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Controllability of Second Order Semilinear Random Differential Equations in Fréchet Spaces

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Abstract

This paper deals with the study of some existence and controllability results for a class of second-order functional differential equations with random effects in Fréchet spaces. The technique used to show the existence of random mild solutions is a generalization of the original Darbo fixed point theorem for Fréchet spaces combined with the notion of measure of noncompactness, and we prove that our problems are controllable.

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References

  1. Abbas, S., Benchohra, M.: Advanced Functional Evolution Equations and Inclusions. Springer, Cham (2015)

    Book  MATH  Google Scholar 

  2. Agarwal, R.P., Meechan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  3. Arara, A., Benchohra, M., Gorniewicz, L., Ouahab, A.: Controllability results for semilinear functional differential inclusions with unbounded delay. Math. Bull. 3, 157–183 (2006)

    MATH  Google Scholar 

  4. Banas, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Marcel Dekker, New York (1980)

    MATH  Google Scholar 

  5. Balachandran, K., Dauer, J.P.: Controllability of nonlinear systems in Banach spaces: a survey. Dedicated to Professor Wolfram Stadler. J. Optim. Theory Appl. 115, 7–28 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Balachandran, K., Anthoni, S.M.: Controllability of second-order semilinear neutral functional differential systems in Banach spaces. Comput. Math. Appl. 41, 1223–1235 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benchohra, M., Gatsori, E.P., Gorniewicz, L., Ntouyas, S.K.: Controllability results for evolution inclusions with non-local conditions. Z. Anal. Anwend. 22(2), 411–431 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Benchohra, M., Gorniewicz, L., Ntouyas, S.K.: Controllability of neutral functional differential and integrodifferential inclusions in Banach spaces with nonlocal conditions. Nonlinear Anal. Forum 7, 39–54 (2002)

    MathSciNet  MATH  Google Scholar 

  9. Benchohra, M., Gorniewicz, L., Ntouyas, S.K.: Controllability results for multivalued semilinear differential equations with nonlocal conditions. Dyn. Syst. Appl. 11, 403–414 (2002)

    MathSciNet  MATH  Google Scholar 

  10. Benchohra, M., Ntouyas, S.K.: Controllability results for multivalued semilinear neutral functional equations. Math. Sci. Res. J. 6, 65–77 (2002)

    MathSciNet  MATH  Google Scholar 

  11. Bharucha-Reid, A.T.: Random Integral Equations. Academic Press, New York (1972)

    MATH  Google Scholar 

  12. Bothe, D.: Multivalued perturbation of m-accretive differential inclusions. Isr. J. Math. 108, 109–138 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dudek, S.: Fixed point theorems in Fréchet algebras and Fréchet spaces and applications to nonlinear integral equations. Appl. Anal. Discrete Math. 11, 340–357 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Engl, H.W.: A general stochastic fixed-point theorem for continuous random operators on stochastic domains. Anal. Appl. 66, 220–231 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fattorini, H.O.: Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies, vol. 108. North-Holland, Amsterdam (1985)

    Google Scholar 

  16. Fu, X.: Controllability of neutral functional differential systems in abstract space. Appl. Math. Comput. 141, 281–296 (2003)

    MathSciNet  MATH  Google Scholar 

  17. Fu, X., Ezzinbi, K.: Existence of solutions for neutral functional differential evolution equations with nonlocal conditions. Nonlinear Anal. 54, 215–227 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Henríquez, H.R., Hernández, E.M.: Approximate controllability of second-order distributed implicit functional systems. Nonlinear Anal. 70, 1023–1039 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Itoh, S.: Random fixed point theorems with an application to random differential equations in Banach space. Anal. Appl. 67, 261–273 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kwun, Y.C., Park, J.Y., Ryu, J.W.: Approximate controllability and controllability for delay Volterra system. Bull. Korean Math. Soc. 28(2), 131–145 (1991)

    MathSciNet  MATH  Google Scholar 

  21. Mönch, H.: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4, 985–999 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  22. Naito, K.: On controllability for a nonlinear Volterra equation. Nonlinear Anal. 18(1), 99–108 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nakagiri, S., Yamamoto, R.: Controllability and observability for linear retarded systems in Banach space. Int. J. Control 49(5), 1489–1504 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  24. Travis, C.C., Webb, G.F.: Compactness, regularity, and uniform continuity properties of strongly continuous cosine families. Houst. J. Math. 3, 555–567 (1977)

    MathSciNet  MATH  Google Scholar 

  25. Travis, C.C., Webb, G.F.: Cosine families and abstract nonlinear second order differential equations. Acta Math. Acad. Sci. Hung. 32, 75–96 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tsokos, C.P., Padgett, W.J.: Random Integral Equations with Applications to Life Sciences and Engineering. Academic Press, New York (1974)

    MATH  Google Scholar 

  27. Tunç, C.: Stability to vector Liénard equation with constant deviating argument. Nonlinear Dynam. 73(3), 1245–1251 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Graef, John R.; Tunç, C.: Continuability and boundedness of multi-delay functional integro-differential equations of the second order. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109(1), 169–173 (2015)

  29. Tunç, C., Tunç, O.: On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order. J. Adv. Res. 7(1), 165–168 (2016)

    Article  Google Scholar 

  30. Tunç, C., Tunç, O.: A note on the stability and boundedness of solutions to non-linear differential systems of second order. J. Assoc. Arab Univ. Basic Appl. Sci. 24, 169–175 (2017)

  31. Tunç, O., TunçC.: On the asymptotic stability of solutions of stochastic differential delay equations of second order. J. Taibah Univ. Sci. 13(1), 875–882 (2019)

  32. Tunç, C., Golmankhaneh, A.K.: On stability of a class of second alpha-order fractal differential equations. AIMS Math. 5(3), 2126–2142 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P.O. Box 89, 22000, Sidi Bel-Abbès, Algeria

    Abdelkrim Salim, Fatima Mesri & Mouffak Benchohra

  2. Faculty of Technology, Hassiba Benbouali University, P.O. Box 151, 02000, Chlef, Algeria

    Abdelkrim Salim

  3. Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, Campus, 65080, Van, Turkey

    Cemil Tunç

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  1. Abdelkrim Salim
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  2. Fatima Mesri
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  3. Mouffak Benchohra
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  4. Cemil Tunç
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Correspondence to Cemil Tunç.

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Salim, A., Mesri, F., Benchohra, M. et al. Controllability of Second Order Semilinear Random Differential Equations in Fréchet Spaces. Mediterr. J. Math. 20, 84 (2023). https://doi.org/10.1007/s00009-023-02299-0

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  • DOI: https://doi.org/10.1007/s00009-023-02299-0

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