scholarly journals Principal Eigenvalues of a Second-Order Difference Operator with Sign-Changing Weight and Its Applications

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Ruyun Ma ◽  
Man Xu ◽  
Yan Long
Keyword(s):  
Eigenvalue Problem ◽  
Weight Function ◽  
Nonlinear Problem ◽  
Bifurcation Theory ◽  
Difference Operator ◽  
Global Bifurcation ◽  
Order Difference ◽  
Principal Eigenvalues ◽  
Existence Of Positive Solutions ◽  
Global Bifurcation Theory

Let T>2 be an integer and T={1,2,…,T}. We show the existence of the principal eigenvalues of linear periodic eigenvalue problem -Δ2u(j-1)+q(j)u(j)=λg(j)u(j),  j∈T, u(0)=u(T),  u(1)=u(T+1), and we determine the sign of the corresponding eigenfunctions, where λ is a parameter, q(j)≥0 and q(j)≢0 in T, and the weight function g changes its sign in T. As an application of our spectrum results, we use the global bifurcation theory to study the existence of positive solutions for the corresponding nonlinear problem.

scholarly journals The existence of positive solutions for Kirchhoff-type problems via the sub-supersolution method

2018 ◽  
Vol 26 (1) ◽  
pp. 5-41 ◽  
Author(s):  
Baoqiang Yan ◽  
Donal O’Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Positive Solutions ◽  
Bifurcation Theory ◽  
Sufficient Conditions ◽  
Global Bifurcation ◽  
Existence Of A Solution ◽  
Kirchhoff Type ◽  
Existence Of Positive Solutions ◽  
Global Bifurcation Theory ◽  
Necessary And Sufficient ◽  
Kirchhoff Type Problems

Abstract In this paper we discuss the existence of a solution between wellordered subsolution and supersolution of the Kirchhoff equation. Using the sub-supersolution method together with a Rabinowitz-type global bifurcation theory, we establish the existence of positive solutions for Kirchhoff-type problems when the nonlinearity is singular or sign-changing. Moreover, we obtain some necessary and sufficient conditions for the existence of positive solutions for the problem when N = 1.

Global Bifurcation of Stationary Solutions for a Volume-Filling Chemotaxis Model with Logistic Growth

2020 ◽  
Vol 30 (13) ◽  
pp. 2050182
Author(s):  
Yaying Dong ◽  
Shanbing Li
Keyword(s):  
Bifurcation Theory ◽  
Global Bifurcation ◽  
Neumann Boundary ◽  
Stationary Solutions ◽  
Logistic Growth ◽  
Fredholm Operators ◽  
Chemotaxis Model ◽  
Volume Filling ◽  
Constant Solution ◽  
Global Bifurcation Theory

In this paper, we show how the global bifurcation theory for nonlinear Fredholm operators (Theorem 4.3 of [Shi & Wang, 2009]) and for compact operators (Theorem 1.3 of [Rabinowitz, 1971]) can be used in the study of the nonconstant stationary solutions for a volume-filling chemotaxis model with logistic growth under Neumann boundary conditions. Our results show that infinitely many local branches of nonconstant solutions bifurcate from the positive constant solution [Formula: see text] at [Formula: see text]. Moreover, for each [Formula: see text], we prove that each [Formula: see text] can be extended into a global curve, and the projection of the bifurcation curve [Formula: see text] onto the [Formula: see text]-axis contains [Formula: see text].

scholarly journals Existence of Positive Solutions of a Discrete Elastic Beam Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Ruyun Ma ◽  
Jiemei Li ◽  
Chenghua Gao
Keyword(s):  
Positive Solutions ◽  
Fourth Order ◽  
Nonlinear Boundary ◽  
Global Bifurcation ◽  
Elastic Beam ◽  
Beam Equation ◽  
Nonlinear Boundary Value Problems ◽  
Order Difference ◽  
Bifurcation Theorem ◽  
Existence Of Positive Solutions

LetTbe an integer withT≥5and letT2={2,3,…,T}. We consider the existence of positive solutions of the nonlinear boundary value problems of fourth-order difference equationsΔ4u(t−2)−ra(t)f(u(t))=0,t∈T2,u(1)=u(T+1)=Δ2u(0)=Δ2u(T)=0, whereris a constant,a:T2→(0,∞),  and  f:[0,∞)→[0,∞)is continuous. Our approaches are based on the Krein-Rutman theorem and the global bifurcation theorem.

Bifurcation of steady-state solutions in predator-prey and competition systems

1984 ◽  
Vol 97 ◽  
pp. 21-34 ◽  
Author(s):  
J. Blat ◽  
K. J. Brown
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Existence Of Solutions ◽  
Global Bifurcation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Predator Prey ◽  
Interacting Species ◽  
Global Bifurcation Theory ◽  
Steady State Solutions

SynopsisWe discuss steady-state solutions of systems of semilinear reaction-diffusion equations which model situations in which two interacting species u and v inhabit the same bounded region. It is easy to find solutions to the systems such that either u or v is identically zero; such solutions correspond to the case where one of the species is extinct. By using decoupling and global bifurcation theory techniques, we prove the existence of solutions which are positive in both u and v corresponding to the case where the populations can co-exist.

Nonlinear eigenvalue problems for the whirling of heavy elastic strings

1980 ◽  
Vol 85 (1-2) ◽  
pp. 59-85 ◽  
Author(s):  
Stuart S. Antman
Keyword(s):  
Material Properties ◽  
Bifurcation Theory ◽  
Eigenvalue Problems ◽  
Global Bifurcation ◽  
Qualitative Description ◽  
Nonlinear Eigenvalue Problems ◽  
Elastic Strings ◽  
Global Bifurcation Theory ◽  
Nonlinearly Elastic ◽  
Nonlinear Eigenvalue

SynopsisThis paper combines the global bifurcation theory of Rabinowitz with Sturmian theory and careful estimates to obtain a detailed qualitative description of bifurcating branches of solutions to the equations for whirling nonuniform, nonlinearly elastic strings. These results generalize earlier work of Kolodner and Stuart on inextensible strings. It is shown that the location of solution branches for the generalization of Kolodner's problem is especially sensitive to the material properties of the string, whereas that for Stuart's problem is not. The analysis of a third problem illuminates the source of this dichotomy.

A SYSTEM OF REACTION–DIFFUSION EQUATIONS IN THE UNSTIRRED CHEMOSTAT WITH AN INHIBITOR

2006 ◽  
Vol 16 (04) ◽  
pp. 989-1009 ◽  
Author(s):  
HUA NIE ◽  
JIANHUA WU
Keyword(s):  
Bifurcation Theory ◽  
Global Bifurcation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Monotone Method ◽  
The Maximum Principle ◽  
Global Bifurcation Theory ◽  
Steady State Solutions ◽  
Competing Populations

A system of reaction–diffusion equations is considered in the unstirred chemostat with an inhibitor. Global structure of the coexistence solutions and their local stability are established. The asymptotic behavior of the system is given as a function of the parameters, and it is determined when neither, one, or both competing populations survive. Finally, the results of some numerical simulations indicate that the global stability of the steady-state solutions is possible. The main tools for our investigations are the maximum principle, monotone method and global bifurcation theory.

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