Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems

We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem $$ \left \{ \begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,\quad0 where \(0\leq\alpha_{i}\leq\sum_{i=1}^{m_{1}}\alpha_{i}<1\) , i=1,2,…,m ₁, \(0<\xi_{1}< \xi_{2}< \cdots<\xi_{m_{1}}<1\) , \(0\leq\beta_{j}\leq\sum_{i=1}^{m_{2}}\beta_{i}<1\) , j=1,2,…,m ₂, \(0<\eta_{1}< \eta_{2}< \cdots<\eta_{m_{2}}<1\) . And we obtain some necessary and sufficient conditions for the existence of C ¹[0,1] and C ²[0,1] positive solutions by means of the fixed point theorems on a special cone. Our nonlinearity f(t,x,y) may be singular at t=0 and t=1.     

Existence of multiple positive solutions for one-dimensional p-Laplacian operator

In this paper, we consider the multipoint boundary value problem for one-dimensional p-Laplacian $$(\phi_{p}(u'))'+f(t,u,u')=0,\quad t\in [0,1],$$ subject to the boundary value conditions: $$u'(0)=\sum_{i=1}^{n-2}\alpha_{i}u'(\xi_{i}),\qquad u(1)=\sum_{i=1}^{n-2}\beta_{i}u(\xi_{i}),$$ where φ _p (s)=|s| ^p?2?s,p>1;ξ _i ∈(0,1) with 0<ξ ₁<ξ ₂<???<ξ _n?2<1 and α _i ,β _i satisfy α _i ,β _i ∈[0,∞),0≤∑ _i=1 ^n?2 α _i <1 and 0≤∑ _i=1 ^n?2 β _i <1. Using a fixed point theorem for operators in a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.     

A nonclassical law of the iterated logarithm for self-normalized partial sums

Let {X _n ,?n≧1} be a sequence of nondegenerate, symmetric, i.i.d. random variables which are in the domain of attraction of the normal?law?with zero means and possibly infinite variances. Denote ${S_{n}=\sum_{i=1}^{n} X_{i}}$ , ${V_{n}^{2}=\sum_{i=1}^{n} X_{i}^{2}}$ . Then we prove that there is a sequence of positive constants {b(n),?n≧1} which is defined by Klesov and Rosalsky [11], is monotonically approaching infinity and is not asymptotically equivalent to loglogn but is such that $\displaystyle \limsup_{n\to\infty} \frac{|S_n|}{\sqrt{2V_n^2b(n)}}= 1$ almost surely if some additional technical assumptions are imposed.     

Modular forms,hypergeometric functions and congruences

Let $$A_k(n)=\sum_{\substack{i_1,i_2,\ldots, i_k \ge0\\ i_1+i_2+\cdots+ i_k=n}}\binom{2i_1}{i_1}^2 \binom{2i_2}{i_2}^2\cdots \binom{2i_k}{i_k}^2, \quad \textrm{for } k,n\in\mathbb{N}. $$ Using the theory of Stienstra and Beukers (Math. Ann., 271:269–304, 1985), we prove that the numbers A ₃(n) and A ₃(n?1)?16A ₃(n?2) satisfy three term congruence relations similar to those satisfied by Apery numbers. Moreover, for k≥3 and p prime, we prove divisibility by p of some simple linear combinations of the numbers A _k (n), for $n\in \mathbb{N}$ . To obtain this, we study the arithmetic properties of the Fourier coefficients of certain holomorphic and weakly holomorphic modular forms.     

On the rational approximation of the sum of the reciprocals of the Fermat numbers

Let $\mathcal{G}(z):=\sum_{n\geqslant0} z^{2^{n}}(1-z^{2^{n}})^{-1}$ denote the generating function of the ruler function, and $\mathcal {F}(z):=\sum_{n\geqslant} z^{2^{n}}(1+z^{2^{n}})^{-1}$ ; note that the special value $\mathcal{F}(1/2)$ is the sum of the reciprocals of the Fermat numbers $F_{n}:=2^{2^{n}}+1$ . The functions $\mathcal{F}(z)$ and $\mathcal{G}(z)$ as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers $\mathcal {F}(\alpha)$ and $\mathcal{G}(\alpha)$ are transcendental for all algebraic numbers α which satisfy 0<α<1. For a sequence u, denote the Hankel matrix $H_{n}^{p}(\mathbf {u}):=(u({p+i+j-2}))_{1\leqslant i,j\leqslant n}$ . Let α be a real number. The irrationality exponent μ(α) is defined as the supremum of the set of real numbers μ such that the inequality |α?p/q|<q ^?μ has infinitely many solutions (p,q)∈?×?. In this paper, we first prove that the determinants of $H_{n}^{1}(\mathbf {g})$ and $H_{n}^{1}(\mathbf{f})$ are nonzero for every n?1. We then use this result to prove that for b?2 the irrationality exponents $\mu(\mathcal{F}(1/b))$ and $\mu(\mathcal{G}(1/b))$ are equal to 2; in particular, the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2.     

