Abstract
The Volterra integral equation of the second order with a regular singularity is considered. Under the conditions that a kernel K(x,t) is a real matrix function of order n×n with continuous partial derivatives up to order N+1 inclusively and K(0,0) has complex eigenvalues ν±i μ (ν>0), it is shown that if ν>2|‖K|‖ C -N-1, then a given equation has two linearly independent solutions.
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References
T. Sato, “Sur l’equation integrale,” J. Math. Soc. Jpn., 5, No. 2, 145–153 (1953).
T. Takesada, “On the singular point of integral equations of Volterra type,” J. Math. Soc. Jpn., 7, No. 2, 123–136 (1955).
L. I. Panov, “About integral equations with kernels having not integrable singularity of arbitrary order,” Dokl. Akad. Nauk Tadzh. SSR, 10, No. 6, 3–7 (1967).
N. A. Magnitsky, “Multiparameter families of solutions of Volterra integral equations,” Dokl. Akad. Nauk SSSR, 240, No. 2, 268–271 (1978).
N. A. Magnitsky, Basic Analytical Theory of Volterra Integral Equations [in Russian], Institute for System Studies, Moscow (1986).
N. A. Magnitsky, Integral and Integro-Differential Volterra Equations with Singular Points (Analytical Theory) [in Russian], Institute for System Studies, Moscow (1978).
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Voronezh Forest Industry Academy, Voronezh. Published in Ukrainskii Matematicheskii Zhurnal Vol. 49, No. 3, pp. 424–432, March, 1997.
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Krein, S.G., Sapronov, I.V. One class of solutions of Volterra equations with regular singularity. Ukr Math J 49, 467–476 (1997). https://doi.org/10.1007/BF02487243
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DOI: https://doi.org/10.1007/BF02487243