Positive Solutions of a Third-Order Boundary Value Problem with Full Nonlinearity

This paper is concerned with the existence of positive solutions of the third-order boundary value problem with full nonlinearity

$$\begin{aligned} \left\{ \begin{array}{lll} u'''(t)&{}=f(t,u(t),u'(t),u''(t)),\quad t\in [0,1],\\ u(0)&{}=u'(1)=u''(1)=0, \end{array}\right. \end{aligned}$$

where \(f:[0,1]\times \mathbb {R}^+\times \mathbb {R}^+\times \mathbb {R}^-\rightarrow \mathbb {R}^+\) is continuous. Under some inequality conditions on f as |(x, y, z)| small or large enough, the existence results of positive solution are obtained. These inequality conditions allow that f(t, x, y, z) may be superlinear, sublinear or asymptotically linear on x, y and z as \(|(x,y,z)|\rightarrow 0\) and \(|(x,y,z)|\rightarrow \infty \). For the superlinear case as \(|(x,y,z)|\rightarrow \infty \), a Nagumo-type growth condition is presented to restrict the growth of f on y and z. Our discussion is based on the fixed point index theory in cones.

     

Uniform in bandwidth exact rates for a class of kernel estimators

Given an i.i.d sample (Y _i , Z _i ), taking values in \({\mathbb{R}^{d'}\times\mathbb{R}^d}\), we consider a collection Nadarya–Watson kernel estimators of the conditional expectations \({\mathbb{E}( <\,c_g(z),g(Y)>+d_g(z)\mid Z=z)}\), where z belongs to a compact set \({H\subset \mathbb{R}^d}\), g a Borel function on \({\mathbb{R}^{d'}}\) and c _g (·), d _g (·) are continuous functions on \({\mathbb{R}^d}\). Given two bandwidth sequences \({h_n<\mathfrak{h}_n}\) fulfilling mild conditions, we obtain an exact and explicit almost sure limit bounds for the deviations of these estimators around their expectations, uniformly in \({g\in\mathcal{G},\;z\in H}\) and \({h_n\le h\le \mathfrak{h}_n}\) under mild conditions on the density f _Z , the class \({\mathcal{G}}\), the kernel K and the functions c _g (·), d _g (·). We apply this result to prove that smoothed empirical likelihood can be used to build confidence intervals for conditional probabilities \({\mathbb{P}( Y\in C\mid Z=z)}\), that hold uniformly in \({z\in H,\; C\in \mathcal{C},\; h\in [h_n,\mathfrak{h}_n]}\). Here \({\mathcal{C}}\) is a Vapnik–Chervonenkis class of sets.     

Exit Time of a Hyperbolic α-Stable Process from a Halfspace or a Ball

For a hyperbolic α-stable process in the hyperbolic space \(\mathbb {H}^{d}, d\ge 2\), we prove that the mean exit time from a halfspace \(H(a)=\{x_{d}>a\}\subset \mathbb {H}^{d} \) is equal to \(\mathbb {E}^{x} \tau _{H(a)} = c(\alpha , d) \delta ^{\alpha /2}_{H(a)} (x),\) where δ_D(x) is the (hyperbolic) distance of x to D^c. Based on this exact result we provide a sharp estimate of the mean exit time from a hyperbolic ball B(x₀,R) of radius R and center x₀: \(\mathbb {E}^{x}\tau _{B(x_{0},R)}\approx (\delta _{B(x_{0},R)}(x) \tanh R)^{\alpha /2}, x\in \mathbb {H}^{d}\). By usual isomorphism argument the same estimate holds in any other model of real hyperbolic space.     

