Some properties of strong and exponential solutions of neutral-type differential-difference equations (Q1594295)

This paper is concerned with the asymptotic behavior of solutions to neutral-type differential-difference equations of the form \[ \sum_{j=0}^{n} \left( B_ju(t-h_j)+D_j\dfrac{du}{dt}(t-h_{j})\right) +\int_{0}^{h} K(s)u(t-s)ds=0,\quad t\in R_{+},\tag{1} \] \[ u(t)=y(t),\quad t\in \left[ -h,0\right) ,\quad u(+0)=y(-0).\tag{2} \] Here, \(B_j\) and \(D_j\), \(j=0,1,\dots,n\), are \(m\times m\)-matrices with constant complex elements, the numbers \(h_j\) are such that \(0=h_0<h_1<\dots <h_n=h\), the elements \(K_{ij}(s)\) with \(i,j=1,2,\dots,m\), of the matrix \(K\) belong to the space \(L_2(0,h)\), and \(y\in W_2^1((0,h),C^m)\) is a given vector-function. The author considers solutions to problem (1), (2) as vector functions from the weighted Sobolev spaces \(W_{2,\gamma }^1((-h,+\infty),C^m)\) with the norms \[ \left\|u\right\|_{W_{2,\gamma }^1(a,b)}\equiv \left( \int \exp (-2\gamma t)\left( \sum \left\|u^{(j)}\right\|_{C^m}^2\right) dt\right) ^{1/2},\qquad \gamma\in \mathbb{R}. \] It is a solution to equation (1) at almost all \(t\in \mathbb{R}_{+}\) and satisfies condition (2). The author provides the assertion of existence and uniqueness of the solution with the estimate of the form \[ \|u\|_{W_{2,\gamma }^1(-h,+\infty)} \leq d\|y\|_{W_{2,\gamma }^1(-h,+\infty)}, \] where the constant \(d\) is independent of the function \(y\). Spectral properties of the operator generated by problem (1), (2) and properties of the system of exponential (elementary) solutions to these equations under the conditions \(\det D_0\neq 0\) and \(\det D_n\neq 0\) are studied.