We investigate the uniqueness of solutions to cordial Volterra integral equations in the sense of Vainikko in the case where the kernel function $\mathcal K(\theta ) \equiv \mathcal K(y/x)$ vanishes on the diagonal $x=y$. When, in addition, $\mathcal K$ is sufficiently regular, is strictly positive on $(0,1)$, and $\theta ^{-k} \, \mathcal K'(\theta )$ is nonincreasing for some $k\in \mathbb{R} $, we prove that the solution to the corresponding Volterra integral equation of the first kind is unique in the class of functions which are continuous on the positive real axis and locally integrable at the origin. Alternatively, we obtain uniqueness in the class of locally integrable functions with locally integrable mean. We further discuss a uniqueness-of-continuation problem where the conditions on the kernel need only be satisfied in some neighborhood of the diagonal. We illustrate with examples the necessity of the conditions on the kernel and on the uniqueness class, and sketch the application of the theory in the context of a nonlinear model.

We consider controllability of nonlinear impulsive integro-differential fractional time-invariant systems. We first give a solution expression for linear impulsive fractional time-invariant systems. By the Schauder fixed point theorem, sufficient conditions for the controllability of nonlinear impulsive integro-differential fractional time-invariant systems are obtained.

We investigate the asymptotic solution beha\-vior for the partial neutral functional differential equation$$ \begin{cases} & \frac {d}{dt}\mathcal {D}u_t=(A+B(t))\mathcal {D}u_t+f(t,u_t), \quad \hbox {$t\geq s\geq 0$,} \\ & u_s=\phi \in \mathcal {C}:=C([-r,0],X), \end{cases} $$ where the linear operator $A$ is not necessarily densely defined and satisfies the Hille-Yosida condition, and the delayed part $f$ is assumed to satisfy the $\varphi $-Lipschitz condition, i.e., $\|f(t,\phi )-f(t,\psi )\|\leq \varphi (t)\|\phi -\psi \|_{\mathcal {C}}$. Here $\varphi $ belongs to some admissible spaces and $\phi ,\ \psi \in \mathcal {C}:=C([-r,0],X)$.

More precisely, we prove the existence of stable (respectively, center stable) manifolds when the linear part generates an evolution family having an exponential dichotomy (respectively, trichotomy) on the half positive line. Furthermore, we show that such stable manifold attracts all mild solutions of the considered neutral equation. An example is given to assimilate our theory.

Motivated by the state-based peridynamic framework, we introduce a new nonlocal Laplacian that exhibits double nonlocality through the use of iterated integral operators. The operator introduces additional degrees of flexibility that can allow for better representation of physical phenomena at different scales and in materials with different properties. We study mathematical properties of this state-based Laplacian, including connections with other nonlocal and local counterparts. Finally, we obtain explicit rates of convergence for this doubly nonlocal operator to the classical Laplacian as the radii for the horizons of interaction kernels shrink to zero.

We consider an integral operator with a kernel of the Green's function type. We prove the existence of asymptotic expansion of an eigenvalue of multiplicity $m>1$, when the integral operator is approximated by the iterated Galerkin operator. This enables us to use the Richardson extrapolation to increase the order of convergence of the eigenvalue. We consider a numerical example to illustrate our theoretical results.

Applying Laplace transform on a generalized acoustical radiosity equation results in a special Fredholm integral equation of the second kind with $\lambda =1$ being the integral coefficient. The kernel of the equation contains a varying exponential complex parameter. The values of the parameter that make $\lambda =1$ be an eigenvalue of the kernel are defined in this paper as $L$-eigenvalues of the kernel, and the corresponding eigenfunctions are called $L$-eigenfunctions. The interest of this study is on the properties of the $L$-eigenvalues, $L$-eigenfunctions and the residues of related function at the $L$-eigenvalues. A set of theorems with a series of lemmas as bases are given and proven. They construct an integrated ensemble to reveal the decay structure of the generalized acoustical radiosity system with finite nonzero initial excitation.