Fixed-point theorem for an infinite Toeplitz matrix and its extension to general infinite matrices
Abstract
UDC 517.9
In [V. M. Abramov, Bull. Austral. Math. Soc., 104, 108–117 (2021)], the fixed-point equation was studied for an infinite nonnegative particular Toeplitz matrix. In the present paper, we provide an alternative proof for the existence of a positive solution in the general case. The presented proof is based on the application of a version of the M. A. Krasnosel'ski fixed-point theorem. The results are then extended to the equations with infinite matrices of a general type.
References
V. M. Abramov, Fixed point theorem for an infinite Toeplitz matrix, Bull. Aust. Math. Soc., 104, 108–117 (2021). DOI: https://doi.org/10.1017/S0004972720001215
V. M. Abramov, Optimal control of a large dam with compound Poisson input and costs depending on water levels, Stochastics, 91, № 3, 433–483 (2019). DOI: https://doi.org/10.1080/17442508.2018.1551395
F. Başar, Summability theory and its applications, 2nd ed., CRC Press/Taylor & Francis Group, Boca Raton etc. (2022). DOI: https://doi.org/10.1201/9781003294153
T. A. Burton, A fixed point theorem of Krasnosel'skii, Appl. Math. Lett., 11, 85–88 (1998). DOI: https://doi.org/10.1016/S0893-9659(97)00138-9
T. A. Burton, T. Furumochi, Krasnosel'skii fixed point theorem and stability, Nonlinear Anal., 49, 445–454 (2002). DOI: https://doi.org/10.1016/S0362-546X(01)00111-0
R. A. Horn, C. R. Johnson, Matrix analysis, Cambridge Univ. Press, Cambridge (1988).
M. A. Krasnosel'skii, Some problems of nonlinear analysis, Amer. Math. Soc. Transl. Ser. 2, 10, 345–409 (1958). DOI: https://doi.org/10.1090/trans2/010/13
M. A. Krasnosel'skii, J. A. Lifshits, A. V. Sobolev, Positive linear systems, Heldermann Verlag, Berlin (1989).
M. A. Krasnosel'skii, G. M. Vainikko, P. P. Zabreiko, Y. B. Rutitskii, V. Y. Stetsenko, Approximate solutions of operator equations, Wolters-Noordhoff, Groningen (1972). DOI: https://doi.org/10.1007/978-94-010-2715-1
W. Leontief, Input-output economics, 2nd ed., Oxford Univ. Press, Oxford (1986).
L. Liu, Z. Li, Krasnosel'skii type fixed point theorems and applications, Proc. Amer. Math. Soc., 136, 1213–1220 (2008). DOI: https://doi.org/10.1090/S0002-9939-07-09190-3
D. O'Regan, Fixed point theory for the sum of two operators, Appl. Math. Lett., 9, 1–8 (1996). DOI: https://doi.org/10.1016/0893-9659(95)00093-3
V. M. Sehgal, S. P. Singh, On a fixed point theorem of Krasnosel'skii for locally convex spaces, Pacific J. Math., 62, 561–567 (1976). DOI: https://doi.org/10.2140/pjm.1976.62.561
E. K. Shah, M. Sarvar, D. Beleanu, Study of Krasnoselkii's fixed point theorem for Caputo–Fabrizio fractional differentional equations, Adv. Difference Equat., 2020, Article 178 (2020). DOI: https://doi.org/10.1186/s13662-020-02624-x
D. R. Smart, Fixed point theorems, Cambridge Univ. Press, Cambridge (1980).
L. Takács, Combinatorial methods in the theory of stochastic processes, John Wiley, New York (1967).
L. Takács, On the busy periods of single-server queues with Poisson input and general service times, Oper. Res., 24, № 3, 564–571 (1976). DOI: https://doi.org/10.1287/opre.24.3.564
F. Trèves, Topological vector spaces, distributions and kernels, Dover Publ., New York (2006).
F. Wang, F. Wang, Krasnosel'skii type fixed point theorem for nonlinear expansion, Fixed Point Theory, 13, 285–291 (2012). DOI: https://doi.org/10.1186/1687-1812-2012-107
W. K. Williams, V. Vijayakumar, R. Udhayakumar, K. S. Nisar, A new study on existence and uniqueness of nonlocal fractional delay differential systems of order $1, Numer. Methods Partial Different. Equat., 37, 949–961 (2021). DOI: https://doi.org/10.1002/num.22560
T. Xiang, R. Yuan, A class of expansive-type Krasnosel'skii fixed point theorems, Nonlinear Anal., 71, 3229–3239 (2009). DOI: https://doi.org/10.1016/j.na.2009.01.197
T. Xiang, R. Yuan, Critical type of Krasnosel'skii fixed point theorem, Proc. Amer. Math. Soc., 139, 1033–1044 (2011). DOI: https://doi.org/10.1090/S0002-9939-2010-10517-8
T. Xiang, R. Yuan, A note on Krasnosel'skii fixed point theorem, Fixed Point Theory and Appl., 2015, Article 99 (2015). DOI: https://doi.org/10.1186/s13663-015-0351-0
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