We are interested in the convergence of a system of integrodifferential equations with respect to an asymptotic parameter $𝜀$. It appears in the context of cell adhesion modeling [Oelz and Schmeiser 2010; Oelz, Schmeiser and Small 2008]. We extend the framework from [Milišić and Oelz 2011; 2015], strongly depending on the hypothesis that the external load $f$ is in $\text{Lip}([0,T])$ to the case where $f\in \text{BV}(0,T)$ only. We show how results presented in [Milišić and Oelz 2015] naturally extend to this new setting, while only partial results can be obtained following the comparison principle introduced in [Milišić and Oelz 2011].

A numerical algorithm via a modified hat functions (MHFs) has been proposed to solve a class of nonlinear fractional Volterra integral equations of the second kind. A fractional-order operational matrix of integration is introduced. In a new methodology, the operational matrices of MHFs and the powers of weakly singular kernels of integral equations are used as a structure for transforming the main problem into a number of systems consisting of two equations for two unknowns. Relative errors for the approximated solutions are investigated. Convergence analysis of the proposed method is evaluated and convergence rate is addressed. Finally, the extraordinary accuracy of the utilized approach is illustrated by a few examples. The results, absolute and relative errors are illustrated in some tables and diagrams. In addition, a comparison is made between the absolute errors obtained by the proposed method and two other methods; one uses a hybrid approach and the other applies second Chebyshev wavelet.

We investigate properties of a family of integral operators $\mathcal{ℬ}$ with a weakly singular compactly supported zonal kernel function on the surface $S$ of the unit 3D sphere. The support is over a spherical cap of height $h\in (0,2]$. Operators like this arise in some common types of approximations of time domain boundary integral equations (TDBIE) describing the scattering of acoustic waves from the surface of the sphere embedded in an infinite homogeneous medium where $h$ is directly related to the time step size.

We show that the Legendre polynomials of degree $\ell \ge 0$ satisfy for all $h\in (0,2]$ and, using spherical harmonics and the Funk–Hecke formula for the eigenvalues of $\mathcal{ℬ}$, that this is a key to unlocking positivity results for a subfamily of these operators. As well as positivity results we give detailed upper and lower bounds on the eigenvalues of $\mathcal{ℬ}$ and on ${\int }_S\hspace{0.17em}u(\mathit{x})(\mathcal{ℬ}u)(\mathit{x})d\mathit{x}$. We give various examples of where these results are useful in numerical approximations of the TDBIE on the sphere and show that positivity of $\mathcal{ℬ}$ is a necessary condition for these approximation schemes to be well defined. We also show the connection between the results for eigenvalues and the separation of variables solution of the TDBIE on the sphere. Finally we show how this relates to scattering from an infinite flat surface and Cooke’s 1937 result ${\int }_0^r\hspace{0.17em}{J}_0(z)dz>0$ for all $r>0$.

We investigate the asymptotic behavior of a viscoelastic neutral differential equation. A stability with an explicit decay result of the energy associated to the problem is established. It is found that the energy decay rate is optimal, in the sense that, it is the same as that of the relaxation function.

We use the Banach fixed point theorem to obtain stability results of the zero solution for a mixed linear Levin–Nohel integrodifferential system. To be more precise, we are concerned with the system

$$x^{\prime }(t)+{\int }_{t-\tau (t)}^{t}C(t,s)x(s)ds+B(t)x(t-h(t))=0,$$

where the importance of studying this system is that it generalizes results due to Burton (2004), Becker and Burton (2006), Jin and Luo (2009) and Dung (2013), from one dimension to $n$ dimensions. The last system with several delays terms is discussed as well.

We prove a new result of resolvent operators generated by the lattice operator, and investigate the existence and asymptotic stability of mild solutions for a class of lattice stochastic integrodifferential equations with infinite delays driven by fractional Brownian motion. With the help of the Banach fixed point theorem and some inequality techniques, the existence of mild solutions are obtained. We give some sufficient conditions to ensure the asymptotic stability of mild solutions in the mean square moment.

We study Tricomi–Gellerstedt–Keldysh-type fractional elliptic equations and obtain results on the well-posedness of fractional elliptic boundary value problems for general positive operators with discrete spectrum and for Fourier multipliers with positive symbols. As examples, we discuss results in half-cylinder, star-shaped graph, half-space and other domains.