A characterization of the normal distribution using stationary max-stable processes

Consider a max-stable process of the form \(\eta (t) = \max _{i\in \mathbb {N}} U_{i} \mathrm {e}^{\langle X_{i}, t\rangle - \kappa (t)}\), \(t\in \mathbb {R}^{d}\), where \(\{U_{i}, i\in \mathbb {N}\}\) are points of the Poisson process with intensity u ^?2du on (0,∞), X _i , \(i\in \mathbb {N}\), are independent copies of a random d-variate vector X (that are independent of the Poisson process), and \(\kappa :\mathbb {R}^{d} \to \mathbb {R}\) is a function. We show that the process η is stationary if and only if X has multivariate normal distribution and κ(t)?κ(0) is the cumulant generating function of X. In this case, η is a max-stable process introduced by R. L. Smith.     

Normal forms of functional-differential equations of Beardon type and Briot–Bouquet type equations in the complex domain

We study local analytic solutions of the functional-differential equation of the form \({h(\psi(z)) = b(z) h(z) h^\prime(z) + d(z)h(z)^{2}}\) which are called Beardon type functional-differential equations. All functions involved are supposed to be holomorphic in a neighbourhood of zero. Special cases are the equations f(kz) =  kf(z) f′(z) where k is a complex number, \({k \neq 0}\), and \({f(\varphi(z)) = a(z) f(z) f'(z)}\) with given \({\varphi}\) and a. The class of these equations is invariant under transformations \({h \to \alpha h, \alpha(z) \neq 0}\) for all z in a neighbourhood of zero, of the unknown function and \({z \to T(z)}\) of the argument z. In particular, we are interested to know under which conditions a Beardon type functional-differential equation can be transformed to the simplified (normal form) \({h(kz) = k h(z) h'(z) + c(z) h(z)^2}\) where \({k \in \mathbb {C} \backslash\left\{0\right\}}\). We solve this normal form by another transfomation to a so-called Briot–Bouquet type functional-differential equation.     

Searching for a Best Least Absolute Deviations Solution of an Overdetermined System of Linear Equations Motivated by Searching for a Best Least Absolute Deviations Hyperplane on the Basis of Given Data

We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations Xa=z, X∈?^m×n, m≥n, \(\mathbf{a}\in \mathbb{R}^{n}, \mathbf {z}\in\mathbb{R}^{m}\). This problem is equivalent to the problem of determining a best LAD-hyperplane x?a ^T x, x∈? ⁿ on the basis of given data \((\mathbf{x}_{i},z_{i}), \mathbf{x}_{i}= (x_{1}^{(i)},\ldots,x_{n}^{(i)})^{T}\in \mathbb{R}^{n}, z_{i}\in\mathbb{R}, i=1,\ldots,m\), whereby the minimizing functional is of the form

$F(\mathbf{a})=\|\mathbf{z}-\mathbf{Xa}\|_1=\sum_{i=1}^m|z_i-\mathbf {a}^T\mathbf{x}_i|.$

An iterative procedure is constructed as a sequence of weighted median problems, which gives the solution in finitely many steps. A criterion of optimality follows from the fact that the minimizing functional F is convex, and therefore the point a ^?∈? ⁿ is the point of a global minimum of the functional F if and only if 0∈?F(a ^?).

Motivation for the construction of the algorithm was found in a geometrically visible algorithm for determining a best LAD-plane (x,y)?αx+βy, passing through the origin of the coordinate system, on the basis of the data (x _i ,y _i ,z _i ),i=1,…,m.     

Homoclinic Solutions for <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)-Laplacian–Hamiltonian Systems Without Coercive Conditions

In this paper, we study the existence and multiplicity of homoclinic solutions for the following second-order p(t)-Laplacian–Hamiltonian systems

$$\frac{{\rm d}}{{\rm d}t}(|\dot{u}(t)|^{p(t)-2}\dot{u}(t))-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=0,$$

where \({t \in \mathbb{R}}\), \({u \in \mathbb{R}^n}\), \({p \in C(\mathbb{R},\mathbb{R})}\) with p(t) > 1, \({a \in C(\mathbb{R},\mathbb{R})}\), \({W\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R})}\) and \({\nabla W(t,u)}\) is the gradient of W(t, u) in u. The point is that, assuming that a(t) is bounded in the sense that there are constants \({0<\tau_1<\tau_2<\infty}\) such that \({\tau_1\leq a(t)\leq \tau_2 }\) for all \({t \in \mathbb{R}}\) and W(t, u) is of super-p(t) growth or sub-p(t) growth as \({|u|\rightarrow \infty}\), we provide two new criteria to ensure the existence and multiplicity of homoclinic solutions, respectively. Recent results in the literature are extended and significantly improved.

