# On the Existence of Solutions of Nonlinear Fredholm Integral Equations from Kantorovich's Technique
## 1 Introduction  
Integral equations have numerous applications in almost all branches of the sciences and many physical processes and mathematical models in Engineering are usually governed by integral equations. The main feature of these equations is that they are usually nonlinear. In particular, nonlinear integral equations arise in fluid mechanics, biological models, solid state physics, kinetics chemistry, etc. In addition, many initial and boundary value problems can be easily turned into integral equations. One type of particularly interesting equation is a nonlinear Fredholm integral equation of the form  
\[x(s)=f(s)+\varpi\int_{a}^{b}\bar{\kappa}(s,t)x(t)^{p}dt,\quad s\in[a,b],\quad p \in\mathbb{R},\quad p\geq 2, \tag{1}\]  
where \(\varpi\in\mathbb{R}\), \(-\infty<a<b<+\infty\), the function \(f(s)\) is continuous on \([a,b]\) and given, the kernel \(\bar{\kappa}(s,t)\) is a known continuous function in \([a,b]\times[a,b]\) and \(x\) is a solution to be determined. As integral equations of the form (1) cannot be solved exactly, we use numerical methods to solve them; we can apply different numerical techniques and some of them can be found in the references of this work. For a general background on numerical methods for integral equations of the form (1), the books of [PERSON] [1] and [PERSON] and [PERSON] [2] are recommended. For a review of less recent methods, we refer the reader to the survey by [PERSON] [3]. There is a great deal of publication on the numerical solution of Equation (1). In recent publications, different mathematical tools and numerical implementations have been applied to solve integral equations (1). In some of these publications, certain authors extensively use methods based on different kinds of wavelets [4, 5]. Polynomial approximation methods using different base functions, such as Chebyshev polynomials, have been introduced; see for example [6, 7]. An approximation with Sinc functions has been developed in [8]. Sinc methods have increasingly been recognized as powerful tools for tackling problems in applied physics and engineering [9]. Several different variants of numerical or theoretical studies on (1) have been developed in the literature. For some examples, see papers [10, 11]. In terms of iterative schemes for solving Equation (1), in [12], we can find an iterative scheme based on the homotopy analysismethod, which is a general analytic approach to obtain series solutions of various types of nonlinear equations and based on homotopy. In particular, by means of the aforementioned method, we can construct a continuous mapping of an initial guess approximation to the exact solution of the equation to be solved. In [13], the authors present an adapted modification to the Newton-Kantorovich method. Finally, in [14], the Newton-Kantorovich method and quadrature methods are combined to develop a new method for solving Equation (1). In this work, we propose using [PERSON]'s method for solving Equation (1). For this, we previously analysed the semilocal convergence of the method and then compared the efficacy of the method with the former techniques for solving a particular integral equation of the form (1). The semilocal convergence results need to know the conditions of the operator involved in the equation to be solved and the starting points of the iterative methods; the results show the existence of solutions of the equation that allow us to obtain the domain of existence of a solution. The main interest of this work is two-fold. On the one hand, we conduct a qualitative study of Equation (1) and obtain results on the existence and uniqueness of a solution. On the other hand, we obtain the numerical resolution of the equation. For this, we previously consider a separable kernel \(\mathfrak{K}(s,t)\) and we directly approximate a solution of Equation (1). Secondly, by means of Taylor series, we consider the case of a non-separable kernel. For both aims, we use [PERSON]'s method, which is the most well-known iterative method for solving nonlinear equations.