×

zbMATH — the first resource for mathematics

Nonlinear second-order multivalued boundary value problems. (English) Zbl 1052.34022
The authors study the following nonlinear multivalued boundary value problem \[ \varphi(x'(t))'\in A(x(t))+ F(t,x(t))\quad \text{ a.e. on}\,\,\, t\in[0,T], \]
\[ (\varphi(x'(0)),-\varphi(x'(T)))\in\xi(x(0),x(T)), \] where \(\varphi:{\mathbb R}^{N}\to {\mathbb R}^{N}\) is the function defined by \(\varphi (\xi):= \| \xi\| ^{p-2}\xi\), \(p \geq 2\), \(A:D(A) \subseteq \mathbb{R}^N\to 2^{{\mathbb R}^{N}}\) is a maximal monotone map, \(F:[0,T]\times {\mathbb R}^{N}\to 2^{{\mathbb R}^{N}}\) is a multivalued vector field and \(\xi:{\mathbb R}^{N}\times {\mathbb R}^{N}\to 2^{{\mathbb R}^{N}\times {\mathbb R}^{N}}\) is a maximal monotone map describing the boundary conditions. Using notions and techniques from nonlinear operator theory and multivalued analysis, some existence theorems for both convex and nonconvex problems are obtained under the hypothesis that \(F\) satisfies the well-known Hartman condition on a priori bound. Their framework is general and unifying and incorporates gradient systems, evolutionary variational inequalities, and the classical boundary value problems, namely the Dirichlet, the Neumann and the periodic problems. Finally, they present some special cases for illustrating the generality and unifying character of their results.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34A60 Ordinary differential inclusions
PDF BibTeX Cite
Full Text: DOI
References:
[1] Bader R, A topological fixed point theory for evolution inclusions,Zeitshrift für Anal. Anwend,20 (2001) 3–15 · Zbl 0985.34053
[2] Boccardo L, Drabek P, Giachetti D and Kucera M, Generalizations of Fredholm alternative for nonlinear differential operators,Nonlinear Anal. 10 (1986) 1083–1103 · Zbl 0623.34031
[3] Dang H and Oppenheimer S F, Existence and uniqueness results for some nonlinear boundary value problems,J. Math. Anal. Appl. 198 (1996) 35–48, doi:10.1006/jmaa. 1996.0066 · Zbl 0855.34021
[4] De Blasi F S, Górniewicz L and Pianigiani G, Topological degree and periodic solutions of differential inclusions,Nonlinear Anal. 37 (1999) 217–245 · Zbl 0936.34009
[5] De Blasi F S and Pianigiani G, The Baire category method in existence problem for a class of multivalued equations with nonconvex right hand side,Funkcialaj Ekvacioj 28 (1985) 139–156 · Zbl 0584.34007
[6] De Blasi F S and Pianigiani G, Nonconvex valued differential inclusions in Banach spaces,J. Math. Anal. Appl. 157 (1991) 469–494 · Zbl 0728.34013
[7] De Blasi F S and Pianigiani G, On the density of extremal solutions of differential inclusions,Ann. Polonici Math. LVI (1992) 133–142 · Zbl 0760.34019
[8] De Blasi F S and Pianigiani G, Topological properties of nonconvex differential inclusions,Nonlinear Anal. 20 (1993) 871–894 · Zbl 0774.34010
[9] De Coster C, Pairs of positive solutions for the one-dimensionalp-Laplacian,Nonlinear Anal. 23 (1994) 669–681 · Zbl 0813.34021
[10] Dugundji J and Granas A, Fixed point theory (Warsaw: Polish Scientific Publishers) (1982) · Zbl 0483.47038
[11] Erbe L and Krawcewicz W, Nonlinear boundary value problems for differential inclusionsy F(t, y,ý),Annales Polonici Math. LIV (1991) 195–226 · Zbl 0731.34078
[12] Frigon M, Theoremes d’existence des solutions d’inclusion differentielles, NATO ASI Series, Section C, Vol. 472 (The Netherlands: Kluwer, Dordrecht) (1995) 51–87
[13] Gaines R and Mawhin J, Coincidence degree and nonlinear differential equations,Lecture Notes in Math. 568 (New York: Springer Verlag) (1977) · Zbl 0339.47031
[14] Guo Z, Boundary value problems of a class of quasilinear ordinary differential equations,Diff. Integral Eqns 6 (1993) 705–719 · Zbl 0784.34018
[15] Halidias N and Papageorgiou N S, Existence and relaxation results for nonlinear second order multivalued boundary value problems in \(\mathbb{R}\) N ,J. Diff. Eqns 147 (1998) 123–154, doi:10.1006/jdeq.1998.3439 · Zbl 0912.34020
[16] Hartman P, On Boundary value problems for systems of ordinary nonlinear second order differential equations,Trans. AMS 96 (1960) 493–509 · Zbl 0098.06101
[17] Hartman P, Ordinary differential equations (New York: Wiley) (1964). · Zbl 0125.32102
[18] Hu S, Kandilakis D and Papageorgiou N S, Periodic solutions for nonconvex differential inclusions,Proc. AMS 127 (1999) 89–94 · Zbl 0905.34036
[19] Hu S and Papageorgiou N S, On the existence of periodic solutions for nonconvex-valued differential inclusions in \(\mathbb{R}\) N ,Proc. AMS 123 (1995) 3043–3050 · Zbl 0851.34014
[20] Hu S and Papageorgiou N S, Handbook of multivalued analysis. Volume I: Theory (The Netherlands: Kluwer, Dordrecht) (1997) · Zbl 0887.47001
[21] Hu S and Papageorgiou N S, Handbook of multivalued analysis. Volume II: Applications, (The Netherlands: Kluwer, Dordrecht) (2000) · Zbl 0943.47037
[22] Kandilakis D and Papageorgiou N S, Existence theorem for nonlinear boundary value problems for second order differential inclusions,J. Diff. Eqns 132 (1996) 107–125, doi:10.1006/jdeq.1996.0173. · Zbl 0859.34011
[23] Kandilakis D and Papageorgiou N S, Neumann problem for a class of quasilinear differential equations,Atti. Sem. Mat. Fisico Univ. di Modena 48 (1999) 1–15
[24] Knobloch W, On the existence of periodic solutions for second order vector differential equations,J. Diff. Eqns 9 (1971) 67–85 · Zbl 0211.11801
[25] Marcus M and Mizel V, Absolute continuity on tracks and mappings of Sobolev spaces,Arch. Rational Mech. Anal. 45 (1972) 294–320 · Zbl 0236.46033
[26] Mawhin J, Some boundary value problems for Hartman-type perturbations of the ordinary vectorp-Laplacian,Nonlinear Anal. 40 (2000) 497–503 · Zbl 0959.34014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.