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In this article, we combine rotating periodic functions with µ ergodic functions to obtain a new class of functions called µ pseudo rotating periodic functions. Then we study the existence and uniqueness of µ pseudo rotating periodic solutions for linear systems, semi-linear systems, and non-linear systems by exponential dichotomy. [ABSTRACT FROM AUTHOR]

In this article, we establish the existence of Lipschitz stable in-variant manifolds for the semiows generated by the delay differential equation x' = L(t)xt+f(t, xt, λ) with impulses at times {τi}∞i =1, assuming that the perturbation f(t, xt, λ) as well as the impulses are small and the corresponding linear delay differential equation admits a nonuniform exponential dichotomy We also show that the obtained manifolds are Lipschitz in the parameter λ. [ABSTRACT FROM AUTHOR]

For a non-autonomous delay difference equation with infinite delay, we construct smooth stable and unstable invariant manifolds for any sufficiently small perturbation of an exponential dichotomy. We consider a general class of norms on the phase space satisfying an axiom considered by Matsunaga and Murakami that goes back to earlier work by Hale and Kato for continuous time. In addition, we show that the invariant manifolds are as regular as the perturbation. Finally, we consider briey the case of center manifolds and we formulate corresponding results. [ABSTRACT FROM AUTHOR]

We consider the notion of an exponential dichotomy in mean for a cocycle, in which the exponential behavior in the classical notion of an exponential dichotomy is replaced by an average with respect to an invariant probability measure. We give a complete characterization of the exponential behavior in mean in terms of Lyapunov functions, both for cocycles over maps and flows. [ABSTRACT FROM AUTHOR]

This article concerns Liénard-type systems with time-varying de-lays. By applying the theory of exponential dichotomies, the properties of pseudo almost periodic function, inequality analysis techniques, and contrac tion mapping principle, new criteria for the existence and uniqueness of pseudo almost periodic solutions are established. An example is given to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

We study the covering dimension of the set of (positive) solutions to various classes of nonlinear equations involving condensing and A-proper maps. It is based on the nontriviality of the fixed point index of a certain condensing map or on oddness of a nonlinear map. Applications to nonlinear singular integral equations and to semilinear ordinary and elliptic partial differential equations are given with finite or infinite dimensional null space of the linear part. [ABSTRACT FROM AUTHOR]

This work concerns the existence of almost periodic solutions for certain differential equations with piecewise constant arguments. The coefficients of these equations are almost periodic and the equation can be seen as perturbations of a linear equation satisfying an exponential dichotomy on a difference equation. The stability of that solution on a semi-axis is also studied. [ABSTRACT FROM AUTHOR]

We study the forms of a range of free energy functionals for materials with memory for two types of strain history, namely sinusoidal and exponential behaviours. The work deals with discrete spectrum materials, which are those with relaxation functions given by sums of decaying exponentials. Various standard free energy functionals are explored, including the minimum free energy and related quantities. It is shown that quite different formulae are obtained, depending on the manner in which these functionals are evaluated, particularly for those related to the minimum free energy. Such differences are resolved. Various numerical plots of free energies are presented and discussed, along with the rates of dissipation associated with them. This is the first comprehensive exploration of quantitative comparisons between the various free energy functionals. The results both illustrate certain properties that have been deduced theoretically and expose an interesting proximity between apparently unconnected functionals. [ABSTRACT FROM AUTHOR]

In this article we introduce a class of discontinuous almost auto- morphic functions which appears naturally in the study of almost automorphic solutions of differential equations with piecewise constant argument. Their fundamental properties are used to prove the almost automorphicity of bounded solutions of a system of differential equations with piecewise constant argument. Due to the strong discrete character of these equations, the existence of a unique discrete almost automorphic solution of a non-autonomous almost automorphic difference system is obtained, for which conditions of exponential dichotomy and discrete Bi-almost automorphicity are fundamental. [ABSTRACT FROM AUTHOR]

