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SynopsisIn this paper we study an ordinary second-order integro-differential equation (IDE) on a finite closed interval. We demonstrate the equivalence of this equation to a certain integral equation, and deduce that the homogeneous IDE may have either 2 or 3 linearly independent solutions, depending on the value of a parameter λ. We study a Cauchy problem for the IDE, both by this integral equation approach and by an independent approach, based on the perturbation theory for linear operators. We give necessary and sufficient conditions for the Cauchy problem to be solvable for arbitrary right-hand sides—these conditions again depend on λ—and specify the behaviour of the IDE when these conditions are not satisfied. At the end of the paper some examples are given of the type of behaviour described.

In a manner similar to that given in a preceding paper (cf. Castiet al., 1970), in this paper we show how to get a Cauchy system for the scattering and transmission matrices of the azimuth-independent term of partially polarized light in accordance with Rayleigh scattering in a homogeneous atmosphere bounded by a specular reflector. This set of integro-differential equations for the scattering and transmission matrices is suitable for numerical computation by high-speed digital computer.

E. Hopf discussed in details on the Cauchy problem of Burgers' equation in his famous paper [5]. Since then, many papers on the equation and its related topics have been published. However, they have not treated the initial-boundary value problem of it. The author previously discussed on the first initial-boundary value problem of this equation in [17]. Recently, N. Itaya has shown the existence and the uniqueness, in a certain sense, of the temporally global solution of the Cauchy problem of the following generalized Burgers' equation

has a (unique) solution on the whole positive real line R+. However, the well known criteria for the convergence of successive approximations, which are all included in the theorem of Coddington-Levinson [3] (cf. Lakshimikanthan-Leela [5, Section 2.31) do not imply that the successive approximamations of (1) converge on the whole interval R+. The aim of this paper is to prove a general theorem ensuring that the convergence of the successive approximations of (1) is really global, thus avoiding any discrepancy between the domain of existence of the solution and the domain of convergence of the successive approximations. The proof works also for the case of infinite dimensional Banach spaces. Moreover, from our argument one can obtain a much easier proof of the theorem of Coddington-Levinson [3] and of its generalization to Banach spaces due to Kisynski [5] and Wazewski [7]. The method inspired a new sufficient condition for the convergence of successive approximations which is independent of the properties of any scalar Cauchy problem and yields a new existence theorem for the Cauchy problem in Banach spaces. The first result of this paper is the following.

In formulating phenomenological models and developing solution algorithms for problems in continuum and structural mechanics, experience with the behavior of actual systems has always played an important guidance role for the engineer concerned with predicting performance of feasible alternatives. By exercising control over various system parameters the engineer attempts to minimize the difference between a desired objective and a predicted approximation thereof. Constitutive modeling of mechanical behavior of materials is an example typical of this activity. In this paper constitutive modeling of an isolated material element is discussed as an identification problem. The mathematical structure of the problem is discussed and limitations arising from experimental sources, measurement errors, and material nonlinearity are noted. As a consequence of the structure of the problem it is shown that an inverse initial-boundary value problem must be considered, in which material identification and numerical analysis of this problem must proceed simultaneously. This paper discusses a discretized form of the inverse problem and illustrates the methodology by citing a number of examples in the literature which appear as special cases of the present treatment.

