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The purpose of this work is to establish a priori $C^{2,\alpha}$ estimates for mesh function solutions of nonlinear difference equations of positive type in fully nonlinear form on a uniform mesh, where the fully nonlinear finite difference operator $\F$ is concave in the second-order variables. The estimate is an analogue of the corresponding estimate for solutions of concave fully nonlinear elliptic partial differential equations. We use the results for the special case that the operator does not depend explicitly upon the independent variables (the so-called frozen case) established in part I to approach the general case of explicit dependence upon the independent variables. We make our approach for the diagonal case via a discretization of the approach of Safonov for fully nonlinear elliptic partial differential equations using the discrete linear theory of Kuo and Trudinger and an especially agreeable mesh function interpolant provided by Kunkle. We generalize to non-diagonal operators using an idea which, to the author’s knowledge, is novel. In this paper we establish the desired Hölder estimate in the large, that is, on the entire mesh $n$-plane. In a subsequent paper a truly interior estimate will be established in a mesh $n$-box.AMS 2000 Mathematics subject classification: Primary 35J60; 35J15; 39A12. Secondary 39A70; 39A10; 65N06; 65N22; 65N12

The aim of this paper is twofold. The first aim is to describe the entire solutions of the partial differential equation (PDE) $u_{z_1}^2+2Bu_{z_1}u_{z_2}+u_{z_2}^2=e^g$, where B is a constant and g is a polynomial or an entire function in $\mathbb {C}^2$. The second aim is to consider the entire solutions of another PDE, which is a generalization of the well-known PDE of tubular surfaces.

In the present paper we deal with a quasilinear problem involving a singular term. By combining truncation techniques with variational methods, we prove the existence of three weak solutions. As far as we know, this is the first contribution in this direction in the high-dimensional case.

In this paper we study a quasi-linear elliptic problem coupled with Dirichlet boundary conditions. We propose a new set of assumptions ensuring the existence of infinitely many solutions.

This paper is concerned with the problem of obtaining explicit expressions of solutions of a system of coupled Lyapunov matrix differential equations of the typewhere Fi, Ai(t), Bi(t), Ci(t) and Dij(t) are m×m complex matrices (members of ℂm×m), for 1≦i, j≦N, and t in the interval [a,b]. When the coefficient matrices of (1.1) are timeinvariant, Dij are scalar multiples of the identity matrix of the type Dij=dijI, where dij are real positive numbers, for 1≦i, j≦N Ci, is the transposed matrix of Bi and Fi = 0, for 1≦i≦N, the Cauchy problem (1.1) arises in control theory of continuous-time jump linear quadratic systems [9–11]. Algorithms for solving the above particular case can be found in [12]]. These methods yield approximations to the solution. Without knowing the explicit expression of the solutions and in order to avoid the error accumulation it is interesting to know an explicit expression for the exact solution. In Section 2, we obtain an explicit expression of the solution of the Cauchy problem (1.1) and of two-point boundary value problems related to the system arising in (1.1). Stability conditions for the solutions of the system of (1.1) are given. Because of developed techniques this paper can be regarded as a continuation of the sequence [3, 4, 5, 6].

In a recent paper Professor A. C. Aitken directed attention to a little known type of inverse factorial series, which he calls inverse central factorial series. He showed that this type of series is a very powerful means of asymptotic representation of functions of a certain type, and concluded his note with the-remark that there is here scope for considerable investigation. As a modest contribution to this investigation, the asymptotic theory, foreshadowed in the last paragraph of Aitken's paper, will be given here: it is hoped to give the convergence theory of inverse central factorial series later.

Earlier this year Professor W. L. Edge drew my attention to the paper (1). Since Black probably wrote his paper about 1890, the integral he considered must have been one of the earliest n-variate integrals to be evaluated. The present paper generalizes Black's result from a column vector as variate to a rectangular matrix—his integral is the case p = m = k = 1 of the integral J below.

The contents of this paper were suggested by a discussion of the equation:in a paper by Glaisher which appears in the Philosophical Transactions, 1881, Part III.The solutions in series of (1) are:and in the paper referred to it is shewn that the coefficients of hp+1 in the expansions of and of satisfy equation (1) when p is a positive integer.

This paper is an appendix to a joint paper with Professor Macbeath. In (3), it was proved that the invariant measure, m(k), of the set of real n × nmatrices τ, with determinant 1 and norm satisfying ∥τ∥≦ k, had the property that

In this paper we consider the problem of existence of mild solutions to semilinear fractional heat equations with non-local initial conditions. We provide sufficient conditions for existence and regularity of such solutions.

