Abstract
In this paper, we are concerned with the existence of traveling wave solutions in nonlocal delayed higher-dimensional lattice systems with quasi-monotone nonlinearities. By using the upper and lower solution method and Schauder’s fixed point theorem, we establish the existence of traveling wave solutions. To illustrate our results, the existence of traveling wave solutions for a nonlocal delayed higher-dimensional lattice cooperative system with two species are considered.
(Communicated by Michal Fečkan)
References
[1] Britton, N. F.: Travelling Wave Front Solutions of a Differential-Difference Equation Arising in the Modelling of Myelinated Nerve Axon. Ordinary and Partial Differential Equations, Lecture Notes in Math. 1151, Springer, Berlin, 1985.10.1007/BFb0074717Search in Google Scholar
[2] Cahn, J. W.—Mallet-Paret, J.—Van Vleck, E. S.: Traveling wave solutions for systems of ODE’s on a two-dimensional spatial lattice, SIAM J. Appl. Math. 59 (1998), 455–493.10.1137/S0036139996312703Search in Google Scholar
[3] Chen, X.—Guo, J.: Uniqueness and existence of travelling waves for discrete quasilinear monostable dynamics, Math. Ann. 326 (2003), 123–146.10.1007/s00208-003-0414-0Search in Google Scholar
[4] Chen, X.—Guo, J.: Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Differential Equations 184 (2002), 549–569.10.1006/jdeq.2001.4153Search in Google Scholar
[5] Cheng, C. P.—Li, W. T.—Wang, Z. C.—Zheng, S. Z.: Traveling waves connecting equilibrium and periodic orbit for a delayed population model on a two-dimensional spatial lattice, Int. J. Bifurcat. Chaos Appl. Sci. Engrg. 26 (2016), Art. No. 1650049.10.1142/S0218127416500498Search in Google Scholar
[6] Chi, H.—Bell, J.—Hassard, B.: Numerical solutions of a nonlinear advanced-delay differential equation from nerve condition theory, J. Math. Biol. 24 (1986), 583–601.10.1007/BF00275686Search in Google Scholar PubMed
[7] Chow, S. N.—Mallet-Paret, J.—Shen, W.: Traveling waves in lattice dynamical systems, J. Differential Equations 149 (1998), 248–291.10.1006/jdeq.1998.3478Search in Google Scholar
[8] Guo, J. S.—Wu, C. H.: Existence and uniqueness of traveling waves for a monostable 2-D lattice dynamical system, Osaka J. Math. 45 (2008), 327–346.Search in Google Scholar
[9] Guo, J. S.—Wu, C. H.: Traveling wave front for a two-component lattice dynamical system arising in competition models, J. Differential Equations 252 (2012), 4357–4391.10.1016/j.jde.2012.01.009Search in Google Scholar
[10] Hankerson, D.—Zinner, B.: Wave fronts for a cooperative triangonal system of differential equations, J. Dynam. Differential Equations 5 (1993), 359–373.10.1007/BF01053165Search in Google Scholar
[11] Huang, J. H.—Lu, G.—Zou, X.: Existence of traveling wave fronts of delayed lattice differential equations, J. Math. Anal. Appl. 298 (2004), 538–558.10.1016/j.jmaa.2004.05.027Search in Google Scholar
[12] Huang, J. H.—Lu, G.—Ruan, S.: Traveling wave solutions in delayed lattice differential equations with partial monotonicity, Nonlinear Anal. TMA 60 (2005), 1331–1350.10.1016/j.na.2004.10.020Search in Google Scholar
[13] Huang, J. H.—Huang, L. H.: Traveling wavefronts in systems of delayed reaction diffusion equations on higher dimentional lattices, Acta Math. Appl. Sin. 28 (2005), 100–113.Search in Google Scholar
[14] Keener, J. P.: Propagation and its failure to coupled systems of discrete excitable cells, SIAM J. Appl. Math. 47 (1987), 556–572.10.1137/0147038Search in Google Scholar
[15] Lin, G.—Li, W. T.: Traveling waves in delayed lattice dynamical systems with competition interactions, Nonlinear Anal. RWA 11 (2010), 3666–3679.10.1016/j.nonrwa.2010.01.013Search in Google Scholar
[16] Lin, G.—Li, W. T.—Pan, S. X.: Travelling wavefronts in delayed lattice dynamical systems with global interaction, J. Difference Equ. Appl. 16 (2010), 1429–1446.10.1080/10236190902828387Search in Google Scholar
[17] Ma, S.—Liao, X.—Wu, J.: Traveling wave solution for planar lattice differential systems with applications to neural networks, J. Differential Equations 182 (2002), 269–297.10.1006/jdeq.2001.4109Search in Google Scholar
[18] Mallet-Paret, J.: The Fredholm alternative for functional-differential equations of mixed type, J. Dynam. Differential Equations 11 (1999), 1–47.10.1023/A:1021889401235Search in Google Scholar
[19] Mallet-Paret, J.: The global structure of traveling waves in spatially discrete dynamical systems, J. Dynam. Differential Equations 11 (1999), 49–127.10.1023/A:1021841618074Search in Google Scholar
