# On the convergence of solutions of boundary value problems for integral equations with operator measures

On a segment $[a,b]$, we consider integral equations

y_k(t)=y_k(t_0)+\int\limits_{t_0}^t (d\mathbf{p}_k)y_k(s)+\int\limits_{t_0}^t f_k(s) ds,\: k=0,1,2,\dots ,

where $\int_{t_0}^t$ stands for $\int_{[t_0,t)}$ if $t_0\,{<}\,t$; and for $0$ if $t_0=t$. Here $\mathbf{p_k}$ are operator-valued measures defined on Borel sets $\triangle \subset [a,b]$ and taking values in the set of linear bounded operators acting in a separable Hilbert space $H$; $f_k \in L_1(H; a,b)$; $y_k$ are unknown functions. Measures $\mathbf{p_k}$ are assumed to have bounded variations on $[a,b]$. For these equations we consider the boundary conditions

\Gamma_k y_k=c_k,\: k=0,1,2,\dots ,

where $\Gamma_k:\widetilde C\rightarrow B$ are linear continuous mappings; $c_k \in B, \widetilde C$ is a space of functions continuous from the left on $[a,b]$ and taking values in $H$; $B$ is a Banach space; $k=0,1,2,\dots .$
We obtain sufficient conditions under which $||y_n(t)-y_0(t)||\to 0$ uniformly with respect to $t\in [a,b]$. The main assumptions are as follows: the solution of the homogeneous equation is only for $k=0$; $\mathbf{V}_{[a,b]}(\mathbf{p}_n-\mathbf{p}_0)\to 0$, where $\mathbf{V}_{[a,b]}(\mathbf{p}_n-\mathbf{p}_0)$ is a variation of $\mathbf{p}_n-\mathbf{p}_0$ on $[a,b]$; $||\Gamma_n-\Gamma_0|| \to 0$ as $n \to \infty$; the operator $\Gamma_0$ maps bijectively a set of solutions of the homogeneous boundary problem for $k=0$ onto the space $B$.
Key words: integral equation, operator measure, boundary value problem, Hilbert space, linear operator, linear relation.

UDC:
517.983