On the Solvability of a Neumann Boundary Value Problem at Resonance

1997 ◽  
Vol 40 (4) ◽  
pp. 464-470 ◽  
Author(s):  
Chung-Cheng Kuo
Keyword(s):  
Boundary Value Problem ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Semilinear Equations ◽  
Neumann Boundary Value Problem ◽  
At Resonance ◽  
Problem At Resonance

AbstractWe study the existence of solutions of the semilinear equations (1) in which the non-linearity g may grow superlinearly in u in one of directions u → ∞ and u → −∞, and (2) −Δu + g(x, u) = h, in which the nonlinear term g may grow superlinearly in u as |u| → ∞. The purpose of this paper is to obtain solvability theorems for (1) and (2) when the Landesman-Lazer condition does not hold. More precisely, we require that h may satisfy are arbitrarily nonnegative constants, . The proofs are based upon degree theoretic arguments.

scholarly journals POSITIVE SOLUTIONS OF THE SEMIPOSITONE NEUMANN BOUNDARY VALUE PROBLEM

2015 ◽  
Vol 20 (5) ◽  
pp. 578-584 ◽  
Author(s):  
Johnny Henderson ◽  
Nickolai Kosmatov
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Neumann Boundary Value Problem ◽  
At Resonance ◽  
Krasnoselskii Fixed Point Theorem ◽  
Problem At Resonance

In this paper we consider the Neumann boundary value problem at resonance −u''(t) = f t, u(t) , 0 < t < 1, u' (0) = u' (1) = 0. We assume that the nonlinear term satisfies the inequality f(t, z) + α2z + β(t) ≥ 0, t ∈ [0, 1], z ≥ 0, where β : [0, 1] → R + , and α ≠ 0. The problem is transformed into a non-resonant positone problem and positive solutions are obtained by means of a Guo–Krasnoselskii fixed point theorem.

scholarly journals On second order nonlocal boundary value problem at resonance

2016 ◽  
Vol 56 (1) ◽  
pp. 143-153 ◽  
Author(s):  
Katarzyna Szymańska-Dębowska
Keyword(s):  
Boundary Value Problem ◽  
Bounded Variation ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Second Order ◽  
Function Of Bounded Variation ◽  
Nonlocal Boundary Value Problem ◽  
At Resonance ◽  
Problem At Resonance

Abstract This work is devoted to the existence of solutions for a system of nonlocal resonant boundary value problem $$\matrix{{x'' = f(t,x),} \hfill & {x'(0) = 0,} \hfill & {x'(1) = {\int_0^1 {x(s)dg(s)},} }} $$ where f : [0, 1] × ℝk → ℝk is continuous and g : [0, 1] → ℝk is a function of bounded variation.

Existence results for a second order nonlocal boundary value problem at resonance

2016 ◽  
Vol 53 (1) ◽  
pp. 42-52
Author(s):  
Katarzyna Szymańska-Dȩbowska
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Existence Results ◽  
Function Of Bounded Variation ◽  
Nonlocal Boundary Value Problem ◽  
At Resonance ◽  
Problem At Resonance

The paper focuses on existence of solutions of a system of nonlocal resonant boundary value problems , where f : [0, 1] × ℝk → ℝk is continuous and g : [0, 1] → ℝk is a function of bounded variation. Imposing on the function f the following condition: the limit limλ→∞f(t, λ a) exists uniformly in a ∈ Sk−1, we have shown that the problem has at least one solution.

scholarly journals ON SOLUTIONS OF NEUMANN BOUNDARY VALUE PROBLEM FOR THE LIÉNARD TYPE EQUATION

2008 ◽  
Vol 13 (2) ◽  
pp. 161-169
Author(s):  
Svetlana Atslega
Keyword(s):  
Boundary Value Problem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Type Equation ◽  
Neumann Boundary ◽  
Neumann Boundary Value Problem ◽  
Liénard Type Equation

We provide conditions on the functions f(x) and g(x), which ensure the existence of solutions to the Neumann boundary value problem for the equation x'' + f(x)x'2+g(x)=0.

scholarly journals On a Fourth-Order Boundary Value Problem at Resonance

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Man Xu ◽  
Ruyun Ma
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Elastic Beam ◽  
First Eigenvalue ◽  
At Resonance ◽  
Problem At Resonance ◽  
Spectrum Structure

We investigate the spectrum structure of the eigenvalue problem u4x=λux,  x∈0,1;  u0=u1=u′0=u′1=0. As for the application of the spectrum structure, we show the existence of solutions of the fourth-order boundary value problem at resonance -u4x+λ1ux+gx,ux=hx,  x∈0,1;  u0=u1=u′0=u′1=0, which models a statically elastic beam with both end-points being cantilevered or fixed, where λ1 is the first eigenvalue of the corresponding eigenvalue problem and nonlinearity g may be unbounded.

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