THE results of a numerical solution of a boundary value problem for a strongly elliptic system of second-order equations are described. The difference equations of the mesh method for the original problem are solved by a two-step iterative method with Chebyshev parameters and by the alternating directions method as an internal iterative process. The results of numerical experiments are quoted, and compared with results obtained by other methods. The original problem describes the displacement of a flexible reticulate shell, loaded by internal pressure and external forces distributed over the shell surface. Structures of this kind are used in pneumatic tyres. The computational results relate to loading of the shell over a part of its surface. The present paper deals with the numerical solution of boundary value problems for systems of equations of a strongly elliptic type, describing the equilibrium of flexible reticulate shells (toroidal shells of revolution and cylindrical shells), loaded by external forces and by internal pressure (see [1] ). Such problems arise in the design of structures which are armoured by extensible threads, e.g., tyres. The unknowns are the components of the shell displacements. While the boundary conditions and external loads are common to problems for shells of this kind, the coefficients of the equation of equilibrium depend on the type of shell: they are constants in the case of a cylindrical shell, and variables along the meridian in the toroidal case. Since the equations retain the same structure and type, it is possible to use a unified algorithm for their solution. The systems in question are the systems of Euler equations for the variational problem of minimizing a functional, where the functional represents the total energy of deformation by the internal pressure and the external forces on the shell. As the approximate method of solution we used a finite-difference method (mesh method), in which the system of equations is the system of Euler equations for minimizing a discrete functional on a mesh, approximating the initial functional. The systems of difference equations thus obtained were solved by a two-step iterative method with Chebyshev parameters (see [2,3]) and by the alternating directions method as an internal iterative process. The numerical results quoted may be of interest, not only from the point of view of solving such problems in reticulate shell theory, but also from the point of view of checking experimentally iterative methods for solving the difference analogues of strongly elliptic systems. The iterative methods employed are analyzed theoretically in [2, 3]. The effectiveness of the methods depends on the choice of difference operators, spectrally equivalent to the difference analogue of the initial differential operator. Such problems were investigated for the equations of equilibrium of cylindrical shells in [4, 5]. Methods similar to those of [4, 5] may also be used for shells of revolution. However, in the present paper the required equivalence conditions, which determine the iteration parameters and hence the convergence rate of the iterations, were found experimentally, since the coefficients of the equations can themselves be found approximately for a shell of revolution by solving a Cauchy problem for a system of ordinary differential equations [6].

The subject of this paper is uniqueness and non-uniqueness in the Cauchy problem at simple characteristic points of the initial hypersurface. We only consider linear partial differential equations with analytic coefficients. We use the notation from the book [1] by H6rmander with the exception that D~ = t3/~xj, 1 0 in a neighbourhood of x °, and if H is of a type covered by [6, Theorem 3] then there is a neighbourhood V of x ° and a function u e C ~ ( V ) such that P ( x , D ) u = O , x ° ~ suppu C {x; ~0(x) =>0, x ~ V}." When we want to construct characteristic hypersurfaces that are not hyperplanes we have to rely on integration of Hamilton's equations. So we restrict ourselves to operators with real principal part and to simple characteristic points on the initial hypersurface. The three points of departure of this paper is the method to prove non-uniqueness by Malgrange [3], the uniqueness theorem [2, Theorem 8.1] by H6rmander, see Theorem 3 below, and the technique of proving uniqueness taken from Persson [5]. The part played by Theorem 3 is a passive one leaving to us to study the case when a priori the bicharacteristic curve does not leave the support of the solution. In [9] Zachmanoglou proves the following theorem.

In this paper we examine a problem connected with Cauchy's problem for linear elliptic equations.Let be a bounded region of , and let be its boundary. In we consider the elliptic equation (1)where is a regular elliptic expression with complex coefficients. Let , be a piece of the surface . The coefficients of the expression , the surface , and the boundary are assumed to be infinitely smooth. We are concerned with Cauchy's problem on with the initial conditions , , where designates the direction normal to . In this paper we prove that under our assumptions the set of Cauchy initial data for solutions of (1) in is dense in for any integer if Cauchy's problem is unique for the formal conjugate operator , as is the case, for example, when has no multiple complex characteristics.In addition, in this paper we give conditions under which the analogous assertion holds for certain elliptic systems.Bibliography: 4 items.

