Let $a₁,...,a_k$ be arbitrary nonzero real numbers. An $(a₁,...,a_k)$-decomposition of a function f:ℝ → ℝ is a sum $f₁+\cdots +f_k=f$ where $f_i:ℝ\to ℝ$ is an $a_i$-periodic function. Such a decomposition is not unique because there are several solutions of the equation $h₁+\cdots +h_k=0$ with $h_i:ℝ\to ℝa_i$-periodic. We will give solutions of this equation with a certain simple structure (trivial solutions) and study whether there exist other solutions or not. If not, we say that the $(a₁,...,a_k)$-decomposition is essentially unique. We characterize those periods for which essential...

The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let $\alpha$ and $\beta$ be two multiplicatively independent Perron numbers. Then a sequence $x\in A^{\mathbb{N}}$, where $A$ is a finite alphabet, is both $\alpha$-substitutive and $\beta$-substitutive if and only if $x$ is ultimately...

We study the existence of positive solutions of the quasilinear problem ⎧ $-{\Delta }_{N}u+{V\left(x\right)\left|u\right|}^{N-2}u={f(u,|\nabla u|}^{N-2}\nabla u)$, $x\in {ℝ}^N$, ⎨ ⎩ u(x) > 0, $x\in {ℝ}^N$, where ${\Delta }_{N}u={div\left(\right|\nabla u|}^{N-2}\nabla u)$ is the N-Laplacian operator, $V:{ℝ}^N\to ℝ$ is a continuous potential, $f:ℝ\times {ℝ}^N\to ℝ$ is a continuous function. The main result follows from an iterative method based on Mountain Pass techniques.

Consider the following higher order difference equation $$x(n+1)=f(n,x(n\left)\right)+g(n,x(n-k\left)\right),n=0,1,\cdots$$ where $f(n,x)$ and $g(n,x):\{0,1,\cdots \}\times [0,\infty )\to [0,\infty )$ are continuous functions in $x$ and periodic functions in $n$ with period $p$, and $k$ is a nonnegative integer. We show the existence of a periodic solution $\left\{\tilde{x}\left(n\right)\right\}$ under certain conditions, and then establish a sufficient condition for $\left\{\tilde{x}\left(n\right)\right\}$ to be a global attractor of all nonnegative solutions of the equation. Applications to Riccati difference equation and some other difference equations derived from mathematical biology are also...

Let $G$ be a graph of order $n$ and $\lambda \left(G\right)$ the spectral radius of its adjacency matrix. We extend some recent results on sufficient conditions for Hamiltonian paths and cycles in $G$. One of the main results of the paper is the following theorem: Let $k\ge 2,$ $n\ge k^3+k+4,$ and let $G$ be a graph of order $n$, with minimum degree $\delta \left(G\right)\ge k.$ If $$\lambda \left(G\right)\ge n-k-1,$$ then $G$ has a Hamiltonian cycle, unless $G=K_1\vee (K_{n-k-1}+K_k)$ or $G=K_k\vee (K_{n-2k}+{\overline{K}}_k).$

Let $X$ be a connected closed manifold and $f$ a self-map on $X$. We say that $f$ is almost quasi-unipotent if every eigenvalue $\lambda$ of the map $f_{*k}$ (the induced map on the $k$-th homology group of $X$) which is neither a root of unity, nor a zero, satisfies that the sum of the multiplicities of $\lambda$ as eigenvalue of all the maps $f_{*k}$ with $k$ odd is equal to the sum of the multiplicities of $\lambda$ as eigenvalue of all the maps $f_{*k}$ with $k$ even. We prove that if $f$ is $C^1$ having finitely many periodic points all of them...

We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots $t_j$ = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let $V_{n,\nu }$ be the space of piecewise linear continuous functions on the torus with knots $t_j:0\le j\le N-1$. Finally, let $P_{n,\nu }$ be the orthogonal projection operator from L²([0,1)) onto $V_{n,\nu }$. The main result is $li{m}_{n\to \infty ,\nu =1}\left|\right|P_{n,\nu }:L^{\infty }\to L^{\infty }\left|\right|=su{p}_{n\in \mathbb{N},0\le \nu \le n}\left|\right|P_{n,\nu }:L^{\infty }\to L^{\infty }\left|\right|=2+(33-18\surd 3)/13$. This shows in particular that the Lebesgue constant of the classical...

