Lyapunov exponents of solutions to linear differential equations with periodic forcing functions

We give Lyapunov exponents of solutions to linear differential equations of the form x^′=Ax+f(t), where A is a complex matrix and f(t) is a τ-periodic continuous function. Notice that f(t) is not “small” as t→∞. The proof is essentially based on a representation [J. Kato, T. Naito, J.S. Shin, A characterization of solutions in linear differential equations with periodic forcing functions, J. Difference Equ. Appl. 11 (2005) 1-19] of solutions to the above equation.     

A Massera type theorem for almost automorphic solutions of differential equations

In this note, we present a Massera type theorem for the existence of almost automorphic solutions of periodic linear evolution equations of the form x^′(t)=A(t)x(t)+f(t), where A(t) is unbounded linear operator depending on t periodically and generates a τ-periodic evolutionary process, f is almost automorphic. The main results are stated in terms of the almost automorphy of solutions and their Carleman spectra.     

Existence of quasibounded solutions for the higher order dynamic equations on measure chains

By using the exponential dichotomy and Schauder’s fixed point theorem, some new criteria are established for the existence of quasibounded solutions of the inhomogeneous system x^Δ = A(t)x + g(t, x) + h(t), which generalize the previous results in [15], [19].     

Bounded and periodic solutions of infinite delay evolution equations

For A(t) and f(t,x,y) T-periodic in t, we consider the following evolution equation with infinite delay in a general Banach space X:

(0.1)     

Conjugacies for linear and nonlinear perturbations of nonuniform behavior

We construct topological conjugacies between linear and nonlinear evolution operators that admit either a nonuniform exponential contraction or a nonuniform exponential dichotomy. We consider evolution operators defined by nonautonomous differential equations x^′=A(t)x+f(t,x) in a Banach space. The conjugacies are obtained by first considering sufficiently small linear and nonlinear perturbations of linear equations x^′=A(t)x. In the case of linear perturbations, we construct in a more or less explicit manner topological conjugacies between the two linear flows. In the case of nonlinear perturbations, we obtain a version of the Grobman-Hartman theorem for nonuniformly hyperbolic dynamics. Furthermore, all the conjugacies that we construct are locally Hölder continuous provided that the vectors fields are of class C¹. As a byproduct of our approach, we give conditions for the robustness of strong nonuniform exponential behavior, in the sense that under sufficiently small perturbations the structure determined by the stable and unstable bundles persists up to small variations. We also show that the constants determining the nonuniform exponential contraction or nonuniform exponential dichotomy vary continuously with the perturbation. All the results are obtained in Banach spaces.     

On the instability of differential systems with varying delay

The unstable properties of the null solution of the nonautonomous delay system x′(t)=A(t)x(t)+B(t)x(t−r₁(t))+f(t,x(t),x(t−r₂(t))) are examined; the nonconstant delays r₁, r₂ are assumed to be continuous bounded functions. The case A=constant is reviewed, where a theorem, recalling the Perron instability theorem for ordinary differential equations, is obtained.     

Periodic solutions for a nonautonomous ordinary differential equation

We consider the nonautonomous differential equation of second order x^″+a(t)x−b(t)x²+c(t)x³=0, where a(t),b(t),c(t) are T-periodic functions. This is a biomathematical model of an aneurysm in the circle of Willis. We prove the existence of at least two positive T-periodic solutions for this equation, using coincidence degree theories.     

Periodic behavior of a nonlinear third order vibrating system

A theorem is proved to show that the third order differential equation x^‴+f(t,x,x^′,x^″)=0 has nontrivial solutions characterized by x^′(0)=x^′(τ)=0 when x,x^′,x^″ and f(t,x,x^′,x^″) are bounded. A second condition is introduced to prove the existence of periodic solution for this equation. It is shown that the equation has a τ-periodic solution if f(t,x,x^′,x^″) is an even function with respect to x^′. The existence and periodicity conditions would be applied to third order systems such as viscoelastic mechanical vibration isolator system. The concepts of Green’s function and the Schauder’s fixed-point theorem have been used for proving the third-order-existence theorem.     

Existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for the first order neutral functional differential equation of the form

(x(t)+Bx(t−δ)^)′=g₁(t,x(t))+g₂(t,x(t−τ))+p(t).     