     

Solutions of perturbed Schrödinger equations with critical nonlinearity

We consider the perturbed Schrödinger equation

$\left\{\begin{array}{ll}{- \varepsilon ^2 \Delta u + V(x)u = P(x)|u|^{p - 2} u + k(x)|u|^{2* - 2} u} &; {\text{for}}\, x \in {\mathbb{R}}^N\\ \qquad \qquad \quad {u(x) \rightarrow 0} &; \text{as}\, {|x| \rightarrow \infty} \end{array} \right.$

where \(N\geq 3, \ 2^*=2N/(N-2)\) is the Sobolev critical exponent, \(p\in (2, 2^*)\) , P(x) and K(x) are bounded positive functions. Under proper conditions on V we show that it has at least one positive solution provided that \(\varepsilon\leq{\mathcal{E}}\) ; for any \(m\in{\mathbb{N}}\) , it has m pairs of solutions if \(\varepsilon\leq{\mathcal{E}}_{m}\) ; and suppose there exists an orthogonal involution \(\tau:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}\) such that V(x), P(x) and K(x) are τ -invariant, then it has at least one pair of solutions which change sign exactly once provided that \(\varepsilon\leq{\mathcal{E}}\) , where \({\mathcal{E}}\) and \({\mathcal{E}}_{m}\) are sufficiently small positive numbers. Moreover, these solutions \(u_\varepsilon\to 0\) in \(H^1({\mathbb{R}}^N)\) as \(\varepsilon\to 0\) .

     

Infinitely many solutions for Kirchhoff equations with sign-changing potential and Hartree nonlinearity

This paper is concerned with the following Kirchhoff-type equations:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\big (a+b\int _{\mathbb {R}^{3}}|\nabla u|^{2}\mathrm {d}x\big )\Delta u+ V(x)u+\mu \phi |u|^{p-2}u=f(x, u)+g(x,u), &{} \text{ in } \mathbb {R}^{3},\\ (-\Delta )^{\frac{\alpha }{2}} \phi = \mu |u|^{p}, &{} \text{ in } \mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$

where \(a>0,~b,~\mu \ge 0\) are constants, \(\alpha \in (0,3)\), \(p\in [2,3+2\alpha )\), the potential V(x) may be unbounded from below and \(\phi |u|^{p-2}u\) is a Hartree-type nonlinearity. Under some mild conditions on V(x), f(x, u) and g(x, u), we prove that the above system has infinitely many nontrivial solutions. Specially, our results cover the general Schrödinger equations, the Kirchhoff equations and the Schrödinger–Poisson system.

     

A Schwarz Lemma at the Boundary of Hilbert Balls

In this paper, the authors prove a general Schwarz lemma at the boundary for the holomorphic mapping f between unit balls B and B′in separable complex Hilbert spaces H and H′, respectively. It is found that if the mapping f ∈ C~(1+α)at z_0∈ ?B with f(z_0) = w_0∈ ?B′, then the Fr′echet derivative operator Df(z_0) maps the tangent space Tz_0(?B~n) to Tw_0(?B′), the holomorphic tangent space T_(z_0)~(1,0)(?B~n) to T_(w_0)~(1,0)(?B′),respectively.     

