On abstract Volterra equations with kernels having a positive resolvent

We consider the nonlinear abstract Volterra equation of convolution type: V $$u(t) + b * Au(t) = u_0 + b * g(t), t \geqq 0,$$ whereA ism-accretive in Banach spaceX, b is a given real kernel,u ₀ andg are given. Boundedness and asymptotic properties of the solutions are established under the assumption that the kernel satisfies certain natural positivity conditions.     

Existence of Solutions for a Wave Equation with Non-monotone Nonlinearity and a Small Parameter

We provide sufficient conditions for the existence of solutions to a semilinear wave equation with non-monotone nonlinearity involving a small parameter. Our results are based on the analysis of a an operator that characterizes the projection onto the kernel of the wave operator subject to periodic-Dirichlet boundary conditions. Such a kernel is infinite dimensional which makes standard compactness arguments inapplicable.     

Solving the generalized regularized long wave equation on the basis of a reproducing kernel space

On the basis of a reproducing kernel space, an iterative algorithm for solving the generalized regularized long wave equation is presented. The analytical solution in the reproducing kernel space is shown in a series form and the approximate solution u_n is constructed by truncating the series to n terms. The convergence of u_n to the analytical solution is also proved. Results obtained by the proposed method imply that it can be considered as a simple and accurate method for solving such evolution equations.     

Fredholm-Volterra integral equation with singular kernel

The purpose of this paper is to obtain the solution of Fredholm-Volterra integral equation with singular kernel in the space L₂(?1, 1) × C(0,T), 0 ≤t ≤T < ∞, under certain conditions. The numerical method is used to solve the Fredholm integral equation of the second kind with weak singular kernel using the Toeplitz matrices. Also, the error estimate is computed and some numerical examples are computed using the MathCad package.     

On systems of linear functional equations of the second kind in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

We consider a general system of functional equations of the second kind in L ₂ with a continuous linear operator T satisfying the condition that zero lies in the limit spectrum of the adjoint operator T*. We show that this condition holds for the operators of a wide class containing, in particular, all integral operators. The system under study is reduced by means of a unitary transformation to an equivalent system of linear integral equations of the second kind in L ₂ with Carleman matrix kernel of a special kind. By a linear continuous invertible change, this system is reduced to an equivalent integral equation of the second kind in L ₂ with quasidegenerate Carleman kernel. It is possible to apply various approximate methods of solution for such an equation.     

Stability of global and exponential attractors for a three‐dimensional conserved phase‐field system with memory

We consider a conserved phase‐field system on a tri‐dimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature ?, which is represented through a convolution integral whose relaxation kernel k is a summable and decreasing function. Therefore, the system consists of a linear integrodifferential equation for ?, which is coupled with a viscous Cahn–Hilliard type equation governing the order parameter χ. The latter equation contains a nonmonotone nonlinearity ? and the viscosity effects are taken into account by a term ?αΔ?_tχ, for some α?0. Rescaling the kernel k with a relaxation time ε>0, we formulate a Cauchy–Neumann problem depending on ε and α. Assuming a suitable decay of k, we prove the existence of a family of exponential attractors {?_α,ε} for our problem, whose basin of attraction can be extended to the whole phase–space in the viscous case (i.e. when α>0). Moreover, we prove that the symmetric Hausdorff distance of ?_α,ε from a proper lifting of ?_α,0 tends to 0 in an explicitly controlled way, for any fixed α?0. In addition, the upper semicontinuity of the family of global attractors {??_α,ε} as ε→0 is achieved for any fixed α>0. Copyright © 2009 John Wiley & Sons, Ltd.     

Existence of optimal control without convexity and a bang-bang theorem for linear volterra equations

We consider a Mayer problem of optimal control monitored by an integral equation of Volterra type: $$x(t) = x(t_1 ) + \int_{t_1 }^t { [h(t,s)x(s) + g(t, s)f(s, u(s))] ds,} $$ where the measurable control functionu satisfies a constraint of the formu(t) ∈U(t) ?E ^m,t ₁≤t≤t ₂, andg is a continuous kernel. Using the resolvent kernel associated with the kernelh, we prove the existence of an optimal usual solution for orientor fields without convexity assumptions. Further, ifU is a fixed compact set, we show the existence of an optimal bang-bang control.     

Resolvent kernels that constitute an approximation of the identity and linear heat-transfer problems

Sufficient conditions are obtained for a Volterra integral equation whose kernel depends on an increasing parameter a to admit an approximation of the identity with respect to a in the form of a resolvent kernel. In this case, the solution of the integral equation tends to zero as a tends to infinity, and we establish estimates of this convergence in L. These results are used for obtaining estimates of the convergence of linear heat-transfer boundary conditions to Dirichlet ones as the heat-transfer coefficient tends to infinity.     

On a representation of an operator equation with first time derivative in the form of a B u -Hamiltonian equation

We obtain necessary and sufficient conditions for the representability of an evolution operator equation with the first derivative with respect to time in the form of a B _u -Hamiltonian equation.     

