Solution of Volterra operator-integral equations in the nonregular case by the successive approximation method
The authors consider the operators \(G\) and \(K\) acting from a Banach space \(E_1\) to the Banach space \(E_2\), defined by \[ \begin{aligned} G(u,t) & = \sum^N_{i+k\geq 1} G_{ik}(u) t^k+ R_1(u, t),\\ K(t,s,u) & = \sum^N_{i+ k+ j\geq 0} K_{ikj}(u) t^k s^j+ R_2(u,t,s)\end{aligned} \] and the equation \[ G(u,t)+ \int^t_0 K(t,s,u(s))\,ds= 0,\qquad 0\leq s\leq t\leq\rho,\tag{1} \] where \(G_{ik}(u)\) and \(K_{ikj}(u)\) are \(i\)-power operators and the operators \(R_1\) and \(R_2\) are continuous and Fréchet differentiable with respect to \(u\) and satisfy the estimates \[ \begin{aligned} \| R_1(u,t)\| & = o((\| u\|+|t|)^N),\\ \| R_2(u,t,s)\| & = o((\| u\|+|t|+ |s|)^N).\end{aligned} \] Under some hypotheses A and B, the authors prove that the equation (1) has a continuous solution \(u\). They consider also the equation \[ Bu= b(t)+ \int^t_0 K(t,s,u(s))\,ds,\qquad 0\leq t\leq\rho, \] where \(B\) is a Fredholm operator of index \(0\), \(b(t)\) is a differentiable function ranging in \(E_2\) and mapping \(K(t,s,u)\) is differentiable with respect to \(t\) and continuous with respect to \(s\) and satisfies the Lipschitz condition with respect to \(u\).
scientific article; zbMATH DE number 5824138