A Nonlinear Integral Operator Arising from a Model in Population Genetics III. Heterozygote Inferior Case

1985 ◽  
Vol 16 (6) ◽  
pp. 1180-1206 ◽  
Author(s):  
Roger Lui
Keyword(s):  
Population Genetics ◽  
Integral Operator ◽  
Nonlinear Integral Operator

scholarly journals Bifurcation of Bounded Solutions of Impulsive Differential Equations

2016 ◽  
Vol 26 (14) ◽  
pp. 1650242 ◽  
Author(s):  
Kevin E. M. Church ◽  
Xinzhi Liu
Keyword(s):  
Differential Equation ◽  
Integral Operator ◽  
Differential Equations ◽  
Impulsive Differential Equation ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Necessary Condition ◽  
Bounded Solutions ◽  
Pitchfork Bifurcations ◽  
Nonlinear Integral Operator

In this article, we examine nonautonomous bifurcation patterns in nonlinear systems of impulsive differential equations. The approach is based on Lyapunov–Schmidt reduction applied to the linearization of a particular nonlinear integral operator whose zeroes coincide with bounded solutions of the impulsive differential equation in question. This leads to sufficient conditions for the presence of fold, transcritical and pitchfork bifurcations. Additionally, we provide a computable necessary condition for bifurcation in nonlinear scalar impulsive differential equations. Several examples are provided illustrating the results.

scholarly journals On Existence and Uniqueness of Solutions of a Nonlinear Integral Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-7 ◽  
Author(s):  
M. Eshaghi Gordji ◽  
H. Baghani ◽  
O. Baghani
Keyword(s):  
Banach Space ◽  
Integral Equation ◽  
Fixed Point ◽  
Integral Operator ◽  
Existence And Uniqueness ◽  
Nonlinear Integral Equation ◽  
Uniqueness Of Solutions ◽  
Nonlinear Integral Operator

The purpose of this paper is to study the existence of fixed point for a nonlinear integral operator in the framework of Banach space . Later on, we give some examples of applications of this type of results.

scholarly journals Blow-up and global existence for a kinetic equation of swarm formation

2017 ◽  
Vol 27 (06) ◽  
pp. 1153-1175 ◽  
Author(s):  
Mirosław Lachowicz ◽  
Henryk Leszczyński ◽  
Martin Parisot
Keyword(s):  
Asymptotic Behavior ◽  
Kinetic Equation ◽  
Global Existence ◽  
Integral Operator ◽  
Blow Up ◽  
Existence Of Solutions ◽  
Interaction Rate ◽  
Global Existence Of Solutions ◽  
Nonlinear Integral Operator ◽  
The Right

In this paper we study a kinetic equation that describes swarm formations. The right-hand side of this equation contains nonlinear integro-differential terms responsible for two opposite tendencies: dissipation and swarming. The nonlinear integral operator describes the changes of velocities (orientations) of interacting individuals. The interaction rate is assumed to be dependent of velocities of interacting individuals. Although the equation seems to be rather simple it leads to very complicated dynamics. In this paper, we study possible blow-ups versus global existence of solutions and provide results on the asymptotic behavior. The complicated dynamics and possibility of blow-ups can be directly related to creation of swarms.

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