A Functional Equation Of J.A. Baker Involving Integral Means

The functional equation $$f(x)={1\over 2}\int^{x+1}_{x-1}f(t)\ dt\ \ \ {\rm for}\ \ \ x\ \in\ {\rm R}$$ has the linear functions ?(x) = a + bx (a, b ∈ ?) as trivial solutions. It is shown that there are two kinds of nontrivial solutions, (i) ?(x) = eλ_i ^x (i = 1, 2, …), where the λ_i∈ ? are the fixed points of the map z ? sinh z, and (ii) C^∞-solutions ? for which the values in the interval [?1,1] can be prescribed arbitrarily, but with the provision that ?^(j)(? 1) = ?^(j)(0) = ?^(j)(1) = 0 for all j = 0, 1, 2 …     

Parameter Dependence of Stable Manifolds for Nonuniform (μ, ν)-dichotomies

We construct stable invariant manifolds for semiflows generated by the nonlinear impulsive differential equation with parameters x'= A(t)x + f(t, x, λ), t≠τi and x(τ+i) = Bix(τi) + gi(x(τi), λ), i ∈ N in Banach spaces, assuming that the linear impulsive differential equation x'= A(t)x, t≠τi and x(τ+i) = Bix(τi), i ∈ N admits a nonuniform (μ, ν)-dichotomy. It is shown that the stable invariant manifolds are Lipschitz continuous in the parameter λ and the initial values provided that the nonlinear perturbations f, g are sufficiently small Lipschitz perturbations.     

A uniqueness result for a semilinear elliptic problem: A computer-assisted proof

Starting with the famous article [A. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243], many papers have been devoted to the uniqueness question for positive solutions of −Δu=λu+up in Ω, u=0 on ∂Ω, where p>1 and λ ranges between 0 and the first Dirichlet eigenvalue λ₁(Ω) of −Δ. For the case when Ω is a ball, uniqueness could be proved, mainly by ODE techniques. But very little is known when Ω is not a ball, and then only for λ=0. In this article, we prove uniqueness, for all λ∈[0,λ₁(Ω)), in the case Ω=²(0,1) and p=2. This constitutes the first positive answer to the uniqueness question in a domain different from a ball. Our proof makes heavy use of computer assistance: we compute a branch of approximate solutions and prove existence of a true solution branch close to it, using fixed point techniques. By eigenvalue enclosure methods, and an additional analytical argument for λ close to λ₁(Ω), we deduce the non-degeneracy of all solutions along this branch, whence uniqueness follows from the known bifurcation structure of the problem.     

Note on forced oscillation of nth-order sublinear differential equations

In the case of oscillatory potentials, we establish an oscillation theorem for the forced sublinear differential equation x(n)+q(t)λ|x|sgnx=e(t), t∈[t₀,∞). No restriction is imposed on the forcing term e(t) to be the nth derivative of an oscillatory function. In particular, we show that all solutions of the equation x^″+tαsintλ|x|sgnx=mtβcost, t?0, 0<λ<1 are oscillatory for all m≠0 if β>(α+2)/(1−λ). This provides an analogue of a result of Nasr [Proc. Amer. Math. Soc. 126 (1998) 123] for the forced superlinear equation and answers a question raised in an earlier paper [J.S.W. Wong, SIAM J. Math. Anal. 19 (1988) 673].     

Normal Forms for Nonautonomous Differential Equations

We extend Henry Poincarés normal form theory for autonomous differential equations x=f(x) to nonautonomous differential equations x=f(t, x). Poincarés nonresonance condition λ_j−∑ⁿ_i=1 ?_iλ_i≠0 for eigenvalues is generalized to the new nonresonance condition λ_j∩∑ⁿ_i=1 ?_iλ_i=∅ for spectral intervals.     

