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Asymptotic behaviour of solutions of third order nonlinear difference equations of neutral type

Anna Andruch-Sobiło, Andrzej Drozdowicz (2008)

Mathematica Bohemica

In the paper we consider the difference equation of neutral type Δ 3 [ x ( n ) - p ( n ) x ( σ ( n ) ) ] + q ( n ) f ( x ( τ ( n ) ) ) = 0 , n ∈ ℕ ( n 0 ) , where p , q : ℕ ( n 0 ) → ℝ + ; σ , τ : ℕ → ℤ , σ is strictly increasing and lim n → ∞ σ ( n ) = ∞ ; τ is nondecreasing and lim n → ∞ τ ( n ) = ∞ , f : ℝ → ℝ , x f ( x ) > 0 . We examine the following two cases: 0 < p ( n ) ≤ λ * < 1 , σ ( n ) = n - k , τ ( n ) = n - l , and 1 < λ * ≤ p ( n ) , σ ( n ) = n + k , τ ( n ) = n + l , where k , l are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as n → ∞ with a weaker assumption on q than the...

Cauchy problems for discrete affine minimal surfaces

Marcos Craizer, Thomas Lewiner, Ralph Teixeira (2012)

Archivum Mathematicum

In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes,...

Cheeger inequalities for unbounded graph Laplacians

Frank Bauer, Matthias Keller, Radosław K. Wojciechowski (2015)

Journal of the European Mathematical Society

We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.

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