On the existence of positive solutions for a class of singular boundary value problems

In this paper, by the use of a fixed point theorem, many new necessary and sufficient conditions for the existence of positive solutions in C[0,1]∩C¹[0,1]∩C²(0,1) or C[0,1]∩C²(0,1) are presented for singular superlinear and sublinear second-order boundary value problems. Singularities at t=0, t=1 will be discussed.     

Positive solution of singular Dirichlet boundary value problems for second order differential equation system

This paper investigates the existence of positive solutions of singular Dirichlet boundary value problems for second order differential system. A necessary and sufficient condition for the existence of C[0,1]×C[0,1] positive solutions as well as C¹[0,1]×C¹[0,1] positive solutions is given by means of the method of lower and upper solutions and the fixed point theorems. Our nonlinearity fi(t,x₁,x₂) may be singular at x₁=0, x₂=0, t=0 and/or t=1, i=1,2.     

On the equivalence of the centered Gaussian measure in L 2 with the correlation operator (−d 2/dx 2)−1 and the conditional Wiener measure

Let w and µ be respectively the conditional Wiener measure in C ₀([0, 1]) and the centered Gaussian measure in L ₂[0, 1] with the correlation operator (?d ²/dx ²)^?1. We prove the equivalence of these two measures in the following sense: for any Borel set A ? L ₂[0, 1] the set A ∩ C ₀([0, 1]) is a Borel subset of C ₀([0, 1]) and µ(A) = w(A∩C ₀([0, 1])).     

The method of lower and upper solutions for fourth order singular m-point boundary value problems

This paper investigates the existence of positive solutions for fourth order singular m-point boundary value problems. Firstly, we establish a comparison theorem, then we define a partial ordering in C²[0,1]∩C⁴(0,1) and construct lower and upper solutions to give a necessary and sufficient condition for the existence of C²[0,1] as well as C³[0,1] positive solutions. Our nonlinearity f(t,x,y) may be singular at x, y, t=0 and/or t=1.     

Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions

We characterize finite codimensional linear isometries on two spaces, C ⁽ⁿ⁾[0; 1] and Lip [0; 1], where C ⁽ⁿ⁾[0; 1] is the Banach space of n-times continuously differentiable functions on [0; 1] and Lip [0; 1] is the Banach space of Lipschitz continuous functions on [0; 1]. We will see they are exactly surjective isometries. Also, we show that C ⁽ⁿ⁾[0; 1] and Lip [0; 1] admit neither isometric shifts nor backward shifts.     

Wentzell-Robin boundary conditions on C[0,1]

Given a ∈ C ¹[0,1], a(x) ≥ α > 0, we prove that the second order differential operator on C[0,1] defined by A _W u := (au′)′ with Wentzell-Robin boundary conditions $$\left( {au'} \right)^\prime \left( j \right) + \beta _j u'\left( j \right) + \gamma _j u\left( j \right) = 0, j = 0,1,$$ where β _j and γ _j are real numbers, generates a holomorphic C ₀-semigroup on C[0,1].     

On a multiplicative inequaluty for derived functions

В работе доказываетс я следующее неравенс тво. Пусть α₀, α₁, α₂, - произво льные неотрицательн ые числа α₀≠α₂. Тогда, еслиx(t) люб ая функция, для которой п роизводнаяx непреры вна и функция \(x^{a_0 } (\dot x)^{a_1 } (\ddot x)^{a_2 } \) принадлежи т пространствуL _∞[0, 1], то (*) $$\left\| x \right\|_{H^r [0,1]} \leqq c\left\| {x^{a_0 } (\dot x)^{a_1 } (\ddot x)^{a_2 } } \right\|L_{\infty [0,1]} ,$$ где ∥ · ∥H ^r [0,1] - норма в кл ас се функций на отрезке [0, 1], обладающих в простра нствеL _∞[0, 1] дробной производной гельдеровского типа порядкаr=(α₁+2α₂)/(α₀+α₁+α₂);с - конста нта, зависящая только от α₀, α₁, α₂. Это неравенство является точным в том смысле, что показател ьr есть максимальный, п ри котором неравенст во (*) имеет место с конечной конс тантойс. При α₀=α_? появляются логарифмические доб авки. Хорошо известно, что д ля непрерывной на [0, 1] фу нкции частные суммы Фурье п о тригонометрической системе равномерно с уммируются к ней методом (С, 1). И. Пра йс доказал, что для любой неограниченной последовательности целых положительных чисел {P _k} _k ^∞ =1 и Для любогоa∈[0, 1] существует непрерыв ная на [0, 1] функция, ряд Фурье которой по ортонормированно й мультипликативной системе (OHMC) не суммируется методом (С, 1) в точкеx=a. СССР, МОСКВА 103 055 УЛ. ОБРАЗЦОВА 15 МОСКОВСКИЙ ИНСТИТУТ ИНЖЕНЕРОВ ЖЕЛЕЗНОДО РОЖНОГО ТРАНСТПОРТА     

