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On fractional integro-differential equations with state-dependent delay. (English) Zbl 1228.35262

Summary: We provide sufficient conditions for the existence of mild solutions for a class of fractional integro-differential equations with state-dependent delay. A concrete application in the theory of heat conduction in materials with memory is also given.

MSC:

35R11 Fractional partial differential equations
45K05 Integro-partial differential equations
35Q79 PDEs in connection with classical thermodynamics and heat transfer
26A33 Fractional derivatives and integrals
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[1] Agarwal, R. P.; Belmekki, M.; Benchohra, M., A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. Difference Equ., 47 (2009), Article ID 981728 · Zbl 1182.34103
[2] Agarwal, R. P.; Benchohra, M.; Hamani, S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109, 3, 973-1033 (2010) · Zbl 1198.26004
[3] Agarwal, R. P.; Benchohra, M.; Hamani, S., Boundary value problems for fractional differential equations, Georgian Math. J., 16, 3, 401-411 (2009) · Zbl 1179.26011
[4] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equations, J. Math. Anal. Appl., 204, 609-625 (1996) · Zbl 0881.34005
[5] He, J. H., Some applications of non linear fractional differential equations and their approximations, Bull. Sci. Technol., 15, 2, 86-90 (1999)
[6] He, J. H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Eng., 167, 57-58 (1998) · Zbl 0942.76077
[7] Jaradat, O. K.; Al-Omari, A.; Momani, S., Existence of mild solutions for fractional semilinear initial value problems, Nonlinear Anal., 69, 3153-3159 (2008) · Zbl 1160.34300
[8] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., (Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204 (2006), Elsevier) · Zbl 1092.45003
[9] Adimy, M.; Crauste, F.; Hbid, M. L.; Qesmi, R., Stability and Hopf bifurcation for a cell population model with state-dependent delay, SIAM J. Appl. Math., 70, 5, 1611-1633 (2009/10) · Zbl 1206.34102
[10] Arino, O.; Boushaba, K.; Boussouar, A., A mathematical model of the dynamics of the phytoplankton-nutrient system, Nonlinear Anal. RWA, 1, 69-87 (2000), Spatial heterogeneity in ecological models (Alcalá de Henares, 1998) · Zbl 0984.92032
[11] Arjunan, M. M.; Kavitha, V., Existence results for impulsive neutral functional differential equations with state-dependent delay, Electron J. Qual. Theory Differ. Equ., 26, 1-13 (2009) · Zbl 1183.34121
[12] Bartha, M., Periodic solutions for differential equations with state-dependent delay and positive feedback, Nonlinear Anal. TMA, 53, 6, 839-857 (2003) · Zbl 1028.34062
[13] Cao, Y.; Fan, J.; Gard, T. C., The effects of state-dependent time delay on a stage-structured population growth model, Nonlinear Anal. TMA, 19, 2, 95-105 (1992) · Zbl 0777.92014
[14] Domoshnitsky, A.; Drakhlin, M.; Litsyn, E., On equations with delay depending on solution, Nonlinear Anal. TMA, 49, 5, 689-701 (2002) · Zbl 1012.34066
[15] Dos Santos, J. P.C., On state-dependent delay partial neutral functional integro-differential equations, Appl. Math. Comput., 216, 1637-1644 (2010) · Zbl 1196.45013
[16] Dos Santos, J. P.C., Existence results for a partial neutral integro-differential equation with state-dependent delay, Electron. J. Qual. Theory Differ. Equ., 29, 1-12 (2010) · Zbl 1208.45009
[17] Fengde, C.; Dexian, S.; Jinlin, S., Periodicity in a food-limited population model with toxicants and state dependent delays, J. Math. Anal. Appl., 288, 1, 136-146 (2003) · Zbl 1087.34045
[18] Hartung, F., Linearized stability in periodic functional differential equations with state-dependent delays, J. Comput. Appl. Math., 174, 2, 201-211 (2005) · Zbl 1077.34074
[19] Hartung, F., Parameter estimation by quasilinearization in functional differential equations with state-dependent delays: a numerical study, Nonlinear Anal. TMA, 47, 7, 4557-4566 (2001), Proceedings of the Third World Congress of Nonlinear Analysts, Part 7 (Catania, 2000) · Zbl 1042.34582
[20] Hartung, F.; Herdman, T.; Turi, J., Parameter identification in classes of neutral differential equations with state-dependent delays, Nonlinear Anal. TMA, 39, 3, 305-325 (2000) · Zbl 0955.34067
[21] Hartung, F.; Turi, J., Identification of parameters in delay equations with state-dependent delays, Nonlinear Anal. TMA, 29, 11, 1303-1318 (1997) · Zbl 0894.34071
[22] Hernández, E.; Ladeira, L.; Prokopczyk, A., A note on state dependent partial functional differential equations with unbounded delay, Nonlinear Anal., RWA, 7, 4, 510-519 (2006) · Zbl 1109.34060
