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December  2016, 21(10): 3637-3654. doi: 10.3934/dcdsb.2016114

## Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays

 1 Vietnam Education Publishing House, 81 Tran Hung Dao, Hanoi, Vietnam 2 Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam 3 Department of Mathematics, Electric Power University, 235 Hoang Quoc Viet, Hanoi, Vietnam

Received  November 2015 Revised  August 2016 Published  November 2016

We deal with the Cauchy problem associated with integro-differential inclusions of diffusion-wave type involving infinite delays. Based on the behavior of resolvent operator associated with the linear part, an explicit estimate for solutions will be established. As a consequence, the weak stability of zero solution is proved in case the resolvent operator is asymptotically stable.
Citation: Thanh-Anh Nguyen, Dinh-Ke Tran, Nhu-Quan Nguyen. Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3637-3654. doi: 10.3934/dcdsb.2016114
##### References:
 [1] B. de Andrade and C. Cuevas, $S$-asymptotically $\omega$-periodic and asymptotically $\omega$-periodic solutions to semi-linear Cauchy problems with non-dense domain, Nonlinear Anal., 72 (2010), 3190-3208. doi: 10.1016/j.na.2009.12.016.  Google Scholar [2] N. T. Anh and T. D. Ke, Decay integral solutions for neutral fractional differential equations with infinite delays, Math. Methods Appl. Sci., 38 (2015), 1601-1622. doi: 10.1002/mma.3172.  Google Scholar [3] E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001. Google Scholar [4] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.  Google Scholar [5] T. A. Burton and T. Furumochi, Fixed points and problems in stability theory for ordinary and functional differential equations, Dyn. Syst. Appl., 10 (2001), 89-116.  Google Scholar [6] J. P. Carvalho dos Santos and C. Cuevas, Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations, Appl. Math. Lett., 23 (2010), 960-965. doi: 10.1016/j.aml.2010.04.016.  Google Scholar [7] N. M. Chuong, T. D. Ke and N. N. Quan, Stability for a class of fractional partial integro-differential equations, J. Integral Equations Appl., 26 (2014), 145-170. doi: 10.1216/JIE-2014-26-2-145.  Google Scholar [8] C. Cuevas and J. César de Souza, Existence of $S$-asymptotically $\omega$-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear Anal., 72 (2010), 1683-1689. doi: 10.1016/j.na.2009.09.007.  Google Scholar [9] C. Cuevas and J. César de Souza, $S$-asymptotically $\omega$-periodic solutions of semilinear fractional integro-differential equations, Appl. Math. Lett., 22 (2009), 865-870. doi: 10.1016/j.aml.2008.07.013.  Google Scholar [10] E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin. Dyn. Syst., (Supplement) (2007), 277-285.  Google Scholar [11] J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^{1}(\mu, X)$, Proc. Amer. Math. Soc., 118 (1993), 447-453. doi: 10.2307/2160321.  Google Scholar [12] R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York Inc., 1977.  Google Scholar [13] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.  Google Scholar [14] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Translated from the Russian. Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar [15] Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27 (1990), 309-321.  Google Scholar [16] Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation. II, Osaka J. Math., 27 (1990), 797-804.  Google Scholar [17] C. Gori, V. Obukhovskii, M. Ragni and P. Rubbioni, Existence and continuous dependence results for semilinear functional differential inclusions with infinite delay, Nonlinear Anal., 51 (2002), 765-782. doi: 10.1016/S0362-546X(01)00861-6.  Google Scholar [18] J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.  Google Scholar [19] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar [20] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, vol. 1473, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.  Google Scholar [21] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, in: de Gruyter Series in Nonlinear Analysis and Applications, vol. 7, Walter de Gruyter, Berlin, New York, 2001. doi: 10.1515/9783110870893.  Google Scholar [22] F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoustics, 9 (2001), 1417-1436. doi: 10.1142/S0218396X01000826.  Google Scholar [23] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar [24] R. Metzler and J. Klafter, Accelerating Brownian motion: A fractional dynamics approach to fast diffusion, Europhys. Lett., 51 (2000), 492-498. doi: 10.1209/epl/i2000-00364-5.  Google Scholar [25] J.-S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program. Ser. A, 113 (2008), 345-424. doi: 10.1007/s10107-006-0052-x.  Google Scholar [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [27] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar [28] T. I. Seidman, Invariance of the reachable set under nonlinear perturbations, SIAM J. Control Optim., 25 (1987), 1173-1191. doi: 10.1137/0325064.  Google Scholar

