scholarly journals Sign-Changing Solutions for Nonlinear Operator Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Yanbin Sang
Keyword(s):  
Fixed Point Index ◽  
Nonlinear Operator ◽  
Index Theory ◽  
Elliptic Partial Differential Equations ◽  
Point Index ◽  
Operator Equations ◽  
Nonlinear Operator Equations ◽  
Fixed Point Index Theory ◽  
Sign Changing Solutions ◽  
Theoretical Results

The existence of six solutions for nonlinear operator equations is obtained by using the topological degree and fixed point index theory. These six solutions are all nonzero. Two of them are positive, the other two are negative, and the fifth and sixth ones are both sign-changing solutions. Furthermore, the theoretical results are applied to elliptic partial differential equations.

scholarly journals On fixed point index theory for the sum of operators and applications to a class of ODEs and PDEs

2021 ◽  
Vol 22 (2) ◽  
pp. 259
Author(s):  
Svetlin Georgiev Georgiev ◽  
Karima Mebarki
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Point Index ◽  
Index Theory ◽  
Point Index ◽  
First Order ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions ◽  
Theoretical Results ◽  
Positive Half

The aim of this work is two fold: first  we  extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a $k$-set contraction  obtained in \cite{DjebaMeb, Svet-Meb}, to  the case of the sum $T+F$, where $T$ is a mapping such that $(I-T)$ is Lipschitz invertible and $F$ is a $k$-set contraction.  Secondly, as  illustration of some our theoretical results,  we study  the existence of positive solutions  for two classes of differential equations, covering a class of first-order ordinary differential equations (ODEs for short) posed on the positive half-line as well as  a class of  partial differential equations (PDEs for short).

scholarly journals Sign-Changing Solutions for Discrete Second-Order Three-Point Boundary Value Problems

2010 ◽  
Vol 2010 ◽  
pp. 1-14 ◽  
Author(s):  
Tieshan He ◽  
Wei Yang ◽  
Fengjian Yang
Keyword(s):  
Fixed Point Index ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Degree Theory ◽  
Index Theory ◽  
Second Order ◽  
Topological Degree Theory ◽  
Point Index ◽  
Fixed Point Index Theory ◽  
Sign Changing Solutions

We consider the second-order three-point discrete boundary value problem. By using the topological degree theory and the fixed point index theory, we provide sufficient conditions for the existence of sign-changing solutions, positive solutions, and negative solutions. As an application, an example is given to demonstrate our main results.

scholarly journals Positive Periodic Solutions of Third-Order Ordinary Differential Equations with Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yongxiang Li ◽  
Qiang Li
Keyword(s):  
Periodic Solutions ◽  
Fixed Point Index ◽  
Index Theory ◽  
Existence Results ◽  
Positive Periodic Solutions ◽  
Point Index ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
The Third ◽  
Fixed Point Index Theory

The existence results of positiveω-periodic solutions are obtained for the third-order ordinary differential equation with delaysu′′′(t)+a(t)u(t)=f(t,u(t-τ0),u′(t-τ1),u′′(t-τ2)),t∈ℝ,wherea∈C(ℝ,(0,∞))isω-periodic function andf:ℝ×[0,∞)×ℝ2→[0,∞)is a continuous function which isω-periodic int,and τ0,τ1,τ2are positive constants. The discussion is based on the fixed-point index theory in cones.

scholarly journals Existence Result for Impulsive Differential Equations with Integral Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Peipei Ning ◽  
Qian Huan ◽  
Wei Ding
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fixed Point Index ◽  
Integral Boundary Conditions ◽  
Impulsive Differential Equations ◽  
Index Theory ◽  
Upper And Lower Bounds ◽  
Point Index ◽  
Integral Boundary ◽  
Fixed Point Index Theory

We investigate the following differential equations:-(y[1](x))'+q(x)y(x)=λf(x,y(x)), with impulsive and integral boundary conditions-Δ(y[1](xi))=Ii(y(xi)),i=1,2,…,m,y(0)-ay[1](0)=∫0ωg0(s)y(s)ds,y(ω)-by[1](ω)=∫0ωg1(s)y(s)ds, wherey[1](x)=p(x)y'(x). The expression of Green's function and the existence of positive solution for the system are obtained. Upper and lower bounds for positive solutions are also given. Whenp(t),I(·),g0(s), andg1(s)take different values, the system can be simplified to some forms which has been studied in the works by Guo and LakshmiKantham (1988), Guo et al. (1995), Boucherif (2009), He et al. (2011), and Atici and Guseinov (2001). Our discussion is based on the fixed point index theory in cones.

scholarly journals Eigenvalue of Coupled Systems for Hammerstein Integral Equation with Two Parameters

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Wanjun Li
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Index Theory ◽  
Coupled Systems ◽  
Point Index ◽  
Hammerstein Integral Equation ◽  
Fixed Point Index Theory ◽  
Two Parameters

By using the fixed-point index theory, we discuss the existence, multiplicity, and nonexistence of positive solutions for the coupled systems of Hammerstein integral equation with parameters.

scholarly journals Positive Solutions for a Class of Fourth-Order Boundary Value Problems in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Jingjing Cai ◽  
Guilong Liu
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Positive Solutions ◽  
Fourth Order ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Index Theory ◽  
Point Index ◽  
Fixed Point Index Theory ◽  
Existence And Nonexistence

Using a specially constructed cone and the fixed point index theory, this work shows existence and nonexistence results of positive solutions for fourth-order boundary value problem with two different parameters in Banach spaces.

Positive Radial Solutions for Elliptic Systems with Nonlocal Conditions

2012 ◽  
Vol 204-208 ◽  
pp. 4800-4808
Author(s):  
Sheng Li Xie
Keyword(s):  
Fixed Point ◽  
Elliptic System ◽  
Fixed Point Index ◽  
Nonlocal Conditions ◽  
Elliptic Systems ◽  
Index Theory ◽  
Point Index ◽  
Radial Solutions ◽  
Fixed Point Index Theory ◽  
Positive Radial Solutions

By using fixed point index theory,we study the existence of positive radial solutions and multiple positive radial solutions for the elliptic system with nonlocal conditions. Our results extend and improve some existing ones.

scholarly journals Positive Periodic Solutions of Second-Order Differential Equations with Delays

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Yongxiang Li
Keyword(s):  
Periodic Solutions ◽  
Fixed Point Index ◽  
Order Differential Equation ◽  
Index Theory ◽  
Existence Results ◽  
Second Order ◽  
Positive Periodic Solutions ◽  
Point Index ◽  
Second Order Differential Equations ◽  
Fixed Point Index Theory

The existence results of positiveω-periodic solutions are obtained for the second-order differential equation with delays−u″+a(t)=f(t,u(t−τ1),...,u(t−τn)), wherea∈C(ℝ,(0,∞))is aω-periodic function,f:ℝ×[0,∞)n→[0,∞)is a continuous function, which isω-periodic int, andτ1,τ2,...,τnare positive constants. Our discussion is based on the fixed point index theory in cones.

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