Exponentially Dichotomous Operators and Exponential Dichotomy of Evolution Equations on the Half-Line

To an evolution family on the half-line of bounded operators on a Banach space X we associate operators I_X and I_Z related to the integral equation and a closed subspace Z of X. We characterize the exponential dichotomy of by the exponential dichotomy and the quasi-exponential dichotomy of the operators X we associate operators I_X and I_Z, respectively.     

Projection Methods for Integral Equations on the Half-Line

Convergence results are proved for projection methods for integralequations of the form are such that Wiener-Hopf integral equations are included in our analysis. The convergenceresults indicate that the iterated-projection solution may exhibitsuperconvergence. The case of collocation using piecewise-constantbasis functions applied to an integral equation with kernel is discussed in detail, and numerical results are given. For this example superconvergence of the iteratedsolution, and hence also of the collocation solution at thecollocation points, is both proved theoretically and observednumerically.     

Integral Equations on Function Spaces and Dichotomy on the Real Line

The purpose of this paper is to give new and general characterizations for uniform dichotomy and uniform exponential dichotomy of evolution families on the real line. We consider two general classes denoted and and we prove that if V,W are Banach function spaces with and , then the admissibility of the pair for an evolution family implies the uniform dichotomy of . In addition, we consider a subclass and we prove that if , then the admissibility of the pair implies the uniform exponential dichotomy of the family . This condition becomes necessary if . Finally, we present some applications of the main results.     

Solvability and Fredholm Properties of Integral Equations on the Half-Line in Weighted Spaces

The solvability of integral equations of the form and the behaviour of the solution x at infinity are investigated. Conditions on k and on a weight function w are obtained which ensure that the integral operator K with kernel k is bounded as an operator on X_w, where X_w denotes the weighted space of those continuous functions defined on the half-line which are O(w(s)) as We also derive conditions on w and k which imply that the spectrum and essential spectrum of K on X_w are the same as on BC[0,). In particular, the results apply when when the integral equation is of Wiener-Hopf type. In this case we show that our results are particularly sharp.     

Preconditioned Conjugate Gradient Methods for Integral Equations of the Second Kind Defined on the Half-Line

1.IntroductionInthispaPer,westudynumericalsolutionstointegralequationsofthesecondkinddefinedonthehalfline.Morepreciselyweconsidertheequationy(t)+Iooa(t,s)y(s)ds=g(t),OS相似文献   

Critical Nonlinear Nonlocal Equations on a Half-Line

We study nonlinear nonlocal equations on a half-line in the critical case

Hammerstein型积分方程非零解的存在性及应用

在不假定非线性项非负的情况下,利用半序理论讨论了Ｈａｍｍｅｒｓｔｅｉｎ型积分方程非零解的存在性,并将所得结果应用于常微分方程两点边值问题．     

Integral Inequalities and Global Solutions of Semilinear Evolution Equations

A criterion for the nonexplosion of solutions to semilinear evolution equations on Banach spaces is proved. The result is obtained by applying a modification of the Bihari type inequality to the case of a weakly singular nonlinear integral inequality.     

Absolute and Convective Instability for Evolution PDEs on the Half-Line

We analyze evolution PDEs exhibiting absolute (temporal) as well as convective (spatial) instability. Let  ω( k )  be the associated symbol, i.e., let  exp[ ikx −ω( k ) t ]  be a solution of the PDE. We first study the problem on the infinite line with an arbitrary initial condition   q ₀( x )  , where   q ₀( x )  decays as  | x | → ∞  . By making use of a certain transformation in the complex k -plane, which leaves  ω( k )  invariant, we show that this problem can be analyzed in an elementary manner. We then study the problem on the half-line, a problem physically more realistic but mathematically more difficult. By making use of the above transformation, as well as by employing a general method recently introduced for the solution of initial-boundary value problems, we show that this problem can also be analyzed in a straightforward manner. The analysis is presented for a general PDE and is illustrated for two physically significant evolution PDEs with spatial derivatives up to second order and up to fourth order, respectively. The second-order equation is a linearized Ginzburg–Landau equation arising in Rayleigh–Bénard convection and in the stability of plane Poiseuille flow, while the fourth-order equation is a linearized Kuramoto–Sivashinsky equation, which includes dispersion and which models among other applications, interfacial phenomena in multifluid flows.     