The set of periodic scalar differential equations with cubic nonlinearities

We study the structure induced by the number of periodic solutions on the set of differential equations x^′=f(t,x) where f∈C³(R²) is T-periodic in t, fx³(t,x)<0 for every (t,x)∈R², and f(t,x)→?∞ as x→∞, uniformly on t. We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions. Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold.     

A remark on Liouville's formula on small time scales

We present a new proof of the Liouville formula for a d-dimensional linear dynamic system xΔ=A(t)x on a time scale T, where T is in a sense small. Our proof demonstrates that Liouville's formula on small time scales is a direct consequence of its well-known counterpart for ordinary differential equations.     

Weighted pseudo almost periodic mild solutions of semilinear evolution equations with nonlocal conditions

In this paper, applying the theory of semigroups of operators to evolution families and Banach fixed point theorem, we prove the existence and uniqueness of the weighted pseudo almost periodic mild solution of the semilinear evolution equation x^′(t)=A(t)x(t)+f(t,x(t)) with nonlocal conditions x(0)=x₀+g(x) in Banach space X under some suitable hypotheses.     

Singular and nonsingular first-order initial value problems

Under barrier strip type arguments we investigate the existence of global solutions to the initial value problem x^′=f(t,x,x^′), x(0)=A, where the scalar function f(t,x,p) may be singular at x=A.     

Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations

In this paper, applying the theory of semigroups of operators to evolution family and Banach fixed point theorem, we prove the existence and uniqueness of an (a) almost automorphic (weighted pseudo almost automorphic) mild solution of the semilinear evolution equation x^′(t)=A(t)x(t)+f(t,x(t)) in Banach space under conditions.     

The Lower Bounds of T-Periodic Solutions for the Forced Duffing Equation

This paper is devoted to the discussion of the number of T-periodic solutions for the forced Duffing equation, x″ + kx′ + g(t, x) = s(1 + h(t)), with g(t, x) being a continuous function by using the degree theory, upper and lower solutions method, and the twisting theorem.     

Circular spectrum and bounded solutions of periodic evolution equations

We consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u^′=A(t)u+?H(t,u)+f(t), where A(t) is, in general, an unbounded operator depending 1-periodically on t, H is 1-periodic in t, ? is small, and f is a bounded and continuous function that is not necessarily uniformly continuous. We propose a new approach to the spectral theory of functions via the concept of “circular spectrum” and then apply it to study the linear equations u^′=A(t)u+f(t) with general conditions on f. For small ? we show that the perturbed equation inherits some properties of the linear unperturbed one. The main results extend recent results in the direction, saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of f, then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of f.     

Periodic solutions of semilinear equations of evolution of compact type

The existence of solutions in a weak sense of x′ + (A + B(t, x))x = f(t, x), x(0) = x(T) is established under the conditions that A generates a semigroup of compact type on a Hilbert space H; B(t,x) is a bounded linear operator and f(t, x) a function with values in H; for each square integrable ?(t) the problem with B(t, ?(t)) and f(t, ?(t)) in place of B(t, x) and f(t, x) has a unique solution; and B and f satisfy certain boundedness and continuity conditions.     

Instability results for slowly time varying linear dynamic systems on time scales

We develop eigenvalue criteria under which the solutions of a “slowly” time varying linear dynamic system of the form xΔ(t)=A(t)x(t) are unstable.     

Three positive solutions to a discrete focal boundary value problem

We are concerned with the discrete focal boundary value problem Δ³x(t − k) = f(x(t)), x(a) = Δx(t₂) = Δ²x(b + 1) = 0. Under various assumptions on f and the integers a, t₂, and b we prove the existence of three positive solutions of this boundary value problem. To prove our results we use fixed point theorems concerning cones in a Banach space.     

An existence theorem for probability measures invariant under a Markov kernel

LetP be a Markov kernel defined on a measurable space (X,A). A probability measure μ onA is said to beP-invariant if μ(A=∫P(x,A)dμ(x) for allA ∈A∈A. In this note we prove a criterion for the existence ofP-invariant probabilities which is, in particular, a substantial generalization of a classical theorem due to Oxtoby and Ulam ([5]). As another consequence of our main result, it is shown that every pseudocompact topological space admits aP-invariant Baire probability measure for any Feller kernelP.