IN this paper we consider the non-linear discrete boundary value problem obtained from a finite difference approximation of the boundary value problem for the ordinary differential equation ϑ = y″ + ƒ(x, y) = 0 (0.1) y(0) = y(1) = 0 (0.2) In [1] a continuous analog of Newton's method was used to find an approximate solution of problem (0.1)–(0.2). The existence of these solutions is assumed. The proposed computational scheme is a realization of Euler's method of solving a Cauchy problem for a certain differential equation in B-space. It reduces to the solution for a fixed value of the parameter t of a boundary value problem for a linear ordinary differential equation, and to a subsequent advancement along the parameter by a simple difference relation. The solution of the boundary value problem for fixed t is assumed to be found numerically, for example, by the use of difference approximations. It is easy to see that this computational scheme is a numerical realization of the continuous analog of Newton's method of solving the difference operator equation approximating to the original boundary value problem (0.1)–(0.2). In the present paper a justification of this method is given: a proof is given of the existence of a solution of the discrete boundary value problem which is the difference analog of problem (0.1)–(0.2), and the convergence of the approximate solution to the solution of the original problem is proved. The result obtained in [2] is used in proving the existence of the solution of the discrete boundary value problem. Similar problems were considered, for example, in [3], pp. 347–388, [4], pp. 59–67. Thus, in [4] a proof is given of the existence and method of finding an approximate solution of problem (0.1)–(0.2) by Newton's method. However, by connecting the questions of the existence and uniqueness of the solution of the original problem with the method of the approximate solution, the author obtains extremely restricted conditions for the applicability of the method.

This paper considers the magnetoelasticity equations for a homogeneous and isotropic elastic medium which satisfies the condition eμ e = e 0 μ e 0 . It is assumed that Maxwell's displacement currents are neglected. The equations are written in dimensionless variables. The linearization of the system of equations leads to the obtaining of a single equation for the current density. As a practical application, Cauchy's problem is solved for the perfectly conductive infinite plane in the presence of a homogeneous magnetic field. It is shown that the wave propagation takes place without dispersion. In the final part of the paper Cauchy's problem is solved for the elastic half-plane subjected to a normal pressure on the boundary. The solution is obtained by a continuous superposition of plane waves. ]It is demonstrated that in both cases Cauchy's problem is correctly set.

In this paper we are concerned with the convergence of semigroups of operators. The problem may be stated as follows: Let K&(t); t > %1,2,... b e a sequence of semigroups of operators of a specified class. Under what conditions is there a semigroup {T(t); t > 0} of the specified class such that T,(t) -+ T(t) strongly for each t > O? This problem was first introduced by Trotter [I] and then treated by Hasegawa [2] in the case of (C,)-semigroups, by Miyadera [3] in the case of (1, A)-semigroups on a Banach space, and by Kato and Yosida [4; Chap. IX, 121 and Oharu [5] for (C,)-semigroups on a locally convex space. The purpose of this paper is to establish a theorem on convergence of (A)-semigroups and, as its simple consequences, theorems on convergence of (0, A)and (1, A)-semigroups. The theorems are stated and the relations among them are studied in Section 1. Sections 2 and 3 consist of proof of the main theorem (A). In Section 4, some variations of theorems are studied. It is well known that convergence of semigroups was introduced by considering finite difference approximations of the solution of a Cauchy problem. As noted by Phillips [6], (0, A)-semigroups are related to the abstract Cauchy problem. The theorem of convergence of (0, A)-semigroups is desirable for applications, however we shall not give any in this paper. The authors want to express their deep gratitude to Professor I. Miyadera for his many valuable suggestions.

