Abstract: Let T be a bounded linear operator between two Hilbert spaces with the range of T not necessarily closed, and let T † denote the generalized inverse of T. The method of steepest descent for minimizing is shown to converge monotonically starting with x 0 = 0, to T † y for any y whose orthogonal projection on the closure of the range of T, is in the range of TT *. This set of y's is dense in the domain of T†. The method is also applied to generalized least squares solutions of a class of unbounded linear operator equations

Abstract: An estimate of errors in the numerical quadrature of analytic functions is given in connection with the behavior of the integrand in the complex plane.The Integral is transformed into a contour integral as well as its approximation so that the problem of approximation of the integral is reduced to the one of approximation of a transcendental function by a rational function G n (z)/F n (z). The function which characterizes the error of the numerical quadrature is introduced and the contour maps of in the complex plane are given for several quadurature formulas. By the use of these maps, investigating the analytic behavior of the integrand in the complex plane, one can know which formula is best suited to the given integrand in the prescribed precision.

Abstract: We reconsider here a classical “not well posed” problem, the Dirichlet problem for the wave equation for a rectangle with sides parallel to the coordinate axes. For the same domain, we also consider the Neumann, and many mixed Dirichlet–Neumann boundary value problems, including a “general mixed problem” where, at each point which is not a corner, the boundary condition is either of Dirichlet or of Neumann type.

Abstract: We define here a degree theory for proper analytic Fredholm maps of index zero defined on open subsets of complex Banach spaces, and we prove that the standard properties for a degree theory hold. Our approach avoids the differential geometry tools used by Elworthy and Tromba [8] in similar work. We prove that for analytic maps the degree theory defined by Nussbaum in [11,13] agrees with ours, and similarly Browder and Gupta's degree theory in [3] is a special case of ours. If is an analytic Fredholm map of index zero defined on an open subset of the complexification of a real Banach space B,if commutes with complex conjugation, and if is compact for some , then if N is the number of points in (mod 2). Under further assumptions on and y (see Theorem 10 below), deg and deg (mod 2). Our results generalize some recent work of Jane Cronin [6][7].

Abstract: In a recent paper of the same title, L. A. Medeiros has considered the question of uniqueness for the Cauchy problem for ordinary differential equations in a complex Hilbert space. Section 1 contains a discussion of Medeiros' results, and provides a motivation for the improved uniqueness results in Section 2

Abstract: Two methods are described for the a priori location of singularities of solutions to exterior boundary value problems. One uses an expansion for the solution in a circle centered on a regular exterior point P. A singularity lies on the circle of convergence. The envelope of these circles, generated as P makes a circuit about the closed boundary, circumscribes the singularities. The radius of convergence depends on singularities of the solution u(s) and its normal derivative v(s) on the boundary. The second method employs complex characteristics to relate singularities of the boundary data to real singularities of the solution. Integral equations connecting (y), v(s) and the analytic boundary condition are used to continue the data into the complex s-plane and to locate their singularities. Explicit solution of the integral equations is unnecessary; some nonlinear boundary conditions can be handled.

Abstract: The logarithmic derivative of the gamma function χ(z) can be represented in terms of a generalized hypergeometric function. Using results of our previous studies, we can approximate χ(z) by a sequence of ratios of polynomials. The sequence converges in the half-plane R(z) > 0. Further, numerical computation is facilitated as the numerator and denominator polynomials of the sequence satisfy the same four-term recurrence relation. A similar analysis is developed for χ(z+½)−χ(z). The efficiency of our scheme is illustrated with some numerical examples.

Abstract: Integral operator techniques are used to construct the solution to Cauchy's problem for a class of fourth order elliptic equations in two independent variables. If the Cauchy data is prescribed along an arbitrary analytic arc C, then approximate solutions can be obtained on compact subsets of domains which are conformally symmetric with respect to C. This improves upon results previously obtained by Henrici, Pucci, and Colton.

