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Existence, uniqueness and regularity of the solution for a system of weakly singular Volterra integral equations of the first kind. (English) Zbl 1525.45004

The authors investigate existence, uniqueness, and regularity of solutions for systems of Volterra integral equations of the first and second kind given, respectively, by \[\int^{t}_{0}K_{\alpha}(t,s)u(s)ds=g(t),\ (t,s)\in D,\] and \[u(t)=g(t)+\int^{t}_{0}K_{\alpha}(t,s)u(s)ds,\ (t,s)\in D,\] with \(u(t)=(u_{1}(t),\dots,u_{n}(t))\), \(g(t)=(g_{1}(t),\dots,g_{n}(t))\) and the weakly singular matrix kernel \(K_{\alpha}(t,s)=\left((t-s)^{\alpha_{i,j}-1}k_{i,j}(t,s)\right)_{i,j=1,\dots,n}\), where \(0<\alpha_{i,j}\leq 1\) with \(\alpha_{i,j}<1\) for at least one index pair \((i,j)\). Here \(D := \{(t,s): 0 \leq s \leq t \leq T\}\). To get their results, the authors first state sufficient conditions for the existence and uniqueness of solutions for a system of weakly singular Volterra integral equations of the second kind.

MSC:

45F15 Systems of singular linear integral equations
45D05 Volterra integral equations

Citations:

Zbl 1376.45002
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References:

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