×

zbMATH — the first resource for mathematics

Existence of solutions for quasilinear second order differential inclusions with nonlinear boundary conditions. (English) Zbl 0941.34008
Continuing their previous work [J. Differ. Equations 147, No. 1, 123-154 (1998; Zbl 0912.34020)], the authors prove the existence of a solution \(x(.)\in C^1(T,\mathbb{R}^N), \;\|x'(.)\|^{p-2}x'(.)\in W^{1,q}(T,\mathbb{R}^N)\) to a boundary value problem of the form: \[ (\|x'(t)\|^{p-2}x'(t))'\in F(t,x(t),x'(t)) \quad \text{a.e. }(T=[0,b]), \]
\[ (\|x'(0)\|^{p-2}x'(0),- \|x'(b)\|^{p-2}x'(b))\in \xi (x(0),x(b)), \] defined the convex-valued multifunction \(F(.,.,.)\), by the maximal monotone operator \(\xi (.,.)\) and by \(p\in [2,\infty)\).
The rather technical proofs of the auxiliary results and of the main one rely on the construction of some approximate selectors of Carathéodory type and on the use of the Leray-Schauder fixed point theorem for compact single-valued operators.
The authors show that their general formulation contains as particular cases the Dirichlet, Neumann, periodic solutions and some Sturm-Liouville-type problems.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34B24 Sturm-Liouville theory
34A60 Ordinary differential inclusions
PDF BibTeX Cite
Full Text: DOI
References:
[1] Erbe, L.; Krawcewicz, W., Boundary value problems for differential inclusions y∈ F(t,y,y′), Ann. polon. math., 56, 195, (1990) · Zbl 0731.34078
[2] M. Frigon, Théorèmes d’existence des solutions d’inclusions differentielles, in: Topological Methods in Differential Equations and Inclusions, NATO ASI Series, Section C, Vol. 472, Kluwer Academic, Dordrecht, The Netherlands, 1995, p. 51.
[3] M. Frigon, Application de la theorie de la transversalité topologique a des problèmes non linéaires pour des equations differentielles ordinaires, Dissertationes Math., Vol. 296, Warsaw, 1990.
[4] Frigon, M.; Granas, A., Théorèmes d’existence pour des inclusions differentielles sans convexes, CRAS Paris, t.310, 819, (1990) · Zbl 0731.47048
[5] Halidias, N.; Papageorgiou, N.S., Existence and relaxation results for nonlinear second-order multivalued boundary value problems in \(R\^{}\{N\}\), J. diff. eq., 147, 123, (1998) · Zbl 0912.34020
[6] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964. · Zbl 0125.32102
[7] S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I: Theory, Kluwer, Dordrecht, The Netherlands, 1997. · Zbl 0887.47001
[8] Kravvaritis, D.; Papageorgiou, N.S., Boundary value problems for nonconvex differential inclusions, J. math. anal. appl., 185, 146, (1994) · Zbl 0817.34009
[9] Marano, S., Existence theorems for a multivalued boundary value problem, Bull. austral. math. soc., 45, 249, (1992) · Zbl 0741.34008
[10] T. Pruszko, Some applications of the topological degree theory to multivalued boundary value problems, Dissertationes Math., Vol. 229, Warsaw, 1984. · Zbl 0543.34008
[11] E. Zeidler, Nonlinear Functional Analysis and its Applications II, Springer, New York, 1990. · Zbl 0684.47029
[12] E. Zeidler, Nonlinear Functional Analysis and its Applications I, Springer, New York, 1985. · Zbl 0583.47051
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.