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Solvability of vector integro-differential equations of convolution type on the semiaxis. (English. Russian original) Zbl 1302.45017
J. Contemp. Math. Anal., Armen. Acad. Sci. 43, No. 5, 305-316 (2008); translation from Izv. Nats. Akad. Nauk Armen., Mat. 43, No. 5, 73-82 (2008).
Summary: The paper deals with vector integro-differential equation of convolution type that have the form $- \frac{{d^2 f_i }} {{dx^2 }} + a_i f_i (x) = g_i (x) + \sum\limits_{j = 1}^N {\int\limits_0^\infty {K_{ij} (x - t)f_j (t)dt, } \quad i = 1,2, \dots,N,}$ where $$\vec f = (f_{1}, f_{2}, \dots, f_{N})^{T}$$ is the unknown vector-function, $$a_{i}$$ are nonnegative numbers, $$\vec g = (g_{1}, g_{2}, \dots, g_{N})^{T} \in L_{1}^{{\times}N} (0,+\infty) \equiv L_{1} (0,+\infty) {\times}\dots {\times} L (0,+\infty)$$ is the independent term of the equation with nonnegative components and $$0 \leq K_{ij} \in L_{1} (-\infty,+\infty)$$, $$i, j = 1, 2,\dots, N$$ are the kernel-functions. These equations have significant applications in the wave non-local interaction theory. Using some special factorization methods, solvability of the system is proved in different functional spaces.
##### MSC:
 45J05 Integro-ordinary differential equations 45E05 Integral equations with kernels of Cauchy type 45G05 Singular nonlinear integral equations 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 47G20 Integro-differential operators
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