On the set of solutions of fractional order Riemann-Liouville integral inclusions

• Autorzy: Abbas S. , Benchohra M.
• Języki publikacji: EN

Abstrakty

EN

In this paper, we prove the arcwise connectedness of the solution set of a nonclosed, nonconvex Fredholm type, Riemann–Liouville integral inclusion of fractional order.

Słowa kluczowe

EN

fractional order; Riemann-Liouville integral; solution set

PL

niecałkowity rząd; pochodno-całka Riemanna-Liouville'a; zestaw rozwiązań

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2013
• Tom: Vol. 46, nr 2
• Strony: 271--281
• Opis fizyczny: Bibliogr. 34 poz.

Twórcy

autor

Abbas S.

• Laboratoire de Mathématiques, Université de Saïda, B.P. 138, 20000, Saïda, Algérie

autor

Benchohra M.

• Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, B.P. 89, 22000, Sidi Bel-Abbès, Algérie

Bibliografia

• [1] S. Abbas, R. P. Agarwal, M. Benchohra, Darboux problem for impulsive partial hyperbolic differential equations of fractional order with variable times and infinite delay, Nonlinear Anal. Hybrid Syst. 4 (2010), 818–829.
• [2] S. Abbas, M. Benchohra, Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative, Commun. Math. Anal. 7 (2009), 62–72.
• [3] S. Abbas, M. Benchohra, The method of upper and lower solutions for partial hyperbolic fractional order differential inclusions with impulses, Discuss. Math. Differ. Incl. 30(1) (2010), 141–161.
• [4] S. Abbas, M. Benchohra, Impulsive partial hyperbolic functional differential equations of fractional order with state-dependent delay, Fract. Calc. Appl. Anal. 13(3) (2010), 225–244.
• [5] S. Abbas, M. Benchohra, L. Górniewicz, Existence theory for impulsive partial hyperbolic functional differential equations involving the Caputo fractional derivative, Sci. Math. Jpn. online e- 2010, 271–282.
• [6] S. Abbas, M. Benchohra, J. J. Nieto, Global uniqueness results for fractional order partial hyperbolic functional differential equations, Adv. Differential Equ. 2011, Art. ID 379876, 25 pp.
• [7] A. Belarbi, M. Benchohra, A. Ouahab, Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces, Appl. Anal. 85 (2006), 1459–1470.
• [8] M. Benchohra, J. R. Graef, S. Hamani, Existence results for boundary value problems of nonlinear fractional differential equations with integral conditions, Appl. Anal. 87(7) (2008), 851–863.
• [9] M. Benchohra, S. Hamani, S. K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl. 3 (2008), 1–12.
• [10] M. Benchohra, J. Henderson, S. K. Ntouyas, A. Ouahab, Existence results for functional differential equations of fractional order, J. Math. Anal. Appl. 338 (2008), 1340–1350.
• [11] F. S. De Blasi, J. Myjak, On the set of solutions of a differential inclusion, Bull. Inst. Math. Acad. Sinica 14 (1986), 271–275.
• [12] F. S. De Blasi, G. Pianigiani, V. Staicu, On the solution sets of some nonconvex hyperbolic differential inclusions, Czechoslovak Math. J. 45 (1995), 107–116.
• [13] A. Cernea, On the set of solutions of some nonconvex nonclosed hyperbolic differential inclusions, Czechoslovak Math. J. 52(127)(1) (2002), 215–224.
• [14] A. Cernea, On the solution set of a nonconvex nonclosed second order differential inclusion, Fixed Point Theory 8(1) (2007), 29–37.
• [15] A. Cernea, Arcwise connectedness of the solution set of a nonclosed nonconvex integral inclusion, Miskolc Math. Notes 9(1) (2008), 33–39.
• [16] K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), 229–248.
• [17] A. Fryszkowski, Fixed Point Theory for Decomposable Sets. Topological Fixed Point Theory and Its Applications, 2. Kluwer Academic Publishers, Dordrecht, 2004.
• [18] W. G. Glockle, T. F. Nonnenmacher, A fractional calculus approach of selfsimilar protein dynamics, Biophys. J. 68 (1995), 46–53.
• [19] L. Górniewicz, T. Pruszko, On the set of solutions of the Darboux problem for some hyperbolic equations, Bull. Acad. Polon. Sci. Math. Astronom. Phys. 38 (1980), 279–285.
• [20] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
• [21] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
• [22] 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.
• [23] A. A. Kilbas, S. A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differential Equations 41 (2005), 84–89.
• [24] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, 1991.
• [25] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in “Fractals and Fractional Calculus in Continuum Mechanics” (A. Carpinteri and F. Mainardi, Eds), pp. 291–348, Springer-Verlag, Wien, 1997.
• [26] S. Marano, V. Staicu, On the set of solutions to a class of nonconvex nonclosed differential inclusions, Acta Math. Hungar. 76 (1997), 287–301.
• [27] F. Metzler, W. Schick, H. G. Kilian, T. F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys. 103 (1995), 7180–7186.
• [28] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
• [29] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, London, 1974.
• [30] I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, 1999.
• [31] I. Podlubny, I. Petraš, B. M. Vinagre, P. O’Leary, L. Dorčak, Analogue realizations of fractional-order controllers, fractional order calculus and its applications, Nonlinear Dynam. 29 (2002), 281–296.
• [32] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
• [33] A. N. Vityuk, A. V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscil. 7(3) (2004), 318–325.
• [34] S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional diffrential equations, Electron. J. Differential Equations 36 (2006), 1–12.