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In this paper, a numerical method for solving second kind nonlinear Volterra integral equations is presented. The method is based upon the extension of unknown function by single term Walsh series. Indeed, the interval [0, 1) is divided to m equal subinterval and in each interval, the unknown function is extended by the first term of Walsh series functions. By using a collocation method the coefficients of these extensions are computed and a block-pulse approximation of the unknown function is obtained. By the block-pulse approximation both continuous and pointwise approximations can be obtained. A convergence analysis for continuous approximations are investigated. The numerical examples confirm the ability and accuracy of the method. The method is computationally attractive and can be easily generalized for the systems of nonlinear Volterra equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of solutions for nonlinear Hammerstein, Volterra–Hammerstein, and Volterra equations in the space of functions of bounded second variation in the sense of Shiba, (Λ p 2 B V [ a , b ]) . [ABSTRACT FROM AUTHOR]

We studied in this work a class of nonlinear Volterra integral equations with delay and present a sufficient condition for the bounded solution also we give an illustrative example which support the main result of this paper.

In this paper we study some nonlinear integral equations and give conditions for the functions and kernel involved in such equations under which we assure the existence, uniqueness, and continuity of solutions in the space of functions of bounded K-variation in the sense of Korenblum, (KBV ([a,b])). [ABSTRACT FROM AUTHOR]

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We present a method for calculating the numerical approximation of the two-dimensional mixed Volterra Fredholm integral equations, using the properties of the rationalized Haar (RH) wavelets and the matrix operator. Attaining this purpose, first, an operator and then an orthogonal projection should be defined. Regarding the characteristics of Haar wavelet, we solve the integral equation without using common mathematical methods. An upper bound and the convergence of the mentioned method have been proved, by using the Banach fixed point. Moreover, the rate of the convergence method is O(n(2q)n). Finally, several examples of different kinds of functions are presented and solved by this method. [ABSTRACT FROM AUTHOR]

This study attempts to estimate the volatility of the American options pricing model under jump-diffusion underlying asset model. Therefore, the problem is formulated then inverted, and afterward, direct finance problems are defined. It is demonstrated, then, that the price of this type of options satisfies a free boundary Partial Integral Differential Equation (PIDE). The inverse method for estimating the volatility and the American options price is also described in three phases: first, transformation of the direct problem to a non-linear initial and boundary value problem. Second, finding the solution by using the method of lines and the fourthorder Runge-Kutta method. Third, presenting a minimization function with Tikhonov regularization. [ABSTRACT FROM AUTHOR]

We give an existence theorem for some functional-integral equations which includes many key integral and functional equations that arise in nonlinear analysis and its applications. In particular, we extend the class of characteristic functions appearing in Chandrasekhar's classical integral equation from astrophysics and retain existence of its solutions. Extensive use is made of measures of noncompactness and abstract fixed point theorems such as Darbo's theorem.

Abstract: This paper is concerned with the existence of integral solutions to a general nonlinear integral equationWith the help of Krasnoselskii’s fixed point theorem and the theory of measure of weak noncompactness, we establish a new and general existence theorem for the nonlinear functional integral equation. Moreover, an example, which can not be treated by the related theorems in [5,20,22], is given to illustrate the new existence theorem. [Copyright &y& Elsevier]

A fixed point theorem in ordered spaces and a recently proved monotone convergence theorem are applied to derive existence and comparison results for solutions of a functional integral equation of Volterra type and a functional impulsive Cauchy problem in an ordered Banach space. A novel feature is that equations contain locally Henstock-Kurzweil integrable functions. [ABSTRACT FROM AUTHOR]

We give an existence theorem for some functional-integral equations which includes many key integral and functional equations that arise in nonlinear analysis and its applications. In particular, we extend the class of characteristic functions appearing in Chandrasekhar's classical integral equation from astrophysics and retain existence of its solutions. Extensive use is made of measures of noncompactness and abstract fixed point theorems such as Darbo's theorem. [ABSTRACT FROM AUTHOR]

Sufficient conditions for the existence of a solution to a non-linear Volterra integral equation are given for special cases of the general equation. In the generality given here, this equation has, apparently, not been studied before. The major technique used is the classical fixed point theorem of Banach. An apparent innovation of this article is the use of Banach's theorem to prove both the existence and find the location of a solution to the integral equation and prove the existence and find the location of the derivative to this solution, which exists almost everywhere. Furthermore, it is shown that for some particular choices of the constants, multiple solutions exist to this equation. [ABSTRACT FROM AUTHOR]