Superconvergence results for linear second-kind Volterra integral equations

In this paper, Galerkin method is applied to approximate the solution of Volterra integral equations of second kind with a smooth kernel, using piecewise polynomial bases. We prove that the approximate solutions of the Galerkin method converge to the exact solution with the order \({\mathcal {O}}(h^{r}),\) whereas the iterated Galerkin solutions converge with the order \({\mathcal {O}}(h^{2r})\) in infinity norm, where h is the norm of the partition and r is the smoothness of the kernel. We also consider the multi-Galerkin method and its iterated version, and we prove that the iterated multi-Galerkin solution converges with the order \({\mathcal {O}}(h^{3r})\) in infinity norm. Numerical examples are given to illustrate the theoretical results.     

Some results on linear nabla Riemann-Liouville fractional difference equations

In this paper, we establish some criteria for boundedness, stability properties, and separation of solutions of autonomous nonlinear nabla Riemann-Liouville scalar fractional difference equations. To derive these results, we prove the variation of constants formula for nabla Riemann-Liouville fractional difference equations.     

Some oscillation results for second-order nonlinear delay dynamic equations

This paper is concerned with oscillatory behavior of a class of second-order delay dynamic equations on a time scale. Two new oscillation criteria are presented that improve some known results in the literature. The results obtained are sharp even for the second-order ordinary differential equations.     

Some results on second-order neutral functional differential equations with infinite distributed delay

In this paper, we consider two types of second-order neutral functional differential equations with infinite distributed delay. By choosing available operators and applying Krasnoselskii’s fixed-point theorem, we obtain sufficient conditions for the existence of periodic solutions to such equations.     

Three theorems on third-kind linear integral equations

Consider a solution space $\text{P}$_τ consisting essentially of linear combinations of continuous functions and poles with integration over the poles defined by Cauchy's principal value. In $\text{P}$_τ the linear integral equation of the third kind is shown to have properties similar to those of the Fredholm equation of the second kind. The space $\text{P}$_τ is a subspace of a solution space P_τ previously studied where no unique solution to the non-homogeneous third-kind equation exists. Solutions in $\text{P}$_τ may be explicitly constructed via Fredholm theory. For simplicity full details are presented only for the case involving a single pole. Generalization to the case of several poles is, however, briefly discussed.     

关于时滞差分方程有界性的Razumikhin型定理

对时滞差分方程建立了新的关于有界性的Raxumikhin型定理,其中可以避免采用不易寻找的辅助函数P,它包含了已知的结果,由它推出的一些结论更易于应用。     

Some results related to complex linear and nonlinear difference equations

In this paper, we investigate the complex oscillation problems of meromorphic solutions to some linear difference equations with meromorphic coefficients, and obtain some results about the relationships between the exponent of convergence of zeros, poles and the order of growth of meromorphic solutions to complex linear difference equations. We also study the existence of solution of certain types of nonlinear differential-difference equations, and partially answer a conjecture concerning the above problem posed by Yang and Laine (C.C. Yang and I. Laine, On analogies between nonlinear difference and differential equations, Proc. Japan Acad. Ser. A Math. Sci. 86(1) (2010), pp. 10–14).     

Nonnegative integral solution of linear equations

A method to obtain a nonnegative integral solution of a system of linear equations, if such a solution exists is given. The method writes linear equations as an integer programming problem and then solves the problem using a combination of artificial basis technique and a method of integer forms.     

Some results on integral sum graphs

Let Z denote the set of all integers. The integral sum graph of a finite subset S of Z is the graph (S,E) with vertex set S and edge set E such that for u,vS, uvE if and only if u+vS. A graph G is called an integral sum graph if it is isomorphic to the integral sum graph of some finite subset S of Z. The integral sum number of a given graph G, denoted by ζ(G), is the smallest number of isolated vertices which when added to G result in an integral sum graph. Let x denote the least integer not less than the real x. In this paper, we (i) determine the value of ζ(K_n−E(K_r)) for r2n/3−1, (ii) obtain a lower bound for ζ(K_n−E(K_r)) when 2r<2n/3−1 and n5, showing by construction that the bound is sharp when r=2, and (iii) determine the value of ζ(K_r,r) for r2. These results provide partial solutions to two problems posed by Harary (Discrete Math. 124 (1994) 101–108). Finally, we furnish a counterexample to a result on the sum number of K_r,s given by Hartsfiedl and Smyth (Graphs and Matrices, R. Rees (Ed.), Marcel, Dekker, New York, 1992, pp. 205–211).     

Further results on linear interval equations

As a continuation of my paper “New techniques for the analysis of linear interval equations” [Linear Algebra Appl. 58:273–325 (1984)], the interrelation between interval Gauss elimination and interval iteration is investigated. Main results are a new existence theorem for interval Gauss elimination (in the guise of a perturbation theorem), a convergence and comparison theorem for a general family of interval iteration schemes, and a new method for the calculation of the hull of the solution set of linear interval equations with inverse positive coefficient matrix.     

Some results on linear arboricity

The linear arboricity of the graph G is the minimum number of linear forests whose union is G. In the paper exact values and bounds of linear arboricity for some additional classes of graphs are determined.     

Blow-up behavior of Hammerstein-type delay Volterra integral equations

We consider the blow-up behavior of Hammerstein-type delay Volterra integral equations (DVIEs). Two types of delays, i.e., vanishing delay (pantograph delay) and non-vanishing delay (constant delay), are considered. With the same assumptions of Volterra integral equations (VIEs), in a similar technology to VIEs, the blow-up conditions of the two types of DVIEs are given. The blow-up behaviors of DVIEs with non-vanishing delay vary with different initial functions and the length of the lag, while DVIEs with pantograph delay own the same blow-up behavior of VIEs. Some examples and applications to delay differential equations illustrate this influence.     

Solutions of Volterra integral equations with infinite delay

We present several new existence results for a Volterra integral equation with infinite delay. We discuss periodic and bounded solutions. Sufficient conditions for the existence of positive periodic solutions are also provided. The techniques we employ have not been used for this equation before. Our results generalize and complement those in the literature and several examples are presented to show their applicability. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic behaviour of linear evolutionary integral equations

The asymptotic behaviour of bounded solutions of evolutionary integral equations in a Banach spaceX

Superconvergence results of iterated projection methods for linear Volterra integral equations of second kind

In this paper, we develop the iteration techniques for Galerkin and collocation methods for linear Volterra integral equations of the second kind with a smooth kernel, using piecewise constant functions. We prove that the convergence rates for every step of iteration improve by order \({\mathcal {O}}(h^{2})\) for Galerkin method, whereas in collocation method, it is improved by \({\mathcal {O}}(h)\) in infinity norm. We also show that the system to be inverted remains same for every iteration as in the original projection methods. We illustrate our results by numerical examples.     

General linear methods for Volterra integral equations

We investigate the class of general linear methods of order p and stage order q=p for the numerical solution of Volterra integral equations of the second kind. Construction of highly stable methods based on the Schur criterion is described and examples of methods of order one and two which have good stability properties with respect to the basic test equation and the convolution one are given.