Some Problems for Cubic Nonlinear Equations on a Half-Line

The paper deals with a problem of developing an inverse-scattering based formalism for solving problems for the cubic nonlinear (or the modified Korteweg–de Vries (KdV)) equations: q _t +q _xxx +6q ² q _x =0, 0x<, –<t<,q _t +q _xxx –6q ² q _x =0, with the given initial and boundary conditions: q(x,0)=q(x),q(0,t)=p(t), p(t)L ₁(–,). The relation between the solution of the initial-boundary value problem (1), (3), (4) and that of the KdV equation on the half-line is shown. The Cauchy problem for the cubic nonlinear equation: q _t +q _xxx –6|q|² q _x =0, 0x<, –<t<, with the given initial condition (3) is considered also. Here we solve the above problems on the half-line 0x< but with –<t<.     

Hammerstein非线性积分方程组的非平凡解及应用

该文利用拓扑方法和锥理论研究下列Hammerstein非线性积分方程组u(x)=∫_G k(x,y)f(y,u(y),v(y)) dy,v(x)=∫_Gk(x,y)g(y,u(y),v(y)) dy.在适当的条件下,证明了上述方程组非平解的存在性,并把所得结果应用于研究非线性二阶常微分方程组边值问题的非平凡解的存在性.     

Banach空间中超线性Hammerstein型积分方程的解及其应用

本文利用不动点指数理论研究Ｂａｎａｃｈ空间中超线性Ｈａｍｍｅｒｓｔｅｉｎ型积分方程正解及非零解的存在性，并应用于Ｂａｎａｃｈ空间中超线性常微分方程的Ｓｔｕｒｍ－Ｌｉｏｕｖｉｌｌｅ问题，最后，本文给出了一个非线性常微分方程无穷组存在正解的例子．     

Integral Equations on Multi-Screens

In the present article, we develop a new functional framework for the study of scalar wave scattering by objects, called multi-screens, that are arbitrary arrangements of thin panels of impenetrable materials. From a geometric point of view, multi-screens are a priori non-orientable non-Lipschitz surfaces. We use our new framework to study boundary integral formulations of the scattering by such objects.     

Approximation Technique for Fractional Evolution Equations with Nonlocal Integral Conditions

We consider a class of fractional evolution equations with nonlocal integral conditions in Banach spaces. New existence of mild solutions to such a problem are established using Schauder fixed-point theorem, diagonal argument and approximation techniques under the hypotheses that the nonlinear term is Carathéodory continuous and satisfies some weak growth condition, the nonlocal term depends on all the value of independent variable on the whole interval and satisfies some weak growth condition. This work may be viewed as an attempt to develop a general existence theory for fractional evolution equations with general nonlocal integral conditions. Finally, as a sample of application, the results are applied to a fractional parabolic partial differential equation with nonlocal integral condition. The results obtained in this paper essentially extend some existing results in this area.     

The Inverse Problem Method on the Half-Line and a Nested System of Riccati Equations

A mixed problem for the nonlinear Bogoyavlenskii system on the half-line is studied by the inverse problem method. The solution of the mixed problem is reduced to the solution of the inverse spectral problem of recovering a forth-order differential operator on the half-line from the Weyl matrix. We derive evolution equations for the elements of the Weyl matrix and give an algorithm for the solution of the mixed problem. Evolution equations of the elements of the Weyl matrix are nonlinear. It is shown that they can be reduced to a nested system of three successively solvable matrix Riccati equations.     

A Generalization of Bihari''s Inequality and Its Applications to Nonlinear Volterra Integral Equations

In a recent paper of Dhongade, U. D. and Deo, S. G.^[2], the well-known improtant integral inequality due to Bihari^[1] was generalized to the case of haying finite terms of nonlinear integral functionals. Certainly, the generalizations of this type are very useful in treating many problems. Unfortunately the theorems given in [2] are not quite correct. The purpose of the present paper is first to prove the validity of another generalization of Bihari’s inequality, which corrects and extends all of the results in [2], and then as a further application of the obtained inequality, we consider here the perturbations of nonlinear Yolterra integral equations by combining with the nonlinear variation of constants formula established by Brauer, F.^[5] for the Yolterra equations.