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SIAM J. on Mathematical Analysis

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1978

Volume 9, Issue 6, pp. 979-1191


A Multi-Time Scale Method in Almost-Periodic Stability Problems

Jon H. Davis

SIAM J. Math. Anal. 9, pp. 979-995 (17 pages) | Cited 1 time

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This paper considers the problem of stability of a class of almost-periodic ordinary differential equations. Necessary and sufficient stability conditions are derived by use of a method of “multiple time scales” suggested by similar techniques in perturbation theory. The conditions obtained involve a spectral analysis of a compact operator determined by the original system coefficients, and in principle are computable to any desired degree of accuracy by finite dimensional methods.

Analytic Representations and Fourier Transforms of Analytic Functionals in $Z'$ Carried by the Real Space

J. W. de Roever

SIAM J. Math. Anal. 9, pp. 996-1019 (24 pages) | Cited 4 times

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In the space $Z'$, the Fourier transform of the space $\mathcal{D}'$ of Schwartz-distributions, the notion of carrier is introduced. A characterization is given of all distributions $\mathcal{D}'$, the Fourier transform of which is carried by $\mathbb{R}^n $. Both, such distributions and the analytic functionals in $Z'$ carried by $\mathbb{R}^n $, are represented as sum of boundary values of holomorphic functions. This extends the case of tempered distributions which, regarded as elements of $Z'$, are obviously carried by $\mathbb{R}^n $.

On the Solutions of a Class of Nonlinear Sturm–Liouville Problems

P. de Mottoni and A. Tesei

SIAM J. Math. Anal. 9, pp. 1020-1029 (10 pages)

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A class of Sturm–Liouville problems with monomial nonlinearities is studied in a constructive way by explicit integration. Results are obtained concerning the existence and uniqueness of the solutions with a prescribed number of nodes, the location of the nodes, the behavior of the maximum norm of the solutions as functions of a distinguished parameter (bifurcation parameter).

On the Asymptotic Solution of a Partial Differential Equation with an Exponential Nonlinearity

V. H. Weston

SIAM J. Math. Anal. 9, pp. 1030-1053 (24 pages) | Cited 15 times

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The asymptotic behavior of a large norm (maximum) solution of the Dirichlet problem associated with the equation \[ - \Delta u = \lambda e^u \] for a bounded simply-connected domain in $\mathbb{R}^2 $ is investigated for the case of the positive parameter $\lambda $ tending to zero. By means of a conformal transformation function $f(z)$, the problem is transformed to one involving the unit disc. For a class of domains which are described by implicit conditions for $f(z)$, a first and higher asymptotic expressions are developed for the large norm solution characterized by a single maximum proportional to $\ln ({1 / \lambda })$. It is shown that for $\lambda $ sufficiently small, an exact solution can be generated by the modified Newton iteration scheme, if the asymptotic solution of appropriate order is used for the initial step.

A Variational Approach to Multi-Parameter Eigenvalue Problems in Hilbert Space

Paul Binding and Patrick J. Browne

SIAM J. Math. Anal. 9, pp. 1054-1067 (14 pages) | Cited 15 times

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Let $T_r $ and $V_{rs} $ be self-adjoint linear operators on Hilbert spaces $H_r $, $1 \leqq r \leqq k$. Assuming that $T_r $ have compact resolvents and $V_{rs} $ are bounded, we use standard variational characterizations of eigenvalues to treat the multiparameter case $(T_r + \sum_{s = 1}^k {\lambda _s V_{rs} } )x_r = 0$, $1 \leqq r \leqq k$. In particular we establish existence of a “purely point spectrum” satisfying cone monotonicity conditions in $\mathbb{R}^k $. We prove continuous, monotonic and Lipschitz parametric dependence theorems, and we examine $\mathbb{R}^k $-valued generalized Rayleigh quotients. In particular, we interpret the (not necessarily closed) convex hull of the spectrum as the “vectorial range” of these quotients.

Representation of Special Functions by Differintegral and Hyperdifferential Operators

Do Tan Si

SIAM J. Math. Anal. 9, pp. 1068-1075 (8 pages) | Cited 2 times

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The use of differintegral operators $D^\nu $ with $D = {d / {dz}}$ and $\nu \in \mathbb{C}$ and hyperdifferential operators, together with some operational calculus manipulations leads to new formulae of representation of usual special functions. These formulae are useful for obtaining generating functions and differential recurrence relations of orthogonal polynomials.

A Method of Global Blockdiagonalization for Matrix-Valued Functions

H. Gingold

SIAM J. Math. Anal. 9, pp. 1076-1082 (7 pages) | Cited 3 times

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Let $A(x)$ be an $n \times n$ analytic matrix function of the vector variable $x$. Let the eigenvalues of $A(x)$ belong to two disjoint sets for every fixed $x$. Then there exists an invertible analytic matrix function $M(x)$ which takes $A(x)$ by a similarity transformation into a blockdiagonal form. Similar theorems for $A(x)$ being smooth are also proved.

