In the present paper, we introduce some singular integral operators, singular quadrature operators and discretization matrices of singular integral equations with Hilbert kernel. These results both improve the classical theory of singular integral equations and develop the theory of singular quadrature with Hilbert kernel. Then by using them a unified framework for various collocation methods o...

We will also use Fubini’s theorem [2, p. 170–171] and Lebesgue’s dominated convergence theorem [2, p. 91]. Fubini’s theorem allows us to change the order of integration in a double integral of a function, when we know that the function is integrable in one of the orders (i.e., the integral of the absolute value of the function is finite in one of the orders). Lebesgue’s dominated convergence th...

Recommended by Sehie Park We prove that the recent fixed point theorem for contractions of integral type due to Branciari is a corollary of the famous Meir-Keeler fixed point theorem. We also prove that Meir-Keeler contractions of integral type are still Meir-Keeler contractions.

We offer an elementary proof of Pontryagin duality theorem for compact and discrete abelian groups. To this end we make use of an elementary proof of Peter-Weyl theorem due to Prodanov that makes no recourse to Haar integral. As a long series of applications of this approach we obtain proofs of Bohr von Neumann’s theorem on almost periodic functions, Comfort-Ross’ theorem on the description of ...

The discovery of elliptic functions emerged from investigations of integral addition theorems. An addition theorem for a function f is a formula expressing f(u+ v) in terms of f(u) and f(v). For a function defined as a definite integral with a variable upper limit, an addition theorem takes the form of an equation between the sum of two such integrals, with upper limits u and v, and an integral...

where 1 2 1 2 , , , r r θ θ are any real numbers, and n is any positive integer. We can obtain the infinite series forms of these double integrals using Taylor series expansions and integration term by term theorem; these are the major results of this paper (i.e., Theorems 1-3). Adams et al. [1], Nyblom [2], and Oster [3] provided some techniques to solve the integral problems. Yu [4-29], Yu an...

This paper presents a formal proof of the Riesz representation theorem in the PVS theorem prover. The Riemann Stieltjes integral was defined in PVS, and the theorem relies on this integral. In order to prove the Riesz representation theorem, it was necessary to prove that continuous functions on a closed interval are Riemann Stieltjes integrable with respect to any function of bounded variation...

‎In this paper‎, ‎first‎, ‎we investigate the construction of compact sets of $ C^k$ and $ C_0^k$‎ ‎by proving ``$C^k‎, ‎C_0^k-version$‎" ‎of Arzel`{a}-Ascoli theorem‎, ‎and then introduce new measures of noncompactness on these spaces‎. ‎Finally‎, ‎as an application‎, ‎we study the existence of entire solutions for a class of the functional integral-differential equations by using Darbo's fixe...

  In this paper, an existence result for a class of infinite systems of functional-integral equations in the Banach sequence space $c_{0}$ is established via the well-known Schauder fixed-point theorem together with a criterion of compactness in the space $c_{0}$. Furthermore, we include some remarks to show the vastity of the class of infinite systems which can be covered by our result. The a...

In this paper, An effective and simple numerical method is proposed for solving systems of integral equations using radial basis functions (RBFs). We present an algorithm based on interpolation by radial basis functions including multiquadratics (MQs), using Legendre-Gauss-Lobatto nodes and weights. Also a theorem is proved for convergence of the algorithm. Some numerical examples are presented...