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A system of integral-difference equations on a half-line. (Russian) Zbl 0601.45006

A system of convolution integral-difference equations with operator \[ (1)\quad (Au)(x)=\sum^{\infty}_{j=0}[a_ i(x)u(x-h_ j)+b_ j(x)\int^{\infty}_{-\infty}k_ j(x-y)u(y)dy], \] is considered in the Lebesgue space of vector functions \(L_ 2(R^ 1)\); the coefficients \(a_ j(x)\) and \(b_ j(x)\) are continuous \(N\times N\) matrix-functions on \(R^ 1\) and may have different limits at the infinities \(\pm \infty\); \(k_ j(x)\in L_ 1^{N\times N}(R^ 1)\) and the condition \[ \sup_{x}\sum^{\infty}_{j=0}[\| a_ j(x)\|_{R^ N}+\| b_ j(x)\|_{R^ N}\| k_ j(x)\|_{L_ 1^{N\times N}}]<\infty \] holds. Certain necessary conditions on the Noether property of the operator (1) are announced. (The symbol which is a semi-almost periodic matrix-function is non-degenerated and its two periodic indices vanish.) One theorem about the factorization of a positive definite almost-periodic matrix-function is announced as well. In the last section certain boundary conditions are announced for (1) in the case of non-zero index of the symbol, making this equation Noetherian (certain analogs of the Shapiro-Lopatinsky conditions).
Reviewer: R.Dudučava

MSC:

45F15 Systems of singular linear integral equations
47A53 (Semi-) Fredholm operators; index theories
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