Harnack Inequality for Non-Local Schrödinger Operators

Let \(x \in \mathbb {R}^{d}\), d ≥ 3, and \(f: \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let \(a_{ij} : \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated to

$$\mathcal{L}f(x) = \frac12 \sum\limits_{i=1}^{d} \sum\limits_{j=1}^{d} \frac{\partial}{\partial x_{i}} \left( a_{ij}(\cdot) \frac{\partial f}{\partial x_{j}}\right)(x) + {\int}_{\mathbb{R}^{d}\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy $$

where \(J: \mathbb {R}^{d} \times \mathbb {R}^{d} \rightarrow \mathbb {R}\) is a symmetric measurable function. Let \(q: \mathbb {R}^{d} \rightarrow \mathbb {R}.\) We specify assumptions on a, q, and J so that non-negative bounded solutions to

$$\mathcal{L}f + qf = 0 $$

satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to \(\mathcal {L}f = 0.\)

     

Cotorsion and wild homology

We solve the problem of describing the solutions of E-operators of order μ ≥ 1 admitting at z = 0 a basis over C of local solutions which are all holomorphic at z = 0. We prove that the components of such a basis can be taken of the form \(\sum {_{j = 1}^\ell } {P_j}\left( z \right){e^{{\beta _{{j^z}}}}}\), where ? ≤ μ, β ₁,...,β _? ∈ \(\overline {\mathbb{Q}} \) ^x, and P ₁(z),..., P _?(z) ∈ \(\overline {\mathbb{Q}} \)[z].     

An improvement of Ozaki’s P-valent conditions

The old result due to[Ozaki,S.:On the theory of multivalent functions Ⅱ.Sci.Rep.Tokyo Bunrika Daigaku Sect.A,45-87(1941)],says that if f(z) = z~p + ∑_(n=p+1~(a_nz~n))~∞ is analytic in a convex domain D and for some real α we have Re{exp(iα)f~((p))(z)} 0 in D,then f(z) is at most p-valent in ED.In this paper,we consider similar problems in the unit disc B = {z ∈ C:|z| 1}.     

Finite Gap Jacobi Matrices,III. Beyond the Szegő Class

Let \({\frak {e}}\subset {\mathbb {R}}\) be a finite union of ?+1 disjoint closed intervals, and denote by ω _j the harmonic measure of the j left-most bands. The frequency module for \({\frak {e}}\) is the set of all integral combinations of ω ₁,…,ω _? . Let \(\{\tilde{a}_{n}, \tilde{b}_{n}\}_{n=-\infty}^{\infty}\) be a point in the isospectral torus for \({\frak {e}}\) and \(\tilde{p}_{n}\) its orthogonal polynomials. Let \(\{a_{n},b_{n}\}_{n=1}^{\infty}\) be a half-line Jacobi matrix with \(a_{n} = \tilde{a}_{n} + \delta a_{n}\), \(b_{n} = \tilde{b}_{n} +\delta b_{n}\). Suppose

$\sum_{n=1}^\infty \lvert \delta a_n\rvert ^2 + \lvert \delta b_n\rvert ^2 <\infty $

and \(\sum_{n=1}^{N} e^{2\pi i\omega n} \delta a_{n}\), \(\sum_{n=1}^{N} e^{2\pi i\omega n} \delta b_{n}\) have finite limits as N→∞ for all ω in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to ω, then for z∈???, \(p_{n}(z)/\tilde{p}_{n}(z)\) has a limit as n→∞. Moreover, we show that there are non-Szeg? class J’s for which this holds.