THIS paper is devoted to the problem of the unboundedness of the classical solutions of quasilinear hyperbolic systems for n ⩾ 3. An example illustrating this fact is constructed. A class of systems is isolated for which the solution remains bounded. This class includes the system of equations of gas dynamics in the non-isentropic case with some constraints on the equation of state. For model equations of states close to the absolute zero, where these constraints are not satisfied, effects connected with localised unbounded growth of density or specific volume, are assumed, which are confirmed by the results of numerical experiment. According to a statement in [1], the absence of integrating factors of the Pfaffian forms dωk = lαk (t, x, u) duα for n ⩾ 3 (“spin of the eigenvectors”) may lead to unbounded growth of the classical solution of the Cauchy problem for the hyperbolic system of equations (1) ∂u ∂t + A (t, x, u) ∂u ∂x = F(t, x, u), u(0, x) = u 0 (x), xϵ(− ∞,∞) . where t and x are the independent variables; u and F are n-dimensional vector functions; A is a matrix of order n; lk is the left eigenvector of the matrix A corresponding to the eigenvalue ξk; F(t, x, u), A(t, x, u), u0(x) ϵ C1, even, if F  0. (Here and everywhere below repetition of a Greek subscript means summation.) This paper is devoted to a study of the question of unboundedness of solutions. The main results are as follows. An example is constructed illustrating the unboundedness of solutions for n = 3. A narrow class of systems for which the classical solution remains bounded is isolated. These are systems permitting n − 1 or n − 2 “entropic” [1] variables and satisfying constraints on the growth of the coefficients of the characteristic form. For the equations of gas-dynamics in the non-isentropic case, these constraints reduce to conditions on the equation of state which are expressed by the inequalities (2) 0 v 0 ⩽ ¦ c v ¦ ∗ , ¦ ( ∂p ∂ϱ∂s ) ( ∂p ∂ϱ ) s −1 , ¦ ⩽ M 0 , C V 0 , M 0 = const . We will use the following notation: ϱ is the density; u is the velocity; s is the entropy of unit mass; T is the temperature; p = p(ϱ, s) is the pressure (equation of state); c v = T ( ∂s ∂T ) ϱ is the specific heat per unit mass; c = √ ( ∂p ∂ϱ ) s is the speed of sound. For such systems the solution satisfies some non-linear multi-dimensional integral equation of Volterra type where the domain of definition of the hyperbolic equations emerges as the cone of integration, and the boundedness follows from the property of the local absolute integrability of the kernel, which in terms of quasilinear equations indicates the absence of discontinuities of the entropic variables or the possibility of discontinuities with small variations, and implies another remarkable property of the gas-dynamical equations. We consider the case where the constraints (2) are not satisfied. This is the case if the temperature approaches the absolute zero, and then by Nernst's theorem the entropy and specific heat tend to zero. As models we take the equations of state of degenerate Fermi and Bose gases close to the absolute zero. For a Fermi gas ( ∂p ∂s ) ϱ  0 the system of equations of hydrodynamics may be split up and the question of unboundedness of the solution does not arise. For a Bose gas the question requires additional study. In this case we consider a Cauchy problem with the initial functions ϱ(0, x)= ϱ0 = const, u(0, x) = 0, s(0, x) = s0(x) where s0 (x) has at the point x = 0 a local minimum smin, which tends to zero. We have here parabolic degeneracy of the gas-dynamical equations, since the speed of sound tends to zero. We construct a trivial model of degeneracy which assumes the possibility of a gas-dynamical effect which consists of the unbounded increase in density at a point of minimum when smin tends to zero. This assumption is confirmed by the results of a numerical experiment.

