A Liapunov functional for a matrix difference-differential equation

A quadratic positive definite functional that yields necessary and sufficient conditions for the asymptotic stability of the solutions of the matrix difference-differential equation $\text{x}\text{?}\text{(t) = Ax(t) + Bx(t ? τ)}$ is constructed and its structure is analyzed. This functional, a Liapunov functional, provides the best possible estimate for the rates of growth or decay of the solutions of this equation. The functional obtained, and its method of construction, are natural generalizations of the same problem for ordinary differential equations, and this relationship is emphasized. An example illustrates the applicability of the results obtained.     

The Fuller index and global Hopf bifurcation

Using an index for periodic solutions of an autonomous equation defined by Fuller, we prove Alexander and Yorke's global Hopf bifurcation theorem. As the Fuller index can be defined for retarded functional differential equations, the global bifurcation theorem can also be proved in this case. These results imply the existence of periodic solutions for delay equations with several rationally related delays, for example, $\text{x}\text{?}\text{(t) = ?α[ax(t ? 1) + bx(t ? 2)]g(x(t))}$, with a and b non-negative and α greater than some computable quantity ξ(a, b) calculated from the linearized equation.     

Some remarks on the asymptotic behavior of a nonautonomous,nonlinear functional differential equation

The asymptotic behavior of the abstract nonautonomous, nonlinear functional differential equation $\text{x}\text{?}\text{(t) = f(t, x(t)) + g(t, x}{}_{\mathrm{t}}\text{), x}{}_{\mathrm{s}}\text{= ?}$ is considered. Estimates on the growth of solutions are given and these estimates are shown to be the best possible.     

Oscillation theorems for second-order functional differential equations

This paper presents some comparison theorems on the oscillatory behavior of solutions of second-order functional differential equations. Here we state one of the main results in a simplified form: Let q, τ₁, τ₂ be nonnegative continuous functions on (0, ∞) such that τ₁ ? τ₂ is a bounded function on [1, ∞) and t ? τ₁(t) → ∞ if t → ∞. Then $\text{y}\text{?}\text{(t) + q(t) y(t ? τ}{}_1\text{(t)) = 0}$ is oscillatory if and only if $\text{y}\text{?}\text{(t) + q(t) y(t ? τ}{}_2\text{(t)) = 0}$ is oscillatory.     

Periodic solutions of nonlinear delay equations

Existence and uniqueness of 2π-periodic solutions of $\text{d}{}^{\mathrm{j}}\text{x(t)}\text{dt}{}^{\mathrm{j}}\text{+}\text{grad}\text{G(x(t ? τ)) = e(t, x(t), x(t ? τ)) (j = 1, 2)}$, where x(t) is in $\text{R}$ⁿ and e(t, u, v) is a given vector function, 2π-periodic in t, are shown under conditions on the spectrum of the Hessian of G. The equation is studied using a fixed point theorem in the space L²(0, 2π). One feature of this approach is that no relationship between the delay and the period is necessary.     

Stability conditions for linear retarded functional differential equations

The purpose of this note is to study the exponential stability for the linear retarded functional differential equation $\text{x}\text{?}\text{(t) = ∫}{}_{\mathrm{?1}}{}^0\text{[dη(θ)] x(t ? r(θ))}$, where the delay function r(θ) ? 0 is continuous and η(θ) is of bounded variation on the interval [?1, 0]. It is shown that the spectral limit function for the equation above has a continuous dependence on the pair (η, r). The set of all functions of bounded variation η for which the equation above is exponentially stable for every delay function r, the so-called region of stability globally in the delays, is a cone. Therefore for a fixed r, the set of all η which make our equation exponentially stable, that is, the region of stability for the delay function r, contains a cone. A discussion of the characterization of these regions of stability, as well as of the largest cone contained in each region of stability for a fixed delay function r, is given. Some remarks are made with respect to a similar question for the equation $\text{x}\text{?}\text{(t) = Ax(t) + ∫}{}_{\mathrm{?\; 1}}{}^0\text{[dμ(θ)] x(t?r(θ))}$, where A is a real n by n matrix, μ(θ) is bounded variation on [?1, 0] and r(θ) as before. Several examples illustrate the results obtained.     

