In this article, by a concept of Stepanov type $\mu$-pseudo almost automorphic functions developed recently, we investigate some new existence results on bounded solutions to a semilinear integro-differential equation in Banach spaces. We first establish a new composition theorem of such functions, and then we prove the main results via ergodicity and composition theorems of Stepanov type $\mu$-pseudo almost automorphic functions combined with theories of uniformly exponentially stable and strongly continuous family of operators. These bounded solutions can cover (weighted) pseudo almost automorphic solutions with a Stepanov type forcing term as special cases.

We consider the inverse problem of recovering the shape of an extended source of known homogeneous strength within a conducting medium from one voltage and current measurement on the accessible boundary of the medium and present an iterative solution method via boundary integral equations. The main idea of our approach is to equivalently reformulate the inverse source problem as an inverse boundary value problem with a non-local Robin condition on the boundary of the source domain. Following our approach in \cite{KR} for an inverse Dirichlet problem, from Green's representation formula we obtain a nonlinear integral equation for the unknown boundary curve which can be solved by regularized Newton iterations. We present the foundations of the inverse algorithm and illustrate its feasibility by some numerical examples.

We consider a general nonlinear functional integral equation, and we prove the existence of solutions of this equation in the space of Lebesgue integrable functions on $\R^+$. Our analysis uses a recent version of Krasnosel'skii's fixed point theorem (Theorem \ref{2t1}) and the concept of the measure of weak noncompactness. In the appendix, we give an extension of Theorem~\ref{2t1} to expansive mappings.

By using the theory of resolvent families, fixed point theorems and measures of noncompactness, we prove the existence of mild solutions on a compact interval for a semilinear Volterra equation with state-dependent delay. An example is given.

A numerical technique for solving scattering problems is presented. It is based on a boundary integral equation idea, so the unknowns are localized on the contour (in 2D case) or the surface (in 3D case) of the scattering object. Two major difficulties of traditional boundary integral methods (the appearance of spurious resonances and the necessity to perform numerical integration of singular functions) are overcome by studying the problem in an approximate discrete formulation from the very beginning. The space is filled by cubic blocks, and the shape of the scatterer is formed by a set of blocks removed from the space. Thus, the formulation of the problem is discrete, and the continuous Green's function is substituted by a discrete mesh Green's function. An analogue of combined field boundary integral equation (CFIE) is developed for this formulation.

This paper is concerned with existence results for a quite general nonlinear functional integral equation in $L^1$ spaces. For this purpose, making use of the De Blasi measure of weak noncompactness, we first establish a new fixed point theorem of the nonautonomous superposition operators. After that, our theorem is applied to prove the solvability of the mentioned nonlinear functional integral equation.

In this note, we study the boundedness and compactness of integral operators $I_g$ and $T_g $ from analytic Morrey spaces to Bloch space. Furthermore, the norm and essential norm of those operators are given.