Asymptotic behavior of solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0

The author discusses the asymptotic behavior of the solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0 (1) where x(t) is an n-dimensional column vector and A, B are n × n matrices with complex constant entries. He obtains the following results for the case 0 < λ < 1: (i) If B is diagonalizable with eigenvalues b_i such that Re b_i < 0 for all i, then there is a constant α such that every solution of (1) is O(t^α) as t → ∞. (ii) If B is diagonalizable with eigenvalues b_i such that 0 < Re b₁ ? Re b₂ ? ··· ? Re b_n and λ times Re b_n < Re b₁, then every solution of (1) is O(e^bn^t) as t → ∞. For the case λ>1, he has the following results: (i) If B is diagonalizable with eigenvalues b_i such that Re b_i>0 for all i, then there is a constant α such that no solution x(t) of (1), except the identically zero solution, is 0(t^α) as t → ∞. (ii) If B is diagonalizable with eigenvalues b_i such that Re b₁ ? Re b₂ ? ··· ? Re b_n < 0 and λ Re b_n < Re b₁, then no solution x(t) of (1), except the identically zero solution, is 0(e^b1^t) as t → ∞.     

Numerical computation and analysis of the Titchmarsh–Weyl mα(λ) function for some simple potentials

This article is concerned with the Titchmarsh–Weyl m_α(λ) function for the differential equation d²y/dx²+[λ−q(x)]y=0. The test potential q(x)=x², for which the relevant m_α(λ) functions are meromorphic, having simple poles at the points λ=4k+1 and λ=4k+3, is studied in detail. We are able to calculate the m_α(λ) function both far from and near to these poles. The calculation is then extended to several other potentials, some of which do not have analytical solutions. Numerical data are given for the Titchmarsh–Weyl m_α(λ) function for these potentials to illustrate the computational effectiveness of the method used.     

A stability theorem for entropy solutions of initial value problems for first order quasilinear hyperbolic systems in several space variables

In this paper we prove an uniqueness and stability theorem for the solutions of Cauchy problem for the systems $$\frac{\partial }{{\partial t}}u + \sum\limits_{i = 1}^n { \frac{\partial }{{\partial x_i }} } f^i (x,t,u) = g(x,t,u),$$ whereu is a vector function (u ₁ (x, t),..., u _r (x, t)),f ⁱ =(a ₁ ⁱ (x, t, u),..., a _r ⁱ (x, t, u)), i=1,...,n, g=(g ₁ (x, t, u),...,g _r (x, t, u),i G ? ⁿ and t≥0. We use the concept of entropy solution introduced by Kruskov and improved by Lax, Dafermos and others autors. We assume that the Jacobian matricesf _u ⁱ are symmetric and the Hessian(a _j ⁱ ) _uu (i=1,...,n; j=1,...,r) are positive. We obtain uniqueness and stability inL _loc ² within the class of those entropy solutions which satisfy $$\frac{{u_j (---,x_i ,---,t)---u_j (---,y_i ,---,t)}}{{x_i - y_i }} \geqslant - K(t),$$ (i=1,...,n; j=1,...,r) for (?,x _i ,?,t), (?,y _i ,?,t) on a compact setD ? ? ⁿ x (0, ∞) and a functionK(t)∈L _loc ¹ ([0, ∞)) depending onD. Here we denote by (?,x _i ,?,t) and (?,y _i ,?,t) two points whose coordinates only differ in thei-th space variable. At the end we relax the hypotheses of symmetry and convexity on the system and give a theorem of uniqueness and stability for entropy solutions which are locally Lipschitz continuous on a strip ? ⁿ x [0,T].     

Eigenvalue Intervals for 2mth Order Sturm—Liouville Boundary Value Problems

Values of λ are determined for which there exist positive solutions of the 2mth order differential equation on a measure chain, (-1)^m x ^?2m (t)=λa(t)f(u(σ(t))), y? [0,1], satisfying α _i+1 u ^?21(0)+0, γ _i+1 u ^?21(σ(1))=0, 0≤i≤m?1 with αi,βi,γi,δi≥0, where a and f are positive valued, and both lim _x-0+ (f(x)/x) and lim _x-0+ (f(x)/x) exist.     