Finite differences versus finite elements for solving nonlinear integro-differential equations

Consider the nonlinear integro-differential equation $\text{u}{}_{\mathrm{t}}\text{(x, t) = ∝}{}_0{}^{\mathrm{t}}\text{a(t?τ)}\text{?}\text{?x}\text{σ(u}{}_{\mathrm{x}}\text{(x, τ)) dτ + f(x, t)}$, 0 < x <, 0 < t < T, with appropriate initial and boundary conditions. This problem serves as a model for one-dimensional heat flow in materials with memory. The numerical solution via finite elements was discussed in B. Neta [J. Math. Anal. Appl.89 (1982), 598–611]. In this paper we compare the results obtained there with finite difference approximation from the point of view of accuracy and computer storage. It turns out that the finite difference method yields comparable results for the same mesh spacing using less computer storage.     

On a neutral functional differential equation in a fading memory space

The linear autonomous, neutral system of functional differential equations $\text{d}\text{dt}\text{(μ 1 x(t) + ?(t)) = v 1 s(t) + g(t) (t ? o)}$, (1) x(t) = ?(t) (t ? 0), in a fading memory space is studied. Here μ and ν are matrix-valued measures supported on [0, ∞), finite with respect to a weight function, and $\text{?, g,}\text{and}\text{?}$ are Cⁿ-valued, continuous or locaily integrable functions, bounded with respect to a fading memory norm. Conditions which imply that solutions of (1) can be decomposed into a stable part and an unstable part are given. These conditions are of frequency domain type. The usual assumption that the singular part of μ vanishes is not needed. The results can be used to decompose the semigroup generated by (1) into a stable part and an unstable part.     

Periodic solutions for perturbed differential equations

A variant of the Alekseev variation of constants integral equation is obtained relating the solutions of systems of the form $\text{x}\text{?}\text{= f(t, x, λ)}$ and $\text{y}\text{?}\text{= f(t, y, ψ(t, y)) + g(t, y)}$. For the case when f, g, and ψ have period P in t several theorems are given for the existence of periodic solutions extending known results when f is linear in x and does not depend on the parameter m-vector λ. Comparison with an older technique gives hypotheses where the method above is advantageous for establishing periodic solutions. An example is given for constructing limit cycles of autonomous second-order systems.     

Numerical solution of a nonlinear integro-differential equation

Galerkin's method with appropriate discretization in time is considered for approximating the solution of the nonlinear integro-differential equation $\text{u}{}_{\mathrm{t}}\text{(x, t) = ∝}{}_0{}^{\mathrm{t}}\text{a(t ? τ)}\text{?}\text{?x}\text{σ(u}{}_{\mathrm{x}}\text{(x, τ)) dτ + f(x, t)}$, 0 < x < 1, 0 < t < T.An error estimate in a suitable norm will be derived for the difference u ? u^h between the exact solution u and the approximant u^h. It turns out that the rate of convergence of u^h to u as h → 0 is optimal. This result was confirmed by the numerical experiments.     

On forced nonlinear oscillations

The existence of a 1-periodic solution of the generalized Liénard equation x″ + g(x)x′ + f(t, x) = e(t), where g(x) is continuous, e(t) is continuous, periodic of period 1 and with mean value 0 and f is continuous, periodic of period 1 in t, is proved under one of the following conditions: (i) there exists M ? 0 such that $\text{f(t, x)x ? 0}\text{for}\text{¦x¦? M}$ and

$\text{lim sup}\text{|x|?+∞}\text{|f(t,x)|}\text{| x |}\text{<}\text{4π}{}^2\text{2π + 1}$

(ii) there exists M ? 0 such that $\text{f(t, x)x ? 0}\text{for}\text{¦x¦? M}$. Earlier results of A. C. Lazer, J. Mawhin and R. Reissig are obtained as particular cases.     

Effect of delays on functional differential equations

The perturbed functional differential equation $\text{x}\text{?}\text{(t) = L(x}{}_{\mathrm{t}}\text{) + h(x}{}_{\mathrm{t}}\text{)}$ is considered with the assumption that h is Lipschitzian in W_1,∞. Using integral manifold techniques, this equation is reduced to the equivalent ordinary differential equation $\text{u}\text{?}\text{= Bu + Ψ(0)h(ΛΦu)}$. A bifurcation problem is considered for the former equation. Illustrative examples are worked.     

Approximate solutions of functional differential equations

The author discusses the best approximate solution of the functional differential equation x′(t) = F(t, x(t), x(h(t))), 0 < t < l satisfying the initial condition x(0) = x₀, where x(t) is an n-dimensional real vector. He shows that, under certain conditions, the above initial value problem has a unique solution y(t) and a unique best approximate solution $\text{p}\text{?}{}_{\mathrm{k}}\text{(t)}$ of degree k (cf. [1]) for a given positive integer k. Furthermore, $\text{sup}{}_{\mathrm{0?t?l}}\text{¦}\text{p}\text{?}{}_{\mathrm{k}}\text{(t) ? y(t)¦ → 0}\text{as}\text{k → ∞}$, where ¦ · ¦ is any norm in Rⁿ.     

