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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.

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.

Positive solutions of a fourth-order differential equation with integral boundary conditions

Seshadev PadhiJohn R. Graef — 2023

Mathematica Bohemica

We study the existence of positive solutions to the fourth-order two-point boundary value problem u ' ' ' ' ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ' ( 0 ) = u ' ( 1 ) = u ' ' ( 0 ) = 0 , u ( 0 ) = α [ u ] , where α [ u ] = 0 1 u ( t ) d A ( t ) is a Riemann-Stieltjes integral with A 0 being a nondecreasing function of bounded variation and f 𝒞 ( [ 0 , 1 ] × + , + ) . The sufficient conditions obtained are new and easy to apply. Their approach is based on Krasnoselskii’s fixed point theorem and the Avery-Peterson fixed point theorem.

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