677 results
Search Results
102. Some nonexistence theorems for semilinear fourth-order equations
- Author
-
Jorge García-Melián, Alexander Quaas, and M. Á. Burgos-Pérez
- Subjects
Pure mathematics ,Elliptic systems ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Regular polygon ,Monotonic function ,Function (mathematics) ,Infinity ,01 natural sciences ,010101 applied mathematics ,Fourth order ,Maximum principle ,0101 mathematics ,media_common ,Mathematics - Abstract
In this paper, we analyse the semilinear fourth-order problem ( − Δ)2 u = g(u) in exterior domains of ℝN. Assuming the function g is nondecreasing and continuous in [0, + ∞) and positive in (0, + ∞), we show that positive classical supersolutions u of the problem which additionally verify − Δu > 0 exist if and only if N ≥ 5 and $$\int_0^\delta \displaystyle{{g(s)}\over{s^{(({2(N-2)})/({N-4}))}}} {\rm d}s \lt + \infty$$ for some δ > 0. When only radially symmetric solutions are taken into account, we also show that the monotonicity of g is not needed in this result. Finally, we consider the same problem posed in ℝN and show that if g is additionally convex and lies above a power greater than one at infinity, then all positive supersolutions u automatically verify − Δu > 0 in ℝN, and they do not exist when the previous condition fails.
- Published
- 2018
103. Continuous solutions and approximating scheme for fractional Dirichlet problems on Lipschitz domains
- Author
-
Erwin Topp and Patricio Felmer
- Subjects
Dirichlet problem ,General Mathematics ,Hölder condition ,Lipschitz continuity ,Equicontinuity ,01 natural sciences ,Dirichlet distribution ,010305 fluids & plasmas ,symbols.namesake ,Bounded function ,0103 physical sciences ,symbols ,Applied mathematics ,Ball (mathematics) ,Uniqueness ,010306 general physics ,Mathematics - Abstract
In this paper, we study the fractional Dirichlet problem with the homogeneous exterior data posed on a bounded domain with Lipschitz continuous boundary. Under an extra assumption on the domain, slightly weaker than the exterior ball condition, we are able to prove existence and uniqueness of solutions which are Hölder continuous on the boundary. In proving this result, we use appropriate barrier functions obtained by an approximation procedure based on a suitable family of zero-th order problems. This procedure, in turn, allows us to obtain an approximation scheme for the Dirichlet problem through an equicontinuous family of solutions of the approximating zero-th order problems on ${\bar \Omega}$. Both results are extended to an ample class of fully non-linear operators.
- Published
- 2018
104. On the well-posedness and asymptotic behaviour of the generalized Korteweg–de Vries–Burgers equation
- Author
-
Fernando A. Gallego and Ademir F. Pazoto
- Subjects
General Mathematics ,Mathematical analysis ,01 natural sciences ,010305 fluids & plasmas ,Burgers' equation ,Term (time) ,Multiplier (Fourier analysis) ,Compact space ,0103 physical sciences ,Exponent ,Exponential decay ,010306 general physics ,Real line ,Mathematics ,Interpolation theory - Abstract
In this paper we are concerned with the well-posedness and the exponential stabilization of the generalized Korteweg–de Vries–Burgers equation, posed on the whole real line, under the effect of a damping term. Both problems are investigated when the exponent p in the nonlinear term ranges over the interval [1, 5). We first prove the global well-posedness in Hs(ℝ) for 0 ≤ s ≤ 3 and 1 ≤ p < 2, and in H3(ℝ) when p ≥ 2. For 2 ≤ p < 5, we prove the existence of global solutions in the L2-setting. Then, by using multiplier techniques and interpolation theory, the exponential stabilization is obtained with an indefinite damping term and 1 ≤ p < 2. Under the effect of a localized damping term the result is obtained when 2 ≤ p < 5. Combining multiplier techniques and compactness arguments, we show that the problem of exponential decay is reduced to proving the unique continuation property of weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.
- Published
- 2018
105. On the log-concavity of the sequence for some combinatorial sequences
- Author
-
Ernest X. W. Xia
- Subjects
Combinatorics ,Sequence ,Conjecture ,Series (mathematics) ,Heuristic ,General Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Monotonic function ,Function (mathematics) ,GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries) ,Mathematics - Abstract
Recently, Sun posed a series of conjectures on the log-concavity of the sequence , where is a familiar combinatorial sequence of positive integers. Luca and Stănică, Hou et al. and Chen et al. proved some of Sun's conjectures. In this paper, we present a criterion on the log-concavity of the sequence . The criterion is based on the existence of a function f(n) that satisfies some inequalities involving terms related to the sequence . Furthermore, we present a heuristic approach to compute f(n). As applications, we prove that, for the Zagier numbers , the sequences are strictly log-concave, which confirms a conjecture of Sun. We also prove the log-concavity of the sequence of Cohen–Rhin numbers.
- Published
- 2018
106. Sharp decay estimates in a bioconvection model with quadratic degradation in bounded domains
- Author
-
Xinru Cao and Michael Winkler
- Subjects
Cauchy problem ,Solenoidal vector field ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Quadratic equation ,Bounded function ,Neumann boundary condition ,Fluid dynamics ,0101 mathematics ,Mathematics ,Degradation (telecommunications) - Abstract
The paper studies large time behaviour of solutions to the Keller–Segel system with quadratic degradation in a liquid environment, as given byunder Neumann boundary conditions in a bounded domain Ω ⊂ ℝn, where n ≥ 1 is arbitrary. It is shown that whenever U : Ω × (0,∞) → ℝn is a bounded and sufficiently regular solenoidal vector field any non-trivial global bounded solution of (⋆) approaches the trivial equilibrium at a rate that, with respect to the norm in either of the spaces L1(Ω) and L∞(Ω), can be controlled from above and below by appropriate multiples of 1/(t + 1). This underlines that, even up to this quantitative level of accuracy, the large time behaviour in (⋆) is essentially independent not only of the particular fluid flow, but also of any effect originating from chemotactic cross-diffusion. The latter is in contrast to the corresponding Cauchy problem, for which known results show that in the n = 2 case the presence of chemotaxis can significantly enhance biomixing by reducing the respective spatial L1 norms of solutions.
