This paper concerns with a class of delayed Nicholson’s blowflies model with a nonlinear density-dependent mortality term. Under appropriate conditions, we establish some criteria to ensure that the solutions of this model converge globally exponentially to a positive almost periodic solution. Moreover, we give some examples and numerical simulations to illustrate our main results.

This paper presents a new nonlinear least-squares algorithm for fitting band-limited periodic signals with unknown frequency and harmonic content. The new solution features a model-based recursive calculation method that requires less memory space and has smaller computational demand than the known matrix-based algorithms.

In this paper, by using an initial value problem method and a global inverse function theorem, we give some existence and uniqueness results of periodic solution for a class of nth-order nonlinear ordinary differential equations. AMS (MOS) Subject Classification. 34B10, 34B15, 34L16, 65L05, 65L10

In recent years, we have established the iteration theory of the index for symplectic matrix paths and applied it to periodic solution problems of nonlinear Hamiltonian systems. This paper is a survey on these results. 2000 Mathematics Subject Classification: 58E05, 70H05, 34C25.

We analyze the non-degeneracy of the linear 2n-order differential equation u(2n) + 2n−1 ∑ m=1 amu = q(t)u with potential q(t) ∈ Lp(R/TZ), by means of new forms of the optimal Sobolev and Wirtinger inequalities. The results is applied to obtain existence and uniqueness of periodic solution for the prescribed nonlinear problem in the semilinear and superlinear case.

We use Krasnoselskii’s fixed point theorem to show that the nonlinear neutral differential equation with delay d dt [x(t)− ax(t− τ)] = r(t)x(t)− f(t, x(t− τ)) has a positive periodic solution. An example will be provided as an application to our theorems. AMS Subject Classifications: 34K20, 45J05, 45D05

Using ergodicity of functions, we prove the existence and uniqueness of (asymptotically) almost periodic solution for some nonlinear differential equations. As a consequence, we generalize a Massera’s result. A counterexample is given to show that the ergodic condition cannot be dropped. 2000 Mathematics Subject Classification. Primary 34C27, 43A60, 37Axx, 28Dxx.

We study the J−holomorphic curves in the symplectization of the contact manifolds and prove that there exists at least one periodic Reeb orbits in any closed contact manifold with any contact form by using the well-known Gromov's nonlinear Fredholm alternative for J−holomorphic curves. As a corollary, we give a complete solution on the well-known Weinstein conjecture.

We study the existence of periodic solutions for a nonlinear system of n-th order di erential equations on time-scales. Assuming a suitable Nirenberg type condition, we prove the existence of at least one solution of the problem using Mawhin's coincidence degree.

Nonlinear vector integral equations are considered. Solution estimates and solvability conditions are derived. Applications to the periodic boundary value problem are also discussed. Under some restrictions our results improve the well-known ones. The main tool in the paper is the recent estimates for the resolvent of Hilbert-Schmidt operators.