The well-posedness and regularity of a peridynamic model with a special kernel is studied. The differential-integral equation describing the model is first converted to an operator valued Volterra integral equation. Then the existence and regularity of the solution of the peridynamics problem are established through the study of the Volterra integral equation. The regularity results improve the previous known results for more general peridynamics models.

In this paper, we investigate the existence of $\mu$-pseudo almost automorphic solutions to the semilinear integral equation $x(t)=\int_{-\infty}^{t}a(t-s)[Ax(s)+f(s,x(s))]\,ds$, $t\in\mathbf{R}$ in a Banach space $\mathbf{X}$, where $a\in L^{1}(\mathbf{R}_{+})$, $A$ is the generator of an integral resolvent family of linear bounded operators defined on the Banach space $\mathbf{X}$, and $f:\mathbf{R}\times\mathbf{X}\rightarrow\mathbf{X}$ is a $\mu$-pseudo almost automorphic function. The main results are proved by using integral resolvent families combined with the theory of $\mu$-pseudo almost automorphic functions.

In this paper, we study the problem of the rate of approximation for the family of sampling Kantorovich operators in the uniform norm, for uniformly continuous and bounded functions belonging to Lipschitz classes (Zygmund-type classes), and for functions in Orlicz spaces. The general setting of Orlicz spaces allows us to directly deduce the results concerning the order of approximation in $L^p$-spaces, $1 \le p \lt \infty$, very useful in applications to Signal Processing, in Zygmund spaces and in exponential spaces. Particular cases of the sampling Kantorovich series based on Fej\'er's kernel and B-spline kernels are studied in detail.

We introduce a new ``convolution spline'' temporal approximation of time domain boundary integral equations (TDBIEs). It shares some properties of convolution quadrature (CQ) but, instead of being based on an underlying ODE solver, the approximation is explicitly constructed in terms of compactly supported basis functions. This results in sparse system matrices and makes it computationally more efficient than using the linear multistep version of CQ for TDBIE time-stepping. We use a Volterra integral equation (VIE) to illustrate the derivation of this new approach: at time step $t_n = n\dt$ the VIE solution is approximated in a backwards-in-time manner in terms of basis functions $\phi_j$ by $u(t_n-t) \approx \sum_{j=0}^n u_{n-j}\,\phi_j(t/\dt)$ for $t \in [0,t_n]$. We show that using isogeometric B-splines of degree $m\ge 1$ on $[0,\infty)$ in this framework gives a second order accurate scheme, but cubic splines with the parabolic runout conditions at $t=0$ are fourth order accurate. We establish a methodology for the stability analysis of VIEs and demonstrate that the new methods are stable for non-smooth kernels which are related to convergence analysis for TDBIEs, including the case of a Bessel function kernel oscillating at frequency $\oo(1/\dt)$. Numerical results for VIEs and for TDBIE problems on both open and closed surfaces confirm the theoretical predictions.

To tackle a nonlinear equation in a functional space, two numerical processes are involved: discretization and linearization. In this paper we study the differences between applying them in one or in the other order. Linearize first and discretize the linear problem will be in the sequel called option~(A). Discretize first and linearize the discrete problem will be called option~(B). As a linearization scheme, we consider the Newton method. It will be shown that, under certain assumptions on the discretization method, option~(A) converges to the exact solution, contrarily to option~(B) which converges to a finite dimensional solution. These assumptions are not satisfied by the classical Galerkin, Petrov-Galerkin and collocation methods, but they are fulfilled by the Kantorovich projection method. The problem to be solved is a nonlinear Fredholm equation of the second kind involving a compact operator. Numerical evidence is provided with a nonlinear integral equation.