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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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