Fractional evolution equation nonlocal problems with noncompact semigroups

• Autorzy: Zhang X , Chen P.

Identyfikatory

DOI

http://dx.doi.Org/10.7494/OpMath.2016.36.l.123

• Języki publikacji: EN

Abstrakty

EN

This paper is concerned with the existence results of mild solutions to the nonlocal problem of fractional semilinear integro-differential evolution equations. New existence theorems are obtained by means of the fixed point theorem for condensing maps. The results extend and improve some related results in this direction.

Słowa kluczowe

EN

fractional evolution equation; mild solution; nonlocal condition; Co-semigroup; condensing maps; measure of noncompactness

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2016
• Tom: Vol. 36, no. 1
• Strony: 123--137
• Opis fizyczny: Bibliogr. 32 poz.

Twórcy

autor

Zhang X

• Northwest Normal University Department ol Mathematics Lanzhou 730 070, P.R. China Zhixing College ol Northwest Normal University Department ol Mathematics Lanzhou 730 070, P.R. China

autor

Chen P.

• Northwest Normal University Department of Mathematics Lanzhou 730 070, P.R. China

Bibliografia

• [1] R.P. Agarwal, V. Lakshmikantham, J.J. Nieto, On the concept of solutions for fractional differential equations with uncertainly, Nonlinear Anal. 72 (2010), 2859-2862.
• [2] K. Balachandran, J.J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear Anal. 72 (2010), 4587-4593.
• [3] K. Balachandran, S. Kiruthika, J.J. Tmjillo, On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces, Comput. Math. Appl. 62 (2011), 1157-1165.
• [4] J. Banas, K. Goebel, Measure of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math., vol. 60, Marcel Dekker, New York, 1980.
• [5] M. Benchohra, G.M. N'Guerekata, D. Seba, Measure of noncompactness and nondensly definded se.miline.ar functional differential equations with fractional order, Cubo 12 (2010), 35-48.
• [6] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991), 494-505.
• [7] L. Byszewski, Application of properties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems, Nonlinear Anal. 33 (1998), 2413-2426.
• [8] L. Byszewski, V. Lakshmikantham, Theorem about the existence and uniqueness of solutions of a nonlocal Cauchy problem in a Banach space, Appl. Anal. 40 (1990), 11-19.
• [9] P. Chen, Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys. 65 (2014), 711-728.
• [10] P. Chen, Y. Li, Nonlocal problem for fractional evolution equations of mixed type with the measure of noncompactness, Abst. Appl. Anal. 2013, Article ID 784 816, 12.
• [11] P. Chen, Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math. 63 (2013), 731-744.
• [12] P. Chen, Y. Li, Q. Li, Existence of mild solutions for fractional evolution equations with nonlocal initial conditions, Ann. Polon. Math. 110 (2014), 13-24.
• [13] P. Chen, Y. Li, H. Yang, Perturbation method for nonlocal impulsive evolution equations, Nonlinear Anal. Hybrid Syst. 8 (2013), 22-30.
• [14] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985.
• [15] M.M. EI-Borai, Some probability densities and funddamental solutions of fractional evolution equations, Chaos Solitons Fractals 14 (2002), 433-440.
• [16] M.M. EI-Borai, On some stochastic fractional integro-differential equations, Adv. Dyn. Syst. Appl. 1 (2006), 49-57.
• [17] K. Ezzinbi, X. Fu, K. Hilal, Existence and regularity in the a-norrn for some neu­tral partial differential equations with nonlocal conditions, Nonlinear Anal. 67 (2007), 1613-1622.
• [18] L. Fang, G.M. N'Guerekata, An existence result for neutral delay integrodifferentual equations with fractional order nonlocal conditions, Abst. Appl. Anal. 2011, Article ID 952 782, 20 pp.
• [19] H.P. Heinz, On the behaviour of measure of noncompactness with respect to differenti­ation and integration of rector-valued functions, Nonlinear Anal. 7 (1983), 1351-1371.
• [20] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Dif­ferential Equations, [in:] North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.
• [21] F. Li, J. Liang, H.K. Xu, Existence of mild solutions for fractional integrodifjerential equations of Sobolev type with nonlocal conditions, J. Math. Anal. Appl. 391 (2012), 510-525.
• [22] J. Liang, J.H. Liu, T.J. Xiao, Nonlocal impulsive problems for integrodifjerential equa­tions, Math. Comput. Modelling 49 (2009), 798-804.
• [23] B. Lundstrom, M. Higgs, W. Spain, A. Fairhall, Fractional by neocortial pyramidal neurons, Nat Neurosci. 11 (2008), 1335-1342.
• [24] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equa­tions, Springer-Verlag, Berlin, 1983.
• [25] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [26] T. Poinot, J.C. Trigeassou, Identification of fractional systems using an output-error technique, Nonlinear Dynam. 38 (2004), 133-154.
• [27] Y. Rossikhin, M. Shitikova, Application of fractional derivatives to the analysis of damped vibrationsof viscoelaticsingle mass system, Acta Mech. 120 (1997), 109-125.
• [28] N. Tatar, Existence results for an evolution problem with fractional nonlocal conditions, Comput. Math. Appl. 60 (2010), 2971-2982.
• [29] J. Wang, Y. Zhou, W. Wei, H. Xu, Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls, Comput. Math. Appl. 62 (2011), 1427-1441.
• [30] J. Wang, Y. Zhou, A class of fractional evolution equations and optimal controls, Non­linear Anal. Real World Appl. 12 (2011), 263-272.
• [31] R. Wang, T.J. Xiao, J. Liang, A note on the fractional Cauchy problems with nonlocal conditions, Appl. Math. Lett. 24 (2011), 1435-1442.
• [32] Y. Zhou, F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal. Real World Appl. 11 (2010), 4465-4475.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.