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Global attractivity of the equilibrium of a nonlinear difference equation

John R. GraefC. Qian — 2002

Czechoslovak Mathematical Journal

The authors consider the nonlinear difference equation x n + 1 = α x n + x n - k f ( x n - k ) , n = 0 , 1 , . 1 where α ( 0 , 1 ) , k { 0 , 1 , } and f C 1 [ [ 0 , ) , [ 0 , ) ] ( 0 ) with f ' ( x ) < 0 . They give sufficient conditions for the unique positive equilibrium of (0.1) to be a global attractor of all positive solutions. The results here are somewhat easier to apply than those of other authors. An application to a model of blood cell production is given.

Nonoscillation theorems for forced second order non linear differential equations

John R. GraefPaul W. Spikes — 1975

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Gli Autori provano alcuni nuovi criteri sufficienti, indipendenti da altri criteri da loro ottenuti in precedenza, perché gli integrali dell'equazione ( a ( t ) x ) + q ( t ) f ( x ) g ( x ) = r ( t ) siano tutti non oscillatori.

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

John R. GraefLingju KongQingkai KongBo Yang — 2011

Mathematica Bohemica

The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition u ' ' + g ( t ) f ( t , u ) = 0 , t ( 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 ± , 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 .

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