Optimal Sobolev Type Inequalities in Lorentz Spaces

It is well known that the classical Sobolev embeddings may be improved within the framework of Lorentz spaces L ^p,q : the space $\mathcal{D}^{1,p}(\mathbb R^n)$ , 1?<?p?<?n, embeds into $L^{p^*,q}(\mathbb R^n)$ , p?≤?q?≤?∞. However, the value of the best possible embedding constants in the corresponding inequalities is known just in the case $L^{p^*,p}(\mathbb R^n)$ . Here, we determine optimal constants for the embedding of the space $\mathcal{D}^{1,p}(\mathbb R^n)$ , 1?<?p?<?n, into the whole Lorentz space scale $L^{p^{\ast}, q}(\mathbb R^n)$ , p?≤?q?≤?∞, including the limiting case q?=?p of which we give a new proof. We also exhibit extremal functions for these embedding inequalities by solving related elliptic problems.     

On the entire self-shrinking solutions to Lagrangian mean curvature flow

The authors prove that the logarithmic Monge?CAmpère flow with uniformly bound and convex initial data satisfies uniform decay estimates away from time t?=?0. Then applying the decay estimates, we conclude that every entire classical strictly convex solution of the equation $$ \det D^{2}u=\exp\left\{n\left(-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} \right)\right\}, $$ should be a quadratic polynomial if the inferior limit of the smallest eigenvalue of the function |x|² D ² u at infinity has an uniform positive lower bound larger than 2(1 ? 1/n). Using a similar method, we can prove that every classical convex or concave solution of the equation $$ \sum_{i=1}^{n} \arctan\lambda_{i}=-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} $$ must be a quadratic polynomial, where ?? _i are the eigenvalues of the Hessian D ² u.     

Covering a Connected Curve on the Torus with Squares

We throw i.i.d. random squares S ₁,S ₂,… with respective side lengths l ₁,l ₂,… uniformly on the two-dimensional torus ?/?×?/?, where $\{l_{n}\}_{n=1}^{\infty}$ is a nonincreasing sequence with 0<l _n <1 and lim _n→∞ l _n =0. A necessary and sufficient condition for covering the connected curve {0}×?/? is $$\sum_{n=1}^{\infty}\frac{l_n}{(\sum_{i=1}^{n}l_i)^2}\exp{\Biggl(\sum _{i=1}^{n}l_i^2\Biggr)}=\infty.$$      

Limit ultraspherical series and their approximation properties

We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x ₂)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S _n ^?1 (f) = S _n ^?1 (f, x) is the projection onto the subspace of algebraic polynomials p _n = p _n (x) of degree at most n, i.e., S _n (p _n ) = p _n ; in addition, S _n ^?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S _n ^?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ _n (x) of the partial sums S _n ^?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

The ternary Goldbach–Vinogradov theorem with almost equal primes from the Beatty sequence

Let α ₁, α ₂, α ₃, β ₁, β ₂, β ₃ be real numbers with α ₁, α ₂, α ₃ >1. Suppose that each individual α _i is of a finite type and that at least one pair $\alpha_{i}^{-1}$ , $\alpha_{j}^{-1}$ is also of a finite type. In this paper we prove that every large odd integer n can be represented as $$n=p_{1}+p_{2}+p_{3}, $$ with p _i =n/3+O(n ^2/3(logn) ^c ) and $p_{i}\in\mathcal{B}_{i}$ , where c>0 is an absolute constant and $\mathcal{B}_{i}$ denotes the so-called Beatty sequence, i.e. $$\mathcal{B}_{i}=\bigl\{n\in\mathbb{N}: n=[\alpha_{i}m+ \beta_{i}] \mbox { for some } m\in\mathbb{Z}\bigr\}. $$      

