It is well known that imposing a global Lipschitz condition on nonlinear composition operators leads to a strong degeneracy phenomenon in many function spaces. In contrast to this, we show that a local version of Banach's contraction mapping principle is less restrictive and applies to a large variety of nonlinear problems. We illustrate this by means of applications to nonlinear integral equations with bounded or weakly singular kernels.

We consider a surface integral formulation of the so-called interior transmission problem that appears in the study of inverse scattering problems from dielectric inclusions. In the case where the magnetic permeability contrast is zero, the main originality of our approach consists in still using classical potentials for the Helmholtz equation but in weaker trace space solutions. One major outcome of this study is to establish Fredholm properties of the problem for relaxed assumptions on the material coefficients. For instance, we allow the contrast to change sign inside the medium. We also show how one can retrieve discreteness results for transmission eigenvalues in some particular situations.

A pair of coupled nonlinear Volterra equations are examined for solutions that can have either global or blow-up behavior. The results represent an extension of the work in [{\bf8}], where the analysis was restricted to cases in which the integral equations had identical kernels. Sufficient conditions are provided for global solutions as well as for blow-up solutions. In the case of blow-up solutions, bounds on the blow-up time are derived. To demonstrate the applicability of the results, two examples are analyzed.

In this paper, we consider controllability of nonlinear fractional impulsive evolution systems. We firstly give a mild solution expression for nonlinear fractional impulsive evolution systems. Sufficient conditions for controllability results are obtained by Krasnoselskii's fixed point theorem in the infinite-dimensional spaces.

In this paper, we study the numerical approximation of the Urysohn integral equation with two methods. The methods are developed by means of the sinc approximation with the Single Exponential (SE) and Double Exponential (DE) transformations. These numerical methods combine a sinc Nystr\"{o}m method with the {N}ewton iterative process that involves solving a nonlinear system of equations. We provide an error analysis for the methods. These methods improve conventional results and achieve exponential convergence. Some numerical examples are given to confirm the accuracy and ease of implementation of the methods.

In this paper, we consider the inverse scattering problem of recovering the shape of a cavity or the surface impedance from one source and a knowledge of measurements placed on a curve inside the cavity. Based on a potential approach the inverse problem is equivalent to a system of nonlinear and ill-posed integral equations, a regularized Newton iterative approach is applied to reconstruct the boundary and the injectivity for the linearized system is established. Numerical examples are provided showing the viability of our method.