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

Smaïl Djebali; Abdelghani Ouahab

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2010)

- Volume: 30, Issue: 1, page 23-49
- ISSN: 1509-9407

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topSmaïl Djebali, and Abdelghani Ouahab. "Existence results for ϕ-Laplacian Dirichlet BVP of differential inclusions with application to control theory." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 30.1 (2010): 23-49. <http://eudml.org/doc/271135>.

@article{SmaïlDjebali2010,

abstract = {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.},

author = {Smaïl Djebali, Abdelghani Ouahab},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {differential inclusions; boundary value problem; fixed point; compact; convex; nonconvex; decomposable; continuous selection; controllability},

language = {eng},

number = {1},

pages = {23-49},

title = {Existence results for ϕ-Laplacian Dirichlet BVP of differential inclusions with application to control theory},

url = {http://eudml.org/doc/271135},

volume = {30},

year = {2010},

}

TY - JOUR

AU - Smaïl Djebali

AU - Abdelghani Ouahab

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

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2010

VL - 30

IS - 1

SP - 23

EP - 49

AB - 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.

LA - eng

KW - differential inclusions; boundary value problem; fixed point; compact; convex; nonconvex; decomposable; continuous selection; controllability

UR - http://eudml.org/doc/271135

ER -

## References

top- [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. Zbl0538.34007
- [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. Zbl0759.26012
- [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. Zbl1079.34004
- [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. Zbl1059.49001
- [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. Zbl1079.34004
- [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. Zbl1086.34062
- [12] A. Benmezaï, S. Djebali and T. Moussaoui, Positive solutions for ϕ-Laplacian Dirichlet BVPs, Fixed point Theory 8 (2) (2007), 167-186. Zbl1156.34015
- [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. Zbl1168.34010
- [14] S. Bernfeld and V. Lakshmikantham, An Introduction to Nonlinear Boundary Value Problems, Academic Press, New York, 1974. Zbl0286.34018
- [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. Zbl0041.25701
- [16] A. Bressan and G. Colombo, Extensions and selections of maps with decomposable values, Studia Math. 90 (1988), 69-86. Zbl0677.54013
- [17] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Springer-Verlag, Berlin-Heidelberg-New York, 580 1977. Zbl0346.46038
- [18] F.S. De Blasi and J. Myjak, On continuous approximations for multifunctions, Pacific J. Math. 123 (1986), 9-31. Zbl0595.47037
- [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. Zbl0731.34078
- [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. Zbl1112.34008
- [23] H. Frankowska, A priori estimates for operational differential inclusions, J. Diff. Eqns. 84 (1990), 100-128. Zbl0705.34016
- [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. Zbl0728.34017
- [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. Zbl1052.34022
- [27] L. Górniewicz, Topological Fixed Point Theory of Multi-valued Mappings, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht 495 1999. Zbl0937.55001
- [28] J. Henderson, Boundary Value Problems for Functional Differential Equations, World Scientific, Singapore, 1995. Zbl0834.00035
- [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. Zbl0988.34001
- [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. Zbl0859.34011
- [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. Zbl0151.10703
- [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. Zbl1151.34019
- [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 Zbl0981.34013
- [39] J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, AMS Regional Conf. Series in Math. Providence, RI, 40 1979. Zbl0414.34025
- [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. Zbl1172.34014
- [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. Zbl1143.35324
- [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. Zbl1122.34008
- [43] R. Precup, Fixed point theorems for decomposable multi-valued maps ans applications, J. Anal. and Appl. 22 (4) (2003), 843-861. Zbl1057.54032
- [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. Zbl0992.34001
- [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. Zbl1021.34002
- [48] H. Wang, On the number of positive solutions of nonlinear systems, J. Math. Anal. Appl. 281 (2003), 287-306. Zbl1036.34032

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