Existence results for ϕ-Laplacian Dirichlet BVP of differential inclusions with application to control theory

Djebali, Smaïl ; Ouahab, Abdelghani

Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 30 (2010) no. 1, pp. 23-49.

Résumé

In this paper, we study ϕ-Laplacian problems for differential inclusions with Dirichlet boundary conditions. We prove the existence of solutions under both convexity and nonconvexity conditions on the multi-valued right-hand side. The nonlinearity satisfies either a Nagumo-type growth condition or an integrably boundedness one. The proofs rely on the Bonhnenblust-Karlin fixed point theorem and the Bressan-Colombo selection theorem respectively. Two applications to a problem from control theory are provided.

Détail

Keywords: differential inclusions, boundary value problem, fixed point, compact, convex, nonconvex, decomposable, continuous selection, controllability

Bibliographie

[1] R.P. Agarwal, H. Lü and D. O'Regan, Eigenvalues and the One-Dimensional p-Laplacian, J. Math. Anal. Appl. 266 (2002), 383-400. doi:10.1006/jmaa.2001.7742

[2] J. Appell, E. De Pascal, N.H. Thái and P.P. Zabreiko, Multi-valued superpositions, Dissertationaes Mathematicae 345 1995.

[3] J.P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin-Heidelberg, New York, 1984.

[4] J.P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990.

[5] S.A. Aysagaliev, K.O. Onaybar and T.G. Mazakov, The controllability of nonlinear systems, Izv. Akad. Nauk. Kazakh-SSR.-Ser. Fiz-Mat. 1 (1985), 307-314.

[6] D. Bainov and P. Simeonov, Integral Inequalities and Applications, Mathematics and its Applications, Vol. 57, Kluwer Academic Publishers, Dordrecht, 1992.

[7] S. Barnet, Introduction to Mathematical Control Theory, Clarendon Press, Oxford, 1975.

[8] M. Benchohra and S.K. Ntouyas, Multi-point boundary value problems for lower semicontinuous differential inclusions, Miskolc Math. Notes 3 (2) (2005), 19-26.

[9] M. Benchohra, S.K. Ntouyas and L. Górniewicz, Controllability of Some Nonlinear Systems in Banach Spaces (The fixed point theory approch), Plock University Press, 2003.

[10] M. Benchohra, S.K. Ntouyas and A. Ouahab, A note on a nonlinear m-point boundary value problem for p-Laplacian differential inclusions, Miskolc Math. Notes 6 (1) (2005), 19-26.

[11] M. Benchohra and A. Ouahab, Controllability results for functional semilinear differential inclusions in Fréchet Spaces, Nonlin. Anal., T.M.A. 61 (2005), 405-423.

[12] A. Benmezaï, S. Djebali and T. Moussaoui, Positive solutions for ϕ-Laplacian Dirichlet BVPs, Fixed point Theory 8 (2) (2007), 167-186.

[13] A. Benmezaï, S. Djebali and T. Moussaoui, Existence Results for One-dimensional Dirichlet ϕ-Laplacian BVPs: a fixed point approach, Dyn. Syst. and Appli. 17 (2008), 149-166.

[14] S. Bernfeld and V. Lakshmikantham, An Introduction to Nonlinear Boundary Value Problems, Academic Press, New York, 1974.

[15] H.F. Bohnenblust and S. Karlin, On a theorem of Ville, in: Contribution to the theory of Games, Ann. of Math. Stud. (1950) 155-160.

[16] A. Bressan and G. Colombo, Extensions and selections of maps with decomposable values, Studia Math. 90 (1988), 69-86.

[17] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Springer-Verlag, Berlin-Heidelberg-New York, 580 1977.

[18] F.S. De Blasi and J. Myjak, On continuous approximations for multifunctions, Pacific J. Math. 123 (1986), 9-31.

[19] K. Deimling, Multi-valued Differential Equations, De Gruyter, Berlin-New York, 1992.

[20] J. Dugundji and A. Granas, Fixed point Theory, Springer-Verlag, New York, 2003.

[21] L. Erbe and W. Krawcewicz, Nonlinear boundary value problems for differential inclusions y''(t)∈ F(t,y,y'), Ann. Polon. Math. 54 (1991), 195-226.

