电报方程的正双周期解: 存在性、唯一性、多重性和渐近性

Abstract

Abstract:

By using topological degree theory and the theory of convex operator, this paper discusses the existence, uniqueness and multiplicity of positive doubly periodic solutions to a class of telegraph equations, as well as the asymptotic behavior of positive doubly periodic solutions for telegraph equations.

Key words: Telegraph equation, Positive doubly periodic solution, Existence, uniqueness and multiplicity, Asymptotic behavior, Convex operator

CLC Number:

O175.2

Nan Deng,Meiqiang Feng. Positive Doubly Periodic Solutions To Telegraph Equations: Existence, Uniqueness, Multiplicity and Asymptotic Behavior[J].Acta mathematica scientia,Series A, 2022, 42(5): 1360-1380.

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References 25

1	Roussy G , Pearcy J A . Foundations and Industrial Applications of Microwaves and Radio Frequency Fields. New York: Wiley, 1995
2	Barbu V . Nonlinear boundary value problems for a class of hyperbolic systems. Rev Roum Math Pures Appl, 1977, 22, 155- 168
3	Deng N , Feng M . New results of positive doubly periodic solutions to telegraph equations. Electron Res Arch, 2022, 30, 1104- 1125 doi: 10.3934/era.2022059
4	Li Y . Positive doubly periodic solutions of nonlinear telegraph equations. Nonlinear Anal, 2003, 55, 245- 254 doi: 10.1016/S0362-546X(03)00227-X
5	Wang F , An Y . Existence and multiplicity results of positive doubly periodic solutions for nonlinear telegraph system. J Math Anal Appl, 2009, 349, 30- 42 doi: 10.1016/j.jmaa.2008.08.003
6	Wang F , An Y . Doubly periodic solutions to a coupled telegraph system. Nonlinear Anal, 2012, 75, 1887- 1894 doi: 10.1016/j.na.2011.09.039
7	Li Y . Maximum principles and the method of upper and lower solutions for time-periodic problems of the telegraph equations. J Math Anal Appl, 2007, 327, 997- 1009 doi: 10.1016/j.jmaa.2006.04.066
8	Ortega R , Robles-Pérez A M . A maximum principle for periodic solutions of the telegraph equations. J Math Anal Appl, 1998, 221, 625- 651 doi: 10.1006/jmaa.1998.5921
9	Mawhin J , Ortega R , Robles-Pérez A M . Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications. J Differential Equations, 2005, 208, 42- 63 doi: 10.1016/j.jde.2003.11.003
10	Mawhin J , Ortega R , Robles-Pérez A M . A maximum principle for bounded solutions of the telegraph equation in space dimension three. C R Acad Sci Paris Ser I, 2002, 334, 1089- 1094 doi: 10.1016/S1631-073X(02)02406-8
11	Mawhin J , Ortega R , Robles-Pérez A M . A maximum principle for bounded solutions of the telegraph equations and applications to nonlinear forcings. J Math Anal Appl, 2000, 251, 695- 709 doi: 10.1006/jmaa.2000.7038
12	Gilding B H , Kersner R . Wavefront solutions of a nonlinear telegraph equation. J Differential Equations, 2013, 254, 599- 636 doi: 10.1016/j.jde.2012.09.007
13	Hilbert D . Über die gerade Linie als küzeste Verbindung zweier Punkte. Math Ann, 1895, 46, 91- 96 doi: 10.1007/BF02096204
14	Birkhoff G . Extensions of Jentzch's theorem. Trans Amer Math Soc, 1957, 85, 219- 227
15	Bushell P J . Hilbert's metric and positive contraction mappings in a Banach space. Arch Rational Mech Anal, 1973, 52, 330- 338 doi: 10.1007/BF00247467
16	Feng M . Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior. Adv Nonlinear Anal, 2021, 10, 371- 399
17	Feng M , Zhang X . On a $ k $-Hessian equation with a weakly superlinear nonlinearity and singular weights. Nonlinear Anal, 2020, 190, 111601 doi: 10.1016/j.na.2019.111601
18	Zhang X , Feng M . The existence and asymptotic behavior of boundary blow-up solutions to the $ k $-Hessian equation. J Differential Equations, 2019, 267, 4626- 4672 doi: 10.1016/j.jde.2019.05.004
19	Zhang X , Feng M . Boundary blow-up solutions to the Monge-Ampère equation: Sharp conditions and asymptotic behavior. Adv Nonlinear Anal, 2020, 9, 729- 744
20	Aviles P . On isolated singularities in some nonlinear partial differential equations. Indiana Univ Math J, 1983, 32, 773- 791 doi: 10.1512/iumj.1983.32.32051
21	Gidas B , Spruck J . Global and local behavior of positive solutions of nonlinear elliptic equations. Comm Pure Appl Math, 1981, 34, 525- 598 doi: 10.1002/cpa.3160340406
22	Amann H . Fixed point equations and nonlinear eigenvalue problems in order Banach spaces. SIAM Rev, 1976, 18, 620- 709 doi: 10.1137/1018114
23	Guo D , Lakshmikantham V . Nonlinear Problems in Abstract Cones. New York: Academic Press, 1988
24	郭大钧. 非线性泛函分析. 山东: 山东科学技术出版社, 1985
	Guo D J . Nonlinear Functional Analysis. Shandong: Shandong Science and Technology Press, 1985
25	Hardy G H , Littlewood J E , Pólya G . Inequalities. Second Edition. New York: Cambridge University Press, 1952

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Positive Doubly Periodic Solutions To Telegraph Equations: Existence, Uniqueness, Multiplicity and Asymptotic Behavior

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