The Generalized Inverse of an Unbounded Linear Operator with Unbounded Constraints

W. F. Langford

SIAM J. Math. Anal. 9, pp. 1083-1095 (13 pages) | Cited 3 times

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The concept of best least squares solution is used to define a generalized inverse of an unbounded linear operator between two inner product spaces, subject to unbounded linear functional constraints. The nullspace of the operator is assumed finite dimensional. A necessary and sufficient condition for the existence of this generalized inverse is established. When the condition holds, the calculation of the best least squares solution is reduced to an explicit algebraic formula. The theory is illustrated by application to a general linear two-point boundary value problem, for which a new proof of the existence and uniqueness of the best least squares solution is obtained, without the use of Green’s functions.

Analytic Continuation via Hadamard’s Product

Barbara Fromm Chambers

SIAM J. Math. Anal. 9, pp. 1096-1104 (9 pages)

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This paper presents an operational procedure derived from Hadamard’s convolution product which is used to construct continuations of analytic functions in the form of integral functional representations. These representations are more useful in the study of analytic properties than the underlying Taylor’s series, and the method extends the previously well-established continuation results of Borel andMittag–Leffler.

Asymptotic Behavior for Semilinear Differential Equations in Banach Spaces

R. H. Martin, Jr.

SIAM J. Math. Anal. 9, pp. 1105-1119 (15 pages) | Cited 1 time

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Suppose that $X$ is a Banach space and consider the integral equation \[(1)\qquad u(t) = T(t,a)(z - \alpha (a)) + \alpha (t) + \int_a^t {T(t,r)} B(r,u(r))dr,\quad t \geqq a \geqq 0,\] where $\{ T(t,s):t \geqq s \geqq 0\} $ is a linear evolution system on $X$, $\alpha :[0,\infty ) \to X$ is continuous, $E \subset X$, and $B:[0,\infty ) \times E \to X$ is continuous. In this paper we develop methods for studying the behavior as $t \to \infty $ of solutions $u$ to (1). These results are based on Lyapunov-like methods and the proofs use standard techniques. The abstract theorems are presented in the first section and some examples indicating the applicability of these ideas are indicated in the second section. In particular, these methods are used to study the behavior of solutions to systems to semilinear parabolic equations.

A Runge Approximation and Unique Continuation Theorem for Pseudoparabolic Equations

William Rundell and Michael Stecher

SIAM J. Math. Anal. 9, pp. 1120-1125 (6 pages)

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In this paper we construct an integral operator for solutions of the pseudoparabolic equation $\Delta _n u_t - A(r)u_t + B(r)u = f$. This is then used to obtain a Runge approximation and unique continuation result for pseudoparabolic equations.

Estimates for the Green’s Functions of Elliptic Operators

Catherine Bandle

SIAM J. Math. Anal. 9, pp. 1126-1136 (11 pages) | Cited 3 times

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Inequalities for the Green’s functions of $Lu = - \Delta u - pu$ are derived by means of the level line technique and bounds for linear boundary value problems are constructed.

An $N$-Dimensional Extension of the Sturm Separation and Comparison Theory to a Class of Nonselfadjoint Systems

Shair Ahmad and A. C. Lazer

SIAM J. Math. Anal. 9, pp. 1137-1150 (14 pages) | Cited 6 times

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Sturmian theory is extended to nonselfadjoint second order linear homogeneous systems. Almost all the results obtained are new even in the selfadjoint case.

Numerical Approximation of Nonlinear Functional Differential Equations with $L^2 $ Initial Functions

G. W. Reddien and G. F. Webb

SIAM J. Math. Anal. 9, pp. 1151-1171 (21 pages)

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Nonlinear operator semigroup theory is used to treat the numerical approximation of autonomous functional differential equations with $L^2 $ initial functions. The consistency, stability, and convergence of both explicit and implicit schemes are demonstrated and error estimates are established. The stability is obtained easily as a consequence of a renorming of the underlying space.

Monotonic Properties of Analytic Functions

A. D. Rawlins and J. D. Morgan

SIAM J. Math. Anal. 9, pp. 1172-1178 (7 pages)

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A theorem is proved which enables one to obtain monotonic properties of the real part, imaginary part, modules and phase of an arbitrary analytic function in the complex plane. The monotonic properties are established from the behavior of the analytic function and its derivative on the boundary of the domain in which the monotonic properties are required. As an applicationsome monotonic results are derived for the Bessel functions $J_\nu (z)$ and $H_\nu ^{(1)} (z)$, where $z$ is complex and $\nu$ is real and positive. The theorem and a corollary can be used to obtain monotonic properties of many other special functions of mathematical physics.

A Free Boundary Optimization Problem

Andrew Acker

SIAM J. Math. Anal. 9, pp. 1179-1191 (13 pages) | Cited 13 times

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Given a convex set $Q \subset R^2 $ (bounded by a simple closed curve) and a constant $A > 0$, we determine the doubly-connected region $\Omega $ encircling (but not intersecting) $Q$, with area $| \Omega | \leqq A$, which has the least capacitance.
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