Upper and Lower Solution Methods for Fully Nonlinear Boundary Value Problems

Abstract

Sufficient conditions are given for the existence of a solution of a fourth order nonlinear boundary value problem with nonlinear boundary conditions. The conditions assume the existence of a strong upper solution-lower solution pair, a concept that is defined in the paper. The differential equation has nonlinear dependence on all lower order derivatives of the unknown; in particular, appropriate Nagumo conditions are obtained and employed. © 2002 Elsevier Science (USA)

Positive Solutions of Higher-Order Boundary-Value Problems

“…[5] discussed the existence of at least three positive symmetric concave solutions for the BVP consisting of Equation (1.1) and BC (1.4); Ehme et al . [10] obtained sufficient conditions for the existence of solutions of fourthorder BVPs based on the existence of a pair of strong lower and upper solutions. BVPs with special nonlinear BCs have also been studied in the literature (see [8][9][10][11][22][23][24].…”

“…[10] obtained sufficient conditions for the existence of solutions of fourthorder BVPs based on the existence of a pair of strong lower and upper solutions. BVPs with special nonlinear BCs have also been studied in the literature (see [8][9][10][11][22][23][24]. We remark that the BVPs in the general form (1.1), (1.2) are important because of their applications to physical, biological and chemical phenomena (see [2,7,21]).…”

Multiple unbounded solutions for a boundary value problem on infinite intervals

“…It is well known that nonlinear problems always have at least one solution in the ordered interval defined by one pair of well-ordered upper and lower solutions. To show this kind of result, we can employ the topological degree theory or monotone iterative technique, etc, see [1][2][3][4][5] and the reference therein.…”

Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions

“…For example, nonlinear BCs are considered in [5,6,33]. Although each of these uses the technique involving upper and lower solutions, their results are not usually comparable because each considers a perturbation of a different linear problem.…”