987414
Chicago Vivek, Dvivek, Elsayed Elsayed, and Kangarajan K. “Existence Results for Hybrid Stochastic Differential Equations Involving $\psi$-Hilfer Fractional Derivative”. Turkish Journal of Mathematics and Computer Science 14, no. 1 (June 2022): 138-44. https://doi.org/10.47000/tjmcs.987414.
EndNote Vivek D, Elsayed E, K. K (June 1, 2022) Existence Results for Hybrid Stochastic Differential Equations Involving $\psi$-Hilfer Fractional Derivative. Turkish Journal of Mathematics and Computer Science 14 1 138–144.
IEEE D. Vivek, E. Elsayed, and K. K., “Existence Results for Hybrid Stochastic Differential Equations Involving $\psi$-Hilfer Fractional Derivative”, TJMCS, vol. 14, no. 1, pp. 138–144, 2022, doi: 10.47000/tjmcs.987414.
ISNAD Vivek, Dvivek et al. “Existence Results for Hybrid Stochastic Differential Equations Involving $\psi$-Hilfer Fractional Derivative”. Turkish Journal of Mathematics and Computer Science 14/1 (June 2022), 138-144. https://doi.org/10.47000/tjmcs.987414.
JAMA Vivek D, Elsayed E, K. K. Existence Results for Hybrid Stochastic Differential Equations Involving $\psi$-Hilfer Fractional Derivative. TJMCS. 2022;14:138–144.
MLA Vivek, Dvivek et al. “Existence Results for Hybrid Stochastic Differential Equations Involving $\psi$-Hilfer Fractional Derivative”. Turkish Journal of Mathematics and Computer Science, vol. 14, no. 1, 2022, pp. 138-44, doi:10.47000/tjmcs.987414.
Vancouver Vivek D, Elsayed E, K. K. Existence Results for Hybrid Stochastic Differential Equations Involving $\psi$-Hilfer Fractional Derivative. TJMCS. 2022;14(1):138-44.