Abstract
We give a simple geometrical criterion for the nonexistence of constant-sign solutions for a certain type of third-order two-point boundary value problem in terms of the behavior of nonlinearity in the equation. We also provide examples to illustrate the applicability of our results.
Highlights
IntroductionSome additional assumptions on the function p will be introduced later
We study boundary value problem consisting of the nonlinear third-order differential equation x = −p(t)f (x), (1)
Some additional assumptions on the function p will be introduced later. It is the intent of this paper to provide a simple geometrical criterion for nonexistence of constant-sign solutions to boundary value problem (1), (2)
Summary
Some additional assumptions on the function p will be introduced later It is the intent of this paper to provide a simple geometrical criterion for nonexistence of constant-sign solutions to boundary value problem (1), (2). The nonexistence theorems are formulated there in terms of inequalities of the type we use To prove their results, the authors use so-called Guo–Krasnosel’skii fixed point theorem [3, 6], but in our proofs, we employ comparison methods for the first zero functions. For some equations (for example, linear equations, Emden–Fowler-type equations [9]), it is possible to find analytic expressions for the first zero function, but mostly it is impossible In these cases, we can employ comparison methods.
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