[22] L. Erbe, R. Ma and C.C. Tisdell, On two point boundary value problems for second order differential inclusions, Dyn. Syst. and Appl. 16 (1) (2006), 79-88.

[23] H. Frankowska, A priori estimates for operational differential inclusions, J. Diff. Eqns. 84 (1990), 100-128.

[24] M. Frigon, Application de la Théorie de la Transversalité Topologique à des Problèmes non Linéaires pour des Équations Différentielles Ordinaires, Dissertationes Mathematicae Warszawa, Vol. CCXCVI, 1990.

[25] M. Frigon, Théorèmes d'existence de solutions d'inclusions différentielles, Topological Methods in Differential Equations and Inclusions (edited by A. Granas and M. Frigon), 51-87, NATO ASI Series C, Kluwer Acad. Publ., Dordrecht, 472 1995.

[26] L. Gasinski and N.S. Papageorgiou, Nonlinear second order multi-valued boundary value problems, Proc. Indian Acad. Sci. Math. Sci. 113 (2003), 293-319.

[27] L. Górniewicz, Topological Fixed Point Theory of Multi-valued Mappings, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht 495 1999.

[28] J. Henderson, Boundary Value Problems for Functional Differential Equations, World Scientific, Singapore, 1995.

[29] Sh. Hu and N.S. Papageorgiou, Handbook of Multi-valued Analysis, Volume I: Theory, Kluwer, Dordrecht, The Netherlands, 1997.

[30] Sh. Hu and N.S. Papageorgiou, Handbook of Multi-valued Analysis, Volume II: Applications, Kluwer, Dordrecht, The Netherlands, 2000.

[31] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multi-valued Maps and Semilinear Differential Inclusions in Banach Spaces, Walter de Gruyter Co. Berlin, 2001.

[32] D. Kandilakis and N.S. Papageorgiou, Existence theorem for nonlinear boundary value problems for second order differential inclusions, J. Diff. Eqns 132 (1996), 107-125.

[33] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, 1991.

[34] V.I. Korobov, Reduction of a controllability problem to a boundary value problem, Different. Uranen. 12 (1976), 1310-1312.

[35] N.N. Krasovsky, Theory of Motion Control, Linear Systems, Nauka, Moscow, 1973.

[36] A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781-786.

[37] H. Lian and W. Ge, Positive solutions for a four-point boundary value problem with the p-Laplacian, Nonlin. Anal., T.M.A. 68 (11) (2008), 3493-3503.

[38] H. Lü and C. Zhong, A Note on singular nonlinear boundary value problem for the one-Dimensional p-Laplacian, Appl. Math. Lett. 14 (2001), 189-194. doi:10.1016/S0893-9659(00)00134-8

[39] J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, AMS Regional Conf. Series in Math. Providence, RI, 40 1979.

[40] E.H. Papageorgiou and N.S. Papageorgiou, Nonlinear boundary value problems involving the p-Laplacian and p-Laplacian-like operators, J. for Anal. and its Appl. 24 (4) (2005), 691-707.

[41] N.S. Papageorgiou, S.R.A. Santos and V. Staicu, Three nontrivial solutions for the p-Laplacian with a nonsmooth potential, Nonlin. Anal., T.M.A. 68 (12) (2008), 3812-3827.

[42] N.S. Papageorgiou and V. Staicu, The method of upper-lower solutions for nonlinear second order differential inclusions, Nonlin. Anal., T.M.A. 67 (2007), 708-726.

[43] R. Precup, Fixed point theorems for decomposable multi-valued maps ans applications, J. Anal. and Appl. 22 (4) (2003), 843-861.

[44] I. Rachunkova and M. Tvrdy, Periodic problems with ϕ-Laplacian involving non-ordered lower and upper solutions, Fixed Point Theory 6 (2005), 99-112.

[45] G.V. Smirnov, Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics 41, American Mathematical Society, Providence, 2002.

[46] N. Thihoai and N. Van Loi, Positive solutions and continuous branches for boundary-value problems of diffrential inclusions, Elec. J. Diff. Eqns. 98 (2007), 1-8.

[47] A.A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer, Dordrecht, The Netherlands, 2000.

[48] H. Wang, On the number of positive solutions of nonlinear systems, J. Math. Anal. Appl. 281 (2003), 287-306.