Analysis of a two-phase cell division model

In this paper we mainly study an equal mitosis two-phase cell division model. By using the C₀-semigroup theory, we prove that this model is the well-posed in L¹[0, 1] × L¹[0, 1], and exhibits asynchronous exponential growth phenomenon as time goes to infinity. We also give a comparison between this two-phase model with the classical one-phase model. Finally, we briefly study the asymmetric two-phase cell division model, and show that similar results hold for it.     

Schauder estimates for oblique derivative problems

We prove that the solution of the oblique derivative parabolic problem in a noncylindrical domain Ω^T belongs to the anisotropic Holder space C^{2+α, 1+α/2}(gwT) 0 < α < 1, even if the nonsmooth “lateral boundary” of Ω^T is only of class C^{1+α, (1+α)/2}). As a corollary, we also obtain an a priori estimate in the Hölder space C^2+α(Ω₀) for a solution of the oblique derivative elliptic problem in a domain Ω₀ whose boundary belongs only to the classe C^1+α.     

Fourth order singular p-Laplacian differential equations with integral boundary conditions

In this paper, we investigate the existence of positive solutions for fourth order singular p-Laplacian differential equations with integral boundary conditions and non-monotonic function terms. Firstly, we establish a comparison theorem, then we define a partial ordering in E ₀ and construct lower and upper solutions to give a necessary and sufficient condition for the existence of C ²[0,1] as well as pseudo-C ³[0,1] positive solutions. Our nonlinearity f(t,x,y) may be singular at x=0, y=0, t=0 and t=1. Finally, we give some the dual results for the other cases of fourth order singular integral boundary value problems and an example to demonstrate the corresponding main results.     

A straightforward generalization of Diliberto and Straus' algorithm does not work

An algorithm for best approximating in the sup-norm a function f C[0, 1]² by functions from tensor-product spaces of the form π_k C[0, 1] + C[0, 1] π_l, is considered. For the case k = L = 0 the Diliberto and Straus algorithm is known to converge. A straightforward generalization of this algorithm to general k, l is formulated, and an example is constructed demonstrating that this algorithm does not, in general, converge for k² + l² > 0.     

Minimal Positive Solutions to Some Singular Second-Order Differential Equations

The existence of a minimal C¹[0, 1] positive solution is established for some second-order singular boundary value and initial value problems by new schemes, which are related to x′. Our nonlinearity may be singular at t = 0, 1, x = 0, or x′ = 0.     

Lower semicontinuity and the theorem of Datko and Pazy

Let {T(t)} _t≥0 be aC ₀-semigroup on a real or complex Banach spaceX and letJ:C ⁺[0,∞)→[0,∞] be a lower semicontinuous and nondecreasing functional onC ⁺[0,∞), the positive cone ofC[0,∞), satisfyingJ(c 1)=∞ for allc>0. We prove the following result: if {T(t)} _t≥0 is not uniformly exponentially stable, then the set $\{ x \in X: J(||T( \cdot )x||) = \infty \}$ is residual inX.     

