Semilinear fractional order integro-differential equations with infinite delay in Banach spaces

Aissani, Khalida   ; Benchohra, Mouffak

Archivum mathematicum, Tome 49 (2013) no. 2, pp. 105-117

Abstract (VO)

This paper concerns the existence of mild solutions for fractional order integro-differential equations with infinite delay. Our analysis is based on the technique of Kuratowski’s measure of noncompactness and Mönch’s fixed point theorem. An example to illustrate the applications of main results is given.

Détail

DOI : 10.5817/AM2013-2-105

Classification : 26A33, 34G20, 34K30, 34K37
Keywords: semilinear differential equations; Caputo fractional derivative; mild solution; measure of noncompactness; fixed point; semigroup; Banach space

Bibliographie

[1] Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations. Springer, New York, 2012. | MR | Zbl

[2] Agarwal, R. P., Belmekki, M., Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Differential Equations 2009 (2009), 1–47, Article ID 981728. | MR | Zbl

[3] 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 (2010), 973–1033. | DOI | MR | Zbl

[4] Agarwal, R. P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, 2001. | DOI | MR | Zbl

[5] Appell, J. M., Kalitvin, A. S., Zabrejko, P. P.: Partial Integral Operators and Integrodifferential Equations. vol. 230, Marcel Dekker, Inc., New York, 2000. | MR

[6] Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J. J.: Fractional Calculus Models and Numerical Methods. World Scientific Publishing, New York, 2012. | MR | Zbl

[7] Banaś, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 1980. | MR

[8] Benchohra, M., Henderson, J., Ntouyas, S., Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338 (2008), 1340–1350. | DOI | MR | Zbl

[9] Bothe, D.: Multivalued perturbations of $m$–accretive differential inclusions. Israel J. Math. 108 (1998), 109–138. | DOI | MR | Zbl

[10] Case, K. M., Zweifel, P. F.: Linear Transport Theory. Addison-Wesley, Reading, MA, 1967. | MR | Zbl

[11] Chandrasekhe, S.: Radiative Transfer. Dover Publications, New York, 1960. | MR

[12] Corduneanu, C., Lakshmikantham, V.: Equations with unbounded delay. Nonlinear Anal. 4 (1980), 831–877. | DOI | MR | Zbl

[13] Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin, 2010. | MR | Zbl

[14] El–Borai, M.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos, Solitons & Fractals 14 (2002), 433–440. | DOI | MR | Zbl

[15] Hale, J. K., Kato, J.: Phase space for retarded equations with infinite delay. Funkcial. Ekvac. 21 (1) (1978), 11–41. | MR | Zbl

[16] Hale, J. K., Lunel, S. Verduyn: Introduction to Functional–Differential Equations. Applied Mathematical Sciences, vol. 99, Springer-Verlag, New York, 1993. | MR

[17] Heinz, H.–P.: On the behaviour of measures of noncompactness with respect to differentiation and integration of vector–valued functions. Nonlinear Anal. 7 (12) (1983), 1351–1371. | DOI | MR | Zbl

[18] Hilfer, R.: Applications of fractional calculus in physics. Singapore, World Scientific, 2000. | MR | Zbl

[19] Hino, Y., Murakami, S., Naito, T.: Functional Differential Equations with Infinite Delay. Lecture Notes in Mathematics, vol. 1473, Springer-Verlag, Berlin, 1991. | MR | Zbl

[20] Kilbas, A. A., Srivastava, Hari M., Trujillo, Juan J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam, 2006. | MR | Zbl

[21] Kolmanovskii, V., Myshkis, A.: Introduction to the Theory and Applications of Functional–Differential Equations. Kluwer Academic Publishers, Dordrecht, 1999. | MR | Zbl

[22] Lakshmikantham, V., Wen, L., Zhang, B.: Theory of Differential Equations with Unbounded Delay. Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1994. | MR | Zbl

[23] Li, F., Zhang, J.: Existence of mild solutions to fractional integrodifferential equations of neutral type with infinite delay. Adv. Differential Equations 2011 (2011), 1–15, Article ID 963463. | MR | Zbl

[24] Liang, J., Xiao, T.–J., van Casteren, J.: A note on semilinear abstract functional differential and integrodifferential equations with infinite delay. Appl. Math. Lett. 17 (4) (2004), 473–477. | DOI | MR | Zbl

[25] Mainardi, F., Paradisi, P., Gorenflo, R.: Probability distributions generated by fractional diffusion equations. Econophysics: An Emerging Science (Kertesz, J., Kondor, I., eds.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.

[26] Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. John Wiley, New York, 1993. | MR

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

[28] Mophou, G. M., Nakoulima, O., N’Guérékata, G. M.: Existence results for some fractional differential equations with nonlocal conditions. Nonlinear Stud. 17 (2010), 15–22. | MR | Zbl

[29] Mophou, G. M., N’Guérékata, G. M.: Existence of mild solution for some fractional differential equations with nonlocal conditions. Semigroup Forum 79 (2009), 315–322. | DOI | MR

[30] Obukhovskii, V., Yao, J.–C.: Some existence results for fractional functional differential equations. Fixed Point Theory 11 (2010), 85–96. | MR | Zbl

[31] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego, 1999. | MR | Zbl

[32] Samko, S. G., Kilbas, A. A., Marichev, O. I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon, 1993. | MR | Zbl

[33] Tarasov, V. E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Heidelberg; Higher Education Press, Beijing, 2010. | MR

[34] Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer Verlag, New York, 1996. | MR | Zbl