Regularity of functional equations on manifolds

Summary. In this paper the regularity properties of the functional equation¶¶ f (t) = h(t, y, f (g₁(t, y)), ... , f (g_n(t,y))) f (t) = h(t, y, f (g_{1}(t, y)), ... , f (g_{n}(t,y))) ¶ on a \Cal C^￥ {\Cal C}^\infty manifold for the unknown function f are treated. Under general conditions it is proved that solutions which are measurable or have the Baire property are in \Cal C^￥ {\Cal C}^\infty .     

An analysis of the solution set to a homotopy equation between polynomials with real coefficients

We investigate the structure of the solution setS to a homotopy equationH(Z,t)=0 between two polynomialsF andG with real coefficients in one complex variableZ. The mapH is represented asH(x+iy, t)=h ₁(x, y, t)+ih ₂(x, y, t), whereh ₁ andh ₂ are polynomials from ℝ² × [0,1] into ℝ and i is the imaginary unit. Since all the coefficients ofF andG are real, there is a polynomialh ₃ such thath ₂(x, y, t)=yh₃(x, y, t). Then the solution setS is divided into two sets {(x, t)∶h ₁(x, 0, t)=0} and {(x+iy, t)∶h ₁(x, y, t)=0,h ₃(x, y, t)=0}. Using this division, we make the structure ofS clear. Finally we briefly explain the structure of the solution set to a homotopy equation between polynomial systems with real coefficients in several variables.     

Solutions of functional equations having bounded variation

Summary. We prove that a solution f of the functional equation¶¶f(t)=h(t,y,f(g₁(t,y)),...,f(g_n(t,y))) f(t)=h(t,y,f(g_1(t,y)),\dots,f(g_n(t,y))) ¶ having locally bounded variation is a C^￥ {\cal C}^\infty -function.     

On affine functions with respect to some means

The purpose of the present paper is to investigate the functional equation

On the mixed problem for hyperbolic partial differential-functional equations of the first order

We consider the mixed problem for the hyperbolic partial differential-functional equation of the first order where is a function defined by z _(x,y)(t, s) = z(x + t, y + s), (t, s) [–, 0] × [0, h]. Using the method of bicharacteristics and the method of successive approximations for a certain integral-functional system we prove, under suitable assumptions, a theorem of the local existence of generalized solutions of this problem.     

Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients

The following system considered in this paper:

Existence and global attractivity of a positive periodic solution of a class of delay differential equation

The existence and the global attractivity of a positive periodic solution of the delay differential equationy(t)=y(t) F[t, y](t-τ ₁ (t)),⋯,y(t−τ _n (t))] are studied by using some techniques of the Mawhin coincidence degree theory and the constructing suitable Liapunov functionals. When these results are applied to some special delay bio-mathematics models, some new results are obtained, and many known results are improved. Project partially supported by the National Natural Science Foundation of China (Grant No. 10572057) and the Applied Basic Research Foundation of Yunnan Province.     

齐型空间中分数次积分交换子的加权端点估计

该文得到齐型空间中分数次积分交换子[b,I_α]的加权端点估计ω({x∈X:|[b,I_α]f(x)|t})≤Cψ(∫_xA(||b||_*(|f(x)|/t)■(ω(x))dμ(x))其中b∈BMO(X,d,μ),A(t)=tlog(e+t),ψ(t)=[tlog(e+t~α)]~(1/(1-α)),■(t)=t~(1-α)log(e+t~(-α)).     

Essential and discrete spectra of partially integral operators

Let Ω₁, Ω₂ ⊂ ℝ^ν be compact sets. In the Hilbert space L ₂(Ω₁ × Ω₂), we study the spectral properties of selfadjoint partially integral operators T ₁, T ₂, and T ₁ + T ₂, with

A note on a problem by H. S. Shapiro

В РАБОтЕ ДОкАжАНО, ЧтО limk _a *f(x)=f(x) пОЧтИ ВсУДУ, гДЕk _a(t)=a^?n k(a^?1t), t?Rⁿ, Для Дль ДОВОльНО шИРОкОг О клАссА ФУНкцИИk(t). ДАНы УслОВИь, пРИ кОтО Рых пОлУЧЕННыИ РЕжУл ьтАт РАспРОстРАНьЕтсь НА ФУНкцИУ $$k(x,y) = \gamma \frac{1}{{1 + |x|^\alpha }} \cdot \frac{1}{{1 + |y|^\beta }},$$ гДЕ α, β>1, А γ — НОРМИРУУЩ ИИ МНОжИтЕль тАкОИ, Чт О∫∫k(x, y) dx dy=1.     

On second order functional differential equations and inequalities with deviating arguments

A necessary and sufficient condition is established in order that (i) the retarded differential equation $$y''(t) = p_0 y(t) + f(y(t - \tau _1 ),...,y(t - \tau _N ))$$ has no bounded nonoscillatory solution and (ii) the advanced differential equation $$y''(t) = p_0 y(t) + f(y(t + \tau _1 ),...,y(t + \tau _N ))$$ has no unbounded nonoscillatory solution, wherep ₀≥0 and τ _j > 0,1 ?i ?N, are constants. Differential inequalities related to (^*) and (^**) are also studied. Finally, an oscillation criterion is given for a class of differential equations containing both retarded and advanced arguments.     