The functional differential equation x′(t) = x(x(t))

A classification of the solutions of the functional differential equation x′(t) = x(x(t)) is given and it is proved that every solution either vanishes identically or is strictly monotonie. For monotonically increasing solutions existence and uniqueness of the solution x are proved with the condition x(t₀) = x₀ where (t₀, x₀) is any given pair of reals in some specified subset of $\text{R}$². Every monotonically increasing solution is thus obtained. It is analytic and depends analytically on t₀ and x₀. Only for t₀ = x₀ = 1 is the question of analyticity still open.     

On the minimal positive homothetic image of a simplex containing a convex body

Let C be a convex body, and let S be a nondegenerate simplex in ? ⁿ . It is proved that the minimal coefficientσ > 0 for which the translate of σS contains C is $$\sum\limits_{j = 1}^{n + 1} {\mathop {\max \left( { - \lambda _j \left( x \right)} \right) + 1,}\limits_{x \in C} }$$ where λ ₁(x), ..., λ _n +1(x) are the barycentric coordinates of the point x ∈ ? ⁿ with respect to S. In the case C = [0, 1] ⁿ , this quantity is reduced to the form Σ _i=1 ⁿ 1/d _i (S), where d _i (S) is the ith axial diameter of S, i.e., the maximal length of the segment from S parallel to the ith coordinate axis.     

Uniqueness of solutions to an Abel type nonlinear integral equation on the half line

We consider a convolution-type integral equation u = k ? g(u) on the half line (???; a), a ?? ?, with kernel k(x) = x ^???1, 0 < ??, and function g(u), continuous and nondecreasing, such that g(0) = 0 and 0 < g(u) for 0 < u. We concentrate on the uniqueness problem for this equation, and we prove that if ?? ?? (1, 4), then for any two nontrivial solutions u ₁, u ₂ there exists a constant c ?? ? such that u ₂(x) = u ₁(x +c), ??? < x. The results are obtained by applying Hilbert projective metrics.     

Elliptic equation with a singular potential in a domain with a conic point

This paper deals with the behavior of the nonnegative solutions of the problem $$- \Delta u = V(x)u, \left. u \right|\partial \Omega = \varphi (x)$$ in a conical domain Ω ? ? ⁿ , n ≥ 3, where 0 ≤ V (x) ∈ L₁(Ω), 0 ≤ ?(x) ∈ L₁(?Ω) and ?(x) is continuous on the boundary ?Ω. It is proved that there exists a constant C _*(n) = (n ? 2)²/4 such that if V ₀(x) = (c + λ ₁)|x|^?2, then, for 0 ≤ c ≤ C _*(n) and V(x) ≤ V ₀(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ?(x) ∈ L ₁(?Ω); for c > C _*(n) and V(x) ≥ V ₀(x) in Ω, this problem has no nonnegative solutions if ?(x) > 0.     

Existence, uniqueness and stability ofC m solutions of iterative functional equations

In this paper we discuss a relatively general kind of iterative functional equation G(x,f(x), ...,f ⁿ (x)) = 0 (for allx ∈J), whereJ is a connected closed subset of the real number axis ?,G∈C ^m (J ⁿ⁺¹, ?) andn ≥ 2. Using the method of approximating fixed points by small shift of maps, choosing suitable metrics on functional spaces and finding a relation between uniqueness and stability of fixed points of maps of general spaces, we prove the existence, uniqueness and stability ofCm solutions of the above equation for any integer m ≥ 0 under relatively weak conditions, and generalize related results in reference in different aspects.     

On Existence for Polynomial-Like Iterative Equations

Many known results on the iterative equation α_i=1 ⁿ λ_i?ⁱ(x) = F(x) require a condition that λ₁ > 0 for technical reasons. A problem on the existence of solutions of this iterative equation with the natural restriction λ_n ≠ 0 is raised. In this paper we study an auxiliary functional equation for its invertible solutions. Then we apply our results on the auxiliary equation to solve the problem in some cases.     