- Published
- 2018
107. Time-dependent attractors for non-autonomous non-local reaction–diffusion equations
- Author
-
Pedro Marín-Rubio, Marta Herrera-Cobos, and Tomás Caraballo
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Pullback attractor ,Non local ,01 natural sciences ,010101 applied mathematics ,Strong solutions ,Bounded function ,Reaction–diffusion system ,Attractor ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
In this paper the existence and uniqueness of weak and strong solutions for a non-autonomous non-local reaction–diffusion equation is proved. Furthermore, the existence of minimal pullback attractors in the L2-norm in the frameworks of universes of fixed bounded sets and those given by a tempered growth condition is established, along with some relationships between them. Finally, we prove the existence of minimal pullback attractors in the H1-norm and study relationships among these new families and those given previously in the L2 context. We also present new results in the autonomous framework that ensure the existence of global compact attractors as a particular case.
- Published
- 2018
108. New relaxation theorems with applications to strong materials
- Author
-
Jean-Philippe Mandallena and Mikhail Sychev
- Subjects
Conjecture ,Matching (graph theory) ,General Mathematics ,Open problem ,010102 general mathematics ,Relaxation (approximation) ,Statistical physics ,0101 mathematics ,Relaxation theory ,01 natural sciences ,Mathematics - Abstract
Recently, Sychev showed that conditions both necessary and sufficient for lower semicontinuity of integral functionals with p-coercive extended-valued integrands are the W1,p-quasi-convexity and the validity of a so-called matching condition (M). Condition (M) is so general that we conjecture whether it always holds in the case of continuous integrands. In this paper we develop the relaxation theory under the validity of condition (M). It turns out that a better relaxation theory is available in this case. This motivates our research since it is an important old open problem to develop the relaxation theory in the case of extended-value integrands. Then we discuss applications of the general relaxation theory to some concrete cases, in particular to the theory of strong materials.
- Published
- 2018
109. Deforming an є-close-to-hyperbolic metric to a hyperbolic metric
- Author
-
Pedro Ontaneda
- Subjects
Pure mathematics ,General Mathematics ,Metric (mathematics) ,Mathematics - Abstract
We show how to deform a metric of the form g = gr + dr2 to a metric = Hr + dr2, which is a hyperbolic metric for r less than some fixed λ, and coincides with g for r large. Here by hyperbolic metric we mean a metric of constant sectional curvature equal to -1. We study the extent to which is close to hyperbolic everywhere, if we assume g is close to hyperbolic. A precise definition of the close to hyperbolic concept is given. We also deal with a one-parameter version of this problem. The results in this paper are used in the problem of smoothing Charney–Davis strict hyperbolizations.
- Published
- 2018
110. On the finite-element approximation of ∞-harmonic functions
- Author
-
Tristan Pryer
- Subjects
Discretization ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Mathematical analysis ,Infinity ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Harmonic function ,Limit (mathematics) ,0101 mathematics ,Galerkin method ,Laplace operator ,media_common ,Mathematics - Abstract
In this paper we show that conforming Galerkin approximations for p-harmonic functions tend to ∞-harmonic functions in the limit p → ∞ and h → 0, where h denotes the Galerkin discretization parameter.
- Published
- 2018
111. Local Hölder estimates for non-uniformly variable exponent elliptic equations in divergence form
- Author
-
Fengping Yao
- Subjects
010101 applied mathematics ,Variable exponent ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,0101 mathematics ,Divergence (statistics) ,01 natural sciences ,Mathematics - Abstract
In this paper we obtain the local Hölder regularity of the gradients of weak solutions for a class of non-uniformly nonlinear variable exponent elliptic equations in divergence formincluding the following special modelunder some proper assumptions on Ai and the Hölder continuous functions f, pi(x) for i = 1, 2.
- Published
- 2017
112. Integrable zero-Hopf singularities and three-dimensional centres
- Author
-
Isaac A. García
- Subjects
Pure mathematics ,Integrable system ,General Mathematics ,010102 general mathematics ,Quadratic function ,Parameter space ,01 natural sciences ,010101 applied mathematics ,Singularity ,Phase space ,Gravitational singularity ,0101 mathematics ,Affine variety ,Mathematics ,Poincaré map - Abstract
In this paper we show that the well-known Poincaré–Lyapunov non-degenerate analytic centre problem in the plane and its higher-dimensional version, expressed as the three-dimensional centre problem at the zero-Hopf singularity, have a lot of common properties. In both cases the existence of a neighbourhood of the singularity in the phase space completely foliated by periodic orbits (including equilibria) is characterized by the fact that the system is analytically completely integrable. Hence its Poincaré–Dulac normal form is analytically orbitally linearizable. There also exists an analytic Poincaré return map and, when the system is polynomial and parametrized by its coefficients, the set of systems with centres corresponds to an affine variety in the parameter space of coefficients. Some quadratic polynomial families are considered.
- Published
- 2017
113. Partial inverse problems for Sturm–Liouville operators on trees
- Author
-
Chung-Tsun Shieh and Natalia P. Bondarenko
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,Inverse ,Sturm–Liouville theory ,Inverse problem ,Edge (geometry) ,01 natural sciences ,Tree (graph theory) ,Constructive ,010101 applied mathematics ,Graph (abstract data type) ,A priori and a posteriori ,0101 mathematics ,Mathematics - Abstract
In this paper, inverse spectral problems for Sturm–Liouville operators on a tree (a graph without cycles) are studied. We show that if the potential on an edge is known a priori, then b – 1 spectral sets uniquely determine the potential functions on a tree with b external edges. Constructive solutions, based on the method of spectral mappings, are provided for the considered inverse problems.
