On Boundary Value Problems for Integral Equations with Operator Measures.

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

\[y_k(t) = y_k(a) + \int \limits_{[a,t)} (d{\bf p}_k)y_k(s) + \int \limits_{[a,t)} (d{\bf m}_k)f_k(s)ds, ~~~k = 0, 1, 2, ... ,\]

where ${\bf p}_k$, ${\bf m}_k$ are operator-valued measures defined on Borel sets $∆ ⊂ [a, b]$ and taking values in a set of linear bounded operators acting in a separable Hilbert space $H$; $f_k ∈ L_1(H, {\bf m}_k; a, b); ~y_k$ are unknown functions. The measures ${\bf p}_k$, ${\bf m}_k$ are assumed to have bounded variations on $[a, b]$. For these equations we consider boundary conditions

\[Γ_ky_k = c_k,~~ k = 0, 1, 2... ,\]

where $Γ_k : \tilde C →B$ are linear continuous mappings; $c_k ∈ B;~ \tilde C$ is a space of functions continuous from the left on $[a, b]$ and taking values in $H$; $B$ is a Banach space; $k = 0, 1, 2, ...$ .
Let $\mathscr {K}_0$ be a set of solutions of integral equation for $k = 0, f_0 = 0$, and $\tilde Γ_0$ the restriction of $Γ_0$ to $\mathscr {K}_0$, and ${\bf V}_{[a,b]} ({\bf p})$ a variation of a measure $\bf p$. The aim of this paper is to prove following statement.

Theorem. Suppose the operator $~\tilde Γ_0$ is a one-to-one mapping of $\mathscr {K}_0$ onto $B$ and $~\bigl| \bigl| Γ_n − Γ_0 \bigr| \bigr| →0, ~{\bf V}_{[a,b]} ({\bf p}_n−{\bf p}_0)→0, ~{\bf V}_{[a,b]} ({\bf m}_n−{\bf m}_0)→0, f_n→f_0$ uniformly on $[a, b], ~c_n→c$ in $B$ as $n→∞$. Then the problem stated above has a unique solution $y_n$ for large enough $n$ and $\bigl| \bigl| y_n(t) − y_0(t) \bigr| \bigr| →0$ as $n→∞$ uniformly with respect to $t$.

We use the following statement to prove the theorem formulated above.

Theorem. Suppose a measure ${\bf p}$ has a bounded variation on $[a, b]$. Then there exists a unique solution of the equation

\[y(t) = \int \limits_{t_0}^t d{\bf p}(s)y(s) + g(t)\]

on the interval $[t_0 − δ, b]$, where $g ∈ \tilde C, δ = δ(t_0)$ is small enough and $δ = 0$ if $t_0 = a$.

Keywords: integral equation, operator measure, boundary value problem, Hilbert space, linear operator.