This paper is concerned with a boundary-value problem on the half-line for nonlinear two-dimensional delay differential systems with positive delays. A theorem is established, which provides sufficient conditions for the existence of positive solutions. The application of this theorem to the special case of second-order nonlinear delay differential equations is given. Also, the application of the theorem to two-dimensional Emden–Fowler-type delay differential systems with constant delays is presented. Moreover, some general examples demonstrating the applicability of the theorem are included.

This paper is devoted to the Dirichlet problem for quasilinear elliptic hemivariational inequalities at resonance as well as at non-resonance. Using Clarke's notion of the generalized gradient and the property of the first eigenfunction, we also build a Landesman–Lazer theory in the non-smooth framework of quasilinear elliptic hemivariational inequalities.

For $a\in(0,\tfrac12]$ and $r\in(0,1)$, let $\mu_a(r)$ be the so-called generalized Grötzsch function which appears in Ramanujan's generalized modular equations. In this paper, several sharp inequalities for $\mu_a(r)$ are obtained and a conjecture on $\mu_a(r)$, which was presented by Qiu and Vuorinen in 1999, is proved.

In this paper we study the existence and nonexistence of multiple positive solutions for the Dirichlet problem:$$ -\Delta{u}-\mu\frac{u}{|x|^2}=\lambda(1+u)^p,\quad u\gt0,\quad u\in H^1_0(\varOmega), \tag{*} $$where $0\leq\mu\lt(\frac{1}{2}(N-2))^2$, $\lambda\gt0$, $1\ltp\leq(N+2)/(N-2)$, $N\geq3$. Using the sub–supersolution method and the variational approach, we prove that there exists a positive number $\lambda^*$ such that problem (*) possesses at least two positive solutions if $\lambda\in(0,\lambda^*)$, a unique positive solution if $\lambda=\lambda^*$, and no positive solution if $\lambda\in(\lambda^*,\infty)$.

The singular boundary-value problem $(g(x'(t)))'=\mu f(t,x(t),x'(t))$, $x(0)=x(T)=0$ and $\max\{x(t):0\le t\le T\}=A$ is considered. Here $\mu$ is the parameter and the negative function $f(t,u,v)$ satisfying local Carathéodory conditions on $[0,T]\times(0,\infty)\times(\mathbb{R}\setminus\{0\})$ may be singular at the values $u=0$ and $v=0$ of the phase variables $u$ and $v$. The paper presents conditions which guarantee that for any $A>0$ there exists $\mu_A>0$ such that the above problem with $\mu=\mu_A$ has a positive solution on $(0,T)$. The proofs are based on the regularization and sequential techniques and use the Leray–Schauder degree and Vitali’s convergence theorem.AMS 2000 Mathematics subject classification: Primary 34B16; 34B18

We consider symplectic difference systems, which contain as special cases linear Hamiltonian difference systems and Sturm–Liouville difference equations of any even order. An associated discrete quadratic functional is important in discrete variational analysis, and while its positive definiteness has been characterized and is well understood, a characterization of its positive semidefiniteness remained an open problem. In this paper we present the solution to this problem and offer necessary and sufficient conditions for such discrete quadratic functionals to be non-negative definite.AMS 2000 Mathematics subject classification: Primary 39A12; 39A13. Secondary 34B24; 49K99

In this paper we study the existence of multiple positive solutions and the bifurcation problem for the following equation:$$ -\Delta u+u=\biggl(\int_{\mathbb{R}^3}\frac{|u(y)|^2}{|x-y|}\,\mathrm{d}y\biggr)u+\mu f(x),\quad x\in\mathbb{R}^3, $$where $f(x)\in H^{-1}(\mathbb{R}^3)$, $f(x)\geq0$, $f(x)\not\equiv0$. We show that there are positive constants $\mu^{*}$ and $\mu^{**}$ such that the above equation possesses at least two positive solutions for $\mu\in(0,\mu^{*})$, and no positive solution for $\mu>\mu^{**}$. Furthermore, we prove that $\mu=\mu^{*}$ is a bifurcation point for the equation under study.AMS 2000 Mathematics subject classification: Primary 35J60; 35J70

In this paper we are dealing with oscillatory and asymptotic behaviour of solutions of second order nonlinear difference equations of the formSome sufficient conditions for all solutions of (E) to be oscillatory are obtained. Asymptotic behaviour of nonoscillatory solutions of (E) is considered also.

Sequence transformations are extrapolation methods. They are used for the purpose of convergence acceleration. In the scalar case, such algorithms can be obtained by two different approaches which are equivalent. The first one is an elimination approach based on the solution of a system of linear equations and it makes use of determinants. The second approach is based on the notion of annihilation difference operators. In this paper, these two approaches are generalized to the matrix and the vector cases.