- Published
- 2017
114. Discreteness of spectrum for Schrödinger operators with δʹ-type conditions on infinite regular trees
- Author
-
Jia Zhao, Guoliang Shi, and Jun Yan
- Subjects
Computer Science::Information Retrieval ,General Mathematics ,Spectral properties ,Spectrum (functional analysis) ,Type (model theory) ,Topology ,Spectral line ,Discrete spectrum ,symbols.namesake ,Operator (computer programming) ,symbols ,Schrödinger's cat ,Mathematical physics ,Mathematics - Abstract
This paper deals with the spectral properties of self-adjoint Schrödinger operators with δʹ-type conditions on infinite regular trees. Firstly, we discuss the semi-boundedness and self-adjointness of this kind of Schrödinger operator. Secondly, by using the form approach, we give the necessary and sufficient condition that ensures that the spectra of the self-adjoint Schrödinger operators with δʹ-type conditions are discrete.
- Published
- 2017
115. Some sharp results about the global existence and blowup of solutions to a class of pseudo-parabolic equations
- Author
-
Fuyi Li, Yuhua Li, and Xiaoli Zhu
- Subjects
Class (set theory) ,Semigroup ,Computer Science::Information Retrieval ,General Mathematics ,Weak solution ,010102 general mathematics ,Mathematical analysis ,Space (mathematics) ,01 natural sciences ,Parabolic partial differential equation ,010101 applied mathematics ,0101 mathematics ,Exponential decay ,Nehari manifold ,Energy functional ,Mathematics - Abstract
In this paper we are interested in a sharp result about the global existence and blowup of solutions to a class of pseudo-parabolic equations. First, we represent a unique local weak solution in a new integral form that does not depend on any semigroup. Second, with the help of the Nehari manifold related to the stationary equation, we separate the whole space into two components S+ and S– via a new method, then a sufficient and necessary condition under which the weak solution blows up is established, that is, a weak solution blows up at a finite time if and only if the initial data belongs to S–. Furthermore, we study the decay behaviour of both the solution and the energy functional, and the decay ratios are given specifically.
- Published
- 2017
116. Morse index and symmetry breaking for an elliptic equation with negative exponent in expanding annuli
- Author
-
Linfeng Mei, Zhitao Zhang, and Zongming Guo
- Subjects
Elliptic curve ,Index (economics) ,Negative exponent ,law ,General Mathematics ,Mathematical analysis ,Symmetry breaking ,Morse code ,Mathematics ,law.invention - Abstract
Bifurcation of non-radial solutions from radial solutions of a semilinear elliptic equation with negative exponent in expanding annuli of ℝ2 is studied. To obtain the main results, we use a blow-up argument via the Morse index of the regular entire solutions of the equationThe main results of this paper can be seen as applications of the results obtained recently for finite Morse index solutions of the equationwith N ⩾ 2 and p > 0.
- Published
- 2017
117. Geometric aspects of self-adjoint Sturm–Liouville problems
- Author
-
Yicao Wang
- Subjects
Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,Orbit (dynamics) ,Boundary (topology) ,Sturm–Liouville theory ,Boundary value problem ,Diffeomorphism ,Mathematics::Spectral Theory ,Principal type ,Self-adjoint operator ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we use U(2), the group of 2 × 2 unitary matrices, to parametrize the space of all self-adjoint boundary conditions for a fixed Sturm–Liouville equation on the interval [0, 1]. The adjoint action of U(2) on itself naturally leads to a refined classification of self-adjoint boundary conditions – each adjoint orbit is a subclass of these boundary conditions. We give explicit parametrizations of those adjoint orbits of principal type, i.e. orbits diffeomorphic to the 2-sphere S2, and investigate the behaviour of the nth eigenvalue λnas a function on such orbits.
- Published
- 2017
118. On the Hardy–Sobolev equation
- Author
-
Francesca Gladiali, E. N. Dancer, and Massimo Grossi
- Subjects
010101 applied mathematics ,Sobolev space ,Pure mathematics ,Singularity ,Bifurcation theory ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Bifurcation ,Mathematics ,Sobolev inequality - Abstract
In this paper we study the problemwhere Ω = ℝN or Ω = B1, N ⩾ 3, p > 1 and . Using a suitable map we transform problem (1) into another one without the singularity 1/|x|2. Then we obtain some bifurcation results from the radial solutions corresponding to some explicit values of λ.
- Published
- 2017
119. Asymptotic behaviour of the lifespan of solutions for a semilinear heat equation in hyperbolic space
- Author
-
Jingxue Yin and Zhiyong Wang
- Subjects
010101 applied mathematics ,General Mathematics ,Hyperbolic space ,010102 general mathematics ,Mathematical analysis ,Geodetic datum ,Heat equation ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
This paper is concerned with the asymptotic behaviour of the lifespan of solutions for a semilinear heat equation with initial datum λφ(x) in hyperbolic space. The growth rates for both λ → 0 and λ → ∞ are determined.
- Published
- 2016
120. Aspects of Hadamard well-posedness for classes of non-Lipschitz semilinear parabolic partial differential equations
- Author
-
D. J. Needham and J. C. Meyer
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Hölder condition ,Lipschitz continuity ,01 natural sciences ,Parabolic partial differential equation ,010101 applied mathematics ,Stochastic partial differential equation ,Elliptic partial differential equation ,Initial value problem ,Uniqueness ,0101 mathematics ,Mathematics ,Numerical partial differential equations - Abstract
We study classical solutions of the Cauchy problem for a class of non-Lipschitz semilinear parabolic partial differential equations in one spatial dimension with sufficiently smooth initial data. When the nonlinearity is Lipschitz continuous, results concerning existence, uniqueness and continuous dependence on initial data are well established (see, for example, the texts of Friedman and Smoller and, in the context of the present paper, see also Meyer), as are the associated results concerning Hadamard well-posedness. We consider the situations when the nonlinearity is Hölder continuous and when the nonlinearity is upper Lipschitz continuous. Finally, we consider the situation when the nonlinearity is both Hölder continuous and upper Lipschitz continuous. In each case we focus upon the question of existence, uniqueness and continuous dependence on initial data, and thus upon aspects of Hadamard well-posedness.
