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On linear relations generated by an integro-differential equation with Nevanlinna measure in the infinite-dimensional case. (English. Russian original) Zbl 1316.45008
Math. Notes 96, No. 1, 10-25 (2014); translation from Mat. Zametki 96, No. 1, 5-21 (2014).
The author considers the following systems of integral equations with unknown functions $$y$$ and $$y_{1}$$ on $$[a,b]$$: $y(t)=c_{1}+\int_{t_{0}}^{t}y(s)ds \tag{1}$ and $y_{1}(t)=c_{2}-\int_{t_{0}}^{t}(d\widetilde{M}_{\lambda})y(s)-\int_{t_{0}}^{t}(d\widetilde{V})f(s), \tag{2}$ where $$\widetilde{M}_{\lambda}$$ is the Nevanlinna measure whose values are bounded operators in a separable Hilbert space $$H$$, $$\lambda\in\mathcal{C}_{0}\supset \mathcal{C}-\mathbb{R}$$, $$\widetilde{V}=(Im\lambda_{0})^{-1}Im\widetilde{M}_{\lambda_{0}}$$ and $$c_{1}$$ and $$c_{2}\in H$$. If $$H$$ is one-dimensional and the generating function $$t\rightarrow M_{\lambda}(t)$$ of the measure $$\widetilde{M}_{\lambda}$$is of the form $$M_{\lambda}(t)=-q(t)+\lambda m(t)$$, where $$q$$ is a function of bounded variation and the function $$m$$ is nondecreasing, then the system (1) and (2) takes the form of an integro-differential equation $-y'(t)+y'(t_{0})+\int_{t_{0}}^{t}y(s)dq(s)-\lambda\int_{t_{0}}^{t}y(s) dm(s)=\int_{t_{0}}^{t}f(s)dm(s).$ The author determines the families of maximal and minimal relations generated by the system(1) and (2). The holomorphy of these families is established. Moreover, the operators inverse to continuously invertible restrictions of the maximal relations are described. This description is used to derive a formula for the generalized resolvents of the symmetric relation corresponding to (1) and (2).
##### MSC:
 45J05 Integro-ordinary differential equations 45F05 Systems of nonsingular linear integral equations
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