# Two nonnegative solutions for two-dimensional nonlinear wave equations

- Svetlin Georgiev svetlingeorgiev1@gmail.com
- Mohamed Majdoub mmajdoub@iau.edu.sa

## Downloads

## DOI:

https://doi.org/10.56754/0719-0646.2403.0393## Abstract

We study a class of initial value problems for two-dimensional nonlinear wave equations. A new topological approach is applied to prove the existence of at least two nonnegative classical solutions. The arguments are based upon a recent theoretical result.

## Keywords

S. Alinhac, “The null condition for quasilinear wave equations in two space dimensions I”, Invent. Math., vol. 145, no. 3, pp. 597–618, 2001.

L. Benzenati and K. Mebarki, “Multiple positive fixed points for the sum of expansive map- pings and k-set contractions”, Math. Methods Appl. Sci., vol. 42, no. 13, pp. 4412–4426, 2019.

L. Benzenati, K. Mebarki and R. Precup, “A vector version of the fixed point theorem of cone compression and expansion for a sum of two operators”, Nonlinear Stud., vol. 27, no. 3, pp. 563–575, 2020.

L. Benzenati, S. G. Georgiev and K. Mebarki, “Existence of positive solutions for some kinds of BVPs in Banach spaces”, submitted for publication.

D. Christodoulou, “Global solutions of nonlinear hyperbolic equations for small data”, Comm. Pure Appl. Math., vol. 39, no. 2, pp. 267–282, 1986.

K. Deimling, Nonlinear functional Analysis, Heidelberg: Springer Berlin, 1985.

S. Djebali and K. Mebarki, “Fixed Point Theory for Sums of Operators”, J. Nonlinear Convex Anal., vol. 19, no. 6, pp. 1029–1040, 2018.

S. Djebali and K. Mebarki, “Fixed point index for expansive perturbation of k-set contraction mappings”, Topol. Methods Nonlinear Anal., vol. 54, no. 2, pp. 613–640, 2019.

D. Duffy, Green‘s function with Applications, 1st edition, Boca Raton: Chapman & Hall/CRC Press, 2001.

S. G. Georgiev and Z. Khaled, Multiple fixed-point theorems and applications in the theory of ODEs, FDEs and PDEs, Monographs and research notes in mathematics, Boca Raton: CRC Press, 2020.

S. G. Georgiev and K. Mebarki, “Existence of positive solutions for a class ODEs, FDEs and PDEs via fixed point index theory for the sum of operators”, Comm. Appl. Nonlinear Anal., vol. 26, no. 4, pp. 16–40, 2019.

S. G. Georgiev and K. Mebarki, “On fixed point index theory for the sum of operators and applications in a class ODEs and PDEs”, submitted for publication.

S. G. Georgiev, K. Mebarki and Kh. Zennir, “Existence of solutions for a class of nonlinear hyperbolic equations”, submitted for publication.

S. G. Georgiev, K. Mebarki and Kh. Zennir, “Existence of solutions for a class IVP for nonlinear wave equations”, submitted for publication.

S. Georgiev, A. Kheloufi and K. Mebarki, “Classical solutions for the Korteweg-De Vries equation”, New Trends in Nonlinear Analysis and Applications, to be published.

S. Georgiev and K. Mebarki, “Leggett-Williams fixed point theorem type for sums of two operators and application in PDEs”, Differ. Equ. Appl., vol. 13, no. 3, pp. 321–344, 2021.

N. M. Hung, “Asymptotic behaviour of solutions of the first boundary-value problem for strongly hyperbolic systems near a conical point at the boundary of the domain”, Sb. Math., vol. 190, no. 7, pp. 1035–1058, 1999.

S. Ibrahim, M. Majdoub and N. Masmoudi, “Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity”, Comm. Pure Appl. Math., vol. 59, no. 11, pp. 1639–1658, 2006.

F. John and S. Klainerman, “Almost global existence to nonlinear wave equations in three space dimensions”, Comm. Pure Appl. Math., vol. 37, no. 4, pp. 443–455, 1984.

S. Klainerman, “The null condition and global existence to nonlinear wave equations”, Lectures in Appl. Math., vol. 23, pp. 293–326, 1986.

Z. Lei, T. C. Sideris and Y. Zhou, “Almost Global existence for two dimensional incompressible isotropic elastodynamics”, Trans. Amer. Math. Soc., vol. 367, no. 11, pp. 8175–8197, 2015.

A. Polyanin and A. Manzhirov, Handbook of integral equations, Boca Raton: CRC Press, 1998.

Christopher D. Sogge, Lectures on nonlinear wave equations, 2nd Edition, Boston: International press, Inc., 2013.

### Downloads

## Published

## How to Cite

*CUBO*, vol. 24, no. 3, pp. 393–412, Dec. 2022.