Showing papers by "Patricia J. Y. Wong published in 2004"
TL;DR: In this article, the existence of constant-sign solutions for boundary value problems was investigated and the generality of the results obtained is illustrated through applications to several well known boundary value problem problems.
Abstract: We consider the following system of Fredholm intergral equations u
i
(t)=∫0
1
g
i
(t,s)f
i
(s,u
1(s),u
2(s),...,u
n
(s)) ds, t∈[0,1], 1≤i≤n. Criteria are offered for the existence of single, double and multiple solutions of the system that are of constant signs. The generality of the results obtained is illustrated through applications to several well known boundary value problems. We also extend the above system of Fredholm intergral equations to that on the half-line [0,∞) u
i
(t)=∫0
∞
g
i
(t,s)f
i
(s,u
1(s),u
2(s),...,u
n
(s)) ds, t∈[0,∞), 1≤i≤n and investigate the existence of constant-sign solutions.
37 citations
TL;DR: In this paper, the authors considered the problem of determining the values of @l such that the above system has a constant-sign solution, and the generality of the results obtained is illustrated through applications to several well-known boundary value problems.
33 citations
TL;DR: In this paper, the existence of positive periodic solutions for a class of non-autonomous scalar equations with deviating arguments is studied. But the authors focus on the continuous time case and the discrete time case.
Abstract: We offer sufficient and realistic criteria for the existence of positive periodic solutions for a class of non-autonomous scalar equations with deviating arguments. Both the continuous time case and the discrete time case are discussed. Our approach is based on the theory of coincidence degree and the related continuation theorem. To show the usefulness of the results obtained, we include applications to several famous population models.
11 citations
TL;DR: In this paper, the authors considered the problem of finding a constant-sign solution for a system of integral equations with constant sign solution (u1, u2, u3, un) ∈ (Lp[0, 1])n, where the integer 1 ≤ p < ∞ is fixed.
Abstract: We consider the following system of integral equations
$$u_i(t)=\int^1_0g_i(t,s)f(s,u_1(s),u_2(s),\cdots,u_n(s))ds,\quad t\in \lbrack 0,1\rbrack,1\leq i\leq n.$$
Our aim is to establish criteria such that the above system has a constant-sign solution (u1, u2, …, un) ∈ (Lp[0, 1])n, where the integer 1 ≤ p < ∞ is fixed. We shall tackle the case when f is ‘nonnegative’ as well as the case when f is ‘semipositone’. The above problem is also extended to that on the half-line [0, ∞)
$$u_i(t)=\int^1_0g_i(t,s)f(s,u_1(s),u_2(s),\cdots,u_n(s))ds,\quad t\in \lbrack 0,\infty ),1\leq i\leq n.$$
9 citations
TL;DR: In this paper, the authors considered the system of difference equations with constant-sign solution, where λ > 0 and T ≥ N ≥ 0, and gave explicit intervals for λ.
Abstract: We consider the system of difference equations
$$u_i (k) = \lambda \mathop \sum \limits_{\ell = 0}^N g_i (k,\ell )P_i (\ell ,u_1 (\ell ),u_2 (\ell ),...,u_n (\ell )), k \in \{ 0,1,...,T\} , 1 \leqslant i \leqslant n,$$
where λ > 0 and T ≥ N ≥ 0. Our aim is to determine the values of λ for which the above system has a constant-sign solution. In addition, explicit intervals for λ are presented. The generality of the results obtained is illustrated through applications to several well-known boundary-value problems. We also extend the above problem to that on {0, 1, ...}:
$$u_i (k) = \lambda \mathop \sum \limits_{\ell = 0}^\infty g_i (k,\ell )P_i (\ell ,u_1 (\ell ),u_2 (\ell ),...,u_n (\ell )), k \in \{ 0,1,...,T\} , 1 \leqslant i \leqslant n.$$
Finally, both systems above are extended to the general case where λ is replaced by λ i .
7 citations
TL;DR: In this article, the Sturm-Liouville boundary conditions were considered for a system of differential equations, and the existence of three solutions of the system which are of fixed signs on the interval [ 0, 1 ] was investigated.
5 citations
TL;DR: In this article, the spread of infectious disease is modeled for time t in the real ( ), discrete ( ) and time scale (T) domains, and criteria for the existence of a nontrivial and nonnegative periodic solution for the model in all the three domains are presented.
Abstract: The modelling of the spread of infectious disease is discussed for time t in the real ( ), discrete ( ) and time scale (T) domains. We shall offer criteria for the existence of a nontrivial and nonnegative periodic solution for the model in all the three domains. These criteria can be implemented numerically and an algorithm is given.
5 citations
TL;DR: In this article, the existence of three constant-sign solutions of the boundary value problem with constant-size discrete equations was investigated. But the results were not applied to boundary value problems with discrete equations.
Abstract: We consider the following system of discrete equations
$$u_i \left( k \right) = \sum\limits_{\ell = 0}^N {g_i \left( {k,\ell } \right)P_i \left( {\ell ,u_1 \left( \ell \right),u_2 \left( \ell \right),...,u_n \left( \ell \right)} \right),k \in \left\{ {0,1,...,T} \right\},1 \leqslant i \leqslant n} $$
. Criteria for the existence of three constant-sign solutions of the system will be developed. To illustrate the generality of the results obtained, we include applications to several well-known boundary value problems. Parallel results are also established for a system on {0,1,...}
$$u_i \left( k \right) = \sum\limits_{\ell = 0}^\infty {g_i \left( {k,\ell } \right)P_i \left( {\ell ,u_1 \left( \ell \right),u_2 \left( \ell \right),...,u_n \left( \ell \right)} \right),k \in \left\{ {0,1,...} \right\},1 \leqslant i \leqslant n} $$
.
4 citations
TL;DR: In this paper, the Hermite boundary conditions were considered for the discrete system Δ m u i (k) = P i (p,u 1 (k),u 2 (k)), k ∈{0, 1, N, 1⩽i⩾n, together with Hermite conditions, Δ j u i(k ν ) = 0, j=0, m ν −1, ν=1, r, ∑ν=1rmν=m, and kν's are integers such that 0=k1
3 citations
TL;DR: In this article, the authors discuss the Abel-Gontscharoff interpolation problem in the continuous, discrete and time scale cases, and present the best possible error bounds under different settings of ai's.
2 citations