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A model for the transport of charged particles in a random magnetic field is a Volterra integrodifferential equation with a long-range kernel. The integrodifferential equation is solved numerically with the method of Bellman, Kalaba, and Lockett (“Numerical Inversion of the Laplace Transform,” Elsevier, New York, 1966). The results are shown to be in excellent agreement with analytical asymptotic results.

The problem of existence of periodic solutions for some nonautonomous neutral functional differential equations is examined. It is an application of a basic theorem on the Fredholm alternative for periodic solutions of some linear neutral equations and of a generalized Leray-Schauder theory. Although proofs are simple, the results are nontrivial extensions to the neutral case of existence theorems for periodic solutions of functional differential equations.

A system of three quasi-linear partial differential equations is considered, as a simplified model of the transport of mobile carriers in a semiconductor device. Assuming a convenient form of the boundary conditions, it is shown that the initial value problem is well-posed, and that the steady state solution is unique and stable. A finite difference approximation preserving reasonable bounds on the numerical solutions is also described.

in the xixs plane, and assume throughout that fi , fi E Co in Q and fi , fi E Cl in QO. Notice that every trajectory of (1) starting in Q at t = 0 lies entirely either in Q” or on one of the positive semiaxes. Indeed, if x(t) = (xl(t), x2(t)) is a solution of (1) originating in Q” and if t, > 0 were the time of its first intersection with one of the axes, then f,(x(t)) and f,(x(t)) would be bounded for 0 0, m2 > 0); thus 3ii 3 -m,x, and 2s > -msxa , and therefore xl(t) > xi(O) e-“lt and x2(t) 3 x,(O) emzt for all 0 < t < to . A similar argument holds for to < 0, and hence a trajectory of (1) starting in QO cannot reach the x1 or xs axis in finite time. For the same reason, a trajectory originating on one of the positive semiaxes cannot enter Q”. Finally, we assume below conditions on fi and fi that eliminate the possibility of finite escape times.

Certain nonlinear optimal control problems on a manifold can be represented in terms of a switchable family of regulated paths in the space of continuously differentiable vector fields. The methods also yield a theorem on the Dieudonne-Albrecht integrability conditions for non-autonomous first order differential equations on a Banach space.

This paper contains a definition of a linear operator T, referred to as a differential transform. The differential transform T commutes with differentiation and as a result, T can be applied for the solving of some linear partial differential equations. Two illustrative examples for the application of T are introduced.

The bound state energies and scattering phase shifts for the Coulomb potential are obtained from both the Schrodinger and Dirac equations by taking a Laplace transform. Inversion of transforms is not required, and the nonrelativistic eigenvalue problem is solved without even obtaining the transforms explicitly. The nonrelativistic scattering amplitude appears after solving a first-order differential equation.

We consider finding the best possible constants $C = C(n,\alpha ,\beta ,p,q)$ such that $\| f \|_p \leqq C(b - a)^{n + 1/p -1/q} \| {f^{(n)} } \|_q $ under several differen sets of boundary conditions. Specifically in one case, f is required to have a zeros at $a $ and $\beta $ zeros at b where $n \leqq \alpha + \beta \leqq 2n$, $\alpha \leqq n$ and $\beta \leqq n$. In the other case, f has $\alpha $ zeros at $a $, $\beta $ zeros at b and $\int _a^b f(x)x^r (x - a)^\alpha (x - \beta )^\beta dx = 0$, $r = 0, \cdots ,n - 1 - \alpha - \beta $, where $0 \leqq \alpha $, $0 \leqq \beta $ and $\alpha + \beta < n$. The various problems are related and the numbers C are calculated exactly in some cases. Finally, we show how some of these can be applied to disconjugacy problems in differential equations.

