Abstract:
In this article, we give some results concerning the continuity of the nonlinear Volterra and Fredholm integral operators on the space $L^{1}[0,infty)$. Then by using the concept of measure of weak noncompactness, we prove an existence result for a functional integral equation which includes several classes of nonlinear integral equations. Our results extend some previous works.

Abstract:
We study the existence and uniqueness of the solutions of mixed Volterra-Fredholm type integral equations with integral boundary condition in Banach space. Our analysis is based on an application of the Krasnosel'skii fixed-point theorem.

Abstract:
In this paper, under certain conditions, the solution of mixed type of Fredholm-Volterra integral equation is discussed and obtained in the space L_2 ( 1, 1) × C[0, T ], T < ∞. Here, the singular part of kernel of Fredholm-Volterra integral term is established in a logarithmic form, while the kernel of Fredholm-Volterra integral term is a positive continuous function in time and belongs to the class C[0, T ], T < ∞. The solution, when the mixed type integral, takes a system form of Fredholm integral equation of the first or second kind are discussed.

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In this paper we study some basic properties of solutions of a general mixed Volterra Fredholm integral equation. A variant of a certain integral inequality with explicit estimate is obtained and used to establish the results.

Abstract:
This work presents an analysis of the error that is committed upon having obtained the approximate solution of the nonlinear Fredholm-Volterra-Hammerstein integral equation by means of a method for its numerical resolution. The main tools used in the study of the error are the properties of Schauder bases in a Banach space. 1. Introduction In this paper we consider the following nonlinear mixed Fredholm-Volterra-Hammerstein integral equation: where , and the kernels are assumed to be known continuous functions, and is the unknown function to be determined. Equation (1.1) arises in a variety of applications in many fields, including continuum mechanics, potential theory, electricity and magnetism, three-dimensional contact problems, and fluid mechanics, and so forth (see, e.g., [1–4]). Several numerical methods for approximating the solution of integral, and integrodifferential equations are known (see, e.g., [5–8]). For Fredholm-Volterra-Hammerstein integral equations, the classical method of successive approximations was introduced in [9]. An optimal control problem method was presented in [10], and a collocation-type method was developed in [11–13]. Computational methods based on Bernstein operational matrices and the Chebyshev approximation method were presented in [14, 15], respectively. The use of fixed point techniques and Schauder bases, in the field of numerical resolution of differential, integral and integro-differential equations, allows for the development of new methods providing significant improvements upon other known methods (see [16–23]). In this work we make an analysis of the error committed upon having obtained the approximate solution of the nonlinear Fredholm-Volterra-Hammerstein integral equation, using the theorem of Banach fixed point and Schauder bases (see [21], for a detailed description of the numerical method used in a more general equation). In order to recall the aforementioned numerical method, let and be the Banach spaces of all continuous and real-valued functions on and endowed with their usual supnorms. Throughout this paper we will make the following assumptions on and for . (i)Since , there exists such that for all .(ii) are functions such that there exists such that for and for all . (iii) . We organize this paper as follows. In Section 2, we reformulate (1.1) in terms of a convenient integral operator and we describe the numerical method used. The study of the error is described in Section 3. Finally, in Section 4 we show some illustrative examples. 2. Analytical Preliminaries In this section we recall, in a

Abstract:
In this paper, we investigate the existence of global solutions to first-order initial-value problems, with nonlocal condition for nonlinear mixed Volterra-Fredholm integrodifferential equations in Banach spaces. The technique used in our analysis is based on an application of the topological transversality theorem known as Leray-Schauder alternative and rely on a priori bounds of solution.

Abstract:
In this article we study the fractional semilinear mixed Volterra-Fredholm integrodifferential equation $$ frac{d^{alpha }x(t)}{dt^{alpha }} =Ax(t)+fBig(t,x(t), int_{t_0}^tk(t,s,x(s))ds,int_{t_0}^{T}h(t,s,x(s))dsBig) , $$ where $tin [t_0,T]$, $t_0geq 0$, $0

Abstract:
This study considers an integral equation of Fredholm-Volterra type, where the Fredholm integral term is measured with respect to the position, while Volterra integral term is measured with respect to the time. Also, we obtained the solution of Fredholm-Volterra integral equation in series form.