We consider the system of Fredholm integral equations $$\begin{array}{l}\displaystyle u_{i}(t)=\int_{0}^{T}g_{i}(t,s)[h_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))+k_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))]ds,\\[8pt]\quad t\in[0,T],~1\leq i\leq n\end{array}$$
 and also the system of Volterra integral equations $$\begin{array}{l}\displaystyle u_{i}(t)=\int_{0}^{t}g_{i}(t,s)[h_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))+k_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))]ds,\\[8pt]\quad t\in[0,T],~1\leq i\leq n\end{array}$$
 where T>0 is fixed and the nonlinearities h
 i
 (t,u
 1,u
 2,…,u
 n
 ) can be singular at t=0 and u
 j
 =0 where j∈{1,2,…,n}. Criteria are offered for the existence of constant-sign solutions, i.e., θ
 i
 u
 i
 (t)≥0 for t∈[0,1] and 1≤i≤n, where θ
 i
 ∈{1,−1} is fixed. We also include examples to illustrate the usefulness of the results obtained.

Constant-Sign Solutions for Systems of Fredholm and Volterra Integral Equations: The Singular Case

We consider the following system of Volterra integral equations: u"i(t)=@!"0^tg"i(t,s)[f"i(s,u"1(s),u"2(s),...,u"n(s))+h"i(s,u"1(s),u"2(s),...,u"n(s))]ds,t@?[0,T],1@?i@?n and some of its particular cases that arise from physical problems. Criteria are offered for the existence of one and more constant-sign solutions u=(u"1,u"2,...,u"n) of the system in (C[0,T])^n. We say u is of constant sign if for each 1@?i@?n,@q"iu"i(t)>=0 for all t@?[0,T], where @q"i@?{1,-1} is fixed. Examples are also included to illustrate the results obtained.

Constant-sign solutions of a system of Volterra integral equations

The cardinal trigonometric splines on small compact supports are employed to solve integral equations. The unknown function is expressed as a linear combination of cardinal trigonometric splines functions. Then a simple system of equations on the coefficients is deducted. When solving the Volterra integral equations, the system is triangular, so it is relatively straight forward to solve the nonlinear system of the coefficients and a good approximation of the original solution is obtained. The sufficient condition for the existence of the solution is discussed and the convergence rate is investigated.

/pdf/the-applications-of-cardinal-trigonometric-splines-in-1843sqvho5.pdf

The Applications of Cardinal Trigonometric Splines in Solving Nonlinear Integral Equations

Krasnoselskii’s fixed point theorem and time scale inequalities are used to establish the existence of positive solutions to some time scale equations

Existence of positive solutions to time scale equations using time scale inequalities

0 gi(t, s)[Pi(s, u1(s), u2(s), · · · , un(s)) + Qi(s, u1(s), u2(s), · · · , un(s))]ds, t ∈ [0, T ], 1 ≤ i ≤ n where T > 0 is fixed and the nonlinearities Pi(t, u1, u2, · · · , un) can be singular at t = 0 and uj = 0 where j ∈ {1, 2, · · · , n}. Criteria are offered for the existence of fixed-sign solutions (u∗1, u ∗ 2, · · · , u ∗ n) to the system of Volterra integral equations, i.e., θiu ∗ i (t) ≥ 0 for t ∈ [0, 1] and 1 ≤ i ≤ n, where θi ∈ {1,−1} is fixed. We also include an example to illustrate the usefulness of the results obtained.

/pdf/solutions-for-singular-volterra-integral-equations-4ji64qi4xa.pdf

Solutions for singular volterra integral equations

In this paper, the fixed point theorem of increasing operator with non-continuity is utilized to discuss the existence and uniqueness of positive solution for a class of nonlinear Volterra integral equations. An important condition of continuity can be replaced by weak condition.

Dong Wei

Papers

Uniqueness of solutions for a class of non-linear volterra integral equations without continuity