Geodesic
Extremal solutions and strong relaxation for second order multivalued boundary value problems
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 4, pp. 827-844
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In this paper we study semilinear second order differential inclusions involving a multivalued maximal monotone operator. Using notions and techniques from the nonlinear operator theory and from multivalued analysis, we obtain “extremal” solutions and we prove a strong relaxation theorem.
In this paper we study semilinear second order differential inclusions involving a multivalued maximal monotone operator. Using notions and techniques from the nonlinear operator theory and from multivalued analysis, we obtain “extremal” solutions and we prove a strong relaxation theorem.
Classification : 34A60, 34B15
Keywords: maximal monotone operator; pseudomonotone operator; Hartman condition; convex and nonconvex problems; extremal solutions; strong relaxation 
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Gasiński, Leszek; Papageorgiou, Nikolaos S. Extremal solutions and strong relaxation for second order multivalued boundary value problems. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 4, pp. 827-844. http://geodesic.mathdoc.fr/item/CMJ_2005_55_4_a1/

[1] L. Boccardo, P. Drábek, D. Giachetti, and M. Kučera: Generalizations of Fredholm alternative for nonlinear differential operators. Nonlin. Anal. 10 (1986), 1083–1103. | DOI | MR

[2] H. Dang, S. F. Oppenheimer: Existence and uniqueness results for some nonlinear boundary value problems. J.  Math. Anal. Appl. 198 (1996), 35–48. | DOI | MR

[3] F. S.  De Blasi, L. Górniewicz, and G. Pianigiani: Topological degree and periodic solutions of differential inclusions. Nonlin. Anal. 37 (1999), 217–245. | DOI | MR

[4] F. S. De Blasi, G. Pianigiani: The Baire category method in existence problem for a class of multivalued equations with nonconvex right hand side. Funkcialaj Ekvacioj 28 (1985), 139–156. | MR

[5] F. S. De Blasi, G. Pianigiani: Nonconvex valued differential inclusions in Banach spaces. J.  Math. Anal. Appl. 157 (1991), 469–494. | DOI | MR

[6] F. S. De Blasi, G. Pianigiani: On the density of extremal solutions of differential inclusions. Annales Polon. Math.  LVI (1992), 133–142. | MR

[7] F. S. De Blasi, G. Pianigiani: Topological properties of nonconvex differential inclusions. Nonlin. Anal. 20 (1993), 871–894. | DOI | MR

[8] C. De Coster: Pairs of positive solutions for the one-dimensional $p$-Laplacian. Nonlin. Anal. 23 (1994), 669–681. | DOI | MR | Zbl

[9] L. Erbe, W. Krawcewicz: Nonlinear boundary value problems for differential inclusions $y^{\prime \prime }\in F(t,y,y^{\prime })$. Annales Polon. Math.  LIV (1991), 195–226. | MR

[10] M. Frigon: Theoremes d’existence des solutions d’inclusion differentielles. In: NATO ASI Series, Section C,  472, Kluwer, Dordrecht, 1995, pp. 51–87. | MR

[11] R. Gaines, J. Mawhin: Coincidence Degree and Nonlinear Differential Equations. Lecture Notes in Math.  568. Springer-Verlag, New York, 1977. | MR

[12] L. Gasiński, N. S.  Papageorgiou: Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems. Chapman and Hall/CRC Press, Boca Raton, 2005. | MR

[13] Z. Guo: Boundary value problems of a class of quasilinear ordinary differential equations. Diff. Integ. Eqns. 6 (1993), 705–719. | Zbl

[14] N.  Halidias, N. S. Papageorgiou: Existence and relaxation results for nonlinear second order multivalued boundary value problems in  $\mathbb{R}^N$. J.  Differ. Equations 147 (1998), 123–154. | DOI | MR

[15] P. Hartman: On boundary value problems for systems of ordinary nonlinear second order differential equations. Trans. AMS 96 (1960), 493–509. | DOI | MR | Zbl

[16] P. Hartman: Ordinary Differential Equations. Willey, New York, 1964. | MR | Zbl

[17] S. Hu, D. Kandilakis, N. S. Papageorgiou: Periodic solutions for nonconvex differential inclusions. Proc. AMS 127 (1999), 89–94. | MR

[18] S. Hu, N. S. Papageorgiou: On the existence of periodic solutions for nonconvex-valued differential inclusions in $\mathbb{R}^N$. Proc. AMS 123 (1995), 3043–3050. | MR

[19] S. Hu, N. S. Papageorgiou: Handbook of Multivalued Analysis. Volume  I: Theory. Kluwer, Dordrecht, 1997. | MR

[20] S. Hu, N. S. Papageorgiou: Handbook of Multivalued Analysis. Volume  II: Applications. Kluwer, Dordrecht, 2000. | MR

[21] D. Kandilakis, N. S. Papageorgiou: Existence theorem for nonlinear boundary value problems for second order differential inclusions. J.  Differ. Equations 132 (1996), 107–125. | DOI | MR

[22] D. Kandilakis, N. S. Papageorgiou: Neumann problem for a class of quasilinear differential equations. Atti. Sem. Mat. Fisico Univ. di Modena 48 (2000), 163–177. | MR

[23] H. W. Knobloch: On the existence of periodic solutions for second order vector differential equations. J.  Differ. Equations 9 (1971), 67–85. | MR | Zbl

[24] M. Marcus, V. Mizel: Absolute continuity on tracks and mappings of Sobolev spaces. Arch. Rational Mech. Anal. 45 (1972), 294–320. | DOI | MR

[25] J. Mawhin: Some boundary value problems for Hartman-type perturbations of the ordinary vector $p$-Laplacian. Nonlin. Anal. 40 (2000), 497–503. | DOI | MR | Zbl