Abstract
We consider the second-order nonlinear boundary value problems (BVPs) with Sturm–Liouville boundary conditions. We define types of solutions and show that if there exist solutions of different types then there exist intermediate solutions also.
Highlights
We consider boundary value problem x = f (t, x, x ), f ∈ C1 [a, b] × R × R, R, a1x(a) − a2x (a) = A, b1x(b) + b2x (b) = B, (1.1) (1.2)where A, B ∈ R, a1, b1 ∈ R, a2, b2 ∈ R+ := (0, +∞), a21 +a22 > 0 and b21 +b22 > 0
Our study continues a series of papers devoted to two-point boundary value problems for the second order nonlinear differential equations [2, 3, 4]
Habets [1] for more information) and a lot of papers were written devoted to various boundary value problems (BVPs) [8,9,10]
Summary
Our study continues a series of papers devoted to two-point boundary value problems for the second order nonlinear differential equations [2, 3, 4]. Erbe [5], who studied BVPs for equation (1.1) provided that there exist the so called lower and upper functions. Knobloch, who first observed that more can be said about a solution existing in presence of lower and upper functions. He showed for some BVP that there exists a solution with the property (B) which is described later. The objective of this paper is to consider the Sturm–Liouville conditions for nonlinear second-order boundary value problem. The presence of the lower and upper functions α and β guarantees the existence of solutions of the BVP.
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