On the almost automorphy of bounded solutions of differential equations with piecewise constant argument

In this paper we give sufficient spectral conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x^′(t)=Ax([t])+f(t), t∈R, where A is a bounded linear operator in X and f is an X-valued almost automorphic function.     

Almost automorphic solutions to nonautonomous semilinear evolution equations in Banach spaces

This paper is concerned with almost automorphy of the solutions to a nonautonomous semilinear evolution equation u^′(t)=A(t)u(t)+f(t,u(t)) in a Banach space with a Stepanov-like almost automorphic nonlinear term. We establish a composition theorem for Stepanov-like almost automorphic functions. Furthermore, we obtain some existence and uniqueness theorems for almost automorphic solutions to the nonautonomous evolution equation, by means of the evolution family and the exponential dichotomy. Some results in this paper are new even if A(t) is time independent.     

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.     

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.     

On Peano's theorem in Banach spaces

We show that if X is an infinite-dimensional separable Banach space (or more generally a Banach space with an infinite-dimensional separable quotient) then there is a continuous mapping f:X→X such that the autonomous differential equation x^′=f(x) has no solution at any point.     

Periodic solutions for a higher order p-Laplacian neutral functional differential equation with a deviating argument

In this paper, a higher order p-Laplacian neutral functional differential equation with a deviating argument:

[φ_p([x(t)−c(t)x(t−σ)]⁽ⁿ⁾)]^(m)+f(x(t))x^′(t)+g(t,x(t−τ(t)))=e(t)     

On the numerical characterization of the reachability cone for an essentially nonnegative matrix

Given an n×n real matrix A with nonnegative off-diagonal entries, the solution to , x₀=x(0), t?0 is x(t)=e^tAx₀. The problem of identifying the initial points x₀ for which x(t) becomes and remains entrywise nonnegative is considered. It is known that such x₀ are exactly those vectors for which the iterates x^(k)=(I+hA)^kx₀ become and remain nonnegative, where h is a positive, not necessarily small parameter that depends on the diagonal entries of A. In this paper, this characterization of initial points is extended to a numerical test when A is irreducible: if x^(k) becomes and remains positive, then so does x(t); if x(t) fails to become and remain positive, then either x^(k) becomes and remains negative or it always has a negative and a positive entry. Due to round-off errors, the latter case manifests itself numerically by x^(k) converging with a relatively small convergence ratio to a positive or a negative vector. An algorithm implementing this test is provided, along with its numerical analysis and examples. The reducible case is also discussed and a similar test is described.     

Periodic solutions for a Rayleigh type p-Laplacian equation with sign-variable coefficient of nonlinear term

As p-Laplacian equations have been widely applied in the field of fluid mechanics and nonlinear elastic mechanics, it is necessary to investigate the periodic solutions of functional differential equations involving the scalar p-Laplacian. By using Lu’s continuation theorem, which is an extension of Manásevich-Mawhin, we study the existence of periodic solutions for a Rayleigh type p-Laplacian equation

(φ_p(x^′(t)))^′+f(x^′(t))+g₁(x(t-τ₁(t,|x|_∞)))+β(t)g₂(x(t-τ₂(t,|x|_∞)))=e(t).     

The numerical solution of forward-backward differential equations: Decomposition and related issues

This paper focuses on the decomposition, by numerical methods, of solutions to mixed-type functional differential equations (MFDEs) into sums of “forward” solutions and “backward” solutions. We consider equations of the form x^′(t)=ax(t)+bx(t−1)+cx(t+1) and develop a numerical approach, using a central difference approximation, which leads to the desired decomposition and propagation of the solution. We include illustrative examples to demonstrate the success of our method, along with an indication of its current limitations.     

Relaxation Results for Nonlinear Evolution Inclusions with One-sided Perron Right-hand Side

In a Banach space X, we study the evolution inclusion of the form x ^′(t)∈A x(t)+F(t,x(t)), where A is an m-dissipative operator and F is an almost lower semicontinuous multifunction with nonempty closed values. If F is one-sided Perron with sublinear growth, then, we establish the relation between the solutions of the considered differential inclusion and the solutions of the relaxed one, i.e., \(x^{\prime} (t)\in Ax (t)+\overline{co}F (t,x (t) )\) . A variant of the well known Filippov-Pli? lemma is also proved.     

