scholarly journals Semiconservative Systems of Integral Equations with Two Kernels

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
N. B. Yengibaryan ◽  
A. G. Barseghyan
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Positive Solutions ◽  
Singular Case ◽  
Vector Integral ◽  
The Matrix ◽  
Systems Of Integral Equations ◽  
Nonhomogeneous Equation ◽  
Homogeneous Vector

The solvability and the properties of solutions of nonhomogeneous and homogeneous vector integral equation , where , are matrix valued functions, , with nonnegative integrable elements, are considered in one semiconservative (singular) case, where the matrix is stochastic one and the matrix is substochastic one. It is shown that in certain conditions the nonhomogeneous equation simultaneously with the corresponding homogeneous one possesses positive solutions.

scholarly journals Implicit Vector Integral Equations Associated with Discontinuous Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Paolo Cubiotti ◽  
Jen-Chih Yao
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Existence Result ◽  
Scalar Case ◽  
Selection Theorem ◽  
Vector Integral ◽  
Almost Everywhere ◽  
Null Measure ◽  
Recent Selection

LetI∶=[0,1]. We consider the vector integral equationh(u(t))=ft,∫Ig(t,z),u(z),dzfor a.e.t∈I,wheref:I×J→R, g:I×I→ [0,+∞[,andh:X→Rare given functions andX,Jare suitable subsets ofRn. We prove an existence result for solutionsu∈Ls(I, Rn), where the continuity offwith respect to the second variable is not assumed. More precisely,fis assumed to be a.e. equal (with respect to second variable) to a functionf*:I×J→Rwhich is almost everywhere continuous, where the involved null-measure sets should have a suitable geometry. It is easily seen that such a functionfcan be discontinuous at each pointx∈J. Our result, based on a very recent selection theorem, extends a previous result, valid for scalar casen=1.

Covariant integral equations for Heisenberg operators

1954 ◽  
Vol 50 (4) ◽  
pp. 592-603
Author(s):  
R. J. Eden
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Direct Product ◽  
Bound State ◽  
State Theory ◽  
Coupling Constant ◽  
Symbolic Form ◽  
The Matrix ◽  
Heisenberg Operators ◽  
Approximate Equations

ABSTRACTSets of rules are obtained for writing down directly the exact integral equations which are satisfied by certain functions of Heisenberg operators in quantum field theory. Three kinds of function are considered: the direct product, the chronological product and the M-product. The matrix elements of the M-product are equal to the Feynman amplitudes studied by Matthews & Salam (1) and the corresponding integral equation is called here the Matthews-Salam (M.-S.) equation. These authors have given a symbolic form of the M.-S. equation and a method of repeated differentiation and integration which can be used to obtain the explicit form of the integral equation in any particular example. In practice their method involves an immense amount of calculation even in quite simple examples. The rules obtained in the present paper make it possible to write down directly the M.-S. equation without any of the tedious calculations implied by the M.-S. method.So long as the exact theory is used, the three sets of equations (for direct, chronological and M-products) are completely equivalent. When bound state theory is considered by an approximation based on a power series in the coupling constant different results are obtained. The approximation is inapplicable to the direct product equations, and leads to different approximate equations for the amplitudes obtained from the chronological and the M-products even when these amplitudes are identical. This paradox is explained and it is shown that the equation coming from the M-product corresponds to the Bethe-Salpeter equation.

scholarly journals SYSTEMS OF INTEGRAL EQUATIONS WITH WEIGHTED DIFFERENCE KERNELS

2004 ◽  
Vol 47 (1) ◽  
pp. 205-230
Author(s):  
D. Porter ◽  
N. R. T. Biggs
Keyword(s):  
Integral Equation ◽  
Finite Number ◽  
Integral Equations ◽  
Equation System ◽  
Subject Classification ◽  
Finite Generation ◽  
Primary 45A05 ◽  
Systems Of Integral Equations ◽  
Single Integral ◽  
Explicit Expressions

AbstractExplicit expressions are derived for the inverses of operators of a particular class that includes the operator corresponding to a system of coupled integral equations having weighted difference kernels. The inverses are expressed in terms of a finite number of functions and a systematic way of generating different sets of these functions is devised. The theory generalizes those previously derived for a single integral equation and an integral-equation system with pure difference kernels. The connection is made between the finite generation of inverses and embedding.AMS 2000 Mathematics subject classification: Primary 45A05

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

A new extension of Kannan’s fixed point theorem via F-contraction with application to integral equations

2021 ◽  
pp. 2250123
Author(s):  
Pradip Debnath
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Integral Equations ◽  
Point Theorem ◽  
Extended Version

Our aim is to introduce an updated and real generalization of Kannan’s fixed point theorem with the help of [Formula: see text]-contraction introduced by Wardowski for single-valued mappings. Our result can be useful to ascertain the existence of fixed point for a family of mappings for which neither the Wardowski’s result nor that of Kannan can be applied directly. Our result has been applied to solve a particular type of integral equation. Finally, we establish a Reich-type extended version of the main result.

scholarly journals Integral equations of the first kind of Sonine type

2003 ◽  
Vol 2003 (57) ◽  
pp. 3609-3632 ◽  
Author(s):  
Stefan G. Samko ◽  
Rogério P. Cardoso
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Lebesgue Space ◽  
Inverse Operator ◽  
Orlicz Spaces ◽  
Hypersingular Integral ◽  
The Real ◽  
Integrable Kernel

A Volterra integral equation of the first kindKφ(x):≡∫−∞xk(x−t)φ(t)dt=f(x)with a locally integrable kernelk(x)∈L1loc(ℝ+1)is called Sonine equation if there exists another locally integrable kernelℓ(x)such that∫0xk(x−t)ℓ(t)dt≡1(locally integrable divisors of the unit, with respect to the operation of convolution). The formal inversionφ(x)=(d/dx)∫0xℓ(x−t)f(t)dtis well known, but it does not work, for example, on solutions in the spacesX=Lp(ℝ1)and is not defined on the whole rangeK(X). We develop many properties of Sonine kernels which allow us—in a very general case—to construct the real inverse operator, within the framework of the spacesLp(ℝ1), in Marchaud form:K−1f(x)=ℓ(∞)f(x)+∫0∞ℓ′(t)[f(x−t)−f(x)]dtwith the interpretation of the convergence of this “hypersingular” integral inLp-norm. The description of the rangeK(X)is given; it already requires the language of Orlicz spaces even in the case whenXis the Lebesgue spaceLp(ℝ1).

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