show all references

##### References:
 [1] B. de Andrade and C. Cuevas, $S$-asymptotically $\omega$-periodic and asymptotically $\omega$-periodic solutions to semi-linear Cauchy problems with non-dense domain, Nonlinear Anal., 72 (2010), 3190-3208. doi: 10.1016/j.na.2009.12.016.  Google Scholar [2] N. T. Anh and T. D. Ke, Decay integral solutions for neutral fractional differential equations with infinite delays, Math. Methods Appl. Sci., 38 (2015), 1601-1622. doi: 10.1002/mma.3172.  Google Scholar [3] E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001. Google Scholar [4] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.  Google Scholar [5] T. A. Burton and T. Furumochi, Fixed points and problems in stability theory for ordinary and functional differential equations, Dyn. Syst. Appl., 10 (2001), 89-116.  Google Scholar [6] J. P. Carvalho dos Santos and C. Cuevas, Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations, Appl. Math. Lett., 23 (2010), 960-965. doi: 10.1016/j.aml.2010.04.016.  Google Scholar [7] N. M. Chuong, T. D. Ke and N. N. Quan, Stability for a class of fractional partial integro-differential equations, J. Integral Equations Appl., 26 (2014), 145-170. doi: 10.1216/JIE-2014-26-2-145.  Google Scholar [8] C. Cuevas and J. César de Souza, Existence of $S$-asymptotically $\omega$-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear Anal., 72 (2010), 1683-1689. doi: 10.1016/j.na.2009.09.007.  Google Scholar [9] C. Cuevas and J. César de Souza, $S$-asymptotically $\omega$-periodic solutions of semilinear fractional integro-differential equations, Appl. Math. Lett., 22 (2009), 865-870. doi: 10.1016/j.aml.2008.07.013.  Google Scholar [10] E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin. Dyn. Syst., (Supplement) (2007), 277-285.  Google Scholar [11] J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^{1}(\mu, X)$, Proc. Amer. Math. Soc., 118 (1993), 447-453. doi: 10.2307/2160321.  Google Scholar [12] R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York Inc., 1977.  Google Scholar [13] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.  Google Scholar [14] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Translated from the Russian. Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar [15] Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27 (1990), 309-321.  Google Scholar [16] Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation. II, Osaka J. Math., 27 (1990), 797-804.  Google Scholar [17] C. Gori, V. Obukhovskii, M. Ragni and P. Rubbioni, Existence and continuous dependence results for semilinear functional differential inclusions with infinite delay, Nonlinear Anal., 51 (2002), 765-782. doi: 10.1016/S0362-546X(01)00861-6.  Google Scholar [18] J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.  Google Scholar [19] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar [20] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, vol. 1473, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.  Google Scholar [21] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, in: de Gruyter Series in Nonlinear Analysis and Applications, vol. 7, Walter de Gruyter, Berlin, New York, 2001. doi: 10.1515/9783110870893.  Google Scholar [22] F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoustics, 9 (2001), 1417-1436. doi: 10.1142/S0218396X01000826.  Google Scholar [23] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar [24] R. Metzler and J. Klafter, Accelerating Brownian motion: A fractional dynamics approach to fast diffusion, Europhys. Lett., 51 (2000), 492-498. doi: 10.1209/epl/i2000-00364-5.  Google Scholar [25] J.-S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program. Ser. A, 113 (2008), 345-424. doi: 10.1007/s10107-006-0052-x.  Google Scholar [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [27] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar [28] T. I. Seidman, Invariance of the reachable set under nonlinear perturbations, SIAM J. Control Optim., 25 (1987), 1173-1191. doi: 10.1137/0325064.  Google Scholar
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