Second order linear O.D.E. and Riccati equations

We prove that approximate solutions of the Riccati equation ?′ + ?² = a(x) yield asymptotic solutions y = e^∝x?(s)ds of the second order linear equation y″ = a(x)y. We show that the iterative scheme ?₀ = a, ?_{n + 1}² = a ? ?_n′ leads to asymptotic solutions of the cited linear equation in many interesting cases.     

Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝ d

In the present paper we analyze a class of tensor-structured preconditioners for the multidimensional second-order elliptic operators in ? ^d , d≥2. For equations in a bounded domain, the construction is based on the rank-R tensor-product approximation of the elliptic resolvent ? _R ≈(??λ I)^?1, where ? is the sum of univariate elliptic operators. We prove the explicit estimate on the tensor rank R that ensures the spectral equivalence. For equations in an unbounded domain, one can utilize the tensor-structured approximation of Green’s kernel for the shifted Laplacian in ? ^d , which is well developed in the case of nonoscillatory potentials. For the oscillating kernels e ^{?i κ‖x‖}/‖x‖, x∈? ^d , κ∈?₊, we give constructive proof of the rank-O(κ) separable approximation. This leads to the tensor representation for the discretized 3D Helmholtz kernel on an n×n×n grid that requires only O(κ?|log?ε|²? n) reals for storage. Such representations can be applied to both the 3D volume and boundary calculations with sublinear cost O(n ²), even in the case κ=O(n). Numerical illustrations demonstrate the efficiency of low tensor-rank approximation for Green’s kernels e ^?λ‖x‖/‖x‖, x∈?³, in the case of Newton (λ=0), Yukawa (λ∈?₊), and Helmholtz (λ=i κ,?κ∈?₊) potentials, as well as for the kernel functions 1/‖x‖ and 1/‖x‖ ^d?2, x∈? ^d , in higher dimensions d>3. We present numerical results on the iterative calculation of the minimal eigenvalue for the d-dimensional finite difference Laplacian by the power method with the rank truncation and based on the approximate inverse ? _R ≈(?Δ)^?1, with 3≤d≤50.     

The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function

The Nehari manifold for the equation −Δu(x)=λa(x)u(x)+b(x)|u(x)|^ν−1u(x) for x∈Ω together with Dirichlet boundary conditions is investigated. Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form t→J(tu) where J is the Euler functional associated with the equation) we discuss how the Nehari manifold changes as λ changes and show how existence and non-existence results for positive solutions of the equation are linked to properties of the manifold.     

Potential theory associated with Uhlenbeck-Ornstein process

Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis1 (1967), 123–181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator $\text{N}$ is defined by $\text{N}$f(x) = ?lim_∈←0{E[f($\text{U}$(τ^∈_ξ))] ? f(x)}/E[τ^∈_ξ, where τ_x^? is the first exit time of U(t) starting at x from the ball of radius ? with center x. It is shown that $\text{N}$f(x) = ?trace D²f(x)+〈Df(x),x〉 for a large class of functions f. Let r_t(x, dy) be the transition probabilities of U(t). The λ-potential G_λf, λ > 0, and normalized potential Rf of f are defined by G_λf(X) = ∫₀^∞e^?λtr_tf(x) dt and Rf(x) = ∫₀^∞ [r_tf(x) ? r_tf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D²G_λf(x) ? 〈DG_λf(x), x〉 = ?f(x) + λG_λf(x) and trace D²Rf(x) ? 〈DRf(x), x〉 = ?f(x) + ∫_Bf(y)p₁(dy), where p₁ is the Wiener measure in B with parameter 1. Some approximation theorems are also proved.     

New higher order methods for solving nonlinear equations with multiple roots

For a nonlinear equation f(x)=0 having a multiple root we consider Steffensen’s transformation, T. Using the transformation, say, F_q(x)=T^qf(x) for integer q≥2, repeatedly, we develop higher order iterative methods which require neither derivatives of f(x) nor the multiplicity of the root. It is proved that the convergence order of the proposed iterative method is 1+2^q−2 for any equation having a multiple root of multiplicity m≥2. The efficiency of the new method is shown by the results for some numerical examples.