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 Positive Periodic Solutions for Second-Order Ordinary Differential Equations with Derivative Terms and Singularity in Nonlinearities

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Yongxiang Li ◽  
Xiaoyu Jiang
Keyword(s):  
Differential Equation ◽  
Periodic Solutions ◽  
Fixed Point Index ◽  
Index Theory ◽  
Existence Results ◽  
Second Order ◽  
Positive Periodic Solutions ◽  
Point Index ◽  
Order Ordinary Differential Equation ◽  
Fixed Point Index Theory

The existence results of positiveω-periodic solutions are obtained for the second-order ordinary differential equationu′′(t)=f(t,u(t),u'(t)),t∈ℝwhere,f:ℝ×(0,∞)×ℝ→ℝis a continuous function, which isω-periodic intandf(t,u,v)may be singular atu=0. The discussion is based on the fixed point index theory in cones.

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.

scholarly journals Neutral Operator and Neutral Differential Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-29 ◽  
Author(s):  
Jingli Ren ◽  
Zhibo Cheng ◽  
Stefan Siegmund
Keyword(s):  
Fixed Point Index ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Index Theory ◽  
Positive Periodic Solutions ◽  
Point Index ◽  
Second Order Differential Equations ◽  
Fixed Point Index Theory ◽  
Neutral Operator

In this paper, we discuss the properties of the neutral operator(Ax)(t)=x(t)−cx(t−δ(t)), and by applying coincidence degree theory and fixed point index theory, we obtain sufficient conditions for the existence, multiplicity, and nonexistence of (positive) periodic solutions to two kinds of second-order differential equations with the prescribed neutral operator.

scholarly journals Multiple solutions for singular semipositone boundary value problems of fourth-order differential systems with parameters

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Longfei Lin ◽  
Yansheng Liu ◽  
Daliang Zhao
Keyword(s):  
Fourth Order ◽  
Multiple Solutions ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Index Theory ◽  
Existence Results ◽  
Point Index ◽  
Differential Systems ◽  
Fixed Point Index Theory ◽  
Special Cone

AbstractThe aim of this paper is to establish some results about the existence of multiple solutions for the following singular semipositone boundary value problem of fourth-order differential systems with parameters: $$ \textstyle\begin{cases} u^{(4)}(t)+\beta _{1}u''(t)-\alpha _{1}u(t)=f_{1}(t,u(t),v(t)),\quad 0< t< 1; \\ v^{(4)}(t)+\beta _{2}v''(t)-\alpha _{2}v(t)=f_{2}(t,u(t),v(t)),\quad 0< t< 1; \\ u(0)=u(1)=u''(0)=u''(1)=0; \\ v(0)=v(1)=v''(0)=v''(1)=0, \end{cases} $$ { u ( 4 ) ( t ) + β 1 u ″ ( t ) − α 1 u ( t ) = f 1 ( t , u ( t ) , v ( t ) ) , 0 < t < 1 ; v ( 4 ) ( t ) + β 2 v ″ ( t ) − α 2 v ( t ) = f 2 ( t , u ( t ) , v ( t ) ) , 0 < t < 1 ; u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 ; v ( 0 ) = v ( 1 ) = v ″ ( 0 ) = v ″ ( 1 ) = 0 , where $f_{1},f_{2}\in C[(0,1)\times \mathbb{R}^{+}_{0}\times \mathbb{R}, \mathbb{R}]$ f 1 , f 2 ∈ C [ ( 0 , 1 ) × R 0 + × R , R ] , $\mathbb{R}_{0}^{+}=(0,+\infty )$ R 0 + = ( 0 , + ∞ ) . By constructing a special cone and applying fixed point index theory, some new existence results of multiple solutions for the considered system are obtained under some suitable assumptions. Finally, an example is worked out to illustrate the main results.

Existence of positive solutions to multi-point third order problems with sign changing nonlinearities

2020 ◽  
Vol 24 (1) ◽  
pp. 109-129
Author(s):  
Abdulkadir Dogan ◽  
John R. Graef
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Index Theory ◽  
Point Index ◽  
Third Order ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

In this paper, the authors examine the existence of positive solutions to a third-order boundary value problem having a sign changing nonlinearity. The proof makes use of fixed point index theory. An example is included to illustrate the applicability of the results.

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.

ON THE NUMBER OF POSITIVE PERIODIC SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS AND POPULATION MODELS

2005 ◽  
Vol 15 (04) ◽  
pp. 555-573 ◽  
Author(s):  
DAQING JIANG ◽  
DONAL O'REGAN ◽  
RAVI P. AGARWAL ◽  
XIAOJIE XU
Keyword(s):  
Fixed Point ◽  
Periodic Solutions ◽  
Fixed Point Index ◽  
Growth Models ◽  
Infinite Delay ◽  
Functional Differential Equations ◽  
Positive Periodic Solutions ◽  
Point Index ◽  
Functional Differential ◽  
Population Growth Models

In this paper, we employ the fixed point index on cones to study the existence, multiplicity and nonexistence of positive periodic solutions to a system of infinite delay equations, [Formula: see text] in which λ > 0 is a parameter. We prove some general theorems and establish new periodicity conditions for several population growth models.

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