- Published
- 2016
121. Whitney regularity of the image of the Chevalley mapping
- Author
-
Gérard P. Barbançon
- Subjects
Mathematics - Classical Analysis and ODEs ,business.industry ,General Mathematics ,Image (category theory) ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Pattern recognition ,Artificial intelligence ,business ,Mathematics - Abstract
A compact set K ⊂ ℝn is Whitney 1-regular if the geodesic distance in K is equivalent to the Euclidean distance. Let P be the Chevalley map defined by an integrity basis of the algebra of polynomials invariant by a reflection group. This paper gives the Whitney 1-regularity of the image by P of any closed ball centred at the origin. The proof uses the works of Givental', Kostov and Arnol'd on the symmetric group. It needs a generalization of a property of the Vandermonde determinants to the Jacobian of the Chevalley mappings.
- Published
- 2016
122. A convergence result for the ergodic problem for Hamilton–Jacobi equations with Neumann-type boundary conditions
- Author
-
Hitoshi Ishii, Ebraheem O. Alzahrani, Arshad M. M. Younas, and Eman S. Al-Aidarous
- Subjects
Asymptotic analysis ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Function (mathematics) ,01 natural sciences ,Hamilton–Jacobi equation ,010101 applied mathematics ,Kolmogorov equations (Markov jump process) ,Ergodic theory ,Applied mathematics ,Convergence problem ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
We consider the ergodic (or additive eigenvalue) problem for the Neumann-type boundary-value problem for Hamilton–Jacobi equations and the corresponding discounted problems. Denoting by uλ the solution of the discounted problem with discount factor λ > 0, we establish the convergence of the whole family to a solution of the ergodic problem as λ → 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting by-product, we introduce Mather measures associated with Hamilton–Jacobi equations with the Neumann-type boundary conditions. These results are variants of the main results in a recent paper by Davini et al., who study the same convergence problem on smooth compact manifolds without boundary.
- Published
- 2016
123. The existence of a ground-state solution for a class of Kirchhoff-type equations in ℝN
- Author
-
Chun-Lei Tang, Jia-Feng Liao, and Jiu Liu
- Subjects
010101 applied mathematics ,Pure mathematics ,Class (set theory) ,Kirchhoff type ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Ground state ,Nehari manifold ,01 natural sciences ,Mathematics - Abstract
In this paper, we study the following Kirchhoff-type equation: where a, b are positive constants and N = 1, 2, 3. Under appropriate assumptions on V, K and g, we obtain a ground-state solution by using the approach developed by Szulkin and Weth in 2010.
- Published
- 2016
124. Dynamics in the fundamental solution of a non-convex conservation law
- Author
-
Young-Ran Lee and Yong-Jung Kim
- Subjects
Conservation law ,Astrophysics::High Energy Astrophysical Phenomena ,General Mathematics ,010102 general mathematics ,Scalar (mathematics) ,Mathematical analysis ,Regular polygon ,01 natural sciences ,Convexity ,010101 applied mathematics ,Classical mechanics ,Inflection point ,Jump ,Fundamental solution ,0101 mathematics ,Mathematics - Abstract
There is a huge jump in the theory of conservation laws if the convexity assumption is dropped. In this paper we study a scalar conservation law without the convexity assumption by monitoring the dynamics in the fundamental solution. Three extra shock types are introduced other than the usual genuine shock, which are left, right and double sided contacts. There are three kinds of phenomena of these shocks, which are called branching, merging and transforming. All of these shocks and phenomena can be observed if the flux function has two inflection points. A comprehensive picture of a global dynamics of a nonconvex flux is discussed in terms of characteristic maps and dynamical convex-concave envelopes.
- Published
- 2015
125. An eigenvalue optimization problem for the p-Laplacian
- Author
-
Anisa M. H. Chorwadwala and Rajesh Mahadevan
- Subjects
Combinatorics ,Nonlinear system ,General Mathematics ,p-Laplacian ,Ball (bearing) ,Shape optimization ,Monotonic function ,Uniqueness ,Laplace operator ,Eigenvalues and eigenvectors ,Mathematics - Abstract
It has been shown by Kesavan (Proc. R. Soc. Edinb. A (133) (2003), 617–624) that the first eigenvalue for the Dirichlet Laplacian in a punctured ball, with the puncture having the shape of a ball, is maximum if and only if the balls are concentric. Recently, Emamizadeh and Zivari-Rezapour (Proc. Am. Math. Soc.136 (2007), 1325–1331) have tried to generalize this result to the case of the p-Laplacian but could succeed only in proving a domain monotonicity result for a weighted eigenvalue problem in which the weights need to satisfy some artificial conditions. In this paper we generalize the result of Kesavan to the case of the p-Laplacian (1 < p < ∞) without any artificial restrictions, and in the process we simplify greatly the proof, even in the case of the Laplacian. The uniqueness of the maximizing domain in the nonlinear case is still an open question.
- Published
- 2015
126. A theory of fractional integration for generalised functions II
- Author
-
Adam McBride
- Subjects
General Mathematics ,Applied mathematics ,Fractional calculus ,Mathematics - Abstract
SynopsisIn a previous paper [2], a theory of fractional integration was developed for certain spaces Fp,μ of generalised functions. In this paper we extend this theory by relaxing some of the restrictions on the various parameters involved. In particular we show how a generalised Erdelyi-Kober operator can be defined on Fʹp,μ for 1 ≦ p ≦ ∞ and for all complex numbers μ except for those lying on a countable number of lines of the form Re μ = constant in the complex μ-plane. Mapping properties of these generalised operators are obtained and several applications mentioned.