Existence of multiple positive solutions for m-point fractional boundary value problems with p-Laplacian operator on infinite interval

In this paper we consider the following m-point fractional boundary value problem with p-Laplacian operator on infinite interval where 0<????1, 2<????3, $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative, ?? _p (s)=|s| ^p?2 s,p>1, (?? _p )^?1=?? _q , $\frac{1}{p}+\frac{1}{q}=1$ . 0<?? ₁<?? ₂<?<?? _m?2<+??, ?? _i ??0, i=1,2,??,m?2 satisfies $0 <\sum_{i=1}^{m-2}\beta_{i}\xi_{i}^{\alpha-1} < \Gamma(\alpha)$ . We establish solvability of the above fractional boundary value problems by means of the properties of the Green function and some fixed-point theorems.     

Weak real integral characterizations for exponential stability of semigroups in reflexive spaces

Let (t _n ) be a sequence of nonnegative real numbers tending to ∞, such that 1≤t _n+1?t _n ≤α for all natural numbers n and some positive α. We prove that a strongly continuous semigroup {T(t)} _t≥0, acting on a Hilbert space H, is uniformly exponentially stable if $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, y\bigr\rangle\bigr|\bigr)<\infty, $$ for all unit vectors x, y in H. We obtain the same conclusion under the assumption that the inequality $$\sum_{n=0}^\infty\varphi\bigl(\bigl|\bigl\langle T(t_n)x, x^\ast\bigr\rangle\bigr|\bigr)<\infty, $$ is fulfilled for all unit vectors x∈X and x ^?∈X ^?, X being a reflexive Banach space. These results are stated for functions φ belonging to a special class of functions, such as defined in the second section of this paper. We conclude our paper with a Rolewicz’s type result in the continuous case on Hilbert spaces.     

Approximations of ternary Jordan homomorphisms and derivations in multi-C ? ternary algebras

Using fixed point methods, we prove the generalized Hyers–Ulam stability of homomorphisms in multi-C ^? ternary algebras and of derivations on multi-C ^? ternary algebras for the additive functional equation $$\sum_{i=1}^{m}f \bigg(mx_i+\sum_{j=1,\ j\ne i}^{m}x_j\bigg)+ f\bigg(\sum_{i=1}^{m}x_i\bigg)= 2f\bigg(\sum_{i=1}^{m}mx_i\bigg) \quad (m\in {\mathbb{N}},\ m\geqq2).$$      

Ranges of certain functionals in classes of regular functions

LetP _κ,n (λ,β) be the class of functions \(g(z) = 1 + \sum\nolimits_{v = n}^\infty {c_\gamma z^v }\) , regular in ¦z¦<1 and satisfying the condition $$\int_0^{2\pi } {\left| {\operatorname{Re} \left[ {e^{i\lambda } g(z) - \beta \cos \lambda } \right]} \right|} /\left( {1 - \beta } \right)\cos \lambda \left| {d\theta \leqslant \kappa \pi ,} \right.z = re^{i\theta } ,$$ , 0 < r < 1 (κ?2,n?1, 0?Β<1, -π<λ<π/2;M _κ,n (λ,β,α),n?2, is the class of functions \(f(z) = z + \sum\nolimits_{v = n}^\infty {a_v z^v }\) , regular in¦z¦<1 and such thatF _α(z)∈P _κ,n?1(λ,β), where \(F_\alpha (z) = (1 - \alpha )\frac{{zf'(z)}}{{f(z)}} + \alpha (1 + \frac{{zf'(z)}}{{f'(z)}})\) (0?α?1). Onr considers the problem regarding the range of the system {g ^(v?1)(z_?)/(v?1)!}, ?=1,2,...,m,v=1,2,...,N _?, on the classP _κ,1(λ,β). On the classesP _κ,n (λ,β),M _κ,n (λ,β,α) one finds the ranges of C_v, v?n, a_m, n?m?2n-2, and ofg(?),F _?(?), 0<¦ξ¦<1, ξ is fixed.     

