## versión On-line ISSN 0719-0646

### Cubo vol.12 no.3 Temuco  2010

#### http://dx.doi.org/10.4067/S0719-06462010000300003

CUBO A Mathematical Journal Vol.12, N° 03, (35–48). October 2010

Measure of Noncompactness and Nondensely Defined Semilinear Functional Differential Equations with Fractional Order

MOUFFAK BENCHOHRA1, GASTON M. N’GUÉRÉKATA AND DJAMILA SEBA

Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, B.P. 89, 22000, Sidi Bel-Abbès, Algérie email: benchohra@univ-sba.dz

Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore M.D. 21252, USA email: Gaston.N’Guerekata@morgan.edu

Département de Mathématiques, Université de Boumerdès, Avenue de l’indépendance, 35000 Boumerdès, Algérie email: djam_seba@yahoo.fr

ABSTRACT

This paper is devoted to study the existence of integral solutions for a nondensely defined semilinear functional differential equations involving the Riemann-Liouville derivative in a Banach space. The arguments are based upon Mönch’s fixed point theorem and the technique of measures of noncompactness.

Key words and phrases: Partial differential equations, fractional derivative, fractional integral, fixed point, semigroups, integral solutions, finite delay, measure of noncompactness, fixed point, Banach space.

RESUMEN

Este artículo es dedicado al estudio de existencia de soluciones integrales para ecuaciones diferenciales funcionales semilineales envolviendo la derivada de Riemann-Liouville en un espacio de Banach. Los argumentos se basan en un teorema de punto fijo de Mönch y la técnica de no compacidad.

Math. Subj. Class.: 34G20, 34G25, 26A33, 26A42.

Notas

1Corresponding author

References

[1] AGARWAL, R.P., BENCHOHRA, M. AND HAMANI, S., Boundary value problems for differential inclusions with fractional order, Adv. Stud. Contemp.Math., 12 (2) (2008), 181– 196.

[2] AGARWAL, R.P., MEEHAN, M. AND O’REGAN, D., Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001.

[3] ALVÀREZ, J.C.,Measure of noncompactness and fixed points of nonexpansive condensing mappings in locally convex spaces, Rev. Real. Acad. Cienc. Exact. Fis. Natur. Madrid, 79 (1985), 53–66.

[4] ARENDT, W., Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327–352.

[5] BANAS, K. AND DHAGE, B.C., Global asymptotic stability of solutions of a fractional integral equation, Nonlinear Anal., 69 (2008) 1945–1952.

[6] BANAS, J. AND GOEBEL, K., Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980.         [ Links ]

[7] BANAS, J. AND SADARANGANI, K., On some measures of noncompactness in the space of continuous functions, Nonlinear Anal., 68 (2008), 377–383.

[8] BELMEKKI, M. AND BENCHOHRA, M., Existence results for fractional order semilinear functional differential equations, Proc. A. Razmadze Math. Inst., 146 (2008), 9–20.

[9] BELMEKKI, M., BENCHOHRA, M. AND GORNIEWICZ, L., Semilinear functional differential equations with fractional order and infinite delay, Fixed Point Theory, 9 (2) (2008), 423–439.

[10] BELMEKKI, M., BENCHOHRA, M., GÒRNIEWICZ, L. AND NTOUYAS, S.K., Existence results for semilinear perturbed functional differential inclusions with infinite delay, Nonlinear Anal. Forum, 13 (2) (2008), 135–165.

[11] BENCHOHRA, M., GRAEF, J.R. AND HAMANI, S., Existence results for boundary value problems with nonlinear fractional differential equations, Appl. Anal., 87 (7) (2008), 851–863.

[12] BENCHOHRA, M., HAMANI, S. AND NTOUYAS, S.K., Boundary value problems for differential equations with fractional order, Surv. Math. Appl., 3 (2008), 1–12.

[13] BENCHOHRA, M., HENDERSON, J. AND SEBA, D.,Measure of noncompactness and fractional differential equations in Banach spaces, Commun. Appl. Anal., 12 (4) (2008), 419– 428.

[14] CHANG, Y.-K. AND NIETO, J.J., Some new existence results for fractional differential inclusions with boundary conditions, Math. Comput. Model., 49 (2009), 605–609.

[15] DA PRATO, G. AND SINESTRARI, E., Differential operators with non-dense domains, Ann. Scuola. Norm. Sup. Pisa Sci., 14 (1987), 285–344.

[16] DIETHELM, K. AND FORD, N.J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229–248.

[17] FURATI, K.M. AND TATAR, N.-EDDINE, Behavior of solutions for a weighted Cauchytype fractional differential problem, J. Fract. Calc., 28 (2005), 23–42.

[18] FURATI, K.M. AND TATAR, N.-EDDINE, Power type estimates for a nonlinear fractional differential equation, Nonlinear Anal., 62 (2005), 1025–1036.

[19] GAUL, L., KLEIN, P. AND KEMPFLE, S., Damping description involving fractional operators, Mech. Systems Signal Processing, 5 (1991), 81–88.

[20] GLOCKLE, W.G. AND NONNENMACHER, T.F., A fractional calculus approach of selfsimilar protein dynamics, Biophys. J., 68 (1995), 46–53.

[21] GUO, D., LAKSHMIKANTHAM, V. AND LIU, X., Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers Group, Dordrecht, 1996.         [ Links ]

[22] HALE, J.K. AND LUNEL, S.V., Introduction to Functional -Differential Equations, Springer-Verlag, New York, 1993.         [ Links ]

[23] JARADAT, O.K., AL-OMARI, A. AND MOMANI, S., Existence of mild solution for fractional semilinear initial value problems, Nonlinear Anal., 69 (9) (2008), 3153–3159.

[24] KELLERMANN, H. AND HIEBER, M., Integrated semigroup, J. Funct. Anal., 84 (1989), 160–180.

[25] KILBAS, A.A., SRIVASTAVA, H.M. AND TRUJILLO, J.J., Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006.         [ Links ]

[26] KOLMANOVSKII, V. AND MYSHKIS, A., Introduction to the Theory and Applications of Functional-Differential Equations. Kluwer Academic Publishers, Dordrecht, 1999.         [ Links ]

[27] LAKSHMIKANTHAM, V., LEELA, S. AND VASUNDHARA, J., Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009.         [ Links ]

[28] LAKSHMIKANTHAM, V. AND DEVI, J.V., Theory of fractional differential equations in a Banach space, Eur. J. Pure Appl. Math., 1 (2008), 38–45.

[29] MAINARDI, F., Fractional calculus: Some basic problems in continuum and statistical mechanics, in: “Fractals and Fractional Calculus in Continuum Mechanics” (A. Carpinteri, F. Mainardi Eds.), Springer-Verlag, Wien, 1997, pp. 291–348.

[30] METZLER, F., SCHICK, W., KILIAN, H.G. AND NONNENMACHER, T.F., Relaxation in filled polymers: a fractional calculus approach, J. Chem. Phys., 103 (1995), 7180–7186.

[31] MILLER, K.S. AND ROSS, B., An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.         [ Links ]

[32] MÖNCH, H., Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal., 4 (1980), 985–999.

[33] MOPHOU, G.M., NAKOULIMA, O. AND N’GUÉRÉKATA, G.M., Existence results for some fractional differential equations with nonlocal conditions, Nonlinear Studies, 17(1) (2010), 15–22.

[34] MOPHOU, G.M. AND N’GUÉRÉKATA, G.M., Existence of mild solution for some fractional differential equations with nonlocal conditions, Semigroup Forum, 79(2) (2009), 315–322.

[35] MOPHOU, G.M. AND N’GUÉRÉKATA, G.M., On integral solutions of some fractional differential equations with nondense domain, Nonlinear Analysis, T.M.A., 71(10) (2009), 4668–4675.

[36] MOPHOU, G.M. AND N’GUÉRÉKATA, G.M., Mild solutions for semilinear fractional differential equations, Electron. J. Diff. Equ., Vol. 2009 (2009), No. 21, pp. 1–9.

[37] MÖNCH, H. AND VON HARTEN, G.F., On the Cauchy problem for ordinary differential equations in Banach spaces, Archiv. Math. Basel, 39 (1982), 153–160.

[38] PAZY, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.         [ Links ]

[39] PODLUBNY, I., Fractional Differential Equations, Academic Press, San Diego, 1999.         [ Links ]

[40] SAMKO, S.G., KILBAS, A.A. AND MARICHEV, O.I., Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.         [ Links ]

[41] SZUFLA, S., On the application of measure of noncompactness to existence theorems, Rend. Sem. Mat. Univ. Padova, 75 (1986), 1–14.

[42] WU, J., Theory and Applications of Partial Functional Differential Equations, Springer- Verlag, New York, 1996         [ Links ]

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