Abstract
In this paper, we consider the following second-order nonlinear differential equations’ problem: a.e on Φ=[0, T] with a discontinuous perturbation and multivalued boundary conditions. By combining lower and upper solutions method, theory of monotone operators and theory of topological degree, we show the existence of solutions of the investigated problem in two cases. At first, α andβ are assumed respectively an ordered pair of lower and upper solutions of the problem, secondly α and β are assumed respectively non ordered pair of lower and upper solutions of the problem. Moreover, we show multiplicity results when the problem admits a pair of lower and strict lower solutions and a pair of upper and strict upper solutions. We also show that our method of proof stays true for a periodic problem.
Highlights
IntroductionThe tool of investigation for this problem is lower and upper solution’s method
By combining lower and upper solutions method, theory of monotone operators and theory of topological degree, we show the existence of solutions of the investigated problem in two cases
Other authors have investigated the second-order differential equation with multivalued boundary conditions driven by maximal monotone operators, in this direction, see [5] [6] [7] [8] [9] and references therein
Summary
The tool of investigation for this problem is lower and upper solution’s method This method provides a precious tool to get existence results for first and second order initial and boundary value problems. Later Nagumo [2] used upper and lower solutions to study second-order differential equations with Dirichlet boundary conditions. In 2017, combining lower and upper solutions method and theory of topological degree, Goli-Adjé [3] established existence and multiplicity results for the considered problem under Neumann-Steklov boundary value conditions. In 2013, Khattabi-Frigon-Ayyadi [4], by lower and upper solutions method and the fixed point index theory, obtained existence and multiplicity results for the problem under various boundary value conditions (Dirichlet, periodic or Neumann).
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