LET RN be an N-dimensional Euclidean space, N ⩾ 3, 0 ⩽ τ ⩽ 1, Ω τ a family of open bounded domains in RN, \ gW T the closure of the domain Ω τ , Ω 1 τ a connected component of the set R N βΩ τ , containing the point at infinity, Ω 2 τ = R N β (Ω duτ ∪ Ω 1 τ ) . Let Bτ be an operator defined on the boundary of the set Ω τ and having the form ( B τ u) (x) = u(x)¦ xeΓp \ gW τ , ( B τ u) (x) = ( δu δn + σ(x)u) (x)¦ xeΓp \ gW τ . Let Δ be the Laplace operator, H (τ) the extension in (Bτu) (x) = 0 of the operator (−Δ) corresponding to the boundary conditions l2(RNβωτ) E(λ, H (τ)) the spectral function of the operator H(τ). In l2(RNβωτ) we consider the abstract Cauchy problem δ 2 (τ¦t) δt 2 = −H(τ)v , v(τ¦ + 0) = f 1 , v(τ¦ + 0) = f 2 . (1) It is easy to see that the solution of the problem (1) (which in this case is the same as the generalization of it by S. L. Sobolev) is given by the formula 2 v(τ|t)= ⨍ o ∞ cos VλtdE(λ,H(τ))ƒ2+ ⨍ o ∞ λ − 1 2 sin Vλtde(λ,H(τ))ƒ1 It follows from (2) that v(τ¦t) behaves essentially differently depending on whether or not the operator H(τ) has a discrete spectrum. The purpose of our paper is to estimate the variation of the function v(τ¦t) for so small a variation of the domain Ω τ as transforms it from a simply connected into a non-simply connected domain, and changes the nature of the spectrum of the operator v(τ¦t) . The variation of the discrete spectrum of a Laplace operator when the domain is varied was previously studied in [1]. We have formulated the main results in Lemma 2.3 and Theorem 2.1. We use the method developed for solving similar problems for Schrodinger's equation in [2–4]. This method consists of the following. Let G(τ¦x, y, t) be the kernel of the operator exp (−t0H(τ)), redefined in such a way that G(τ¦x, y, t 0 )= 0 , if either xeΩ τ , or yeΩ τ . It happens that the kernel is so well behaved that it permits the properties of the spectrum of the operator exp (−t0H(τ)), to be studied in detail, and corresponding conclusions about the spectrum of the operator H(τ) follow directly from this. It was shown in [5] that instead of the function exp (−t0H(τ)) it would be possible to consider any other function F (H(τ)) (for example, (1 + H(τ))−1), representable in the form F(λ)= ⨍ o ∞ exp (−λt)dμ(t) where μ(t) is a measure satisfying certain conditions. The paper consists of two sections. In section 1 we present some auxiliary estimates and lemmas. The proofs of these statements differ only negligibly from the proofs of the corresponding statements for the Schrodinger operator, and hence we will not discuss them in detail. The main results of the paper are discussed in section 2.

uo+u+ ?... +ut+un+l+ ... are functions of the same variable x, continuous with respect to this variable in the neighbourhood of a particular value for which the series is convergent, the sum s of the series is also, in the neighbourhood of this particular value, a continuous function of x." (p. I31) Lakatos in a paper delivered at the International Logic Colloquium, Hanover, 1966 (European meeting of the Association for Symbolic Logic) and in a manuscript 'The significance of nonstandard analysis for the history and philosophy of mathematics' gives a penetrating discussion of the importance of non-standard analysis for the understanding of the history of the calculus and, in particular, of this theorem. According to our present-day (Weierstrassian) conception of the continuum, as every student of mathematics knows, the theorem is false--one must impose the condition of uniform convergence to get a correct statement. Lakatos, however, maintains that the theorem is a perfectly correct statement about a Leibnizian continuum-an extension of the Weierstrassian continuum in which there are infinitely large and infinitely small numbers. He shows that if "the neighbourhood of a particular point" is understood as the set of points infinitely close to that value, and if the usual definition of convergence is assumed for sequences of numbers in the extended continuum, then Cauchy's proof of theorem I is correct. The aim of this paper is to examine Lakatos' claim more closely. We show that Cauchy's notions can be comfortably interpreted in terms of non-standard analysis and, in particular, that convergence of a series of functions in the infinitesimal neighbourhood of a point in Cauchy's sense is equivalent to the notion of "point of uniform convergence" in the Weierstrassian sense. If the correctness of the interpretation of Cauchy by non-standard analysis is granted one must therefore concede that the notion of uniform convergence was implicit in Cauchy's work of I82I before it was formulated explicitly in E-S terms by Seidel 1847 or Weierstrass.

In a manner similar to that given in preceding papers by Bellman and Ueno, with the aid of the Bellman‐Krein formula for the resolvent, we show how to solve Fredholm integral equations of the second kind with degenerate kernels. The standard procedure for solution is to convert it into an equivalent matrix equation, but in this paper it is transformed into a Cauchy problem which can be solved effectively by high speed digital computers.

