System of Volterra Integral Equations: Existence Results via Brezis–Browder Arguments

Abstract

In this chapter we shall consider the system of Volterra integral equations

$$\displaystyle\begin{array}{rcl} & & u_{i}(t) = h_{i}(t) +\int _{ 0}^{t}g_{ i}(t,s)f_{i}(s,u_{1}(s),u_{2}(s),\cdots \,,u_{n}(s))ds, \\ & & \qquad \qquad \qquad \qquad \qquad \qquad \qquad t \in [0,T],\ 1 \leq i \leq n {}\end{array}$$

(20.1.1)

where 0 < T < ∞, and also the following system on a half-open interval

$$\displaystyle\begin{array}{rcl} & & u_{i}(t) = h_{i}(t) +\int _{ 0}^{t}g_{ i}(t,s)f_{i}(s,u_{1}(s),u_{2}(s),\cdots \,,u_{n}(s))ds, \\ & & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad t \in [0,T),\ 1 \leq i \leq n {}\end{array}$$

(20.1.2)

where 0 < T ≤∞. Throughout, let $u = (u_{1},u_{2},\cdots \,,u_{n}).$ We are interested in establishing the existence of solutions u of the systems (20.1.1) and (20.1.2), in ${(C[0,T])}^{n} = C[0,T] \times C[0,T] \times \cdots \times C[0,T]$ (n times), and (C[0,T))ⁿ, respectively. In addition, we shall tackle the existence of constant-sign solutions of (20.1.1) and (20.1.2). A solution u of (20.1.1) (or (20.1.2)) is said to be of constant sign if for each 1 ≤ i ≤ n, we have $\theta _{i}u_{i}(t) \geq 0$ for all t ∈ [0,T] (or t ∈ [0,T)), where $\theta _{i} \in \{-1,1\}$ is fixed. Note that when θ _i = 1 for all 1 ≤ i ≤ n, a constant-sign solution reduces to a positive solution, which is the usual consideration in the literature.