On a Nonlocal BVP with Nonlinear Boundary Conditions

We consider the existence of at least one positive solution of the problem ${-y''(t)=f(t,y(t)), y(0)=H_1(\varphi(y))+\int_{E}H_2(s,y(s))\,ds, y(1)=0}$ , where ${y(0)=H_1(\varphi(y))+\int_{E}H_2(s,y(s))\,ds}$ represents a nonlinear, nonlocal boundary condition. We show by imposing some relatively mild structural conditions on f, H ₁, H ₂, and ${\varphi}$ that this problem admits at least one positive solution. Finally, our results generalize and improve existing results, and we give a specific example illustrating these generalizations and improvements.     

A collocation by spline in tension

The collocation method by spline in tension for the problem: −εy"+p(x)y=f(x), y(0)=α₀,y(1)=α₁, p(x)>0, 0<ε<<1, is derived. The method has the second order of the global uniform convergence. For the corresponding difference scheme the optimal estimate: O (h_imin(h_i, ε) is obtained. This research was supported partly by NSF and SIZ for Science of SAP Vojvodina through funds made available to the U.S.—Yugoalav Joint Board on Scientific and Tchnological Cooperation (grants JF554, JF799).     

On the existence and uniqueness of minimizers for a class of integral functionals

We study the solvability of the minimization problem

Extensions of the measurable choice theorem by means of forcing

Using the method of forcing of set theory, we prove the following two theorems on the existence of measurable choice functions: LetT be the closed unit interval [0,1] and letm be the usual Lebesgue measure defined on the Borel subsets ofT. Theorem1. LetS⊂T×T be a Borel set such that for alltεT,S _t ^def={x|(t,x)εS} is countable and non-empty. Then there exists a countable series of Lebesgue-measurable functionsf _n: T→T such thatS _t={f_n(t)|nεω} for alltε[0,1],W _x={y|(x,y)εW} is uncountable. Then there exists a functionh:[0,1]×[0,1]→W with the following properties: (a) for each xε[0,1], the functionh(x,·) is one-one and ontoW _x and is Borel measurable; (b) for eachy, h(·, y) is Lebesgue measurable; (c) the functionh is Lebesgue measurable.     

The domination of g-evaluations and Choquet evaluations

In this paper, we obtain that a convex g evaluation can be dominated by the corresponding Choquet evaluation if and only if g has the form g(t, y, z) = μ t y + h(t, z), where h(t, z) is positively homogeneous and subadditive with respect to z.     

A combinatorial interpretation for two transformations of series that commute with compositional inversion

For a formal power series $\text{g(t) =}\text{1}\text{[1 + ∑}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{h}{}_{\mathrm{n}}\text{t}{}^{\mathrm{n}}\text{]}$ with nonnegative integer coefficients, the compositional inverse $\text{f}\text{(t) = t · f(t)}$ of $\text{g}\text{(t) = t · g(t)}$ is shown to be the generating function for the colored planted plane trees in which each vertex of degree i + 1 is colored one of h_i colors. Since the compositional inverse of the Euler transformation of f(t) is the star transformation [[g(t)]^?1 ? 1]^?1 of g(t), [2], it follows that the Euler transformation of f(t) is the generating function for the colored planted plane trees in which each internal vertex of degree i + 1 is colored one of h_i colors for i > 1, and h₁ ? 1 colors for i = 1.     

On a multiplicative functional transformation arising in nonlinear filtering theory

Summary This paper concerns the nonlinear filtering problem of calculating estimates E[f(x_t)¦y _s, st] where {x _t} is a Markov process with infinitesimal generator A and {y _t} is an observation process given by dy _t=h(x_t)dt +dw_twhere {w _t} is a Brownian motion. If h(x_t) is a semimartingale then an unnormalized version of this estimate can be expressed in terms of a semigroup T _s,t ^y obtained by a certain y-dependent multiplicative functional transformation of the signal process {x _t}. The objective of this paper is to investigate this transformation and in particular to show that under very general conditions its extended generator is A _t ^y f=e^y(t)h(A– 1/2h²)(e^–y(t)h f).Work partially supported by the U.S. Department of Energy (Contract ET-76-C-01-2295) at the Massachusetts Institute of Technology     

Asymptotic expansions of solutions of equations with a deviating argument in Banach spaces

For the equation $$Lu = \frac{1}{i}\frac{{du}}{{dt}}\sum\nolimits_{j = 0}^m {A_j u} (l - h_j^0 - h_j^1 (t)) = f(t),$$ whereh ₀ ^o =0,h ₀ ¹ =0 (t) ≡ 0,h _j ^o = const > 0,h ₁ ^j (t),j= 1, ...,m are nonnegative continuously differentiable functions in [0, ∞), Aj are bounded linear operators, under conditions on the resolvent and on the right hand sidef(t), we have obtained an asymptotic formula for any solution u(t) from L₂ in terms of the exponential solutions u_k(t), k=1, ..., n, of the equation $$\frac{1}{i}\frac{{du}}{{dt}} - A_0 u - \sum\nolimits_{j = 0}^m {A_j u} (t - h_j^0 ) = 0,$$ connected with the poles λ_k, k=1, ..., n, of the resolvent R_λ in a certain strip.