The Dichotomy Theorem for evolution bi-families

We prove that the operator G, the closure of the first-order differential operator −d/dt+D(t) on L₂(R,X), is Fredholm if and only if the not well-posed equation u^′(t)=D(t)u(t), t∈R, has exponential dichotomies on R₊ and R₋ and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of G. Here X is a Hilbert space, D(t)=A+B(t), A is the generator of a bi-semigroup, B(⋅) is a bounded piecewise strongly continuous operator-valued function. Also, we prove some perturbations results and consider various examples of not well-posed problems.     

Impulsive semilinear differential inclusions: Topological structure of the solution set and solutions on non-compact domains

This paper deals with an impulsive Cauchy problem governed by the semilinear evolution differential inclusion x^′(t)∈A(t)x(t)+F(t,x(t))$x^{\prime }\left(t\right)\in A\left(t\right)x\left(t\right)+F(t,x\left(t\right))$, where _{{A(t)}t∈[0,b]}${\left\{A\left(t\right)\right\}}_{t\in [0,b]}$ is a family of linear operators (not necessarily bounded) in a Banach space E$E$ generating an evolution operator and F$F$ is a Carathéodory type multifunction. First a theorem on the compactness of the set of all mild solutions for the problem is given. Then this result is applied to obtain the existence of mild solutions for the impulsive Cauchy problem defined on non-compact domains.     

Asymptotic preconditioning of linear homogeneous systems of differential equations

We consider the asymptotic behavior of solutions of a linear differential system x^′=A(t)x, where A is continuous on an interval ([a,∞). We are interested in the situation where the system may not have a desirable asymptotic property such as stability, strict stability, uniform stability, or linear asymptotic equilibrium, but its solutions can be written as x=Pu, where P is continuously differentiable on [a,∞) and u is a solution of a system u^′=B(t)u that has the property in question. In this case we say that P preconditions the given system for the property in question.     

Asymptotic properties of solutions of certain third-order dynamic equations

In this paper, the well known oscillation criteria due to Hille and Nehari for second-order linear differential equations will be generalized and extended to the third-order nonlinear dynamic equation

(r₂(t)((r₁(t)x^Δ(t))^Δ)^γ)^Δ+q(t)f(x(t))=0     

On the distribution of zeros of solutions of first order delay differential equations

This paper contains new estimates for the distance between adjacent zeros of solutions of the first order delay differential equation

x^′(t)+p(t)x(t−τ)=0     

Periodic solutions for Liénard type p-Laplacian equation with a deviating argument

In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Liénard type p-Laplacian equation with a deviating argument of the form:

(_?p(x^′(t)))^′+f(x(t))x^′(t)+g(t,x(t-τ(t)))=e(t).$({?}_p(x^{\prime }(t)){)}^{\prime }+f(x(t))x^{\prime }(t)+g(t,x(t-\tau (t)))=e(t)\text{.}$

Stability and the nonexistence of blowing-up solutions of nonlinear delay systems with linear parts defined by permutable matrices

Sufficient conditions for the exponential stability of the trivial solution of nonlinear differential equations with delay and with linear parts of the form Ax(t)+Bx(t−τ),τ>0, where AB=BA, are proved. A result on the nonexistence of blowing-up solutions is also proved.     

Compact almost automorphic solutions to integral equations with infinite delay

Given a∈L¹(R) and the generator A of an L¹-integrable resolvent family of linear bounded operators defined on a Banach space X, we prove the existence of compact almost automorphic solutions of the semilinear integral equation for each f:R×X→X compact almost automorphic in t, for each x∈X, and satisfying Lipschitz and Hölder type conditions. In the scalar linear case, we prove that a∈L¹(R) positive, nonincreasing and log-convex is sufficient to obtain the existence of compact almost automorphic solutions.