Semiclassical Wormholes with a Smooth Throat

We propose a procedure for calculating heat kernel coefficients and use it to derive the expression for any heat kernel coefficient B _i for an arbitrary profile of the throat. We give an explicit form of these coefficients up to B ₃. We study domains of possible existence of wormholes for different values of the throat length-to-radius ratio and a nonminimal coupling constant.     

On an integro-differential equation with almost difference kernel on the half-line

We study the solvability of a class of integro-differential equations with almost difference kernel on the positive half-line. Using a special three-factor decomposition of the original integro-differential operator, we obtain sufficient conditions for the solvability of this equation in the class of tempered absolutely continuous functions. Under additional conditions on the kernel of the corresponding homogeneous equation with some value of the parameter occurring in it, we prove the existence of a nontrivial absolutely continuous solution, which, depending on the sign of the first moment of the kernel, is either a bounded function or has the asymptotics O(x), x → ∞.     

Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay

In this paper, we consider initial-boundary value problem of viscoelastic wave equation with a delay term in the interior feedback. Namely, we study the following equation $$u_{tt}(x, t) - \Delta {u}(x, t) + \int_{0}^{t} g(t - s)\,\Delta {u}(x, s){\rm d}s + \mu_{1} u_{t}(x, t) + \mu_{2} u_{t}(x, t -\tau) = 0$$ together with initial-boundary conditions of Dirichlet type in Ω × (0, + ∞) and prove that for arbitrary real numbers  μ ₁ and μ ₂, the above-mentioned problem has a unique global solution under suitable assumptions on the kernel g. This improve the results of the previous literature such as Nicaise and Pignotti (SIAM J. Control Optim 45:1561–1585, 2006) and Kirane and Said-Houari (Z. Angew. Math. Phys. 62:1065–1082, 2011) by removing the restriction imposed on μ ₁ and μ ₂. Furthermore, we also get an exponential decay results for the energy of the concerned problem in the case μ ₁ = 0 which solves an open problem proposed by Kirane and Said-Houari (Z. Angew. Math. Phys. 62:1065–1082, 2011).     

Wiener–Hopf equation whose kernel is a probability distribution

We prove the existence of a solution of an inhomogeneous generalized Wiener–Hopf equation whose kernel is a probability distribution on R generating a random walk drifting to +∞, while the inhomogeneous term f of the equation belongs to the space L ₁(0,∞) or L _∞(0,∞). We establish the asymptotic properties of the solution of this equation under various assumptions about the inhomogeneity f.     

One Goursat problem in a Sobolev space

In this paper we consider a hyperbolic-type differential equation with L _p -coefficients in a three-dimensional space. For this equation we study the Goursat problem with nonclassical boundary constraints not requiringmatched conditions. We prove the equivalence of these boundary conditions to classical ones in the case when one seeks for a solution to the stated problem in an anisotropic space introduced by S. L. Sobolev. In addition, we prove the correct solvability of the Goursat problem by the method of integral equations.     

Central limit theorem for perturbed empirical distribution functions evaluated at a random point

Let be an estimator obtained by integrating a kernel type density estimator based on a random sample of size n from a (smooth) distribution function F. Sufficient conditions are given for the central limit theorem to hold for the target statistic where {U_n} is a sequence of U-statistics.     

The Schr?dinger equation on cylinders and the n-torus

In this paper, we study the solutions to the Schr?dinger equation on some conformally flat cylinders and on the n-torus. First, we apply an appropriate regularization procedure. Using the Clifford algebra calculus with an appropriate Witt basis, the solutions can be expressed as multiperiodic eigensolutions to the regularized parabolic-type Dirac operator. We study their fundamental properties, give representation formulas of all these solutions in terms of multiperiodic generalizations of the elliptic functions in the context of the regularized parabolic-type Dirac operator. Furthermore, we also develop some integral representation formulas. In particular, we set up a Green type integral formula for the solutions to the homogeneous regularized Schr?dinger equation on cylinders and n-tori. Then, we treat the inhomogeneous Schr?dinger equation with prescribed boundary conditions in Lipschitz domains on these manifolds. We present an L _p -decomposition where one of the components is the kernel of the first-order differential operator that factorizes the cylindrical (resp. toroidal) Schr?dinger operator. Finally, we study the behavior of our results in the limit case where the regularization parameter tends to zero.     

Stability conditions to Bresse systems with indefinite memory dissipation

ABSTRACT

We study the asymptotic behavior of Bresse system with non-dissipative kernel memory acting only in the equation of longitudinal displacement. We show that the exponential stability depends on conditions regarding the decay rate of the kernel and a new relationship between the coefficients of the system. Moreover, this new condition on the constants of the system is used to prove strong stability and exponential stability for the homogeneous case (frictional dissipation in the longitudinal equation).     

Exact boundary controllability of an integrodifferential equation

An integrodifferential equation of the Volterra type is considered under the action of anL ₂(0, T, L₂())-boundary control. By harmonic analysis arguments it is shown that the controllability results obtained in [17] for the underlying reference model associated with a trivial convolution kernel, carry over to the model under consideration without any smallness assumption concerning the memory kernel.