A change of variables in the asymptotic theory of differential equations with unbounded delays

In this paper, the author investigates the asymptotic properties of solutions of the nonhomogeneous linear differential equation$\text{x}\text{̇}\text{(t)=ax(τ(t))+bx(t)+f(t)}$with nonzero real scalars a,b and the unbounded lag. Using the change of the independent and dependent variable he relates the asymptotic behaviour of solutions of this equation to the asymptotic behaviour of solutions of auxiliary functional (nondifferential) equations.     

On splitting the area under curves

Two area-splitting problems involving real-valued functions of a real variable are investigated. The second of these is essentially equivalent to finding all functions a?C¹((0, r)) with 0 < a(x) < x which satisfy the functional differential equation $\text{a′ (a(x)) =}\text{a(x)}\text{x}\text{for}\text{x ? (0, r)}$. All solutions analytic at x = 0 (and many which are not) are exhibited in closed form.     

Weak solutions for a nonlinear dispersive equation

In this paper we study the existence, uniqueness, and regularity of the solutions for the Cauchy problem for the evolution equation $\text{u}{}_{\mathrm{t}}\text{+ (f (u))}{}_{\mathrm{x}}\text{? u}{}_{\mathrm{xxt}}\text{= g(x, t), (}{}^1\text{)}\text{where}\text{u = u(x, t), x}$ is in (0, 1), 0 ? t ? T, T is an arbitrary positive real number,f(s)?C¹$\text{R}$, and g(x, t)?L^∞(0, T; L²(0, 1)). We prove the existence and uniqueness of the weak solutions for (¹) using the Galerkin method and a compactness argument such as that of J. L. Lions. We obtain regular solutions using eigenfunctions of the one-dimensional Laplace operator as a basis in the Galerkin method.     

On the existence of a periodic solution of nth-order nonlinear differential equations

This note is a generalization of one of a paper by Mehri and Hamedani. Under suitable conditions of ?, the existence of periodic solutions of the nthorder differential equation $\text{x}{}^{\mathrm{(n)}}\text{+ ?(t, x, x′,…, x}{}^{\mathrm{(n\; ?\; 1)}}\text{) = 0}$ is established.     

Some existence and regularity results for abstract non-autonomous parabolic equations

We study existence, uniqueness and regularity of the strict, classical and strong solutions $\text{u? C(¦0, T ¦,E)}$ of the non-autonomous evolution equation $\text{u′(t) ? A(t)u(t)=?(t)}$, with the initial datum u(0) = x, in a Banach space E, under the classical Kato-Tanabe assumptions. The domains of the operators A(t) are not needed to be dense in E. We prove necessary and sufficient conditions for existence and Hölder regularity of the solution and its derivative.     

Attainable sets for linear stochastic control systems

Let x_t^u(w) be the solution process of the n-dimensional stochastic differential equation dx_t^u = [A(t)x_t^u + B(t) u(t)] dt + C(t) dW_t, where A(t), B(t), C(t) are matrix functions, W_t is a n-dimensional Brownian motion and u is an admissable control function. For fixed ? ? 0 and 1 ? δ ? 0, we say that x?Rⁿ is (?, δ) attainable if there exists an admissable control u such that P{x_t^u?S_?(x)} ? δ, where S_?(x) is the closed ?-ball in Rⁿ centered at x. The set of all (?, δ) attainable points is denoted by $\text{A}$(t). In this paper, we derive various properties of $\text{A}$(t) in terms of K(t), the attainable set of the deterministic control system $\text{x}\text{?}\text{= A(t)x + B(t)u}$. As well a stochastic bang-bang principle is established and three examples presented.     

Asymptotic properties of general symmetric hyperbolic systems

Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem $\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0, u(0, x) = ?(x)}$. Here the A_j's are constant, k × k Hermitian matrices, x = (x₁,…, x_n), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit $\text{E}{}_{\mathrm{M}}\text{(?)}\text{as}\text{¦ t ¦ → ∞}$, where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U₀(t) and such that $\text{U}{}_0\text{(t)? = 0}\text{for}\text{¦ x ¦ ? a ¦ t ¦ ? R, a}\text{and}\text{R}$ depending on ? and that the local energy of nonstatic solutions decays as $\text{¦ t ¦ → ∞}$. More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation $\text{E(x)}\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0,}\text{where}\text{¦ E(x) ? I ¦ = O(¦ x ¦}{}^{\mathrm{?1\; ?\; ?}}\text{)}$ at infinity.