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Maximal regularity of discrete and continuous time evolution equations

Sönke Blunck (2001)

Studia Mathematica

We consider the maximal regularity problem for the discrete time evolution equation u n + 1 - T u ₙ = f ₙ for all n ∈ ℕ₀, u₀ = 0, where T is a bounded operator on a UMD space X. We characterize the discrete maximal regularity of T by two types of conditions: firstly by R-boundedness properties of the discrete time semigroup ( T ⁿ ) n ∈ ℕ ₀ and of the resolvent R(λ,T), secondly by the maximal regularity of the continuous time evolution equation u’(t) - Au(t) = f(t) for all t > 0, u(0) = 0, where A:= T - I. By recent results of...

Motion of spiral-shaped polygonal curves by nonlinear crystalline motion with a rotating tip motion

Tetsuya Ishiwata (2015)

Mathematica Bohemica

We consider a motion of spiral-shaped piecewise linear curves governed by a crystalline curvature flow with a driving force and a tip motion which is a simple model of a step motion of a crystal surface. We extend our previous result on global existence of a spiral-shaped solution to a linear crystalline motion for a power type nonlinear crystalline motion with a given rotating tip motion. We show that self-intersection of the solution curves never occurs and also show that facet extinction never...

Multiplicity and uniqueness for a class of discrete fractional boundary value problems

Lv Zhanmei, Gong Yanping, Chen Yi (2014)

Applications of Mathematics

The paper deals with a class of discrete fractional boundary value problems. We construct the corresponding Green's function, analyse it in detail and establish several of its key properties. Then, by using the fixed point index theory, the existence of multiple positive solutions is obtained, and the uniqueness of the solution is proved by a new theorem on an ordered metric space established by M. Jleli, et al. (2012).

Multipoint boundary value problems for discrete equations

Pavel Drábek, Harold Bevan Thompson, Christopher Tisdell (2001)

Commentationes Mathematicae Universitatis Carolinae

In this work we establish existence results for solutions to multipoint boundary value problems for second order difference equations with fully nonlinear boundary conditions involving two, three and four points. Our results are also applied to systems.

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