Georgiev, Svetlin Georgiev; Majdoub, Mohamed Existence of solutions for a class of IBVP for nonlinear hyperbolic equations. (English) Zbl 1452.35090 SN Partial Differ. Equ. Appl. 1, No. 4, Paper No. 22, 20 p. (2020). Summary: We study a class of initial boundary value problems of hyperbolic type. A new topological approach is applied to prove the existence of non-negative classical solutions. The arguments are based upon a recent theoretical result. MSC: 35L20 Initial-boundary value problems for second-order hyperbolic equations 35L70 Second-order nonlinear hyperbolic equations 47H10 Fixed-point theorems 58J20 Index theory and related fixed-point theorems on manifolds Keywords:hyperbolic equations; positive solution; fixed point; cone; sum of operators PDFBibTeX XMLCite \textit{S. G. Georgiev} and \textit{M. Majdoub}, SN Partial Differ. Equ. Appl. 1, No. 4, Paper No. 22, 20 p. (2020; Zbl 1452.35090) Full Text: DOI arXiv References: [1] Agarwal, RP; O’Regan, D.; Stanek, S., Positive and maximal positive solutions of singular mixed boundary value problem, Cent. Eur. J. Math., 7, 694-716 (2009) · Zbl 1193.34039 [2] Benzenati, L.; Mebarki, K., Multiple positive fixed points for the sum of expansive mappings and \(k\)-set contractions, Math. Methods Appl. Sci., 42, 4412-4426 (2019) · Zbl 1447.47043 [3] Benzenati, L., Mebarki, K., Precup, R.: A vector version of the fixed point theorem of cone compression and expansion for a sum of two operators. Nonlinear Stud. Accepted [4] Bociu, L.; Lasiecka, I., Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping, Appl. Math., 35, 281-304 (2008) · Zbl 1152.35416 [5] Buchukuri, T.; Chkadua, O.; Natroshvili, D., Mixed boundary value problems of thermopiezoelectricity for solids with interior cracks, Integral Equ. Oper. Theory, 64, 495-537 (2009) · Zbl 1183.35263 [6] Brown, R.; Mitrea, I.; Mitrea, M.; Wright, M., Mixed boundary value problems for the Stokes system, Trans. Am. Math. Soc., 362, 1211-1230 (2010) · Zbl 1187.35038 [7] Deimling, K., Nonlinear Functional Analysis (1985), Berlin: Springer, Berlin · Zbl 0559.47040 [8] Djebali, S.; Mebarki, K., Fixed point theory for sums of mappings, J. Nonlinear Convex Anal., 19, 1029-1040 (2018) [9] Djebali, S.; Mebarki, K., Fixed point index for expansive perturbation of \(k-\) set contraction mappings, Topol. Methods Nonlinear Anal., 54, 613-640 (2019) · Zbl 1442.37036 [10] Drabek, P.; Milota, J., Methods in Nonlinear Analysis, Applications to Differential Equations (2007), Basel: Birkhäuser, Basel · Zbl 1176.35002 [11] Guo, D.; Cho, YI; Zhu, J., Partial Ordering Methods in Nonlinear Problems (2004), New York: Nova Science Publishers, New York [12] Georgiev, SG; Mebarki, K., Existence of positive solutions for a class ODEs, FDEs and PDEs via fixed point index theory for the sum of operators, Commun. Appl. Nonlinear Anal., 26, 16-40 (2019) [13] Godin, P., The blow-up curve of solutions of mixed problems for semilinear wave equations with exponential nonlinearities in one space dimension, II, Annales de l’I. H. P., 17, 779-815 (2000) · Zbl 0977.35088 [14] Vitillaro, K., Global existence for the wave equation with nonlinear boundary damping and source term, J. Differ. Equ., 186, 259-298 (2002) [15] Kharibegashvili, S.; Shavlakadze, V.; Jokhadze, O., On the solvability of a mixed problem for an one-dimensional semilinear wave equation with a nonlinear boundary condition, Proc. NAS Armenia Math., 53, 31-51 (2018) · Zbl 1407.35127 [16] Nowakowski, A., Solvability and stability of a semilinear wave equation with nonlinear boundary conditions, Nonlinear Anal. Theory, Methods Appl., 73, 1495-1514 (2010) · Zbl 1195.35200 [17] Polyanin, A.; Manzhirov, A., Handbook of Integral Equations (1998), Boca Raton: CRC Press, Boca Raton · Zbl 0896.45001 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.