Solvability conditions and monotone iterative scheme for boundary-value problems related to nonlinear monotone potential operators

Abstract

This article deals with boundary-value problems (BVPs) for the second-order nonlinear differential equations with monotone potential operators of type Au := −∇(k(|∇u|²)∇u(x)) + q(u ²)u(x), x ∈ Ω ⊂ R ⁿ . An analysis of nonlinear problems shows that the potential of the operator A as well as the potential of related BVP plays an important role not only for solvability of these problems and linearization of the nonlinear operator, but also for the strong convergence of solutions of corresponding linearized problems. A monotone iterative scheme for the considered BVP is proposed.

Keywords:

• nonlinear monotone potential operators
• solvability conditions
• linearization
• convergence

AMS Subject Classifications::

• 35J65
• 47H05
• 47H50

Acknowledgements

The research has been supported by Izmir University of Economics. The author thanks an anonymous referee for valuable suggestions which improved the manuscript.