Zero product preserving maps on

The main result of the paper characterizes continuous bilinear maps from C¹[0,1]×C¹[0,1] into a Banach space X with the property that (f,g)=0 whenever fg=0. This is applied to the study of zero product preserving operators on C¹[0,1], and operators on C¹[0,1] satisfying a version of the condition of the locality of an operator.     

The Exact Value of Normal Structure Coefficients and WCS coefficients in a Class of Orlicz Function Spaces

Let φ be an N-function. Then the normal structure coefficients N and the weakly convergent sequence coefficients WCS of the Orlicz function spaces L ^φ[0, 1] generated by φ and equipped with the Luxemburg and Orlicz norms have the following exact values. (i) If F _φ(t) = t _?(t)/φ(t) is decreasing and 1 < C _φ < 2 (where \(C_\Phi = \lim _{t \to + \infty } t\varphi (t)/\Phi (t)\)), then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1?1/Cφ. (ii) If F _φ(t) is increasing and C _φ > 2, then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1/Cφ.     

Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach

In this article, we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(x) in the linear parabolic equation ut(x,t)=(k(x)uxx(x,t)), with Dirichlet boundary conditions u(0,t)=ψ₀, u(1,t)=ψ₁. Main goal of this study is to investigate the distinguishability of the input-output mappings Φ[⋅]:K→C¹[0,T], Ψ[⋅]:K→C¹[0,T] via semigroup theory. In this paper, we show that if the null space of the semigroup T(t) consists of only zero function, then the input-output mappings Φ[⋅] and Ψ[⋅] have the distinguishability property. Moreover, the values k(0) and k(1) of the unknown diffusion coefficient k(x) at x=0 and x=1, respectively, can be determined explicitly by making use of measured output data (boundary observations) f(t):=k(0)ux(0,t) or/and h(t):=k(1)ux(1,t). In addition to these, the values k^′(0) and k^′(1) of the unknown coefficient k(x) at x=0 and x=1, respectively, are also determined via the input data. Furthermore, it is shown that measured output dataf(t) and h(t) can be determined analytically, by an integral representation. Hence the input-output mappings Φ[⋅]:K→C¹[0,T], Ψ[⋅]:K→C¹[0,T] are given explicitly in terms of the semigroup. Finally by using all these results, we construct the local representations of the unknown coefficient k(x) at the end points x=0 and x=1.     

Theorems on asymptotics for singular Sturm-Liouville operators with various boundary conditions

We consider the Sturm-Liouville operator L(y) = ?d ² y/dx ² + q(x)y in the space L ₂[0, π], where the potential q(x) is a complex-valued distribution of the first order of singularity; i.e., q(x) = u′(x), u ∈ L ²[0, π]. (Here the derivative is understood in the sense of distributions.) We obtain asymptotic formulas for the eigenvalues and eigenfunctions of the operator in the case of the Neumann-Dirichlet conditions [y ^[1](0) = 0, y(π) = 0] and Neumann conditions [y ^[1](0) = 0, y ^[1](π) = 0] and refine similar formulas for all types of boundary conditions. The leading and second terms of asymptotics are found in closed form.     

Spectral asymptotics and trace formulas for differential operators with unbounded coefficients

In the space L ²[0, π], we consider the operators $$ L = L_0 + V, L_0 = - y'' + (\nu ^2 - 1/4)r^{ - 2} y (\nu \geqslant 1/2) $$ with the Dirichlet boundary conditions. The potential V is the operator of multiplication by a function (in general, complex-valued) in L ²[0, π] satisfying the condition $$ \int\limits_0^\pi {r^\varepsilon } (\pi - r)^\varepsilon |V(r)|dr < \infty , \varepsilon \in [0,1] $$ . We prove the trace formula Σ _n=1 ^∞ [µ _n ? λ _n ? Σ _k=1 ^m α _k ⁽ⁿ⁾ ] = 0.     

On vector-valued curves that allow a C 1,α parametrization

Let X be a Banach space and α ∈ (0, 1]. We find equivalent conditions for a function f: [0,1] → X to admit an equivalent parametrization, which is C ^1,α (i.e., has α-Hölder derivative). For X = ?, a characterization is well-known. However, even in the case X = ?² several new ideas are needed.