We model time-harmonic acoustic scattering by an object composed of piecewise homogeneous parts and an arbitrarily heterogeneous part. We propose and analyze new formulations that couple, adopting a Costabel-type approach, boundary integral equations for the homogeneous subdomains with volume variational formulations for the heterogeneous subdomain. This is an extension of the Costabel FEM-BEM coupling to a multidomain configuration, with cross-points allowed, i.e., points where three or more subdomains are adjacent. While generally just the exterior unbounded subdomain is treated with the BEM, here we wish to exploit the advantages of BEM whenever it is applicable, that is, for all the homogeneous parts of the scattering object. Our formulation is based on the multitrace formalism, which initially was introduced for acoustic scattering by piecewise homogeneous objects. Instead, here we allow the wavenumber to vary arbitrarily in a part of the domain. We prove that the bilinear form associated with the proposed formulation satisfies a Gårding coercivity inequality, which ensures stability of the variational problem if it is uniquely solvable. We identify conditions for injectivity and construct modified versions immune to spurious resonances.

We consider compact operators and integral equations in terms of integral kernels in the $B{V}_p$ space, the space of all bounded $p$-variation on a compact interval. The integral used in this paper is of Stieltjes type. The integral is an integration with respect to a function of bounded $p$-variation. A fractional Brownian motion in the stochastic integral is a processes of bounded $p$-variation.

We consider a thermoelastic laminated beam system with thermal dissipation governed by the Gurtin–Pipkin law. The exponential stability problem of this system has been addressed by Djellali (2023) and Liu and Zhao (2021), who showed that the exponential stability of the system depends on complicated stability numbers involving the system wave propagation speeds and several other physical parameters of the system. We improve the existing results in the literature by showing that the exponential decay of the energy associated with the system depends solely on the ratio between the system speeds.

We establish the existence results of the solutions for fractional Volterra-type integral equations of two variables. We use the method of measure of noncompactness and Petryshyn’s fixed point theorem to obtain these results. Our results contain many previously obtained existence results with more relaxed conditions. Finally, we give an example to illustrate our obtained results.

We present existence results of the $p$-Laplacian Hadamard fractional boundary value problem. We analyze the existence of multiple positive solutions for the fractional boundary value problem with the nonlinear term which involves the derivative term with the help of a fixed point theorem. An example is provided to describe the validity of the main result.

We investigate the boundedness of weight Hardy operators $U_{\psi }$ and the corresponding Cesàro type average operators $V_{\psi }$ on Q type spaces $Q_{\mathcal{𝒦},\lambda }^{p,q}({ℝ}^n)$. Moreover, we provide explicit norms for both $U_{\psi }$ and $V_{\psi }$ on $Q_{\mathcal{𝒦},\lambda }^{p,q}({ℝ}^n)$ under the assumption of the integrability of $\psi$.

By the Schauder fixed point theorem, we first investigate the boundedness and monotonicity of solutions of Caputo fractional differential equations. We also study the asymptotic behavior of solutions of Caputo fractional differential equations under some different conditions. We prove that the solutions of the Caputo fractional differential equations converge asymptotically to a constant as $t\to +\infty$. Finally, several examples are given to illustrate our main results.