The paper deals with Markov processes which have both random starting and terminal times. Such processes were suggested by G. A. Hunt, were constructed by L. L. Helms (under the name Markov processes with creation and annihilation) and were treated also by M. Nagasawa and the author. The paper contains a new existence proof by a way of constructing such a process from its given associated semigroup of kernels $\tilde{P}_t, t \geqq 0$, and its (Markov) transition function. This construction is more general than that given by L. L. Helms (in terms of the Markov transition function and the creation measure) and is also more convenient as far as perturbation theory of Markov processes is concerned. Indeed more general relations between this theory and creation of mass processes are established. Finally an application to solving the Cauchy problem in partial differential equations is indicated.

This paper deals with an initial value problem that can be set for any analytic hyperbolic equation (rank 2 or 3) linear or non-linear in three independent variables with any inclination of the initial surface. This initial value problem is a Cauchy problem in which we require that the initial data be analytic in one of their arguments but not in the other. Special instances of this problem have been considered by Volterra2 and Hadamard on the one hand and Hamel3 on the other. The (n 1) dimensional characteristic strips employed in this paper are an extension on the one hand of a concept well known in the theory of the second order equation in two independent variables4 and on the other hand of a concept which occurs in the theory of the first order equation in more than two independent variables and which has not been clearly recognized.5 The method used in handling our initial value problem in a great many places bears a resemblance to the method used by Lewy6 in showing that Cauchy's problem for a non-linear hyperbolic equation in two independent variables has a unique solution. In the proof of our existence theorems we

It is the purpose of this paper to discuss two dimensional stationary and pseudo-stationary flows behind shocks. We shall study the relations between geometric properties of the shock and the values of the flow variables behind the shock at points removed from the shock. The arguments which will be used will be local in character. Thus we shall attempt to describe the value of a flow variable at a point removed from a shock by the values of the derivatives of this variable at the shock. In particular we shall give a simple derivation of Thomas' results [4] on the relations between the geometric properties of the shock front and the geometric properties of a streak-line behind the shock front; that is, a locus of particles which have crossed a fixed point of the shock front at times earlier than the time under consideration. The discussion will first assume that the flow ahead of the shock is known and that the shock itself is a known analytic curve. The integration of the various conservation equations will then be shown to be equivalent to the solution of a Cauchy problem of rather simple form, with the shock front as the curve on which the initial data is given. It then follows that in a neighborhood of the shock the flow behind the shock exists and is uniquely determined. We next consider the problem of determining the geometric properties of the shock front from the geometric properties of the streak-lines, that is, of solving some of the relations referred to above. In view of the existence theorem quoted, it is evident that the existence and uniqueness of solutions of the hydrodynamic equations when shocks are present, subject to the usual boundary condition, namely the specification of a streak-line, is intimately related to the existence and uniqueness of the solutions of the equations determining the geometric properties of the shocks from those of the streak-lines. Finally we consider the case of shocks with singularities, that is, shocks which contain points where the tangent, the curvature or some derivative of the curvature is discontinuous. In a previous paper [3] it was shown that if one assumes that at time t every point in the region behind the shock is occupied by a single fluid particle then the geometric nature of the shock is determined in terms of one parameter. In this paper we examine this result in greater detail than previously.

This paper is concerned with the existence of weak solutions to certain nonlinear hyperbolic Cauchy problems. A condition on the curves of discontinuity is used which guarantees uniqueness in the class of piecewise smooth weak solutions. The method of proof is geometric in nature and is constructive in the manner of A. Doughs and Wu Cho-Chun; that is, for certain types of initial data the method of characteristics is employed to construct piecewise smooth weak solutions. A limiting process is then used to obtain existence for bounded, measurable initial data. The solutions in some cases exhibit interesting, new phenomena. For example, a certain class of initial data having one jump gives rise to a solution having a curving contact discontinuity which does not enter the region of intersecting characteristics. 1. Introduction. In this paper we consider existence of global weak solutions to the Cauchy problem