DCDS-B
Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model
Chunqing Wu Patricia J.Y. Wong
Discrete & Continuous Dynamical Systems - B 2015, 20(9): 3255-3266 doi: 10.3934/dcdsb.2015.20.3255
We shall obtain the parameter region that ensures the global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. The parameter region can be illustrated graphically and examples of such regions are presented. Our result partially answers an open problem proposed by Elaydi and Luís [3] and complements the very recent work by Balreira, Elaydi and Luís [1].
keywords: global asymptotical stability coexistence fixed point Ricker-type competitive model open problem.
PROC
Existence of solutions to singular integral equations
Patricia J.Y. Wong
Conference Publications 2009, 2009(Special): 818-827 doi: 10.3934/proc.2009.2009.818
We consider the system of integral equations


$u_i(t)=int_0^Tg_i(t,s)[a_i(s,u_1(s),u_2(s),...,u_n(s))+b_i(s,u_1(s),u_2(s),...,u_n(s))]ds,$   $t \in [0,T],$   $1<=i<=n,$

where $T>0$ is fixed and the nonlinearities $a_i(t,u_1,u_2,\cdots,u_n)$ can be singular at $t=0$ and $u_j=0$ where $j\in\{1,2,\cdots,n\}.$ Criteria are established for the existence of fixed-sign solutions $(u_1^*,u_2^*,\cdots,u_n^*)$ to the above system, i.e., $\theta_iu_i^*(t)\geq 0$ for $t\in [0,T]$ and $1\leq i\leq n,$ where $\theta_i\in \{1,-1\}$ is fixed. We also include an example to illustrate the usefulness of the results obtained.

keywords: Fixed-sign solutions system of singular integral equations
PROC
On the existence of fixed-sign solutions for a system of generalized right focal problems with deviating arguments
Patricia J.Y. Wong
Conference Publications 2007, 2007(Special): 1042-1051 doi: 10.3934/proc.2007.2007.1042
We consider the following system of third-order three-point generalized right focal boundary value problems

$u^(''')_ i (t) = f_i(t, u_1(\phi_1(t)), u_2(\phi_2(t)), · · · , u_n(\phi_n(t))), t \in [a, b]$ $u_i(a) = u^'_i(z_i) = 0$, \gamma_i u_i(b) + \delta_iu^('')_i (b) = 0$

where $i$ = 1, 2, · · · , $n$, $1/2 (a + b) < z_i < b, \gamma_i > 0$, and $\phi_i$ are deviating arguments. By using some fixed point theorems, we establish the existence of one or more fixed-sign solutions $u = (u_1, u_2, · · · , u_n)$ for the system, i.e., for each 1 $<=$ $i$ $<=$ $n$, $\theta_iui(t) >= 0$ for $t \in [a, b]$, where $\theta_i \in$ {1,−1} is fixed. An example is also presented to illustrate the results obtained.

keywords: Fixed-sign solutions system of boundary value problems right focal.

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