Abstract: In this paper, the first of a series, the space of n-dimensional Bessel potentials Lρ α, 0 < α ≦2, is considered with the aim of describing smoothness properties of its elements. This is achieved by forming norms involving the existence of derivatives or the order of Lipschitz conditions of f or its Riesz transform, and by showing these to be equivalent to the Lα ρ- The method of proof, inspired by Sunouchi and Shapiro, consists in interpreting the characterization itself as a saturation problem with Favard class Lα ρ; thus, the characterizations have only to satisfy the conditions of a general saturation theorem, established in Lρ,1≦ρ≦∞ To obtain more specific results in case 1 < ρ < ∞ the Marcinkiewicz–Mikhlin multiplier theorem is applied. Our general results contain particular ones due to Berens–Nessel, Butzer, Butzer–Trebels, Calderon, Cooper, Gorlich, Nessel–Trebels, and Trebels.

Abstract: Separation and comparison theorems of Sturm's type are given for differential inequalities involving singular elliptic operators of the form and a related class of degenerate elliptic operators. The region considered is an open connected subset of the half-space y > 0 in E m+l with an open subset of the hyperplane y< = 0 as part of the boundary. A notable feature is that data need not be prescribed on y = 0 in contrast to the regular case where data is prescribed on the entire boundary. The results are obtained by the use of Green's identity and a new Picone-type identity.

Abstract: We consider two integral operators, L and L k defined by Let L 2(p)(L 2(q)) be the space of functions defined on [−1, 1] and integrable with respect to the weight function (1−x 2)−½((1−x 2)½) . Let W2 1(q) be the space of functions, f, absolutely continuous on [−1,1] with f ∈ L 2(q) and W 2 −1(q) be its dual. It has previously been shown that L and L k are one to one, continuous maps of L 2(q) onto W 2 l(q). Here we show that these mappings can be extended to mappings L and L k which are one-to-one continuous maps of W 2 −1(q) onto L2(p). These results are applied to the problem of solving the two dimensional Laplace and Helmholtz equations with boundary data given on the interval [−1,1] of the x axis.

Abstract: We extend to perturbed nonlinear differential equations some of the basic integral manifold theorems of Bogoliubov-Mitropolski-Hale for perturbed linear systems. The technique involved uses a generalization of the variation of parameters formula. Implicit in the earlier work on perturbed linear systems is a uniqueness theorem for individual trajectories on half-lines which, to our knowledge, has never been proved. This theorem is critical in the arguments which were used to obtain the stability properties of the integral manifold and to establish its uniqueness. In the present paper an alternative development which avoids these arguments and retains the principal features of the theory is given.

Abstract: For a fixed end point problem in (n+1)-space with integrand function there are discussed certain necessary conditions that are satisfied by a minimizing arc having one or more corners. Firstly, there is clarified the type of minimum for which the continuity of the function across corners is a necessary condition, since various authors have erroneously stated that this is a necessary condition for a weak relative minimum. There are discussed two methods by which one may establish the necessity of the continuity of this function in the case of a strong relative minimum. The more comprehensive method is that employed many years ago by the author in the study of discontinuous solutions for the non-parametric problem of Mayer. For the problem herein considered there is presented the second order condition involving the non-negativeness of the second variation along a non-singular extremaloid on the class of so-called generalized admissible variations vanishing at the end-values, including a discussion of conju...

Abstract: This paper is concerned with an extension of the classical concept of staturation to systems of coupled approximations. Let {T j , ϑ}, j=1,2 be two families of bounded linear operators of the product space Y×Y into (the Banach space) y which constitute a strong simultaneous approximation process on Y×Y in the sense that for each and j=1,2 where . The simultaneous process {T j , ϑ} is then said to be saturated on Y×Y if there exist φ j (ϑ) with such that every for which is an invariant elements under {T j , ϑ} and if the set , the so-called Favard class, contains at least one noninvariant element. On the basis of this concept the boundary behavior of the solution of Dirichlet's problem for a wedge is studied in detail as an application of general results on simultaneous approximation processes of Mellin convolution;matrix-methods are employed.

Abstract: An important technique for determining the stability of a system of ordinary differential equations is to determine whether there are any roots in the positive half-plane of a certain polynomial P(z). Cesari has given a criterion for this in terms of the topological degree of the mapping described by P(z). It is shown here that Cesari's criterion can be reformulated as the problem of approximating the real roots of polynomials which are the real and imaginary parts of the P(z) on certain lines in the z-plane. The roots need only be approxi¬mated closely enough so that their magnitudes can be compared. The derivation of this criterion uses the notion of topological degree but the criterion itself is stated entirely in elementary terms