We study the existence and the stability of solutions for Riemann-Liouville, Volterra-Stieltjes quadratic delay integral equations of fractional order. Our results are obtained by applying some fixed point theorems.

We show that a method of embedding for a class of non-linear Volterra equations can be used in a novel fashion to obtain variation of parameters formulas for Volterra integral equations subjected to a general type of variation of the equation. The approach is of intrinsic interest. Our variation of parameters formulas generalize classical formulas for ordinary differential equations (due to Alekseev) and for linear Volterra integral equations (based on resolvents). Illustrative examples are related to known results.

In this paper, a numerical collocation method is developed for solving linear and nonlinear Volterra integral equations of the second kind. The method is based on the approximation of the (exact) solution by a superposition of sigmoidal functions and allows one to solve a large class of integral equations having either continuous or $L^p$ solutions. Special computational advantages are obtained using unit step functions, and analytical approximations of the solution are also at hand. The numerical errors are discussed, and a priori as well as a posteriori estimates are derived for them. Numerical examples are given for the purpose of illustration.

Deconvolution of appearance potential spectra is an old strategy commonly used to investigate electronic properties of solids in the surface region. Recently, this strategy was found to be effective in the study of nanostructures. In this context, the density of unoccupied states in the surface region of a solid is recovered from the measured AP-spectrum data from the governing equation $k*x*x=g$, where $k$ is a Lorentzian type function, $g$ is a measured APS-signal and $x$ is the density function to be recovered. As an important step in solving for $x$, we need to solve the autoconvolution problem $x*x=f$, which is a nonlinear ill-posed Volterra problem. In this paper, we first improve upon the existing local regularization theory developed in [{\bf9}] for solving the autoconvolution problem, allowing for $L_p$ data, where $1 \le p \le \infty$. We prove the solutions of the regularized equation $x_\alpha^\delta \in L_\infty(0,1)$ (smoother than $x_\alpha^\delta \in L_2(0,1)$ as in [{\bf9}]) converge to the true solution $\overline{x}$ of the autoconvolution equation in $L_\infty$ norm (stronger than $L_2$ norm as in [{\bf9}]) when the noise level in the data shrinks to 0. It is worth noting that we obtain the improved convergence theory while imposing less restrictions on the true solution $\overline{x}$; namely $\overline{x} \in C^1(0,1)$ in contrast to $\overline{x}\in W^{2,\infty}(0,1)$. Further, for the stable deconvolution of appearance potential spectra, we apply the local regularization methods to solve a combination of two ill-posed Volterra problems: the linear problem of determining $f$ from $f*k=g$ and then the nonlinear autoconvolution problem of determining $x$ from $x*x=f$. The results include a convergence theory and a fast sequential numerical method which essentially preserves the causal nature of the combined deconvolution problem. Numerical examples are included to show the effectiveness and efficiency of the methods.

In this work we establish some new estimates for layer potentials of the acoustic wave equation in the time domain, and for their associated retarded integral operators. These estimates are proven using time-domain estimates based on the theory of evolution equations and improve known estimates that use the Laplace transform.

By applying a fixed point theorem of Krasnosel'skii type, we study the solvability and existence of asymptotically stable solutions for a nonlinear Volterra-Hammerstein integral equation on an infinite interval. In order to illustrate the results obtained, two examples are also given.