     

Cardinal interpolation with general multiquadrics

This paper studies the cardinal interpolation operators associated with the general multiquadrics, ? _{α, c} (x)=(∥x∥² + c ²) ^α , \(x\in \mathbb {R}^{d}\). These operators take the form

$$\mathcal{I}_{\alpha,c}\mathbf{y}(x) = \sum\limits_{j\in\mathbb{Z}^{d}}y_{j}L_{\alpha,c}(x-j),\quad\mathbf{y}=(y_{j})_{j\in\mathbb{Z}^{d}},\quad x\in\mathbb{R}^{d}, $$

where L _{α, c} is a fundamental function formed by integer translates of ? _{α, c} which satisfies the interpolatory condition \(L_{\alpha ,c}(k) = \delta _{0,k},\; k\in \mathbb {Z}^{d}\). We consider recovery results for interpolation of bandlimited functions in higher dimensions by limiting the parameter \(c\to \infty \). In the univariate case, we consider the norm of the operator \(\mathcal {I}_{\alpha ,c}\) acting on ? _p spaces as well as prove decay rates for L _{α, c} using a detailed analysis of the derivatives of its Fourier transform, \(\widehat {L_{\alpha ,c}}\).

     

Littlewood-Paley Characterizations of Hajłasz-Sobolev and Triebel-Lizorkin Spaces via Averages on Balls

Let \(p\in (1,\infty )\) and \(q\in [1,\infty )\). In this article, the authors characterize the Triebel-Lizorkin space \({F}^{\alpha }_{p,q}(\mathbb {R}^{n})\) with smoothness order α ∈ (0, 2) via the Lusin-area function and the \(g_{\lambda }^{*}\)-function in terms of difference between f(x) and its ball average \(B_{t}f(x):=\frac 1{|B(x,t)|}{\int }_{B(x,t)}f(y)\,dy\) over the ball B(x, t) centered at \(x\in \mathbb {R}^{n}\) with radius t ∈ (0, 1). As an application, the authors obtain a series of characterizations of \(F^{\alpha }_{p,\infty }(\mathbb {R}^{n})\) via pointwise inequalities, involving ball averages, in spirit close to Haj?asz gradients, here some interesting phenomena naturally appear that, in the end-point case when α = 2, some of these pointwise inequalities characterize the Triebel-Lizorkin spaces \(F^{2}_{p,2}(\mathbb {R}^{n})\), while not \(F^{2}_{p,\infty }(\mathbb {R}^{n})\), and that some of other obtained pointwise characterizations are only known to hold true for \(F^{\alpha }_{p,\infty }(\mathbb {R}^{n})\) with \(p\in (1,\infty )\), α ∈ (0, 2) or α ∈ (n/p, 2). In particular, some new pointwise characterizations of Haj?asz-Sobolev spaces via ball averages are obtained. Since these new characterizations only use ball averages, they can be used as starting points for developing a theory of Triebel-Lizorkin spaces with smoothness orders not less than 1 on spaces of homogeneous type.     

Coefficient estimates of negative powers and inverse coefficients for certain starlike functions

For ?1≤B<A≤1, let \(\mathcal {S}^{*}(A,B)\) denote the class of normalized analytic functions \(f(z)= z+{\sum }_{n=2}^{\infty }a_{n} z^{n}\) in |z|<1 which satisfy the subordination relation z f ^′(z)/f(z)?(1 + A z)/(1 + B z) and Σ^?(A,B) be the corresponding class of meromorphic functions in |z|>1. For \(f\in \mathcal {S}^{*}(A,B)\) and λ>0, we shall estimate the absolute value of the Taylor coefficients a _n (?λ,f) of the analytic function (f(z)/z)^?λ . Using this we shall determine the coefficient estimate for inverses of functions in the classes \(\mathcal {S}^{*}(A,B)\) and Σ^?(A,B).     

