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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.
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
Full Text: DOI
[1] G. A. Baraff, ”Transmission of Electromagnetic Waves Through a conducting slab. The Two-Sided Wiener-Hopf Solution,” J.Math. Phys. 9(3), 372–384 (1968). · Zbl 0164.42903 · doi:10.1063/1.1664589
[2] J. Casti and R. Kalaba, Embedding Methods in Applied Mathematics (Addison-Wesley, Reading, Mass.-London-Amsterdam, 1973). · Zbl 0265.65001
[3] E. Volterra, ”On Elastic Continual Hereditary Characteristics,” Journal of Applied Mechanics 18, 273–279 (1951). · Zbl 0043.39405
[4] B.D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation (Cambridge University Press, 1981).
[5] N. B. Yengibaryan and A. Kh. Khachatryan, ”Integro-Differential Equation of Non-Local Wave Interaction,” Mat. Sbornik 198(3), 839–855 (2007). · Zbl 1155.45003 · doi:10.1070/SM2007v198n06ABEH003863
[6] Kh. A. Khachatryan and E. A. Khachatryan, ”Solvability of Convolution Type Integro-Differential Equation on R,” Izv. NAN Armenii, Matematika [Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)] 42(3) 161–175 (2007). · Zbl 1151.45008
[7] Kh. A. Khachatryan and M. G. Kostanyan, ”Factorization of Convolution Type Vector Integro-Differential Equations,” Izv. NAN Armenii, Matematika [Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)] 43(3) 145–156 (2008). · Zbl 1153.45008
[8] M. A. Krasnoselskii, P. P. Zabreyko, E. I. Pustilnik and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions (Nauka, Moscow, 1966).
[9] M. A. Krasnoselskii, Positive solutions of Operator Equations (Fizmatizdom, Moscow, 1962).
[10] N. B. Yengibaryan and A. A. Harutyunyan, ”Integral Equations on the Half-Line With Difference Kernels and Non-Linear Functional Equations,” Mat. Sbornik 97(1), 35–58 (1975).
[11] L.G. Arabajyan and N. B. Yengibaryan, ”Convolution Equations andNon-Linear functional equations,” Itogi Nauki i Tekhniki, Mat. Analiz 22, 175–242 (1984).
[12] A. N. Kolmogorov and V. S. Fomin, Elements of Function Theory and Functional Analysis (Nauka, Moscow, 1981). · Zbl 0501.46002
[13] N. B. Yengibaryan and L. G. Arabajyan, ”Systems of Wiener-Hopf Integral Equations and Non-Linear Equations of factorization,” Mat. Sbornik 124(6), 189–216 (1984).
[14] N. B. Yengibaryan, ”Renewal Theorems for a Systemof Integral Equations,” Sbornik, Mathematics 189(12), 1795–1808 (1998). · Zbl 0932.45005 · doi:10.1070/SM1998v189n12ABEH000360
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