- Published
- 1977
127. A radical for near-rings
- Author
-
J. F. T. Hartney
- Subjects
Combinatorics ,Identity (mathematics) ,Intersection ,Distributive property ,General Mathematics ,Zero (complex analysis) ,Type (model theory) ,Mathematics - Abstract
SynopsisThroughout this paper the near-ring N is assumed to be zero symmetric and to satisfy the right distributive law. That is, x · 0 = 0 and (x + y)z = xz + yz for all x, y, z ∈ N. In what follows we generalise the notion of s-primitivity first introduced in an earlier paper by the author (1968), where only distributively generated (d.g.) near-rings with identity were considered. We define a Jacobson type radical Js (N) and show that J1(N)⊇Js(N) ⊇ Q(N), where Q(N) is the intersection of all 0-modular left ideals of N (Pilz). In addition we settle some of the problems remaining from Hartney (1968).
- Published
- 1982
128. A note on linear ordinary quasi-differential equations
- Author
-
W. N. Everitt
- Subjects
Differential equation ,General Mathematics ,Mathematical analysis ,First-order partial differential equation ,Linear equation ,Mathematics - Abstract
SynopsisThe theory of differential equations is largely concerned with properties of solutions of individual, or classes of, equations. This paper is given over to the converse problem - that of seeking properties of functions which require them to be, in some respect, solutions of a differential equation, and to determining all possible such differential equations.From this point of view this paper discusses only linear ordinary quasi-differential equations of the second order. However, the methods can be extended to quasi-differential equations of general order.
- Published
- 1985
129. Polynomial interpolation at points of a geometric mesh on a triangle
- Author
-
S. L. Lee and George M. Phillips
- Subjects
Discrete mathematics ,Inverse quadratic interpolation ,General Mathematics ,Mathematical analysis ,Bilinear interpolation ,Linear interpolation ,Birkhoff interpolation ,Polynomial interpolation ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Bicubic interpolation ,Spline interpolation ,ComputingMethodologies_COMPUTERGRAPHICS ,Mathematics ,Interpolation - Abstract
SynopsisIn an earlier paper [8], I. J. Schoenberg discussed polynomial interpolation in one dimension at the points of a geometric progression, which was originally proposed by James Stirling. In the present paper, these ideas are generalised to two-dimensional polynomial interpolation at the points of a geometric mesh on a triangle. A Lagrange form is obtained for this interpolating polynomial and an algorithm is derived for evaluating it efficiently.
- Published
- 1988
130. Pointwise convergence of eigenfunction expansions, associated with a pair of ordinary differential expressions
- Author
-
Hsv Desnoo, Ea Coddington, and Aad Dijksma
- Subjects
Pointwise convergence ,General Mathematics ,Uniform convergence ,Mathematical analysis ,Eigenfunction ,Modes of convergence ,Differential (mathematics) ,Compact convergence ,Mathematics - Abstract
SynopsisFor the differential equation Lf = λMf on an open interval of ℝ, a theory in terms of relations in a Hilbert space associated with M was developed in a paper by Coddington and de Snoo, and eigenfunction expansions were derived in a paper by Dijksma and de Snoo. In the case of a regular problem on a compact interval, pointwise convergence of the expansions was shown in another paper by Coddington and de Snoo. Here, we show pointwise convergence in the general singular case.
- Published
- 1984
131. Inverse semigroups generated by a pair of subgroups
- Author
-
D. B. McAlister
- Subjects
Pure mathematics ,Inverse semigroup ,Section (category theory) ,Free product ,Semigroup ,Group (mathematics) ,General Mathematics ,Category of groups ,Semilattice ,Cyclic group ,Mathematics - Abstract
SynopsisThe aim of this paper is to describe the free product of a pair G, H of groups in the category of inverse semigroups. Since any inverse semigroup generated by G and H is a homomorphic image of this semigroup, this paper can be regarded as asking how large a subcategory, of the category of inverse semigroups, is the category of groups? In this light, we show that every countable inverse semigroup is a homomorphic image of an inverse subsemigroup of the free product of two copies of the infinite cyclic group. A similar result can be obtained for arbitrary cardinalities. Hence, the category of inverse semigroups is generated, using algebraic constructions by the subcategory of groups.The main part of the paper is concerned with obtaining the structure of the free product G inv H, of two groups G, H in the category of inverse semigroups. It is shown in section 1 that G inv H is E-unitary; thus G inv H can be described in terms of its maximum group homomorphic image G gp H, the free product of G and H in the category of groups, and its semilattice of idempotents. The second section considers some properties of the semilattice of idempotents while the third applies these to obtain a representation of G inv H which is faithful except when one group is a non-trivial finite group and the other is trivial. This representation is used in section 4 to give a structure theorem for G inv H. In this section, too, the result described in the first paragraph is proved. The last section, section 5, consists of examples.
- Published
- 1977
132. 22.—A Critical Class of Examples concerning the Integrable-square Classification of Ordinary Differential Equations
- Author
-
W. N. Everitt and M. Giertz
- Subjects
Examples of differential equations ,Pure mathematics ,Class (set theory) ,Integrable system ,General Mathematics ,Ordinary differential equation ,Square (algebra) ,Integrating factor ,Mathematics - Abstract
SynopsisLet the coefficient q be real-valued on the half-line [0, ∞) and let q′ be locally absolutely continuous on [0, ∞). The ordinary symmetric differential expressions M and M2 are determined byIt has been shown in a previous paper by the authors that if for non-negative numbers k and X the coefficient q satisfies the conditionthen M is limit-point and M2 is limit–2 at ∞.This paper is concerned with showing that for powers of the independent variable x the condition (*) is best possible in order that both M and M2 should have the classification at ∞ given above.
- Published
- 1976
133. Some properties of periodic B-spline collocation matrices
- Author
-
C. A. Micchelli, S. L. Lee, A. Sharma, and P. W. Smith
- Subjects
Collocation ,General Mathematics ,Collocation method ,B-spline ,Applied mathematics ,Orthogonal collocation ,Mathematics - Abstract
SynopsisIn three recent papers by Cavaretta et al., progress has been made in understanding the structure of bi-infinite totally positive matrices which have a block Toeplitz structure. The motivation for these papers came from certain problems of infinite spline interpolation where total positivity played an important role.In this paper, we re-examine a class of infinite spline interpolation problems. We derive new results concerning the associated infinite matrices (periodic B-spline collocation matrices) which go beyond consequences of the general theory. Among other things, we identify the dimension of the null space of these matrices as the width of the largest band of strictly positive elements.
