In this paper we consider the interval-valued Volterra integral equations (IVIEs). We study the problem of existence and uniqueness of solutions for IVIEs. Finally, we give some examples for IVIEs.

A numerical collocation method is developed for solving nonlinear Volterra integro-differential equations (VIDEs) of the neutral type, as well as other non-standard and classical VIDEs. A sigmoidal functions approximation is used to suitably represent the solutions. Special computational advantages are obtained using unit step functions, and important applications can also be obtained by using other sigmoidal functions, such as logistic and Gompertz functions. The method allows one to obtain a simultaneous approximation of the solution to a given VIDE and its first derivative, by means of an explicit formula. A priori as well as a posteriori estimates are derived for the numerical errors, and numerical examples are given for the purpose of illustration. A comparison is made with the classical piecewise polynomial collocation method as for accuracy and CPU time.

In this paper we analyze constrained optimal Dirichlet boundary control problems subject to the linear heat equation. We propose using boundary integral equations to solve the coupled optimality system, and we present results on unique solvability and related a~priori error estimates for a symmetric Galerkin boundary element method. A numerical example confirms the analytical results.

In this paper regularized solutions of ill-posed Hammerstein type operator equation $KF(x)=y$, where $K:X\rightarrow Y$ is a bounded linear operator with non-closed range and $F:X\rightarrow X$ is non-linear, are obtained by a two step Newton type iterative method in Hilbert scales, where the available data is $y^{\delta}$ in place of actual data $y$ with $\|y-y^\delta\|\leq\delta$. We require only a weaker assumption $\|F^{\p}(x_0)x\|\sim\|x\|_{-b}$ compared to the usual assumption $\|F^{\p}(\widehat{x})x\|\sim\|x\|_{-b}$, where $\widehat{x}$ is the actual solution of the problem, which is assumed to exist, and $x_0$ is the initial approximation. Two cases, viz-a-viz, (i)~when $F^{\p}(x_0)$ is boundedly invertible and (ii)~$F^{\p}(x_0)$ is non-invertible but $F$ is monotone operator, are considered. We derive error bounds under certain general source conditions by choosing the regularization parameter by an a~priori manner as well as by using a modified form of the adaptive scheme proposed by Perverzev and Schock [{\bf14}].

This paper investigates the existence of solutions for nonlinear fractional differential equations with integral boundary conditions on an unbounded domain. An example illustrating how the theory can be applied in practice is also included.

Using the theory of measures of noncompactness and applying a new method, we prove the existence of solutions of quadratic Volterra integral inclusions.