Set-valued fractional order differential equations in the space of summable functions

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

Abstrakty

EN

In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type
$(D^{αₙ} - ∑_{i=1}^{n-1} a_i D^{α_i})x(t) ∈ F(t,x(φ(t)))$,
a.e. on (0,1), $I^{1 - αₙ} x(0) = c$, αₙ ∈ (0,1),
where F(t,·) is lower semicontinuous from ℝ into ℝ and F(·,·) is measurable. The corresponding single-valued problem will be considered first.

Słowa kluczowe

EN

fractional calculus; set-valued problem

Kategorie tematyczne

• 26A33: Fractional derivatives and integrals
• 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae, Differential Inclusions, Control and Optimization
• Rocznik: 2008
• Tom: 28
• Numer: 1
• Strony: 83-93

Daty

wydano

2008

otrzymano

2006-10-28

Twórcy

autor

Hussein A.H. Salem

• Department of Mathematics, Girls College of Education, P.O. Box 1011, Yanbu, Kingdom of Saudi Arabia
• Department of Mathematics, Faculty of Sciences, Alexandria University, Egypt

Bibliografia

• [1] A. Babakhani and V. Daftardar-Gejji, Existence of positive solutions of nonlinear fractional differential equations, J. Math. Anal. Appl. 278 (2003), 434-442.
• [2] Z. Artstein and K. Prikry, Carathéodory selections and the Scorza Dragoni property, J. Math. Anal. Appl. 127 (1987), 540-547.
• [3] M. Bassam, Some existence theorems on D.E. of generalized order, J. fur die Reine und Angewandte Mathematik 218 (1965), 70-78.
• [Bc] A. Bressan and G. Colombo, Extensions and selections of maps with decomposable values, Studia Math. 90 (1988), 69-86.
• [5] M. Cichoń, Multivalued perturbations of m-accretive differential inclusions in a non-separable Banach space, Commentationes Math. 32 (1992), 11-17.
• [6] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, 1991.
• [4] V. Daftardar-Gejji, Positive solutions of a system of non-autonomous fractional differential equations, J. Math. Anal. Appl. 302 (2005), 56-64.
• [7] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985.
• [8] J. Dugundji and A. Granas, Fixed Point Theory, Monografie Mathematyczne, PWN, Warsaw, 1982.
• [9] A.M.A. El-Sayed and A.G. Ibrahim, Multivalued fractional differential equations, Appl. Math. Comput. 68 (1995), 15-25.
• [AAA] A.M.A. El-Sayed and A.G. Ibrahim, Definite integral of fractional order for set-valued functions, J. Frac. Calculus 11 (1996), 15-25.
• [11] A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary order, Nonlinear Anal. 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.
• [14] V.G. Gutev, Selection theorems under an assumption weaker than lower semi-continuity, Topol. Appl. 50 (1993), 129-138.
• [15] S.B. Hadid, Local and global existence theorems on differential equations of non-integer order, J. Frac. Calculus 7 (1995), 101-105.
• [kil] A.A. Kilbas and J.J. Trujillo, Differential equations of fractional order: methods, results and problems, I, Appl. Anal. 78 (2001), 153-192.
• [kst] A.A. Kilbas, H.M. Srivastava and J.J Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam, 2006.
• [16] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
• [17] K. Przesławski and L.E. Rybiński, Michael selection theorem under weak lower semicontinuity assumption, Proc. Amer. Math. Soc. 109 (1990), 537-543.
• [18] D. Repovs and P.V. Semenov, Continuous Selection of Multivalued Mappings, Kluwer Academic Press, 1998.
• [19] S. Samko, A. Kilbas and O. Marichev, Fraction Integrals and Drivatives, Gordon and Breach Science Publisher, 1993.
• [sz] J.-P. Sun and Y.-H. Zhao, Multiplicity of positive solutions of a class of nonlinear fractional differential equations, J. Computer and Math. Appl. 49 (2005), 73-80.
• [20] C. Swartz, Measure, Integration and Function Spaces, World Scientific, Singapore, 1994.

Identyfikatory

DOI

10.7151/dmdico.1096