- Published
- 1983
134. A note on a resonance problem
- Author
-
Sergio Solimini and Daniela Lupo
- Subjects
Partial differential equation ,Real-valued function ,Simple (abstract algebra) ,General Mathematics ,Calculus ,Zero (complex analysis) ,Applied mathematics ,Existence theorem ,Boundary value problem ,Resonance (particle physics) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
SynopsisIn this paper, we prove the existence of at least one solution to the problemwhere ∆k is an eigenvalue of the linear part, h is orthogonal to the eigenspace corresponding to ∆R and g is a nonlinear perturbation which can be, for instance, a continuous periodic real function with mean value zero. We employ the techniques used by the second author in a previous paper in which the same result was obtained in the case in which ∆R is assumed to be simple. The final result is obtained by using variational methods and in particular a suitable version of the saddle point theorem of P. Rabinowitz.
- Published
- 1986
135. Remark on Hilbert's boundary value problem for Beltrami systems
- Author
-
Heinrich Begehr
- Subjects
Elliptic curve ,Pure mathematics ,Partial differential equation ,Computer Science::Information Retrieval ,General Mathematics ,Mathematical analysis ,Boundary value problem ,Mathematics - Abstract
SynopsisThe Schauder continuation method for nonlinear problems is based on appropriate a priori estimates for related linear equations. Recently, in a paper by the present author and G. C. Hsiao, the Hilbert boundary value problem with positive index for nonlinear elliptic systems in the plane was solved by this method but the constructive derivation of the a priori estimate necessarily required a restriction on the ellipticity condition. This is because the norm of the generalized Hilbert transform in the case of positive index is too big. Here, as in a forthcoming paper by G.C. Wen, an indirect and therefore non-constructive proof of the a priori estimate is given which does not require any further restrictions and allows the Hilbert boundary value problem to be solved for nonlinear elliptic systems in general.
- Published
- 1984
136. 4.—A Cauchy Problem for an Ordinary Integro-differential Equation
- Author
-
E. A. Catchpole
- Subjects
Cauchy problem ,Elliptic partial differential equation ,Integro-differential equation ,General Mathematics ,Riccati equation ,Exact differential equation ,Applied mathematics ,Cauchy boundary condition ,Hyperbolic partial differential equation ,Cauchy matrix ,Mathematics - Abstract
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.
- Published
- 1974
137. On the Vitali–Hahn–Saks theorem
- Author
-
Aníbal Moltó
- Subjects
Pure mathematics ,Picard–Lindelöf theorem ,General Mathematics ,Compactness theorem ,Fixed-point theorem ,Vitali–Hahn–Saks theorem ,Danskin's theorem ,Brouwer fixed-point theorem ,Squeeze theorem ,Mathematics ,Carlson's theorem - Abstract
SynopsisIn this paper, a class of Boolean rings containing the class discussed in papers by Seever (1968) and Faires (1976), is defined in such a way that an extension of the classical Vitali–Hahn–Saks theorem holds for exhausting additive set functions. Some new compact topological spaces K for which C(K) is a Grothendieck space are constructed and a Nikodym type theorem is deduced from it. The Boolean algebras of Seever and Faires and those we study here are defined by ‘interpolation properties’ between disjoint sequences in the algebra. We give an example at the end of the paper that illustrates the difficulties arising when we try to find a larger class of Boolean algebras, defined in terms of such properties, for which the Vitali–Hanh–Saks theorem holds.
- Published
- 1981
138. 8.—Bifurcation and Asymptotic Bifurcation for Non-compact Nonsymmetric Gradient Operators
- Author
-
John Toland
- Subjects
symbols.namesake ,Pure mathematics ,Operator (computer programming) ,Compact space ,Transcritical bifurcation ,General Mathematics ,Hilbert space ,symbols ,Saddle-node bifurcation ,Bifurcation diagram ,Lipschitz continuity ,Bifurcation ,Mathematics - Abstract
SynopsisThe first part of this paper is devoted to a study of the classical bifurcation problem in a Hilbert space, under the assumption that the operators involved are gradient operators, but not necessarily compact. Our approach to the problem was introduced by Krasnosel'skii, but here we show that his assumption about the compactness of the operators can be replaced by a much weaker Lipschitz type condition, without affecting the generality of his conclusions.The rest of the paper is concerned with the analogous problem when the operator is knownto be asymptotically linear rather than Fréchet differentiable. Indeed, we show that this question can always be reduced to the first case, after some manipulation. After this manipulation the new operator is found to be a Fréchet differentiable gradient operator, and so we can invoke the results of the first part. This manipulation is in the spirit of that of [11] but is necessarily different.
- Published
- 1975
139. On the exponential behaviour of eigenfunctions and the essential spectrum of differential operators
- Author
-
F. V. Atkinson and W. D. Evans
- Subjects
Differential equation ,General Mathematics ,Mathematical analysis ,Essential spectrum ,Eigenfunction ,Operator theory ,Type (model theory) ,Differential operator ,Fourier integral operator ,Exponential function ,Mathematics ,Mathematical physics - Abstract
SynopsisThe paper deals with the differential equationon [ 0, ∞) Where λ>0 and the coefficients qm are complex-valued with qn continuous and non-zero, w is positive and continuous and qm for m = 0, 1,…, n − 1. In the first part of the paper the exponential behaviour of any solution of (*) is given in terms of a function ρ(λ) which is roughly the distance of λ from the essential spectrum of a closed, densely denned linear operator T generated by T+ in L2(0, ∞ w). Next, estimates are obtained for the solutions in terms of the coefficients in (*). When the latter results are compared with the estimates established previously in terms of ρ(λ), bounds for ρ(λ) are obtained. From the general result there are two kinds of consequences. In the first, criteria for ρ(λ) = 0 for all All λ > 0 are obtained; this means that [0, ∞) lies in the essential spectrum of T in appropriate circumstances. The second type of consequence concerns bounds of the form ρ(λ) = O(λr) for λ → ∞ and r
- Published
- 1982
140. The Hankel transform of some classes of generalized functions and connections with fractional integration
- Author
-
Adam McBride
- Subjects
Pure mathematics ,Hankel transform ,Generalized function ,Kontorovich–Lebedev transform ,General Mathematics ,Mathematical analysis ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Countable set ,Constant (mathematics) ,Complex number ,Hankel matrix ,Fractional calculus ,Mathematics - Abstract
SynopsisIn previous papers [11,12,13], certain spaces of generalized functions were studied from the point of view of fractional calculus. In this paper, we show how a Hankel transform Hv of order v can be defined on for all complex numbers v except for those lying on a countable number of lines of the form Re v = constant in the complex v-plane. The mapping properties of Hv on are obtained. Various connections between Hv (or modifications of Hv) and operators of fractional integration are examined.
