Landesman–Lazer conditions for difference equations involving sublinear perturbations

Volek, Jonáš

doi:10.1080/10236198.2016.1234617

Cited by 9 publications

(4 citation statements)

References 20 publications

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“…More broadly, note that a large part of our paper considered the problem of finding the stationary solution (2.2). The nonlinear algebraic equations have been studied recently by many authors and via various techniques, e.g., [11,19,20,28]. We believe that applications of some of these techniques could improve our estimates.…”

Section: Final Remarks and Open Problemsmentioning

confidence: 96%

Exponential number of stationary solutions for Nagumo equations on graphs

Stehlı́k¹

2017

Journal of Mathematical Analysis and Applications

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We study the Nagumo reaction-diffusion equation on graphs and its dependence on the underlying graph structure and reaction-diffusion parameters. We provide necessary and sufficient conditions for the existence and nonexistence of spatially heterogeneous stationary solutions. Furthermore, we observe that for sufficiently strong reactions (or sufficiently weak diffusion) there are 3 n stationary solutions out of which 2 n are asymptotically stable. Our analysis reveals interesting relationship between the analytic properties (diffusion and reaction parameters) and various graph characteristics (degree distribution, graph diameter, eigenvalues). We illustrate our results by a detailed analysis of the Nagumo equation on a simple graph and conclude with a list of open questions.

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Section: Final Remarks and Open Problemsmentioning

confidence: 96%

Exponential number of stationary solutions for Nagumo equations on graphs

Stehlı́k¹

2017

Journal of Mathematical Analysis and Applications

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“…Turning 1 We use italic letters for double sequences (e.g., u for solutions of (LDE)) and roman ones for vectors (e.g., u for solutions of (GDE)). 2 Additionally, it is well-known that the problem of finding stationary solutions of graph differential equations on cycles G = Cn is actually equivalent to periodic discrete boundary value problems [15,16]. 3 We omit the case of n = 2.…”

Section: Periodic Solutions and Solutions Of Graph Nagumo Equationmentioning

confidence: 99%

“…2 Additionally, it is well-known that the problem of finding stationary solutions of graph differential equations on cycles G = Cn is actually equivalent to periodic discrete boundary value problems [15,16]. 3 We omit the case of n = 2.…”

Section: Periodic Solutions and Solutions Of Graph Nagumo Equationmentioning

confidence: 99%

Counting and ordering periodic stationary solutions of lattice Nagumo equations

Hupkes

Morelli

Stehlı́k

et al. 2019

Applied Mathematics Letters

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We study the rich structure of periodic stationary solutions of Nagumo reaction diffusion equation on lattices. By exploring the relationship with Nagumo equations on cyclic graphs we are able to divide these periodic solutions into equivalence classes that can be partially ordered and counted. In order to accomplish this, we use combinatorial concepts such as necklaces, bracelets and Lyndon words.

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“…By considering the variational structure of relevant energy functionals, we can use standard techniques for coercive functionals, mountain-pass or saddle-point geometries. Consequently, our approach is similar to the ideas used in the area of nonlinear algebraic and difference equations [16], [11], [18], [27].…”

Section: Introductionmentioning

confidence: 99%

Nonuniqueness of implicit lattice Nagumo equation

Stehlı́k

Volek

2019

Appl.Math.

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We consider the implicit discretization of Nagumo equation on finite lattices and show that its variational formulation corresponds in various parameter settings to convex, mountain-pass or saddle-point geometries. Consequently, we are able to derive conditions under which the implicit discretization yields multiple solutions. Interestingly, for certain parameters we show nonuniqueness for arbitrarily small discretization steps. Finally, we provide a simple example showing that the nonuniqueness can lead to complex dynamics in which the number of bounded solutions grows exponentially in time iterations, which in turn implies infinite number of global trajectories.

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Landesman–Lazer conditions for difference equations involving sublinear perturbations

Cited by 9 publications

References 20 publications

Exponential number of stationary solutions for Nagumo equations on graphs

Exponential number of stationary solutions for Nagumo equations on graphs

Counting and ordering periodic stationary solutions of lattice Nagumo equations

Nonuniqueness of implicit lattice Nagumo equation

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