Frame completions with prescribed norms: local minimizers and applications

Let \(\mathcal {F}_{0}=\{f_{i}\}_{i\in \mathbb {I}_{n_{0}}}\) be a finite sequence of vectors in \(\mathbb {C}^{d}\) and let \(\mathbf {a}=(a_{i})_{i\in \mathbb {I}_{k}}\) be a finite sequence of positive numbers, where \(\mathbb {I}_{n}=\{1,\ldots , n\}\) for \(n\in \mathbb {N}\). We consider the completions of \(\mathcal {F}_{0}\) of the form \(\mathcal {F}=(\mathcal {F}_{0},\mathcal {G})\) obtained by appending a sequence \(\mathcal {G}=\{g_{i}\}_{i\in \mathbb {I}_{k}}\) of vectors in \(\mathbb {C}^{d}\) such that ∥g _i ∥² = a _i for \(i\in \mathbb {I}_{k}\), and endow the set of completions with the metric \(d(\mathcal {F},\tilde {\mathcal {F}}) =\max \{ \,\|g_{i}-\tilde {g}_{i}\|: \ i\in \mathbb {I}_{k}\}\) where \(\tilde {\mathcal {F}}=(\mathcal {F}_{0},\,\tilde {\mathcal {G}})\). In this context we show that local minimizers on the set of completions of a convex potential P _φ , induced by a strictly convex function φ, are also global minimizers. In case that φ(x) = x ² then P _φ is the so-called frame potential introduced by Benedetto and Fickus, and our work generalizes several well known results for this potential. We show that there is an intimate connection between frame completion problems with prescribed norms and frame operator distance (FOD) problems. We use this connection and our results to settle in the affirmative a generalized version of Strawn’s conjecture on the FOD.     

On height isotopy classes of embeddings in the plane of a Morse function of a circle

Let \(\mathrm{SM}_{2n}(S^1,\mathbb {R})\) be a set of stable Morse functions of an oriented circle such that the number of singular points is \(2n\in \mathbb {N}\) and the order of singular values satisfies the particular condition. For an orthogonal projection \(\pi :\mathbb {R}^2\rightarrow \mathbb {R}\), let \({\tilde{f}}_0\) and \({\tilde{f}}_1:S^1\rightarrow \mathbb {R}^2\) be embedding lifts of f. If there is an ambient isotopy \(\tilde{\varphi }_t:\mathbb {R}^2\rightarrow \mathbb {R}^2\) \((t\in [0,1])\) such that \({\pi \circ \tilde{\varphi }}_t(y_1,y_2)=y_1\) and \(\tilde{\varphi }_1\circ {\tilde{f}}_0={\tilde{f}}_1\), we say that \({\tilde{f}}_0\) and \({\tilde{f}}_1\) are height isotopic. We define a function \(I:\mathrm{SM}_{2n}(S^1,\mathbb {R})\rightarrow \mathbb {N}\) as follows: I(f) is the number of height isotopy classes of embeddings such that each rotation number is one. In this paper, we determine the maximal value of the function I equals the n-th Baxter number and the minimal value equals \(2^{n-1}\).     

On the nonexistence of solutions of some anisotropic elliptic and backward parabolic inequalities

We study the nonexistence of weak solutions of higher-order elliptic and parabolic inequalities of the following types: \(\sum {_{i = 1}^N\sum\nolimits_{{e_i} \leqslant {\alpha _i} \leqslant {m_i}} {D_{{x_i}}^{{\alpha _i}}\left( {{A_{{\alpha _i}}}\left( {x,u} \right)} \right)} \geqslant f\left( {x,u} \right),} x \in {\mathbb{R}^N}\), and \({u_t} + \sum {_{i = 1}^N\sum\nolimits_{{k_i} \leqslant {\beta _i} \leqslant {n_i}} {D_{{x_i}}^{{\beta _i}}\left( {{B_{{\beta _i}}}\left( {x,t,u} \right)} \right)} > g\left( {x,t,u} \right),\left( {x,t} \right)} \in {\mathbb{R}^N} \times {\mathbb{R}_ + }\), where l _i , m _i , k _i , n _i ∈ N satisfy the condition l _i , k _i > 1 for all i = 1,..., N, and A _αi (x, u), B _βi (x, t, u), f(x, u), and g(x, t, u) are some given Carathéodory functions. Under appropriate conditions on the functions A _αi , B _βi , f, and g, we prove theorems on the nonexistence of solutions of these inequalities.