- Published
- 1978
141. A fixed point theorem for α-condensing maps on a sphere
- Author
-
Paul Massatt
- Subjects
Inversion in a sphere ,Unit sphere ,Schauder fixed point theorem ,General Mathematics ,Mathematical analysis ,Fixed-point theorem ,Fixed point ,Kakutani fixed-point theorem ,Brouwer fixed-point theorem ,Fixed-point property ,Mathematics - Abstract
SynopsisThis paper shows that ifSis a sphere in a Banach space andf:S→Sis an α-contraction, thenfhas a fixed point. The paper generalizes a result of R. D. Nussbaum which holds for α-contractions only. The proof uses the Browder nonrepulsive fixed point theorem and is motivated by recent work of M. Martelli and G. D. Cooperman.
- Published
- 1983
142. 24.—Mean-square Convergence of Non-harmonic Trigonometrical Series
- Author
-
J. Cossar
- Subjects
Almost periodic function ,Periodic function ,Pure mathematics ,Series (mathematics) ,General Mathematics ,Harmonic (mathematics) ,Function (mathematics) ,Fourier series ,Square (algebra) ,Real number ,Mathematics - Abstract
SynopsisThe series considered are of the form , where Σ | cn |2 is convergent and the real numbers λn (the exponents) are distinct. It is known that if the exponents are integers, the series is the Fourier series of a periodic function of locally integrable square (the Riesz-Fischer theorem); and more generally that if the exponents are not necessarily integers but are such that the difference between any pair exceeds a fixed positive number, the series is the Fourier series of a function of the Stepanov class, S2, of almost periodic functions.We consider in this paper cases where the exponents are subject to less stringent conditions (depending on the coefficients cn). Some of the theorems included here are known but had been proved by other methods. A fuller account of the contents of the paper is given in Sections 1-5.
- Published
- 1976
143. Global regularity and formation of singularities of solutions to first order quasilinear hyperbolic systems
- Author
-
Lee Da-Tsin
- Subjects
General Mathematics ,Mathematical analysis ,Initial value problem ,Gravitational singularity ,Composition (combinatorics) ,Finite time ,First order ,Hyperbolic systems ,Eigenvalues and eigenvectors ,Mathematics - Abstract
SynopsisFor the Cauchy problem for strictly hyperbolic systems with general eigenvalues, we obtain existence of global smooth solutions under certain conditions on the composition of the eigenvalues and the initial data; on the other hand, we give a sufficient condition which guarantees that singularities of the solution must occur in a finite time and describe certain applications. The present paper includes the corresponding results in earlier papers by several authors as special cases.
- Published
- 1981
144. Entropy numbers of embedding maps between Besov spaces with an application to eigenvalue problems
- Author
-
Bernd Carl
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,General Mathematics ,Mathematical analysis ,Besov space ,Embedding ,Eigenvalues and eigenvectors ,Mathematics - Abstract
SynopsisIn this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.The statements of this paper are obtained by results recently proved elsewhere by the author.
- Published
- 1981
145. On a fourth-order singular integral inequality
- Author
-
Alexander Russell
- Subjects
symbols.namesake ,General Mathematics ,Mathematical analysis ,Improper integral ,Line integral ,symbols ,Riemann integral ,Riemann–Stieltjes integral ,Daniell integral ,Singular integral ,Integral equation ,Fourier integral operator ,Mathematics - Abstract
SynopsisThe inequality considered in this paper iswhereNis the real-valued symmetric differential expression defined byGeneral properties of this inequality are considered which result in giving an alternative account of a previously considered inequalityto which (*) reduces in the casep=q= 0,r= 1.Inequality (*) is also an extension of the inequalityas given by Hardy and Littlewood in 1932. This last inequality has been extended by Everitt to second-order differential expressions and the methods in this paper extend it to fourth-order differential expressions. As with many studies of symmetric differential expressions the jump from the second-order to the fourth-order introduces difficulties beyond the extension of technicalities: problems of a new order appear for which complete solutions are not available.
- Published
- 1978
146. 5.—Semi-bounded Dirichlet Integrals and the Invariance of the Essential Spectra of Self-adjoint Operators
- Author
-
William Desmond Evans
- Subjects
Pure mathematics ,General Mathematics ,Mathematical analysis ,Dirichlet L-function ,Dirichlet distribution ,Spectral line ,Dirichlet integral ,symbols.namesake ,Dirichlet's principle ,Bounded function ,symbols ,Self-adjoint operator ,Dirichlet series ,Mathematics - Abstract
SynopsisIn the first part of the paper a criterion is given for two self-adjoint operators T, S in a Hilbert space to have the same essential spectrum, S being given in terms of T and a perturbation P. If P is a symmetric operator and the operator sum T+P is self-adjoint, then S = T+P. Otherwise, T is assumed to be semi-bounded and S is taken to be the form extension of T+P defined in terms of semi-bounded sesquilinear forms. In the case when S = T+P, the result obtained generalises the results of Schechter, and Gustafson and Weidmann for Tm- compact (m> 1) perturbations of T. In the second part of the paper a detailed study is made of the Dirichlet integralassociated with the general second-order (degenerate) elliptic differential expression in a domain Conditions under which t is closed and bounded below are established, the most significant feature of the results being that the restriction of q to suitable subsets of Ω can have large negative singularities on the boundary of Ω and at infinity. Lastly some examples are given to illustrate the abstract theory.