[20] Nekorkin, V. I.—Kazantsev, V. B.—Morfu, S.—Bilbault, J. M.—Marquié, P.: Theoretical and experimental study of two discrete coupled Nagumo chains, Phys. Rev. E 64 (2001), Art No. 036602.10.1103/PhysRevE.64.036602Search in Google Scholar
[21] Shen, W.: Traveling waves in time periodic lattice differential eqations, Nonlinear Analysis 54 (2003), 319–339.10.1016/S0362-546X(03)00065-8Search in Google Scholar
[22] Weng, P. X.: Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip, Discrete Contin. Dyn. Syst. Ser. B 12 (2009), 883–904.10.3934/dcdsb.2009.12.883Search in Google Scholar
[23] Wu, S. L.—Liu, S. Y.: Travelling waves in delayed reaction-diffusion equations on higher-dimensional lattices, J. Difference Equ. Appl. 19 (2013), 384–401.10.1080/10236198.2011.645815Search in Google Scholar
[24] Wu, J.—Zou, X.: Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations, J. Differential Equations 135 (1997), 315–357.10.1006/jdeq.1996.3232Search in Google Scholar
[25] Yu, Z. X.—Yuan, R.: Nonlinear stability of wavefronts for a delayed stage-structured population model on a 2-D lattice, Osaka J. Math. 50 (2013), 963–976.Search in Google Scholar
[26] Yu, Z. X.—Zhang, W. G.—Wang, X. M.: Spreading speeds and travelling waves for non-monotone time-delayed 2D lattice systems, Math. Comput. Model. 58 (2013), 1510–1521.10.1016/j.mcm.2013.06.009Search in Google Scholar
[27] Zhao, H. Q.—Wu, S. L.: Wave propagation for a reaction-diffusion model with a quiescent stage on a 2D spatial lattice, Nonlinear Anal. RWA 12 (2011), 1178–1191.10.1016/j.nonrwa.2010.09.011Search in Google Scholar
[28] Zhao, H. Q.: Asymptotic stability of traveling fronts in delayed reaction-diffusion monostable equations on higher-dimensional lattices, Electron. J. Differential Equations 2013 (2013), 119.Search in Google Scholar
[29] Zinner, B.: Stability of traveling wave fronts for the discrete Nagumo equation, SIAM J. Math. Anal. 22 (1991), 1016–1020.10.1137/0522066Search in Google Scholar
[30] Zinner, B.: Existence of traveling wave front solutions for the discrete Nagumo equation, J. Differential Equations 96 (1992), 1–27.10.1016/0022-0396(92)90142-ASearch in Google Scholar
[31] Zinner, B.—Harris, G.—Hudson, W.: Traveling wave fronts for the discrete Fisher’s equation, J. Differential Equations 105 (1993), 46–62.10.1006/jdeq.1993.1082Search in Google Scholar
[32] Zou, X.: Traveling wave fronts in spatially discrete reaction-diffusion equations on higher-dimensional lattices, Proceedings of the 3rd Mississippi State Conference on Difference Equations and Computational Simulations (Mississippi State, MS, 1997), 211–221, Electron. J. Differ. Equ. Conf. 1, Southwest Texas State Univ., San Marcos, TX, 1998.Search in Google Scholar
[33] Zou, X.—Wu, J.: Local existence and stability of periodic traveling waves of lattice functional-differential equations, Can. Appl. Math. Quart. 6 (1998), 397–418.Search in Google Scholar
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Articles in the same Issue
- Regular Papers
- Semidistributivity and Whitman Property in implication zroupoids
- Composition of binary quadratic forms over number fields
- On Z-Symmetric Rings
- On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences
- Coefficient estimates for Libera type bi-close-to-convex functions
- Oscillation of nonlinear third-order differential equations with several sublinear neutral terms
- On rapidly oscillating solutions of a nonlinear elliptic equation
- Multiplicity of solutions for a class of fourth-order elliptic equations of p(x)-Kirchhoff type
- Existence of traveling wave solutions in nonlocal delayed higher-dimensional lattice systems with quasi-monotone nonlinearities
- On absolute double summability methods with high indices
- On the continuity of lattice isomorphisms on C(X, I)
- New fixed point theorems for countably condensing maps with an application to functional integral inclusions
- Common fixed point results under w-distance with applications to nonlinear integral equations and nonlinear fractional Differential Equations
- The form of locally defined operators in waterman spaces
- Conformal vector fields on almost co-Kähler manifolds
- A certain η-parallelism on real hypersurfaces in a nonflat complex space form
- On log-bimodal alpha-power distributions with application to nickel contents and erosion data
- Univariate and bivariate extensions of the generalized exponential distributions
- Pellian equations of special type