Monotonic solutions of multi term fractional differential equations

• Autorzy: Salem H.A.H.
• Języki publikacji: EN

Abstrakty

EN

In this paper, we present an existence of monotonic solutions for a nonlinear multi term non-autonomous fractional differential equation in the Banach space of summable functions. The concept of measure of noncompactness and a fixed point theorem due to G. Emmanuelle is the main tool in carring out our proof.

Słowa kluczowe

EN

fractional calculus; fixed point theorem; Cauchy problem

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
• Rocznik: 2007
• Tom: Vol. 47, [Z] 1
• Strony: 1--9
• Opis fizyczny: bibliogr. 23 poz.

Twórcy

autor

Salem H.A.H.

• Department of Mathematics Faculty of Science, Alexandria University, Egypt,

Bibliografia

• [1] J. Appell and E. De Pascale, Su alcuni parametri connesi con la misura di noncompacttezza di Hausdoff in spazzi functioni misurabili. Boll. Un. Matm. Ital. 3B(6) (1984), 497-515.
• [2] T.M. Atanackovic and B. Stankovic, On a system of differential equations with fractional derivatives in rod theory J. Phys. A 37 (2004), 1241-1250.
• [3] A. Babakhani and V. Daftardar-Gejji, Existence of positive solutions of nonlinear fractional differential equations, J. Math. Anal. Appl. 278 (2003), 434-442.
• [4] J. Banas, On the superposition operator and integrable solutions of some functional equations, Nonlinear Anal. Vol. 12 (1988), 777-784.
• [5] J. Banas and K. Goebel, Measure of noncompactness in Banach spaces, Lecture Notes in Pure and Appl. Math., Vol. 60, Dekker, New York 1980.
• [6] M. Cichon, A.M.A. El-Sayed and H.A.H. Salem, Existence theorem for nonlinear functional integral equations of fractional orders, Commentationes Math. 41 (2001), 59-67.
• [7] N. Dunford and J.T. Schwartz Linear Operators Part I. Interscience Publ. John Wiley, New York 1964.
• [8] L. Debnath, Recents applications of fractional calculus to scinence and engineering, Int. J. Math. Appl. Sci. 54 (2003), 3413-3442.
• [9] F.S. De Blasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roum. (N. S.) 21 (1977), 259-262.
• [10] A.M.A. El-Sayed, Linear differential equations of fractional order, Appl. Math. Comput. Vol. 55 (1993), 1-12.
• [11] A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary order, Nonlinear Anal. Vol. 33(2) (1998), 181-186.
• [12] A.M.A. El-Sayed and A.G. Ibrahim, Set-valued integral equations of arbitrary (fractional) order, Appl. Math. Comput. 118 (2001), 113-121.
• [13] G. Emmanuele, Measure of weak noncompactness and fixed point theorem, Bull. Math. Soc. Sci. Math. R.S. Roum. 25 (1981), 253-258.
• [14] S.B. Hadid, Local and global existence theorems on differential equations of non-integer order, Journal of Fractional Calculus 7 (1995), 101-105.
• [15] N.K. Karapetyants, A. Kilbas and M. Saig, On the solution of nonlinear Volterra convolution equation with power nonlinearity, J. Int. Equat. Appl., 84 (1996), 429-445.
• [16] A.A. Kilbas and J.J. Trujillo, Differential equations of fractional order: methods, results problems, I Appl. Anal. 78(1-2) (2001), 153-192.
• [17] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York 1993.
• [18] Z. Luszczki and B. Rzepecki, An existence theorem for ordinary differential equations of order _ 2 (0, 1], Demonste. Math. 20(3-4) (1987),471-475.
• [19] I. Podlubny, Fractional Differential Equations. Acad. Press, San Diego-New York-London 1999.
• [20] Yu. Rossikhin and M.V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev. 50(1) (1997), 15-67.
• [21] S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publisher 1993.
• [22] B.J. West, M. Bologna and P. Grigolini (Eds.) Physics of Fractal Operators, Springer Verlag, New York 2003.
• [23] K. Wiener, Solutions of systems of differential equations with derivatives of noninteger order, Wiss. Z. Martin-Luther university Hall-Wittenberg Math. Natur. Reihe, 38 (5) (1989) 13-19.