On some variational algebraic problems

Bisci, Giovanni Molica; Repovš, Dušan

doi:10.1515/anona-2012-0028

Cited by 8 publications

(3 citation statements)

References 23 publications

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“…We can mention the following works without being exhaustive at any level. [16][17][18] We consider the following system:…”

Section: Application To the Partial Difference Equationsmentioning

confidence: 99%

“…There are very many results in this area concerning both existence and multiplicity of results. We can mention the following works without being exhaustive at any level 16‐18 . We consider the following system:

\begin{matrix} left \\ array [u (i + 1, j) - 2 u (i, j) + u (i - 1, j)] + [u (i, j + 1) - 2 u (i, j) + u (i, j - 1)] \\ array + λ f ((i, j), u (i, j)) = 0, \\ array for all i \in {1, . ., m}, j \in {1, \dots, n}, \\ array u (i, 0) = u (i, n + 1) = 0 for all i \in {1, . ., m}, \\ array u (0, j) = u (m + 1, j) = 0 for all j \in {1, \dots, n}, \end{matrix}

which may be viewed as the discrete counterpart of the problem

\begin{array}{l} \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + λ f false(false(x, y false), u false(x, y false) false) = 0, \\ u false(x, 0 false) = u false(x, n ... \end{array}

…”

Section: Applicationsmentioning

confidence: 99%

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Localization properties for nonlinear equations involving monotone operators

Galewski

2020

Math Methods in App Sciences

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Using monotonicity methods, the Lagrange multiplier rule, and some variational arguments, we consider a type of localization results pertaining to the existence of critical points to action functionals on a closed ball. A variant of the Schechter critical point theorem on a ball in Hilbert and Banach spaces is obtained. Applications to nonlinear Dirichlet problem and to partial difference equations are given in the final part of this paper.

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“…We can mention the following works without being exhaustive at any level. [16][17][18] We consider the following system:…”

Section: Application To the Partial Difference Equationsmentioning

confidence: 99%

\begin{matrix} left \\ array [u (i + 1, j) - 2 u (i, j) + u (i - 1, j)] + [u (i, j + 1) - 2 u (i, j) + u (i, j - 1)] \\ array + λ f ((i, j), u (i, j)) = 0, \\ array for all i \in {1, . ., m}, j \in {1, \dots, n}, \\ array u (i, 0) = u (i, n + 1) = 0 for all i \in {1, . ., m}, \\ array u (0, j) = u (m + 1, j) = 0 for all j \in {1, \dots, n}, \end{matrix}

which may be viewed as the discrete counterpart of the problem

\begin{array}{l} \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + λ f false(false(x, y false), u false(x, y false) false) = 0, \\ u false(x, 0 false) = u false(x, n ... \end{array}

…”

Section: Applicationsmentioning

confidence: 99%

Localization properties for nonlinear equations involving monotone operators

Galewski

2020

Math Methods in App Sciences

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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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