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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 and 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.

[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.

[3]

E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001.

[4]

T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York Inc., 1977.

[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.

[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.

[15]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27 (1990), 309-321.

[16]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation. II, Osaka J. Math., 27 (1990), 797-804.

[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.

[18]

J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.

[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.

[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.

[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.

[22]

F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoustics, 9 (2001), 1417-1436. doi: 10.1142/S0218396X01000826.

[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.

[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.

[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.

[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.

[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.

[28]

T. I. Seidman, Invariance of the reachable set under nonlinear perturbations, SIAM J. Control Optim., 25 (1987), 1173-1191. doi: 10.1137/0325064.

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.

[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.

[3]

E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001.

[4]

T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York Inc., 1977.

[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.

[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.

[15]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27 (1990), 309-321.

[16]

Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation. II, Osaka J. Math., 27 (1990), 797-804.

[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.

[18]

J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.

[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.

[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.

[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.

[22]

F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoustics, 9 (2001), 1417-1436. doi: 10.1142/S0218396X01000826.

[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.

[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.

[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.

[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.

[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.

[28]

T. I. Seidman, Invariance of the reachable set under nonlinear perturbations, SIAM J. Control Optim., 25 (1987), 1173-1191. doi: 10.1137/0325064.

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