The concept of strongly positive operator is generalized and the properties of the introduced operators are analyzed. The solutions of the Cauchy problem for a linear inhomogeneous differential equation with generalized strongly positive operator coefficient are obtained.
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References
K. Yosida, Functional Analysis, Springer, Berlin (1980).
S. G. Krein, Linear Differential Equations in Banach Space, Amer. Math. Soc., Providence, RI (1971).
E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Mat. Soc., Providence, RI (1957).
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1995).
V. L. Makarov and I. P. Gavrilyuk, “Exponentially convergent methods for the parallel discretization of the first-order evolution equations,” Dop. Nats. Akad. Nauk Ukr., No. 3, 24–28 (2002).
I. P. Gavrilyuk, W. Hackbusch, and B. N. Khoromskij, “Data-sparse approximation to the operator-valued functions of elliptic operator,” Math. Comput., 73, No. 247, 1297–1324 (2004).
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin (1981).
M. F. Gorodnii and A. V. Chaikovskii, “Generalization of the notion of sectorial operator,” Mat. Sb., 197, No. 7, 29–46 (2006).
I. P. Gavrilyuk and V. L. Makarov, Strongly Positive Operators and Algorithms without Accuracy Saturation [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2004).
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Dover, New York (1999).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 8, pp. 1053–1070, August, 2011.
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Il’chenko, Y.V., Chaikovs’kyi, A.V. Cauchy problem for a differential equation in the Banach space with generalized strongly positive operator coefficient. Ukr Math J 63, 1213–1233 (2012). https://doi.org/10.1007/s11253-012-0573-9
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DOI: https://doi.org/10.1007/s11253-012-0573-9