This paper is concerned with the asymptotic behavior of the solution of a laminated Timoshenko beam system with viscoelastic damping. We extend the work known for this system with finite memory to the case of infinite memory. We use minimal and general conditions on the relaxation function and establish explicit energy decay formula, which gives the best decay rates expected under this level of generality. We assume that the relaxation function $g$ satisfies, for some nonnegative functions $\xi$ and $H$, $g^{\prime }\left(t\right)\le -\xi \left(t\right)H\left(g\left(t\right)\right)$, $\forall t\ge 0$. Our decay results generalize and improve many earlier results in the literature. Moreover, we remove some assumptions on the boundedness of initial data used in many earlier papers in the literature.

In this paper, we consider a transmission problem in a bounded domain with a nonlinear dissipation in the first equation. Under suitable assumptions on the weight of the damping, we show the existence and uniqueness of solution by the Faedo–Galerkin method. Also we prove general stability estimates using some properties of convex functions and Lyaponov functional.

Using operator-valued ${Ċ}^{\alpha }$-Fourier multiplier results on vector-valued Hölder continuous function spaces and the Carleman transform, we characterize the $C^{\alpha }$-well-posedness of second order degenerate differential equations with infinite delay: ${\left(Mu\right)}^{\prime \prime }\left(t\right)=Au\left(t\right)+{\int }_{-\infty }^{t}a\left(t-s\right)Au\left(s\right)ds+f\left(t\right)$ and ${\left(M{u}^{\prime }\right)}^{\prime }\left(t\right)=Au\left(t\right)+{\int }_{-\infty }^{t}a\left(t-s\right)Au\left(s\right)ds+f\left(t\right)$ on $ℝ$, where $A:D\left(A\right)\to X$ and $M:D\left(M\right)\to X$ are closed linear operators in a complex Banach space $X$, and $a\in L^1\left({ℝ}_{+}\right)\cap L^1\left({ℝ}_{+};t^{\alpha }dt\right)$.

A fast multiscale Galerkin method is proposed for solving the boundary integral equation derived from the Dirichlet problem of the Laplace equation in a domain with corners. It is well known that the integral operator in the equation can be split into two operators, one is noncompact, the other is compact. We design two truncation strategies for the representation matrices of these operators, respectively, which compress these two dense matrices to sparse ones having only $\mathcal{𝒪}\left(2^n\right)$ number of nonzero entries, where $2^n$ is the number of the wavelet basis functions used in the method. We prove that the proposed truncation strategies do not ruin the stability and convergence rate of the integral equation. Numerical experiments are presented to verify the theoretical results and demonstrate the effectiveness of the method.

We study two-point boundary value problems (BVP) depending on fractional differential operators, which are motivated by physical considerations. We consider very general boundary conditions. Existence and uniqueness criteria are provided. Our results extend previously known results in the literature.

Under certain assumptions on the functions $f$, $G$ and $h$, we establish one new criterion on the operator $L$ defined on $C\left(I\right)$ by

$$Lu\left(t\right)={\int }_0^{w}G\left(t,s\right)h\left(s\right)f\left(u\left(s\right)\right)ds,t\in I,\omega \in \left\{1,\infty \right\},$$

to guarantee that the operator equation has at least three solutions, where $I=\left[0,1\right]$ if $\omega =1$ and $I=\left[0,\infty \right)$ if $\omega =\infty$, via the Leguette–Williams fixed point theorem. Two examples are given to demonstrate our main result.

Existence of a solution of the functional integral equation in an unbounded interval involving the Riemann–Liouville operator is investigated. Here sufficient conditions in the context of existence and stability are derived by employing hybridized fixed point theory in the Banach algebra setting. Further, an example is presented to showcase the validity of the obtained result. Moreover, the solution of the example in closed form is estimated by the semianalytic technique which is being driven by a modified homotopy perturbation method in conjunction with the Adomian decomposition method.

We examine the existence of solutions for more general integral equations with respect to other work in Caballero Mena, Nalepa and Sadarangani (2014). Our techniques are based on the classical Schauder fixed point theorem and a new result about the relative compactness in Hölder spaces. Moreover, an example is presented to illustrate the obtained results.