Li, Yongxiang Positive doubly periodic solutions of nonlinear telegraph equations. (English) Zbl 1036.35020 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 55, No. 3, 245-254 (2003). This interesting paper discusses the existence of (weak) solutions (i.e. understood in the distribution sense) to the nonlinear telegraph equation \[ u_{tt}- u_{x,x}+cu_t= -a(t, x)u+ b(t, x) f (t, x, u), \] \((t, x)\in \mathbb{R}^2\) being positive doubly periodic, i.e. such that \(u(t + 2\pi, x)= u(t, x + 2\pi)= u(t, x)\) on \(\mathbb{R}^2\), where \(a\), \(b\) and \(f\) are continuous, doubly periodic real functions on \(\mathbb{R}^2\) and \(\mathbb{R}^3\), respectively. The approach involves the use of the well-known Krasnoselskij fixed point theorem in cones. The main result concerning the existence is proved under rather mild assumptions allowing the superlinear as well as sublinear growth of the nonlinearity \(f\). An illustrative example ends the paper. Reviewer: Wojciech Kryszewski (Toruń) Cited in 22 Documents MSC: 35B15 Almost and pseudo-almost periodic solutions to PDEs 47H10 Fixed-point theorems 35L70 Second-order nonlinear hyperbolic equations 35B10 Periodic solutions to PDEs 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces Keywords:Krasnoselskij fixed point theorem; fixed point theorem of cone mapping PDFBibTeX XMLCite \textit{Y. Li}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 55, No. 3, 245--254 (2003; Zbl 1036.35020) Full Text: DOI References: [1] Birnir, B.; Grauer, R., An explicit description of the global attractor of the damped and driven sine-Gordon equation, Comm. Math. Phys., 162, 539-590 (1994) · Zbl 0805.35122 [2] W. Eloe, P.; Henderson, J., Positive solutions for \((n\)−1,1) conjugate boundary value problems, Nonlinear Anal., 28, 1669-1680 (1997) · Zbl 0871.34015 [3] Erbe, L. H.; Wang, H., On the existence of the positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120, 743-748 (1994) · Zbl 0802.34018 [4] Fucik, S.; Mawhin, J., Generated periodic solution of nonlinear telegraph equation, Nonlinear Anal., 2, 609-617 (1978) · Zbl 0381.35056 [5] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press New York · Zbl 0661.47045 [6] Kim, W. S., Double-periodic boundary value problem for nonlinear dissipative hyperbolic equations, J. Math. Anal. Appl., 145, 1-16 (1990) · Zbl 0717.35057 [7] Kim, W. S., Multiple doubly periodic solutions of semilinear dissipative hyperbolic equations, J. Math. Anal. Appl., 197, 735-748 (1996) · Zbl 0870.35069 [8] J. Mawhin, Periodic solution of nonlinear telegraph equations, in: A.R. Bedlarek, L. Cesari (Eds.), Dynamical Systems, Academic Press, New York, 1977.; J. Mawhin, Periodic solution of nonlinear telegraph equations, in: A.R. Bedlarek, L. Cesari (Eds.), Dynamical Systems, Academic Press, New York, 1977. · Zbl 0547.35077 [9] Ortega, R.; Robles-Perez, A. M., A maximum principle for periodic solutions of the telegraph equations, J. Math. Anal. Appl., 221, 625-651 (1998) · Zbl 0932.35016 [10] Ortega, R.; Robles-Perez, A. M., A duality theorem for periodic solutions of a class of second order evolution equations, J. Differential Equations, 172, 409-444 (2001) · Zbl 1010.34054 [11] Vejvoda, O., Partial Differential Equations: Time-Periodic Solutions (1982), Nijhoff: Nijhoff Prague · Zbl 0183.10401 [12] Yongxiang Li, Positive periodic solutions of nonlinear second-order ordinary differential equations, Acta Math. Sinica, 45 (2002) 481-488, (in Chinese).; Yongxiang Li, Positive periodic solutions of nonlinear second-order ordinary differential equations, Acta Math. Sinica, 45 (2002) 481-488, (in Chinese). · Zbl 1018.34046 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.