The recursive sequence xn+1 = g(xn,xn−1)/(A + xn)

In this note, we investigate the periodic character of solutions of the nonlinear, second-order difference equation

where the parameter A and the initial conditions x₀ and x₁ are positive real numbers. We give sufficient conditions under which every positive solution of this equation converges to a period two solution.     

Nonlinear reaction-diffusion models for interacting populations

This paper proves that several initial-boundary value problems for a wide class of nonlinear reaction-diffusion equations have solutions c_i(x, t), 1 ? i ? N (with c_i(x, t) representing the concentration of the ith species at position x in a set $\text{Ω}$ at time t ? 0), which exist for all t ? 0 and are unique, smooth, nonnegative, and strictly positive for t > 0. The Volterra-Lotka predator-prey model with diffusion (to which the results above are proved to apply) is then studied in more detail. It is proved that any bounded solution of this model loses its spatial dependence and behaves like a periodic function of time alone as t → ∞. It is proved that if the spatial dimension is one or if the diffusion coefficients of the two species are equal, then all solutions are bounded.     

Cauchy problem for the Korteweg-de Vries equation in the case of a nonsmooth unbounded initial function

In the strip П = (?1, 0) × ?, we establish the existence of solutions of the Cauchy problem for the Korteweg-de Vries equation u _t + u _xxx + uu _x = 0 with initial condition either 1) u(?1, x) = ?xθ(x), or 2) u(?1, x) = ?xθ(?x), where θ is the Heaviside function. The solutions constructed in this paper are infinitely smooth for t ∈ (?1, 0) and rapidly decreasing as x → +∞. For the case of the first initial condition, we also establish uniqueness in a certain class. Similar special solutions of the KdV equation arise in the study of the asymptotic behavior with respect to small dispersion of the solutions of certain model problems in a neighborhood of lines of weak discontinuity.     

Periodic solutions of the Lyness max equation

We investigate the periodic nature of solutions of a “max-type” difference equation sometimes referred to as the “Lyness max” equation. The equation we consider is x_n+1=max{x_n,A}/x_n−1, n=0,1,…, where A is a positive real parameter and the initial conditions are arbitrary positive numbers. We also present related results for a similar equation sometimes referred to as the “period 7 max” equation.     

Globally smooth solutions of quasilinear hyperbolic systems in diagonal form

We prove the existence of a large class of globally smooth solutions of the Cauchy problem for the system of n equations u_t + Λ(x, t, u)u_x = 0, where Λ is a diagonal matrix. We show that, under certain monotonicity conditions on both Λ and the initial data u₀, the solution u will be locally Lipschitz continuous at positive times. Since u₀ is a function of locally bounded variation, our result thus provides both for the smoothing of discontinuities in u₀ as well as for the global preservation of smoothness. The global existence results from an a priori estimate of $\text{?u}\text{?x}$, which we obtain by a device which enables us to effectively uncouple the system of equations for $\text{?u}\text{?x}$. Finally, we prove a partial converse which demonstrates that our hypotheses are not overly restrictive.     

Positive periodic solutions of nonautonomous functional differential systems

Considered is the periodic functional differential system with a parameter, x^′(t)=A(t,x(t))x(t)+λf(t,xt). Using the eigenvalue problems of completely continuous operators, we establish some criteria on the existence of positive periodic solutions. Moreover, we apply the results to a couple of population models and obtain sufficient conditions for the existence of positive periodic solutions, which are compared with existing ones.     

Boundary value problems for the Vlasov-Maxwell equation in one dimension

Global existence in time and uniqueness of solutions are proved for the Cauchy problem for the Vlasov-Maxwell system of equations in one dimension. The limiting values of the field ±(x, t) as the space variable x → E ∞ are shown to be uniquely determined by the initial data. This result then yields existence of solutions of various boundary value problems. Solutions periodic in x are also discussed in this same framework.     

Permanence and Positive Periodic Solutions for Kolmogorov Competing Species Systems

We prove the existence of periodic solutions in a compact attractor of (R⁺)ⁿ for the Kolmogorov system x′_i = x_if_i(t, x₁, …, x_n), i = l, …, n in the competitive case. Extension to differential delay equations are con- sidered too. Applications are given to Lotka-Volterra systems with periodic coefficients.     

Compactness and gradient bounds for solutions of the mean curvature system in two independent variables

We establish here a priori estimates for the gradient of solutions of the minimal surface system in two independent variables and for the curvature of their graphs. With the intent of extending these results to graphs with nonzero mean curvature vectors, we then analyze the compactness properties of smooth (C ²) solutions of the mean curvature system. Using a geometric measure theory approach we are able to classify the possible behaviors of a sequence {u _?(x)} of such solutions, under the assumption that a uniform bound on the area of the graphs holds and suitable hypotheses on the length of the mean curvature vectorH(x). In particular, this implies the existence of an a priori gradient bound depending on the oscillation of the solutionu(x).     

Exponential growth for the wave equation with compact time‐periodic positive potential

We prove the existence of smooth positive potentials V(t, x), periodic in time and with compact support in x, for which the Cauchy problem for the wave equation u_tt ? Δ_xu + V(t, x)u = 0 has solutions with exponentially growing global and local energy. Moreover, we show that there are resonances, z ∈ ?, |z| > 1, associated to V(t, x). © 2008 Wiley Periodicals, Inc.     