Commutators of BMO functions and degenerate Schr?dinger operators with certain nonnegative potentials

Let ${\mathcal{L}f(x)=-\frac{1}{\omega}\sum_{i,j} \partial_i(a_{i,j}(\cdot)\partial_jf)(x)+V(x)f(x)}$ with the non-negative potential V belonging to reverse H?lder class with respect to the measure ??(x)dx, where ??(x) satisfies the A ₂ condition of Muckenhoupt and a _i,j (x) is a real symmetric matrix satisfying ${\lambda^{-1}\omega(x)|\xi|^2\le \sum^n_{i,j=1}a_{i,j}(x)\xi_i\xi_j\le\lambda\omega(x)|\xi|^2. }$ We obtain some estimates for ${V^{\alpha}\mathcal{L}^{-\alpha}}$ on the weighted L ^p spaces and we study the weighted L ^p boundedness of the commutator ${[b, V^{\alpha} \mathcal{L}^{-\alpha}]}$ when ${b\in BMO_\omega}$ and 0?<??? ?? 1.     

Asymptotics of generalized jacobi polynomials

Polynomialsp ₁,(z),p ₂ (z), of degreen are defined by the relation \(p_1 (z) + p_2 (z)\prod\nolimits_{i = 1}^3 {(z - b_l )^{v_1 } } = O(z^{ - n - 1} ),z \to \infty \) , where \(\sum\nolimits_{i = 1}^3 {v_i = 0} \) . We obtain the asymptotic behavior of these polynomials asn→∞ and show that it agrees with a previous conjecture.     

On averages of Fourier coefficients of Maass cusp forms

Let φ be a primitive Maass cusp form and t _φ (n) be its nth Fourier coefficient at the cusp infinity. In this short note, we are interested in the estimation of the sums ${\sum_{n \leq x}t_{\varphi}(n)}$ and ${\sum_{n \leq x}t_{\varphi}(n^2)}$ . We are able to improve the previous results by showing that for any ${\varepsilon > 0}$ $$\sum_{n \leq x}t_{\varphi}(n) \ll\, _{\varphi, \varepsilon} x^{\frac{1027}{2827} + \varepsilon} \quad {and}\quad\sum_{n \leq x}t_{\varphi}(n^2) \ll\,_{\varphi, \varepsilon} x^{\frac{489}{861} + \varepsilon}.$$      

Absolute convergence of double Walsh?CFourier series and related results

We consider the double Walsh orthonormal system $$\{w_m(x)w_n(y):\, m,n \in \mathbb{N}\}$$ on the unit square $\mathbb{I}^{2}$ , where {w _m (x)} is the ordinary Walsh system on the unit interval $\mathbb{I}:=[0,1)$ in the Paley enumeration. Our aim is to give sufficient conditions for the absolute convergence of the double Walsh?CFourier series of a function $f \in L^{p}(\mathbb{I}^{2})$ for some 1<p?Q2. More generally, we give best possible sufficient conditions for the finiteness of the double series $$\sum_{m=1}^{\infty}\ \sum_{n=1}^{\infty} a_{mn} {|\hat{f}(m,n)|}^r,$$ where {a _mn } is a given double sequence of nonnegative real numbers satisfying a mild assumption and 0<r<2. These sufficient conditions are formulated in terms of (either global or local) dyadic moduli of continuity of?f.     

Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u _t = Δu ^m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u ₀(x) ≈ A|x|^?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t ^α u(t ^β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|^?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g _ij /?t = ?Rg _ij with u(x, 0) = u ₀(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.     

On the completeness of the system \{ {t^{{\lambda _n}}}{\log ^{{m_n}}}t\} in C 0(E)

Let $E = \bigcup\limits_{n = 1}^\infty {{I_n}} $ be the union of infinitely many disjoint closed intervals where In = [a _n , b _n ], 0 < a ₁ < b ₁ < a ₂ < b ₂ < … < b _n < …, $\mathop {\lim }\limits_{n \to \infty } $ b _n = ∞. Let α(t) be a nonnegative function and $\{ {\lambda _n}\} _{n = 1}^\infty $ a sequence of distinct complex numbers. In this paper, a theorem on the completeness of the system $\{ {t^{{\lambda _n}}}{\log ^{{m_n}}}t\} $ in C ₀(E) is obtained where C ₀(E) is the weighted Banach space consists of complex functions continuous on E with f(t)e^?α(t) vanishing at infinity.