Abstract
In this chapter we shall consider the system of Volterra integral equations
where 0 < T < ∞, and also the following system on a half-open interval
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.
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Agarwal, R.P., O’Regan, D., Wong, P.J.Y. (2013). System of Volterra Integral Equations: Existence Results via Brezis–Browder Arguments. In: Constant-Sign Solutions of Systems of Integral Equations. Springer, Cham. https://doi.org/10.1007/978-3-319-01255-1_20
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DOI: https://doi.org/10.1007/978-3-319-01255-1_20
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