On a General Class of Nonlocal Equations

Motivated by recent developments in cosmology and string theory, we introduce a functional calculus appropriate for the study of non-linear nonlocal equations of the form f(Δ)u = U(x, u(x)) on Euclidean space. We prove that under some technical assumptions, these equations admit smooth solutions. We also consider nonlocal equations on compact Riemannian manifolds, and we prove the existence of smooth solutions. Moreover, in the Euclidean case we present conditions on f which guarantee that the solutions we find are, in fact, real-analytic.     

On the recursive sequencex n +1=α-(x n /x n −1)

We study the global asymptotic stability, global attractivity, boundedness character, and periodic nature of all positive solutions and all negative solutions of the difference equation $$x_{n + 1} = \alpha - \frac{{x_n }}{{x_{n - 1} }}, n = 0,1,...,$$ where α∈R is a real number, and the initial conditionsx?1,x ₀ are arbitrary real numbers.     

Global weak solutions for a periodic two-component?μ-Hunter–Saxton system

This paper is concerned with global existence of weak solutions for a periodic two-component?μ-Hunter–Saxton system. We first derive global existence for strong solutions to the system with smooth approximate initial data. Then, we show that the limit of approximate solutions is a global weak solution of the two-component?μ-Hunter–Saxton system.     

On the structure of periodic solutions of differential equations

This paper studies the “internal structure” of the periodic solutions of differential equations with the aim of stating when they are constant functions. Yorke [21] and Lasota and Yorke [10] are the first works which show the existence, uńder certain conditions, of a lower bound for the period of non-constant solutions. As applications of the general results proved in Section 1 we obtain a negative solution to an open problem of Browder, the discovery that the periodic solutions ensured by Vidossich [17, Theorem 3.16], are constant functions, and conditions under which the periodic solutions of hyperbolic and parabolic equations are constant functions. Finally, we note that Li [11] applies the results of Section 1 to differential equations with delay.Various result of this paper point out a strong connection between the existence of periodic solutions of small period of x′ = f(x) and the fact that the origin belongs to the range of f. This situation is explored in [19].     

Locally Lipschitz solutions of a single conservation law in several space variables

We consider the Cauchy problem for a single conservation law in several space variables. Letting u(x, t) denote the solution with initial data u₀, we state necessary and sufficient conditions on u₀ so that u(x, t) is locally Lipschitz continuous in the half space {t > 0}. These conditions allow for the preservation of smoothness of u₀ as well as for the smooth resolution of discontinuities in u₀. One consequence of our result is that u(x, t) cannot be locally Lipschitz unless u₀ has locally bounded variation. Another is that solutions which are bounded and locally Lipschitz continuous in {t > 0} automatically have boundary values u₀ at t = 0 in the sense that u(·, t) → u₀ in L_loc¹. Finally, we give an elementary proof that locally Lipschitz solutions satisfy Kruzkov's uniqueness condition.     

Approximation of entropy on hyperbolic sets for one-dimensional maps and their multidimensional perturbations

We consider piecewise monotone (not necessarily, strictly) piecewise C ² maps on the interval with positive topological entropy. For such a map f we prove that its topological entropy h _top(f) can be approximated (with any required accuracy) by restriction on a compact strictly f-invariant hyperbolic set disjoint from some neighborhood of prescribed set consisting of periodic attractors, nonhyperbolic intervals and endpoints of monotonicity intervals. By using this result we are able to generalize main theorem from [1] on chaotic behavior of multidimensional perturbations of solutions for difference equations which depend on two variables at nonperturbed value of parameter.     

Existence of Solutions for a Class of Nonvariational Quasilinear Periodic Problems

We consider a nonlinear nonvariational periodic problem with a nonsmooth potential. Using the spectrum of the asymptotic (as |x| →?∞) differential operator and degree theoretic methods based on the degree map for multivalued perturbations of (S)_?+? operators, we establish the existence of a nontrivial smooth solution.     

Quasi-sure Flows Associated with Vector Fields of Low Regularity

The authors construct a solution Ut（x） associated with a vector field on the Wiener space for all initial values except in a 1-slim set and obtain the 1-quasi-sure flow property where the vector field is a sum of a skew-adjoint operator not necessarily bounded and a nonlinear part with low regularity, namely one-fold differentiability. Besides, the equivalence of capacities under the transformations of the Wiener space induced by the solutions is obtained.     

Periodicity and convergence for xn+1=|xn−xn−1|

Each solution {x_n} of the equation in the title is either eventually periodic with period 3 or else, it converges to zero—which case occurs depends on whether the ratio of the initial values of {x_n} is rational or irrational. Further, the sequence of ratios {x_n/x_n−1} satisfies a first-order difference equation that has periodic orbits of all integer periods except 3. p-cycles for each p≠3 are explicitly determined in terms of the Fibonacci numbers. In spite of the non-existence of period 3, the unique positive fixed point of the first-order equation is shown to be a snap-back repeller so the irrational ratios behave chaotically.     

Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations

In this paper we consider a class of planar autonomous systems having an isolated limit cycle x₀ of smallest period T>0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1. We consider two functions, defined by means of the eigenfunctions of the adjoint of the linearized system, and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of T-periodic solutions of the autonomous system when it is perturbed by nonsmooth T-periodic nonlinear terms of small amplitude. We also show the convergence of these periodic solutions to x₀ as the perturbation disappears and we provide an estimation of the rate of convergence. The employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach, commonly employed in the existing literature on this subject, based on various versions of the implicit function theorem.