- Published
- 1976
147. Homoclinic and heteroclinic orbits of reversible vectorfields under perturbation
- Author
-
Richard C. Churchill and David L. Rod
- Subjects
Nonlinear Sciences::Chaotic Dynamics ,Mathematics::Dynamical Systems ,Classical mechanics ,General Mathematics ,Perturbation (astronomy) ,Homoclinic orbit ,Mathematics ,Mathematical physics - Abstract
SynopsisAveraging techniques on Hamiltonian dynamical systems can often be used to establish the existence of hyperbolic periodic orbits. In equilibrium situations, it is then often difficult to show that there are homoclinic/heteroclinic connections between these hyperbolic orbits in the original unaveraged system. This existence problem is solved in this paper for a class of Hamiltonian systems admitting a sufficient number of symmetries (including reversing symmetries). Under isoenergetic reduction, the problem is reduced to one involving reversible vector fields under time-dependent perturbations admitting the same reversing symmetries. Applications are made to the one-parameter Hénon-Heiles family. The paper concludes with remarks on the problem of showing transversality of these homoclinic/heteroclinic orbits.
- Published
- 1986
148. Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions
- Author
-
Charles T. Fulton
- Subjects
Inverse iteration ,Constant coefficients ,General Mathematics ,Bounded function ,Mathematical analysis ,Spectrum (functional analysis) ,Boundary value problem ,Divide-and-conquer eigenvalue algorithm ,Eigenvalue perturbation ,Eigenvalues and eigenvectors ,Mathematics - Abstract
SynopsisIn this paper I extend the analysis of regular problems containing the eigenvalue parameter in the boundary conditions given by Walter (1973) and myself (1977) to singular problems which involve the eigenvalue parameter linearly in a regular or a limit-circle boundary condition at the left endpoint. The formulation of the limit-circle boundary conditions follows that given in another paper by the present author in 1977, and has the advantage that a λ-dependent boundary condition at a regular endpoint becomes a special case of a λ-dependent boundary condition at a limit-circle endpoint. The simplicity of the spectrum is also built into the formulation given, and the spectral function is shown to have bounded total variation over (−∞, ∞) which is known in terms of the parameters of the λ-dependent boundary condition independently of the limit-circle/limit-point classification at the right endpoint. The theory is applied to the constant coefficient equation in [0, ∞) and the Bessel equation of order zero in (0, ∞), explicit formulae for the spectral function being obtained in each case. Finally, the question is posed as to whether the classical Weyl theory for problems not involving λ in the boundary conditions can also be formulated so as to involve spectral functions having bounded total variation.
- Published
- 1980
149. 23.—The Linear Transport Equation. The Degenerate Case c = 1. II. Half-range Theory
- Author
-
C. G. Lekkerkerker
- Subjects
Range (mathematics) ,General Mathematics ,Operator (physics) ,Mathematical analysis ,Stellar atmosphere ,Characteristic equation ,Spectral theorem ,Space (mathematics) ,Hermitian matrix ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The aim of this paper is to give a functional analytic treatment of the homogeneous and inhomogeneous linear transport equation in the case that the parameter c occurring in that equation equals 1. The larger part of the paper is devoted to the study of a certain operator T −1 A in the space L 2 (– 1, 1). A peculiarity not arising in the case c T −1 A has a double eigenvalue 0 and that it is no longer hermitian. The Spectral Theorem is used to diagonalise the operator as far as possible, and full-range and half-range formulae are derived. The results are applied inter alia to give a new treatment of the Milne problem concerning the propagation of light in a stellar atmosphere.
- Published
- 1976
150. Non-linear functional differential equations and abstract integral equations
- Author
-
W. Schappacher and F. Kappel
- Subjects
Examples of differential equations ,Stochastic partial differential equation ,Method of characteristics ,Independent equation ,Differential equation ,Computer Science::Information Retrieval ,General Mathematics ,Ordinary differential equation ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Mathematical analysis ,Integral equation ,Differential algebraic equation ,Mathematics - Abstract
SynopsisThe equivalence between solutions of functional differential equations and an abstract integral equation is investigated. Using this result we derive a general approximation result in the state space C and consider as an example approximation by first order spline functions. During the last twenty years C1-semigroups of linear transformations have played an important role in the theory of linear autonomous functional differential equations (cf. for instance the discussion in [9, Section 7.7]). Applications of non-linear semigroup theory to functional differential equations are rather recent beginning with a paper by Webb [17]. Since then a considerable number of papers deal with problems in this direction. A common feature of the majority of these papers is that as a first step with the functional differential equation there is associated a non-linear operator A in a suitable Banach-space. Then appropriate conditions are imposed on the problem such that the conditions of the Crandall-Liggett-Theorem [5] hold for the operator A. This gives a non-linear semigroup. Finally the connection of this semigroup tothe solutions of the original differential equation has to be investigated [c.f. 8, 15, 18]. To solve thislast problem in general is the most difficult part of this approach.In the present paper we consider the given functional differential equation as a perturbation of the simple equationx = 0. The solutions of this equation generate a very simple C1-semigroup. The solutions of the original functional differential equation generate solutions of an integral equation which is the variation of constants formula for the abstract Cauchy problem associated with the equation x = 0. Under very mild conditions we can prove a one-to-one correspondence between solutions of the given functional differential equation and solutions of the integral equation in the Lp-space setting. In the C-space setting the integral equation inthe state space has to be replaced by a ‘pointwise’ integral equation. Using the pointwise integral equation together with a theorem which guarantees continuous dependence of fixed points on parameters we show under rather weak hypotheses that the original functional differential equation can be approximated by a sequence of ordinary differential equations. Using 1st order spline functions we finally get results which are very similar to those obtained in [1 and 11] in the L2-space setting.
- Published
- 1979
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.