In this paper, we investigate the complex oscillation theory ofwhere A, F≢0 are entire functions, and obtain general estimates of the exponent of convergence of the zero-sequence and of the order of growth of solutions for the above equation.

The kernel function method of Bergman and Schiffer (see (1) and (2)) has recentlybeen used by Colton and Gilbert (see (3) and (4)) in connection with approximationtheory and the numerical treatment of elliptic differential equations. In (5) Gilbert andWeinacht have successfully extended the kernel function method to elliptic systemsof differential equations. Essential to their work is the concept of a matrix kernelsatisfying the reproducing property. This reproducing kernel is defined initially as thedifference of the Neumann matrix and the Dirichlet matrix. Thus actually to obtainthe kernel matrix from this definition one has to solve both a Neumann problem and aDirichlet problem. In view of this restriction, Gilbert and Weinacht derive an in-genious representation for the reproducing kernel in terms of purely geometricquantities which are obtained directly from the fundamental matrix for the differentialsystem.In this paper a new series representation for the reproducing kernel is given, theterms of which are more easily calculated than those appearing in the representationin (5). The terms of the series involve a certain parameter a which we introduce. Fora = 0 the representation is shown to reduce to the one given in (5). We also obtain thevalue of a leading to the fastest rate of convergence and make a comparison with thea = 0 case.

In a previous paper (9), we introduced the spaces Fp, μ of testing-functions and the corresponding spaces of generalised functions. For 1≦p≦∞, Fp(=Fp, 0) is the linear space of all complex-valued measurable functions φ defined on (0, ∞) which are infinitely differentiable on (0, ∞) and for which for each k = 0, 1, 2, …. In symbols

In a recent paper Segel (1) points out that the diverse techniques (which “comprise the core of the applied mathematicians art (or craft)”) of the applied mathematician, although in general reliably proven, are “rarely explicitly delineated but rather are transmitted indirectly and informally”. In his article Segel aims to clarify two such techniques, namely:(i) Scaling—or how to choose dimensionless variables in such a way that the relative size of the various terms in an equation is explicitly indicated by the magnitudes of the dimensionless parameters which precede them,(ii) Simplification—a procedure in which a term is neglected under the assumption that it is small, and the consistency of the assumption checked later.

In the present paper two problems on approximation by rational functions will be treated. The one concerns rational functions whose poles are of any order but lie at two preassigned points. The other problem relates to rational functions which have simple poles only.

In a previous paper in these Proceedings the author discussed conditions for a maximum or minimum of functions of integrals of the typeusing the methods of the Calculus of Variations. In the effort to establish a third necessary condition for a minimum—the analogue of Jacobi's condition in the ordinary variational problem—it was found that the analogue of Jacobi's Equation was an integrodifferential equation of the form

The present paper is concerned with formulae by which double integrals of functions of two independent variables may be evaluated approximately. The number of such formulae published hitherto is not great, and it has seemed desirable both to make a systematic search for new formulae, and to test the comparative merits of these, and of those previously known, by computing the numerical values of certain selected integrals.

In investigating the properties of the cubic curve C, we are led to consider its Hessian C′, its Cayleyan Γ and the Hessian of its Cayleyan Γ′. In the present paper we propose to deal with the intersections of the systems C + λC and Γ + μΓ′.

In this paper we consider a function f(x) defined byAll quantities are taken to be real, it is assumed that R is a function of the variable x, b is a constant, N and G are functions of the variable t and all the functions are such that the integral (1) exists when x is large enough. We wish to find an asymptotic representation of f(x) as x → + ∞, assuming that we are given certain information about the limiting behaviours of the functions R, N and G.

Methods for solving boundary value problems in linear, second order, partial differential equations in two variables tend to be somewhat rigidly partitioned in some of the standard text-books. Problems for elliptic equations are sometimes solved by finding the fundamental solution which is defined as a solution with a given singularity at a certain point. Another approach is by way of Green's functions which are usually defined as solutions of the original homogeneous equations now made inhomogeneous by the introduction of adelta function on the right hand side. The Green's function coincides with the fundamental solution for elliptic equations but exhibits a totally different type of singularity for parabolic or hyperbolic equations. Boundary value problems for hyperbolic equations can often by solved by Riemann's method which depends on the existence of an auxiliary function called the Riemann or sometimes the Riemann-Green function. The main object of this paper is to show the close relationship between Riemann's method and the method of Green's functions. This not only serves to unify different methods of solution of boundary value problems but also provides an additional method of determining Riemann functions for given hyperbolic equations. Before establishing these relationships we shall survey the general approach to boundary value problems through the use of the Green's function.