A random discrete-time system { x n }, n = 0, 1, 2, … is called stochastically stable if for every ϵ > 0 there exists a λ > 0 such that the probability P [(sup n ∥ x n ∥) > ϵ ] ϵ whenever P [∥ x 0 ∥ > λ ] λ . A system is shown stochastically stable if some local Lyapunov function V (·) satisfies the supermartingale definition on { V ( x n )} in a neighborhood of the origin; earlier proofs of stochastic stability require additional restrictions. A criterion for x n → 0 almost surely is developed. It consists of a global inequality on { U ( x n )} stronger than the supermartingale defining inequality, but applied to a U (·) that need not be a Lyapunov function. The existence of such a U (·) is exhibited for a stochastically unstable nontrivial stochastic system. This indicates that our criterion for x n → 0 is “tight,” and that the two stability concepts studied are substantially distinct.

Using the theory of analytic representations, it is shown that a generalized Hilbert transformation may be defined on the space $\mathcal{D}'$ of Schwartz distributions. It yields extended Plemelj and dispersion relations for all distributions in $\mathcal{D}'$. Distributions which are limits of functions analytic in the upper or lower half-plane are seen to be completely characterized by a certain Fourier transform property. Corresponding results were previously obtained for distributions in the subspaces $\mathcal{S}'$ and $O'_\alpha $ of $\mathcal{D}'$.

Let X be a Hilbert space with inner product (x, y) for x, y E X. For fixed y the distance from x to y is p&x) = 11 x y 11 = (x y, x y)lls and for a nonempty set KC X the distance from x to K is P&) = inf(p,(x) 1 y E K}. It is known [2] (and it follows from Theorem 1 below by taking K = {y}) that pV is FrCchet differentiable at all x # y. We consider here some circumstances under which pK is FrCchet differentiable. Since

A construction is given which allows the Hilbert space treatment of spline functions to be applied to the case of more than one variable, when the basic operator is a linear partial differential one. The particular case of the tensor product polynomial spline in two variables is then studied using a reproducing kernel, and its main properties, including the minimization ones, are deduced. A stable computational method is then given for this spline function, with certain point evaluation functionals. Finally, extensions are discussed, for more general linear functionals, for more general differential operators, and for more than two variables.

Resume Dans cet article on etudie les points fixes au voisinage d'un point homocline qui est doublement asymptotique a un point fixe du type hyperbolique par l'application de Poincare definie par une equation differentielle dans R 2 . On definit deux classes d'equivalence sur l'ensemble des points homoclines et on montre qu'au voisinage d'un point homocline de chacune de ces classes il existe une infinite de points fixes d'indice + 1 ou − 1 ou la valeur de l'indice et la periode sont liees par une certaine loi.

Let G(t, s) be the Green's functions associated with N, a differential operator restricted to certain boundary conditions. Define (u, v)N = (Nu, v)L2. It is shown that the reproducing kernel Hilbert space generated by G is the same as the Hilbert-space completion with respect to ∥ · ∥N of the set of real valued functions which are in C2n and satisfy the boundary conditions. The concept of Sobolev spaces is used in the proof and examples are given.

A uniqueness theorem is proved for algebraically regular solutions to the unbounded initial value problem P′ = AP, P(0) = diag(1, 1, 1,…) in the real Banach algebra of infinite matrices M with standard norm. It is not assumed that A ∈ M , but it is required that A have an inverse in M , a property which is seen to be implied quite naturally by certain divergent or pathological systems. The conditions for the theorem are motivated by a particular system, previously considered by Hille and Feller, which arises from a divergent, purebirth, time dependent stochastic process, although no restriction requiring the solution matrix to be either stochastic or substochastic is necessary. The theorem may be easily generalized to any Banach algebra with identity.

Let of $a_k^{pq} (x,y)$ be analytic functions for $x,y$ in the $z = x + iy$ plane. Consider the equation $\Delta ^n u + \Sigma _{k = 1}^n L_k (\Delta ^{n - k} u) = 0$, where $L_k = \sum _{p,q = 0}^{p + q \leqq k} a_k^{pq} (x,y){{\partial ^{p + q} /\partial x^p \partial y^q }}$. The following three areas will be investigated: (i) integral representation for solutions of the above equation to the boundary$\partial G$ of a simply connected domain G, (ii) the reflection principle for solutions of the above equation under the complementing boundary conditions, (iii) analytic continuation and nonexistence for solutions of the Cauchy problem.