We prove that a family of q-periodic continuous matrix valued function {A(t)}t∈R has an exponential dichotomy with a projector P if and only if ∫0t eiμsU(t, s)Pds is bounded uniformly with respect to the parameter μ and the solution of the Cauchy operator Problem ...(t) = -Y (t)A(t) + eiμt(I - P), t ≥ s Y (s) = 0, has a limit in L(Cn) as s tends to ∞1 which is bounded uniformly with respect to the parameter μ. Here, {U(t, s) : t, ∈ R} is the evolution family generated by {A(t)}t∈R, μ is a real number and q is a fixed positive number. INSET: X. [ABSTRACT FROM AUTHOR]

A method is presented for proving the existence of solutions for boundary-value problems on the half line. The problems under study are nonlinear, nonautonomous systems of ODEs with the possibility of some prescribed value at t = 0 and with the condition that solutions decay to zero as t grows large. The method relies upon a topological degree for proper Fredholm maps. Specific conditions are given to ensure that the boundary-value problem corresponds to a functional equation that involves an operator with the required smoothness, properness, and Fredholm properties (including a calculable Fredholm index). When the Fredholm index is zero and the solutions are bounded a priori, then a solution exists. The method is applied to obtain new existence results for systems of the form ̇v + g(t,w) = f₁(t) and ẇ + h(t, v) = f₂(t). [ABSTRACT FROM AUTHOR]

In this work, we give sufficient conditions for the existence and uniqueness of a weighted pseudo-almost periodic solutions for some neutral partial functional differential equations. Our working tools are based on the variation of constant formula and the spectral decomposition of the phase space. To illustrate our main result, we propose to study the existence and uniqueness of a weighted pseudo-almost periodic solution for some neutral model arising in physical systems. [ABSTRACT FROM AUTHOR]

We study the existence of solutions to a non-homogeneous system of linear ODEs which has the pole of first order at x = 0; these solutions should vanish at infinity and be continuously differentiable on [0, ∞). The resonant case where the corresponding homogeneous problem has nontrivial solutions is of great interest to us. Under the conditions that the homogeneous system is exponentially dichotomic on [1, ∞) and the residue of system's operator at x = 0 does not have eigenvalues with real part 1, we construct the so-called generalized Green function. We also establish conditions under which the main non-homogeneous problem can be reduced to the Noetherian problem with nonzero index. [ABSTRACT FROM AUTHOR]

This paper studies the existence and uniqueness of almost automorphic mild solutions to the semilinear parabolic evolution equation u'(t) = A(t)u(t) + f(t, u(t)), assuming that the linear operators A(·) satisfy the 'Acquistapace--Terreni' conditions, the evolution family generated by A(·) has an exponential dichotomy, and the resolvent R(ω,A(·)), and f are almost automorphic. [ABSTRACT FROM AUTHOR]

By means of the exponential dichotomy method and the properties of the weighted pseudo almost periodic functions, sufficient conditions are obtained for the existence of weighted pseudo almost periodic solutions of functional differential equations with finite delay. [ABSTRACT FROM AUTHOR]

We show that the existence of a nontrivial bounded uniformly continuous (BUC) complete trajectory for a C0-semigroup TA(t) generated by an operator A in a Banach space X is equivalent to the existence of a solution Π = δ0 to the homogenous operator equation ΠS|M = AΠ. Here S|M generates the shift C0-group TS(t)|M in a closed translation-invariant subspace M of BUC(ℝ, X), and δ0 is the point evaluation at the origin. If, in addition, M is operator-invariant and 0 ≠ Π ∈ Ꮭ(M,X) is any solution of ΠS|M = AΠ, then all functions t → ΠTS(t)|Mf, f ∈ M, are complete trajectories for TA(t) inM. We connect these results to the study of regular admissibility of Banach function spaces for TA(t); among the new results are perturbation theorems for regular admissibility and complete trajectories. Finally, we show how strong stability of a C0-semigroup can be characterized by the nonexistence of nontrivial bounded complete trajectories for the sun-dual semigroup, and by the surjective solvability of an operator equation ΠS|M = AΠ. [ABSTRACT FROM AUTHOR]