# zbMATH — the first resource for mathematics

A variational method for multivalued boundary value problems. (English) Zbl 1465.34032
The paper is concerned with a potential system $-\left [\phi (u') \right ]' +\varepsilon \phi (u) = \nabla_u F(t,u) \quad \mbox{ in } (0,T),$ subject to a boundary condition of type $\left ( \phi(u'(0)), -\phi(u'(T))\right ) \in \partial j (u(0), u(T)),$ where $$\phi: \mathbb{R}^N \to \mathbb{R}^N$$ is a monotone homeomorphism with a particular structure – including the vector $$p$$-Laplacian $$\phi_p$$, and $$j:\mathbb{R}^N \times \mathbb{R}^N \to (-\infty, +\infty]$$ is a convex, lower semicontinuous function. Using the approach introduced by G. Moroşanu and the reviewer [J. Math. Anal. Appl. 313, No. 2, 738–753 (2006; Zbl 1105.34008)] for the $$\phi_p$$ case, the authors obtain the existence of solutions when the corresponding energy functional is coercive.
##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 47J30 Variational methods involving nonlinear operators 37C60 Nonautonomous smooth dynamical systems
Zbl 1105.34008
Full Text:
##### References:
 [1] Bader, R.; Papageorgioua, N. S., Quasilinear vector differential equations with maximal monotone terms and nonlinear boundary conditions, Annales Polonici Mathematici, 73, 1, 69-92 (2000) · Zbl 0991.34013 [2] Béhi, D. A.; Adjé, A.; Goli, K. C., Lower and upper solutions method for nonlinear second-order differential equations involving a ϕ-Laplacian operator, African Diaspora Journal of Mathematics, 22, 1, 22-41 (2019) · Zbl 1430.34028 [3] Brezis, H., Analyse Fonctionnelle: Théorie et Applications (1983), Paris, France: Masson, Paris, France · Zbl 0511.46001 [4] Gasinski, L.; Papageorgiou, N. S., Nonlinear second-order multivalued boundary value problems, Proceedings Mathematical Sciences, 113, 3, 293-319 (2003) · Zbl 1052.34022 [5] Jebelean, P., Variational methods for ordinary p-Laplacian systems with potential boundary conditions, Advances in Differential Equations, 13, 3-4, 273-322 (2008) · Zbl 1177.34021 [6] Jebelean, P.; Morosanu, G., Ordinary p-Laplacian systems with nonlinear boundary conditions, Journal of Mathematical Analysis and Applications, 313, 2, 738-753 (2006) · Zbl 1105.34008 [7] Szulkin, A., Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 3, 2, 77-109 (1986) · Zbl 0612.58011 [8] Rockafellar, R. T., Convex Analysis (1972), Princeton, NJ, USA: Princeton University Press, Princeton, NJ, USA · Zbl 0224.49003
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.