It is shown that the parameters of a best linear Lp approximation are both continuous and differentiable when considered as functions of p. An expression for each derivative is obtained. The result assumes only very mild conditions on the problem.

The aim of this concise paper is to present an extension of the local inverse theorem of polynomial type operators, recently considered by the same authors in [l-4] for nonlinear systems analysis. This extension gives a local inversion of certain operators which are represented by generalized power series. The homogeneous operator terms (i.e., powers) of the series are defined algebraically via multilinear mappings. In this generality, theory of polynomial operators and power series is previously introduced by, e.g., Michal [5] and Cartan [6], by using a somewhat different presentation. However, the results concerning the inversion are not presented anywhere, as far as the authors know. Among others, a general recursive representation is obtained which gives the homogeneous operators of the inverted power series successively with increasing orders. The main theorem is presented here without proof because it can be performed by a pedantic modification of the corresponding one in [l, 31.

Resume Dans ce travail on cherche a cerner les limites de la methode des quotients differentiels, ou methode de Nirenberg (cf. [6, 7]), dans l'etude de la regularite des solutions de problemes de Dirichlet variationnels, non necessairement elliptiques. Pour cela on remplace le groupe des translations par un groupe de diffeomorphismes a un parametre, arbitraire, la derivation par le champs de vecteurs associe, les operateurs elliptiques par des iteres d'operateurs de type principal. On obtient une condition necessaire et suffisante, assez restrictive, sur les champs de vecteurs de laquelle on tire des resultats positifs et negatifs concernant la regularite des solutions des problemes de Dirichlet associes.

The main result can be stated roughly as follows: Let u be a Hilbert-space-valued solution of $u'(t) = Au(t) $, $ - \infty < t < \infty $, where A differs from a symmetric linear operator by a suitable nonlinear perturbation. Then either u is constant or else $\sup _t \| {u(t)} \| = \infty $. More generally, A can depend on t, and the equation $u''(t) = A(t)u(t)$ is also considered.

The asymptotic behavior of the classical equations arising from four-Fermion and Yukawa like interactions is examined. In particular for Dirac equations coupled through and Dirac and Maxwell equations coupled through , the global existence of solutions is proved and their decay and scattering properties are discussed.

A boundary-integral-boundary-sum representation for certain distributional solutions of the heat equation in a finite cylinder with n-dimensional rectangular base is derived. The method employed involves use of a duality principle at the boundary.

Local uniqueness theorems are obtained for higher order elliptic equations using integral inequalities and monotonicity properties for eigenvalues.

The solution $x(t)$ of the Volterra integral equation of the second kind $x(t) = f_1 (t) + \sqrt t f_2 (t) + \int _0^t g(t,s,x(s))(t - s)^{ - {1 / 2}} ds$ is examined. It is shown that $x(t) = u(t) + \sqrt t v(t)$, where $u(t)$ and $v(t)$ are smooth under appropriate smoothness conditions on $f_1 (t)$, $f_2 (t)$ and $g(t,s,x)$ and satisfy a system of Volterra integral equations of the second kind.

In this note we study more general iteration processes in certain Banach spaces with the purpose of extending this theorem. It turns out that sometimes our aim can indeed be (partially) achieved (see, for example, Theorem 2.10). Several related results, as well as some open problems, are also included. In Section 1 we relate the boundedness of the sequence of iterates with the existence of a fixed point. Section 2 is devoted to a discussion of the crucial “minimum property” of cl(R(A)) ( see the definition below). Section 3 contains a result concerning the convergence of a certain sequence of iterates towards a fixed point of T. We shall consider here only real normed linear spaces. This restriction does not cause any loss of generality.