Let {Kn} be a sequence of complex numbers, letand letLet D be the open unit disc {z: |z| be its closure and let .The primary object of this paper is to prove the two theorems stated below, the first of which generalises a result of Copson (1).

The various formulae for the Legendre Functions, and the relations between these formulae, have been studied by Kummer,Riemann, Olbricht, Hobson, Barnes, Whipple, and others. Hobson obtained some of the relations directly, by expres ing the functions as Pochhammer integrals, and expanding in a number of series each with its own region of convergence. To obtain some of the other formulae, such as (i) below, he transformed the differential equation, and then expressed the functions in terms of the solutions of the transformed equation. Barnes succeeded, by means of his wellknown integrals involving Gamma Functions, in deducing all the formulae directly from the formulae which define the functions.Notes on the history of the subject and references to previous work will be found in the papers by Hobson and Barnes.

In this paper we first obtain elementary solutions of the integral equationsandUsing these solutions we then define operators of fractional integration. These operators may be regarded as a generalisation of the operators of fractional integration introduced by Sneddon (1) as a modification of Erdé1yi–Kober operators. In fact Erdélyi–Kober–Sneddon operators may be obtained by multiplying both sides of the equations by α-1½β and considering the limiting case α→0. We employ these operators to find a generalisation of the Mellin transform.

In the series of papers in the early forties summarised in (1), (2), A. I. Lur'e showed how to utilise Liapunov's second or direct method in the investigation of the stability of linear automatic control systems with a single nonlinear actuator. His approach consists of1. the transformation of the original system of differential equations via the so-called Lur'e transformation into canonical coordinates in which the construction of the Liapunov function is direct, and2. the conversion of the differential problem into a purely algebraic problem.We will be concerned here with the questions of the existence and construction of the Lur's transformation.

It is proposed in this paper to show now the well-known Laplace's transformation,which is of great help in finding the solution of linear differential equations, gives also interesting results concerning the theory of integral equations. In §2 we shall study its application to certain differential equations, and find a large class of equations which remain unchanged by this transformation. Then, (§3), taking instead of eazt, a more general function of the product zt, we shall find a solution for some homogeneous integral equations ; in § 4 we shall describe a method of solving a very general type of integral equation of the first kind, namely,a further extension to integral equations with the kernel ef(z)f(t) the object of §5. Then, studying an extension of Euler's transformation, we shall (§ 6) consider equations such aswhich will prove to be singular; and finally, in §7, we shall give other examples of singular integral equations.

§ 1. It is proposed to discuss in this paper partial differential equations involving two independent variables x and y, and a dependent variable z. The method of reduction which is explained can be applied to certain equations involving more than two independent variables, but such application is subject to too many restrictions to be of much general utility.

The strong maximum principle for harmonic functions is usually arrived at by appealing to the mean value theorem (c.f. [2], p. 53). It is also of course possible simply to appeal to the Hopf maximum principle [2], but using sledge hammers to kill flies is generally viewed as aesthetically unpleasing. In contrast to the case of harmonic functions, the only proof of the strong maximum principle for the heat equation that is known to me is to invoke Nirenberg's strong maximum principle for parabolic equations [2]. As in the case of harmonic functions, it seems desirable to provide a direct proof of this result without having to go through the subtle comparison arguments that are employed in the more general case. The purpose of this note is to provide a proof of the strong maximum principle for the heat equation based on a mean value theorem for solutions of the heat equation which we derive below. Such an approach provides a straightforward and simple proof of the strong maximum principle which avoids most of the detailed estimates of the proof of the maximum principle for more general parabolic equations. Unfortunately the proof of the maximum principle for the heat equation using the mean value theorem is not as short as the proof in the corresponding case of harmonic functions. It nevertheless seems worthwhile to show that such an alternate proof is possible, and it is to this purpose that we address this paper.

Introduction. In the present paper a formula will be obtained to express a Ferrers' Associated Legendre Function of any integral degree and order as a sum of a finite number of Associated Legendre Functions of an order reduced by an even number. When the order is reduced by unity, an infinite series of the functions of reduced order is required. Thus a Ferrers' function can be expressed as the sum of a finite or infinite number of zonal harmonics according as the order of the function is even or odd.

In this note a simple application of the Kontorovich-Lebedev transform is described. The problem to which the transform is applied is the one considered in a recent paper (1).

The general linear matrix equation has been investigated by Sylvester in a series of interesting papers published in the Comptes Rendus. His method, however, does not show that the solution can be exhibited as an analytical function of the coefficients. The object of this note is to supply such a solution.