A new uniqueness criterion for the number of periodic orbits of Abel equations

A solution of the Abel equation such that x(0)=x(1) is called a periodic orbit of the equation. Our main result proves that if there exist two real numbers a and b such that the function aA(t)+bB(t) is not identically zero, and does not change sign in [0,1] then the Abel differential equation has at most one non-zero periodic orbit. Furthermore, when this periodic orbit exists, it is hyperbolic. This result extends the known criteria about the Abel equation that only refer to the cases where either A(t)?0 or B(t)?0 does not change sign. We apply this new criterion to study the number of periodic solutions of two simple cases of Abel equations: the one where the functions A(t) and B(t) are 1-periodic trigonometric polynomials of degree one and the case where these two functions are polynomials with three monomials. Finally, we give an upper bound for the number of isolated periodic orbits of the general Abel equation , when A(t), B(t) and C(t) satisfy adequate conditions.     

Homoclinic solutions for nonautonomous second-order Hamiltonian systems with a coercive potential

A new existence result of homoclinic orbits is obtained for the second-order Hamiltonian systems , where F(t,x) is periodic with respect to t. This result generalizes some known results in the literature.     

Doubly nonlinear periodic problems with unbounded operators

The solvability of the evolution system v′(t)+B(t)u(t)∋f(t), v(t)∈A(t)u(t), 0<t<T, with the periodic condition v(0)=v(T) is investigated in the case where are bounded, possibly degenerate, subdifferentials and are unbounded subdifferentials.     

Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation

Let B be the unit ball in Cn with respect to an arbitrary norm and let f(z,t) be a g-Loewner chain such that e−tf(z,t)−z has a zero of order k+1 at z=0. In this paper, we obtain growth and covering theorems for . Moreover, we consider coefficient bounds and examples of mappings in .     

On the period function for a family of complex differential equations

We consider planar differential equations of the form being f(z) and g(z) holomorphic functions and prove that if g(z) is not constant then for any continuum of period orbits the period function has at most one isolated critical period, which is a minimum. Among other implications, the paper extends a well-known result for meromorphic equations, that says that any continuum of periodic orbits has a constant period function.     

Stable extendibility of normal bundles over lens spaces

We study the stable extendibility of R-vector bundles over the (2n+1)-dimensional standard lens space Ln(p) with odd prime p, focusing on the normal bundle to an immersion of Ln(p) in the Euclidean space R2n+1+t. We show several concrete cases in which is stably extendible to Lk(p) for any k with k?n, and in several cases we determine the exact value m for which is stably extendible to Lm(p) but not stably extendible to Lm+1(p).     

Carleson embeddings for Sobolev spaces via heat equation

Let u(t,x) be the solution of the heat equation (∂_t-Δ_x)u(t,x)=0 on subject to u(0,x)=f(x) on Rⁿ. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t²,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞).     

Existence results for some polyharmonic problems in the half-space

We study the existence of positive solutions of the m-polyharmonic nonlinear elliptic equation m(−Δ)u+f(⋅,u)=0 in the half-space , n?2 and m?1. Our purpose is to give two existence results for the above equation subject to some boundary conditions, where the nonlinear term f(x,t) satisfies some appropriate conditions related to a certain Kato class of functions .     

Homoclinic solutions for a class of second-order Hamiltonian systems

A new result for existence of homoclinic orbits is obtained for the second-order Hamiltonian systems , where t∈R, u∈Rn and W₁,W₂∈C¹(R×Rn,R) and f∈C(R,Rn) are not necessary periodic in t. This result generalizes and improves some existing results in the literature.     

Positive periodic solutions of functional differential equations

We consider the existence, multiplicity and nonexistence of positive ω-periodic solutions for the periodic equation x′(t)=a(t)g(x)x(t)−λb(t)f(x(t−τ(t))), where are ω-periodic, , , f,g∈C([0,∞),[0,∞)), and f(u)>0 for u>0, g(x) is bounded, τ(t) is a continuous ω-periodic function. Define , , i₀=number of zeros in the set and i_∞=number of infinities in the set . We show that the equation has i₀ or i_∞ positive ω-periodic solution(s) for sufficiently large or small λ>0, respectively.     

Existence and multiplicity of positive solutions for the p-Laplacian with nonlocal coefficient

We consider the Dirichlet problem with nonlocal coefficient given by in a bounded, smooth domain Ω⊂Rn (n?2), where Δp is the p-Laplacian, w is a weight function and the nonlinearity f(u) satisfies certain local bounds. In contrast with the hypotheses usually made, no asymptotic behavior is assumed on f. We assume that the nonlocal coefficient (q?1) is defined by a continuous and nondecreasing function satisfying a(t)>0 for t>0 and a(0)?0. A positive solution is obtained by applying the Schauder Fixed Point Theorem. The case a(t)=tγ/q (0<γ<p−1) will be considered as an example where asymptotic conditions on the nonlinearity provide the existence of a sequence of positive solutions for the problem with arbitrarily large sup norm.     

Perfectness and imperfectness of unit disk graphs on triangular lattice points

Given a finite set of 2-dimensional points P⊆R² and a positive real d, a unit disk graph, denoted by (P,d), is an undirected graph with vertex set P such that two vertices are adjacent if and only if the Euclidean distance between the pair is less than or equal to d. Given a pair of non-negative integers m and n, P(m,n) denotes a subset of 2-dimensional triangular lattice points defined by where . Let T_m,n(d) be a unit disk graph defined on a vertex set P(m,n) and a positive real d. Let be the kth power of T_m,n(1).In this paper, we show necessary and sufficient conditions that [ is perfect] and/or [ is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring (T_m,n(d),w) and .     

On the residue classes of integer sequences satisfying a linear three term recurrence formula

In this paper we investigate linear three-term recurrence formulae with sequences of integers (T(n))_n?0 and (U(n))_n?0, which are ultimately periodic modulo m, e.g.

     

A spectral condition for odd cycles in graphs

Let G be a graph of sufficiently large order n, and let the largest eigenvalue μ(G) of its adjacency matrix satisfies . Then G contains a cycle of length t for every t?n/320This condition is sharp: the complete bipartite graph T₂(n) with parts of size ⌊n/2⌋ and ⌈n/2⌉ contains no odd cycles and its largest eigenvalue is equal to .This condition is stable: if μ(G) is close to and G fails to contain a cycle of length t for some t?n/321, then G resembles T₂(n).     

Integers with dense divisors 2

Let 1=d₁(n)<d₂(n)<?<d_τ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are t-dense iff max_1?i<τ(n)d_i+1(n)/d_i(n)?t. Let D(x,t) be the number of positive integers not exceeding x whose divisors are t-dense. We show that for x?3, and , we have , where , and d(w) is a continuous function which satisfies d(w)?1/w for w?1. We also consider other counting functions closely related to D(x,t).