We consider the well-posedness of the second-order degenerate differential equations with infinite delay $\left(P_2\right)$: ${\left(M{u}^{\prime }\right)}^{\prime }\left(t\right)+{\left(Lu\right)}^{\prime }\left(t\right)=Au\left(t\right)+{\int }_{-\infty }^{t}a\left(t-s\right)Bu\left(s\right)ds+f\left(t\right)$, $\left(0\le t\le 2\pi \right)$ with periodic boundary conditions $u\left(0\right)=u\left(2\pi \right)$, $\left(M{u}^{\prime }\right)\left(0\right)=\left(M{u}^{\prime }\right)\left(2\pi \right)$, in Lebesgue–Bochner spaces $L^p\left(\mathbb{𝕋};X\right)$ and periodic Besov spaces $B_{p,q}^s\left(\mathbb{𝕋};X\right)$, where $A$, $B$, $L$ and $M$ are closed linear operators in a Banach space $X$ satisfying $D\left(A\right)\cap D\left(B\right)\subset D\left(M\right)\cap D\left(L\right)$, $D\left(A\right)\cap D\left(B\right)\ne \left\{0\right\}$ and $a\in L^1\left({ℝ}_{+}\right)$. We completely characterize the well-posedness of $\left(P_2\right)$ in the above function spaces by using known operator-valued Fourier multiplier theorems. We also give concrete examples to support our abstract results.

This work is devoted to the analysis of the mixed impedance-Neumann–Dirichlet boundary value problem (MIND BVP) for the Laplace–Beltrami equation on a compact smooth surface $\mathcal{𝒞}$ with smooth boundary. We prove, using the Lax–Milgram Lemma, that this MIND BVP has a unique solution in the classical weak setting ${ℍ}^1\left(\mathcal{𝒞}\right)$ when considering positive constants in the impedance condition. The main purpose is to consider the MIND BVP in a nonclassical setting of the Bessel potential space ${ℍ}_p^s\left(\mathcal{𝒞}\right)$, for $s>1∕p$, $1<p<\infty$. We apply a quasilocalization technique to the MIND BVP and obtain model Dirichlet–Neumann, Dirichlet-impedance and Neumann-impedance BVPs for the Laplacian in the half-plane. The model mixed Dirichlet–Neumann BVP was investigated by R. Duduchava and M. Tsaava (2018). The other two are investigated in the present paper. This allows to write a necessary and sufficient condition for the Fredholmness of the MIND BVP and to indicate a large set of the space parameters $s>1∕p$ and $1<p<\infty$ for which the initial BVP is uniquely solvable in the nonclassical setting. As a consequence, we prove that the MIND BVP has a unique solution in the classical weak setting ${ℍ}^1\left(\mathcal{𝒞}\right)$ for arbitrary complex values of the nonzero constant in the impedance condition.

Evaluating the Newton potential is crucial for efficiently solving the boundary integral equation of the Dirichlet boundary value problem of the Poisson equation. In the context of the Fourier–Garlerkin method for solving the boundary integral equation, we propose a fast algorithm for evaluating Fourier coefficients of the Newton potential by using a sparse grid approximation. When the forcing function of the Poisson equation expressed in the polar coordinates has $m$th-order bounded mixed derivatives, the proposed algorithm achieves an accuracy of order $\mathcal{𝒪}\left(n^{-m}{log}^{3}n\right)$, with requiring $\mathcal{𝒪}\left(n{log}^{2}n\right)$ number of arithmetics for the computation, where $n$ is the number of quadrature points used in one coordinate direction. With the help of this algorithm, the boundary integral equation derived from the Poisson equation can be efficiently solved by a fast fully discrete Fourier–Garlerkin method.

The asymptotic behavior of solutions of a class of Volterra integrodifferential equations is studied. Some new sufficient conditions are obtained in the presence of perturbation term. Our approach is based on some estimations on the solutions of the perturbed equation with respect to the solutions of the original unperturbed Volterra integrodifferential equations. We prove that the convergence of solutions to the equilibrium point can be studied provided that the perturbation term is bounded by a suitable function. The idea for this approach is based on a new integral inequality of Gronwall type.

We consider a thermoelastic laminated beam with a finite memory acting on the effective rotation angle. We establish an explicit and optimal stability estimate for the solution energy with minimal conditions on the relaxation function, from which the exponential and polynomial stability are just particular cases. This new result improves substantially many earlier results in the literature.

In 2016, Sonia et al. first considered the convergence order for the third-kind linear Volterra integral equations (VIEs) based on the assumption that solutions are smooth. For the third-kind linear VIEs with nonsmooth solutions, we construct high-order numerical algorithms and discuss the convergence order. By introducing a new suitable independent variable, we obtain a transformed equation with a smooth exact solution. Then the solvability of the transformed equation is investigated on the basis of piecewise polynomial collocation methods. Meanwhile, the convergence order of the collocation solution is given. Furthermore, based on the inverse transformation, we get the convergence order of the original equation. Numerical simulations are finally presented to demonstrate the effectiveness of the theoretical results.

Terminal value problems of a class of nonlinear fractional evolution equations with Weyl–Liouville derivative are considered. By using Fourier transforms, such a problem is converted into a singular integral equation on infinite interval. Some sufficient conditions are obtained to ensure the existence of a mild solution when the semigroup is compact or noncompact. Finally, an example is provided to illustrate our results.

We present a correction of the mistake made in the proof concerning the choice of the test function in our published paper in J. Integral Equations Appl. 30:1 (2018), 67–80.