Higher order multi-point boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions.

Graef, J.R.; Kong, Lingju; Kong, Qingkai; Wong, James S.W.

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Second order boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions

John R. Graef, Lingju Kong, Qingkai Kong, Bo Yang (2011)

Mathematica Bohemica

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The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition $\begin{matrix} u^{''} + g (t) f (t, u) = 0, t \in (0, 1), \\ u (0) = α u (ξ) + λ, u (1) = β u (η) + μ . C r i t e r i a f o r t h e e x i s t e n c e o f n o n t r i v i a l s o l u t i o n s o f t h e p r o b l e m a r e e s t a b l i s h e d . T h e n o n l i n e a r t e r m f (t, x) m a y t a k e n e g a t i v e v a l u e s a n d m a y b e u n b o u n d e d f r o m b e l o w . C o n d i t i o n s a r e d e t e r m i n e d b y t h e r e l a t i o n s h i p b e t w e e n t h e b e h a v i o r o f f (t, x) / x f o r x n e a r 0 a n d \pm \infty, a n d t h e s m a l l e s t p o s i t i v e c h a r a c t e r i s t i c v a l u e o f a n a s s o c i a t e d l i n e a r i n t e g r a l o p e r a t o r . T h e a n a l y s i s m a i n l y r e l i e s o n t o p o l o g i c a l d e g r e e t h e o r y . T h i s w o r k c o m p l e m e n t s s o m e r e c e n t r e s u l t s i n t h e l i t e r a t u r e . T h e r e s u l t s a r e i l l u s t r a t e d w i t h e x a m p l e s . \end{matrix}$