Accessible Requires Authentication Published by De Gruyter April 2, 2021

Existence results for a class of random delay integrodifferential equations

Amadou Diop, Mamadou Abdul Diop and K. Ezzinbi

Abstract

In this paper, we consider a class of random partial integro-differential equations with unbounded delay. Existence of mild solutions are investigated by using a random fixed point theorem with a stochastic domain combined with Schauder’s fixed point theorem and Grimmer’s resolvent operator theory. The results are obtained under Carathéodory conditions. Finally, an example is provided to illustrate our results.

MSC 2010: 45D05; 34G20; 47J35

Communicated by Vyacheslav L. Girko


References

[1] N. U. Ahmed, Semigroup Theory with Applications to Systems and Control, Pitman Res. Notes Math. Ser. 246, Longman Scientific & Technical, Harlow, 1991. Search in Google Scholar

[2] S. Baghli and M. Benchohra, Uniqueness results for partial functional differential equations in Fréchet spaces, Fixed Point Theory 9 (2008), no. 2, 395–406. Search in Google Scholar

[3] S. Baghli and M. Benchohra, Global uniqueness results for partial functional and neutral functional evolution equations with infinite delay, Differential Integral Equations 23 (2010), no. 1–2, 31–50. Search in Google Scholar

[4] A. T. Bharucha-Reid, Random Integral Equations, Math. Sci. Eng. 96, Academic Press, New York, 1972. Search in Google Scholar

[5] W. Desch, R. Grimmer and W. Schappacher, Some considerations for linear integro-differential equations, J. Math. Anal. Appl. 104 (1984), no. 1, 219–234. Search in Google Scholar

[6] B. C. Dhage and S. K. Ntouyas, Existence and attractivity results for nonlinear first order random differential equations, Opuscula Math. 30 (2010), no. 4, 411–429. Search in Google Scholar

[7] R. W. Edsinger, Random Ordinary Differential Equations, ProQuest LLC, Ann Arbor, 1968; Thesis (Ph.D.)–University of California, Berkeley. Search in Google Scholar

[8] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math. 194, Springer, New York, 2000. Search in Google Scholar

[9] H. W. Engl, A general stochastic fixed-point theorem for continuous random operators on stochastic domains, J. Math. Anal. Appl. 66 (1978), no. 1, 220–231. Search in Google Scholar

[10] K. Ezzinbi, G. Degla and P. Ndambomve, Controllability for some partial functional integrodifferential equations with nonlocal conditions in Banach spaces, Discuss. Math. Differ. Incl. Control Optim. 35 (2015), no. 1, 25–46. Search in Google Scholar

[11] A. Granas and J. Dugundji, Fixed Point Theory, Springer Monogr. Math., Springer, New York, 2003. Search in Google Scholar

[12] R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc. 273 (1982), no. 1, 333–349. Search in Google Scholar

[13] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac. 21 (1978), no. 1, 11–41. Search in Google Scholar

[14] Y. Hino, S. Murakami and T. Naito, Functional-Differential Equations with Infinite Delay, Lecture Notes in Math. 1473, Springer, Berlin, 1991. Search in Google Scholar

[15] J. Liang, J. H. Liu and T.-J. Xiao, Nonlocal problems for integrodifferential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 15 (2008), no. 6, 815–824. Search in Google Scholar

[16] C. Lungan and V. Lupulescu, Random dynamical systems on time scales, Electron. J. Differential Equations 2012 (2012), Paper No. 86. Search in Google Scholar

[17] V. Lupulescu and S. K. Ntouyas, Random fractional differential equations, Int. Electron. J. Pure Appl. Math. 4 (2012), no. 2, 119–136. Search in Google Scholar

[18] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts Appl. Math. 57, Springer, New York, 2011. Search in Google Scholar

[19] T. T. Soong, Random Differential Equations in Science and Engineering, Math. Sci. Eng. 103, Academic Press, New York, 1973. Search in Google Scholar

[20] C. P. Tsokos and W. J. Padgett, Random Integral Equations with Applications to Life Sciences and Engineering, Math. Sci. Eng. 108, Academic Press, New York, 1974. Search in Google Scholar

[21] J. Wu, Theory and Applications of Partial Functional-Differential Equations, Appl. Math. Sci. 119, Springer, New York, 1996. Search in Google Scholar

Received: 2020-04-18
Accepted: 2021-01-27
Published Online: 2021-04-02
Published in Print: 2021-06-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston