Weak solutions for nonlinear fractional differential equations with integral boundary conditions in Banach spaces

• Autorzy: Benchohra M. , Mostefai F.
• Języki publikacji: EN

Abstrakty

EN

The aim of this paper is to investigate a class of boundary value problems for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of weak noncompactness.

Słowa kluczowe

EN

boundary value problem; Caputo fractional derivative; measure of weak noncompactness; Pettis integrals; weak solution

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2012
• Tom: Vol. 32, no. 1
• Strony: 31--40
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Benchohra M.

autor

Mostefai F.

• Universit é de Sidi Bel-Abb è s Laboratoire de Math é matiques BP 89, 22000 Sidi Bel-Abb è s, Alg é rie,

Bibliografia

• [1] R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. 109 (2010), 973–1033.
• [2] J. Banas, K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980.
• [3] J. Banas, K. Sadarangani, On some measures of noncompactness in the space of continuous functions, Nonlinear Anal. 68 (2008), 377–383.
• [4] M. Benchohra, J.R. Graef, S. Hamani, Existence results for boundary value problems with non-linear fractional differential equations, Appl. Anal. 87 (2008) 7, 851–863.
• [5] M. Benchohra, J.R. Graef, F. Mostefai, Weak solutions for nonlinear fractional differential equations on reflexive Banach spaces, Electron. J. Qual. Theory Differ. Equ. 54 (2010), 1–10.
• [6] M. Benchohra, S. Hamani, S.K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl. 3 (2008), 1–12.
• [7] M. Benchohra, S. Hamani, S.K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal., TMA 71 (2009), 2391–2396.
• [8] M. Benchohra, F. Ouaar, Existence results for nonlinear fractional differential equations with integral boundary conditions, Bull. Math. Anal. Appl. 2 (2010) 2, 7–15.
• [9] F.S. De Blasi, On the property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 (1977), 259–262.
• [10] J. Diestel, J.J. Uhl (Jr.), Vector Measures, [in:] Math. Surveys, vol. 15, Amer. Math. Soc., Providence, R.I., 1977.
• [11] D. Guo, V. Lakshmikantham, X. Liu, Nonlinear Integral Equations in Abstract Spaces, Mathematics and Its Applications, Kluwer, Dordrecht, 1996.
• [12] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
• [13] S. Krzyska, I. Kubiaczyk, On bounded pseudo and weak solutions of a nonlinear differential equation in Banach spaces, Demonstratio Math. 32 (1999), 323–330.
• [14] V. Lakshmikantham, S. Leela, J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009.
• [15] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
• [16] H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. 4 (1980), 985–999.
• [17] D. O’Regan, Fixed point theory for weakly sequentially continuous mapping, Math. Comput. Modelling 27 (1998) 5, 1–14.
• [18] D. O’Regan, Weak solutions of ordinary differential equations in Banach spaces, Appl. Math. Lett. 12 (1999), 101–105.
• [19] B.J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277–304.
• [20] H.A.H. Salem, A.M.A. El-Sayed, O.L. Moustafa, A note on the fractional calculus in Banach spaces, Studia Sci. Math. Hungar. 42 (2005) 2, 115–130.
• [21] S. Szufla, On the application of measure of noncompactness to existence theorems, Rend. Semin. Mat. Univ. Padova 75 (1986), 1–14.
• [22] S. Szufla, A. Szukała, Existence theorems for weak solutions of n-th order differential equations in Banach spaces, Funct. Approx. Comment. Math. 26 (1998), 313–319.
• [23] S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional diffrential equations, Electron. J. Differential Equations 36 (2006), 1–12.