We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet $p$-Laplacian and the Navier $p$-biharmonic operator on a ball of radius $R$ in ${ℝ}^N$ and its asymptotics for $p$ approaching $1$ and $\infty$. Let $p$ tend to $\infty$. There is a critical radius $R_C$ of the ball such that the principal eigenvalue goes to $\infty$ for $0<R\le R_C$ and to $0$ for $R>R_C$. The critical radius is $R_C=1$ for any $N\in \mathbb{N}$ for the $p$-Laplacian and $R_C=\sqrt{2N}$ in the case of the $p$-biharmonic operator. When $p$ approaches $1$, the principal eigenvalue...

We show that every operator from ${\ell }_s$ to ${\ell }_p\otimes ̂{\ell }_q$ is compact when 1 ≤ p,q < s and that every operator from ${\ell }_s$ to ${\ell }_p\widehat{\otimes ̂}{\ell }_q$ is compact when 1/p + 1/q > 1 + 1/s.

Let $a=(a_1,a_2,...,a_n)$ be a nonincreasing sequence of positive real numbers. Denote by $S=\{1,2,...,n\}$ the index set and by $J_k=\{I=\{r_1,r_2,...,r_k\}$, $1\le r_1<r_2<\cdots <r_k\le n\}$ the set of all subsets of $S$ of cardinality $k$, $1\le k\le n-1$. In addition, denote by $a_I=a_{r_1}+a_{r_2}+\cdots +a_{r_k}$, $1\le k\le n-1$, $1\le r_1<r_2<\cdots <r_k\le n$, the sum of $k$ arbitrary elements of sequence $a$, where $a_{I_1}=a_1+a_2+\cdots +a_k$ and $a_{I_n}=a_{n-k+1}+a_{n-k+2}+\cdots +a_n$. We consider bounds of the quantities $R{S}_k\left(a\right)=a_{I_1}/a_{I_n}$, $L{S}_k\left(a\right)=a_{I_1}-a_{I_n}$ and $S_{k,\alpha }\left(a\right)={\sum }_{I\in J_k}{a}_I^{\alpha }$ in terms of $A={\sum }_{i=1}^n{a}_i$ and $B={\sum }_{i=1}^n{a}_i^2$. Then we use the obtained results to generalize some results regarding Laplacian and normalized Laplacian eigenvalues of graphs.

We report on results we recently obtained in Hebey and Thizy [11, 12] for critical stationary Kirchhoff systems in closed manifolds. Let $(M^n,g)$ be a closed $n$-manifold, $n\ge 3$. The critical Kirchhoff systems we consider are written as $$\left(a+b\sum _{j=1}^p{\int }_M{\left|\nabla u_j\right|}^{2}d{v}_g\right){\Delta }_g{u}_i+\sum _{j=1}^p{A}_{ij}{u}_j={\left|U\right|}^{2^{☆}-2}{u}_i$$ for all $i=1,\cdots ,p$, where ${\Delta }_g$ is the Laplace-Beltrami operator, $A$ is a $C^1$-map from $M$ into the space $M_s^p\left(ℝ\right)$ of symmetric $p\times p$ matrices with real entries, the $A_{ij}$’s are the components of $A$, $U=(u_1,\cdots ,u_p)$, $\left|U\right|:M\to ℝ$ is the Euclidean norm of $U$, $2^{☆}=\frac{2n}{n-2}$ is the critical Sobolev exponent, and...

In this paper, we propose an extension of a periodic $GARCH$ ($PGARCH$) model to a Markov-switching periodic $GARCH$ ($MS$-$PGA$ $RCH$), and provide some probabilistic properties of this class of models. In particular, we address the question of strictly periodically and of weakly periodically stationary solutions. We establish necessary and sufficient conditions ensuring the existence of higher order moments. We further provide closed-form expressions for calculating the even-order moments as well...

Via critical point theory we establish the existence and regularity of solutions for the quasilinear elliptic problem ⎧ $div(x,\nabla u)+{a\left(x\right)\left|u\right|}^{p-2}u={g\left(x\right)\left|u\right|}^{p-2}u+h\left(x\right){\left|u\right|}^{s-1}u$ in ${ℝ}^N$ ⎨ ⎩ u > 0, $li{m}_{\left|x\right|\to \infty }u\left(x\right)=0$, where 1 < p < N; a(x) is assumed to satisfy a coercivity condition; h(x) and g(x) are not necessarily bounded but satisfy some integrability restrictions.

Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.