[23] Hernández, E.; McKibben, M., On state-dependent delay partial neutral functional differential equations, Appl. Math. Comput., 186, 1, 294-301 (2007) · Zbl 1119.35106
[24] Hernández, E.; McKibben, M.; Henrquez, H., Existence results for partial neutral functional differential equations with state-dependent delay, Math. Comput. Modelling, 49, 1260-1267 (2009) · Zbl 1165.34420
[25] Hernández, E.; Pierri, M.; União, G., Existence results for a impulsive abstract partial differential equation with state-dependent delay, Comput. Math. Appl., 52, 411-420 (2006) · Zbl 1153.35396
[26] Kuang, Y.; Smith, H., Slowly oscillating periodic solutions of autonomous state-dependent delay equations, Nonlinear Anal. TMA, 19, 9, 855-872 (1992) · Zbl 0774.34054
[27] Torrejn, R., Positive almost periodic solutions of a state-dependent delay nonlinear integral equation, Nonlinear Anal. TMA, 20, 12, 1383-1416 (1993) · Zbl 0787.45003
[28] Yongkun, L., Periodic solutions for delay Lotka Volterra competition systems, J. Math. Anal. Appl., 246, 1, 230-244 (2000) · Zbl 0972.34057
[29] Püss, J., (Evolutionary Integral Equations and Applications. Evolutionary Integral Equations and Applications, Monographs Math., vol. 87 (1993), Birkhäuser-Verlag)
[30] Cuevas, C.; de Souza, J. C., Existence of \(S\)-asymptotically \(\omega \)-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear Anal. (2009)
[31] Agarwal, R. P.; de Andrade, B.; Cuevas, C., On type of periodicity and ergodicity to a class of fractional order differential equations, Adv. Difference Equ., 1-25 (2010), Article ID 179750 · Zbl 1194.34007
[32] Cuesta, E., Asymptotically behavior of the solutions of fractional integro-differential equations and some discretizations, Discrete Contin. Dyn. Syst., Supplement, 277-285 (2007) · Zbl 1163.45306
[33] Cuevas, C.; de Souza, J. C., \(S\)-asymptotically \(\omega \)-periodic solutions of semilinear fractional integro-differential equations, Appl. Math. Lett., 22, 865-870 (2009) · Zbl 1176.47035
[34] de Andrade, B.; Cuevas, C., \(S\)-asymptotically \(\omega \)-periodic and asymptotically \(\omega \)-periodic solutions to semilinear Cauchy problems with non dense domain, Nonlinear Anal., 72, 3190-3208 (2010) · Zbl 1205.34074
[35] Hino, Y.; Murakami, S.; Naito, T., Functional-differential equations with infinite delay, (Lecture Notes in Mathematics, vol. 1473 (1991), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0732.34051
[36] Gripenberg, G.; Londen, S. O.; Staffans, O., (Volterra Integral and Functional Equations. Volterra Integral and Functional Equations, Encyclopedia of Mathematics and Applications, vol. 34 (1990), Cambridge University Press: Cambridge University Press Cambridge, New York) · Zbl 0695.45002
[37] Arendt, W.; Batty, C.; Hieber, M.; Neubrander, F., (Vector-Valued Laplace Transforms and Cauchy Problems. Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, vol. 96 (2001), Birkhäuser: Birkhäuser Basel) · Zbl 0978.34001
[38] Fattorini, O., (Second Order Differential Equations in Banach Spaces. Second Order Differential Equations in Banach Spaces, North-Holland Math. Studies, vol. 108 (1985), North-Holland: North-Holland Amsterdam, New York, Oxford) · Zbl 0564.34063
[39] Lizama, C., On approximation and representation of \(k\)-regularized resolvent families, Integral Equations Operator Theory, 41, 2, 223-229 (2001) · Zbl 1011.45006
[40] Lizama, C.; Prado, H., Rates of approximation and ergodic limits of regularized operator families, J. Approx. Theory, 122, 1, 42-61 (2003) · Zbl 1032.47024
[41] Lizama, C.; Sanchez, J., On perturbation of \(k\)-regularized resolvent families, Taiwanese J. Math., 7, 2, 217-227 (2003) · Zbl 1051.45009
[42] Shaw, S.; Chen, J., Asymptotic behavior of \((a, k)\)-regularized families at zero, Taiwanese J. Math., 10, 2, 531-542 (2006) · Zbl 1106.45004
[43] Haase, M., The functional calculus for sectorial operators, (Operator Theory: Advances and Applications, vol. 169 (2006), Birkhauser-Verlag: Birkhauser-Verlag Basel) · Zbl 1101.47010
[44] Granas, A.; Dugundji, J., Fixed Point Theory (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1025.47002
[45] Martin, R., Nonlinear Operators and Differential Equations in Banach Spaces (1987), Robert E. Krieger Publ. Co.: Robert E. Krieger Publ. Co. Florida
[46] Simon, J., Compact sets in the space \(L^p(0, T; B)\), Ann. Mat. Pure Appl., CXLVI, 65-96 (1987) · Zbl 0629.46031
[47] J. Necas, Les méthodes directes en théorie des équations elliptiques, Masson, 1967.; J. Necas, Les méthodes directes en théorie des équations elliptiques, Masson, 1967. · Zbl 1225.35003
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