On Some Algebraic Properties of Semi-Discrete Hyperbolic Type Equations

Nonlinear semi-discrete equations of the form t_x(n+1)=f(t(n), t(n+1), t_x(n)) are studied. An adequate algebraic formulation of the Darboux integrability is discussed and the attempt to adopt this notion to the classification of Darboux integrable chains has been undertaken.Comment: 18 page

Existence and uniqueness of functional differential equations with n delay

In this paper we give a necessary and suffcient conditions for the existence and uniqueness of periodic solutions of functional differential equations with n delay d dt x(t) = Ax(t) + n j=1 Bx(t -- r j) + f (t). The conditions are obtained in terms of R-boundedness of operator valued Fourier multipliers

Oscillation and nonoscillation of functional differential equations

A qualitative approach is usually concerned with the behavior of solutions of a given differential equation and usually does not seek specific explicit solutions;This dissertation is the analysis of nonoscillation of even order delay differential equations, and the oscillation of solutions of arbitrary order functional differential equations. This is done mainly in Chapters II and III. Chapter IV deals with the oscillation and nonoscillation of bounded solutions of n-th order delay differential equations. By an oscillatory solution we mean that the solution has infinitely many zeros; otherwise, it is called nonoscillatory solution;The functional differential equations under consideration are:(UNFORMATTED TABLE OR EQUATION FOLLOWS)≤ (A) [r(t)y[superscript]\u27(t)][superscript](2n-1) + [sigma][subscript]spi=1k P[subscript]i(t) F[subscript]i[y[subscript][tau](t),y[subscript]sp[sigma][subscript]1\u27(t),...,y[subscript]sp[sigma][subscript]2n-1(2n-1)&(t)] & = f(t), n ≥ 1& x[superscript](n)(t) + [sigma] [sigma][subscript]spi=1m P[subscript]i(t) x[g[subscript]i(t)] + [delta] q(t) x[h(t)]& = f(t), n ≥ 3& (B) L[subscript]n x(t) + (-1)[superscript]n-1 [sigma][subscript]spi=1m a[subscript]i(t)f[x(g[subscript]i(t)]& = b(t), n ≥ 1& (C) (TABLE/EQUATION ENDS);Equation (A) is considered in Chapter II, where we find sufficient conditions for which all solutions of equation (A) are nonnegative. Equation (B) is studied in Chapter III for the cases: [delta] = 0, [sigma] = 0 and [sigma] = [delta] = ±1 i.e., delay arguments, advanced arguments, and mixed type arguments, respectively. Finally, equation (C) is studied in Chapter IV. Sufficient conditions are found to force all bounded solutions to be oscillatory, and sufficient conditions are also found for which every bounded solution of equation (C) tends to zero as t → [infinity]

Vortex equations in abelian gauged sigma-models

We consider nonlinear gauged sigma-models with Kahler domain and target. For a special choice of potential these models admit Bogomolny (or self-duality) equations -- the so-called vortex equations. We find the moduli space and energy spectrum of the solutions of these equations when the gauge group is a torus T^n, the domain is compact, and the target is C^n or CP^n. We also obtain a large family of solutions when the target is a compact Kahler toric manifold.Comment: v2: 60 pages, more details than in CMP versio

Maximal theorems and square functions for analytic operators on Lp-spaces

Let T : Lp --> Lp be a contraction, with p strictly between 1 and infinity, and assume that T is analytic, that is, there exists a constant K such that n\norm{T^n-T^{n-1}} < K for any positive integer n. Under the assumption that T is positive (or contractively regular), we establish the boundedness of various Littlewood-Paley square functions associated with T. As a consequence we show maximal inequalities of the form \norm{\sup_{n\geq 0}\, (n+1)^m\bigl |T^n(T-I)^m(x) \bigr |}_p\,\lesssim\, \norm{x}_p, for any nonnegative integer m. We prove similar results in the context of noncommutative Lp-spaces. We also give analogs of these maximal inequalities for bounded analytic semigroups, as well as applications to R-boundedness properties