About power absolute integrated of the solution of Volterra system of linear second kind integral equations on half-axis. (English) Zbl 1473.45004
The author investigates the solvability condition of the following system of Volterra integral equation \[ x(t) + \int_{t_0}^t[K(t,\tau) + Q(t,\tau)]x(\tau)d\tau = f(t) + q(t), \qquad t \geqslant t_0, \] with an unknown vector-function \(x(t)=(x_1(t),\ldots,x_n(t))^\top\), vector right-hand sides \(f(t)\), \(q(t)\) and matrix functions (kernels) \(K(t,\tau)\), \(Q(t,\tau)\).
Two lemmas are proved and, by imposing several conditions on the data of the equation, the author states (without proof) the existence of a solution to the above equation in the spaces \(L^1(I)\) and in \(L^2(I)\), \(I=[t_0,\infty)\). A concrete example is presented.
Two lemmas are proved and, by imposing several conditions on the data of the equation, the author states (without proof) the existence of a solution to the above equation in the spaces \(L^1(I)\) and in \(L^2(I)\), \(I=[t_0,\infty)\). A concrete example is presented.
Reviewer: Roland Duduchava (Tbilisi)
Keywords:
system of Volterra type equations of second kind; weight functions; truncating functions; sufficient conditions for solvabilityReferences:
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