Asymptotic behavior of solutions to nonlinear Volterra integral equations

A class of operator Riccati integral equations is associated with a factorization problem in a certain Banach algebra. Recent results concerning factorization in this algebra are used to obtain existence, uniqueness, and continuous dependence results for the Riccati equations.     

Asymptotic properties of solutions to some nonlinear integral equations of convolution type

For a class of nonlinear integral equations of convolution type we give necessary and sufficient conditions for the boundedness of nonnegative solutions. Moreover, conditions for the solution to converge asymptotically to a determined limit are obtained.     

Asymptotic behavior of unbounded solutions of linear Volterra integral equations

The asymptotic behavior as t → ∞ of solutions of ∝₀^tu(t ? s) dA(s) = f(t) is studied when f(t) satisfies a “o” estimate as $\text{t ” ∞}$, and A belongs to a weighted space and its Laplace-Stieltjes transform has finitely many zeros in its closed half-plane of convergence. Results for systems of integral equations as well as for integrodifferential systems are also given.     

Asymptotic stability and asymptotic solutions of second-order differential equations

We improve, simplify, and extend on quasi-linear case some results on asymptotical stability of ordinary second-order differential equations with complex-valued coefficients obtained in our previous paper [G.R. Hovhannisyan, Asymptotic stability for second-order differential equations with complex coefficients, Electron. J. Differential Equations 2004 (85) (2004) 1–20]. To prove asymptotic stability of second-order differential equations, we establish stability estimates using integral representations of solutions via asymptotic solutions and error estimates. Several examples are discussed.     

Asymptotic stability of integral sets

We prove theorems on asymptotic, equiasymptotic, and uniform asymptotic stability of the integral sel of a nonautonomous system of ordinary differential equations.     

Asymptotic stability of exact and discrete solutions for neutral multidelay-integro-differential equations

This paper concerns the long-time behavior of the exact and discrete solutions to a class of nonlinear neutral integro-differential equations with multiple delays. Using a generalized Halanay inequality, we give two sufficient conditions for the asymptotic stability of the exact solution to this class of equations. Runge–Kutta methods with compound quadrature rule are considered to discretize this class of equations with commensurate delays. Nonlinear stability conditions for the presented methods are derived. It is found that, under suitable conditions, this class of numerical methods retain the asymptotic stability of the underlying system. Some numerical examples that illustrate the theoretical results are given.     

Asymptotic stability of solutions for some classes of impulsive differential equations with distributed delay

In this paper we show the asymptotic stability of the solutions of some differential equations with delay and subject to impulses. After proving the existence of mild solutions on the half-line, we give a Gronwall–Bellman-type theorem. These results are prodromes of the theorem on the asymptotic stability of the mild solutions to a semilinear differential equation with functional delay and impulses in Banach spaces and of its application to a parametric differential equation driving a population dynamics model.     

Asymptotic energy concentration in the phase space of the weak solutions to the Navier-Stokes equations

We study the asymptotic behavior of the energy of weak solutions of Navier-Stokes equations as t→∞. We characterize the space of the initial data which causes a concentration of the kinetic energy in the phase space. Moreover, an explicit convergence rate is obtained.     

Approximation of smooth solutions to integral equations with degenerate kernels

A method is proposed for finding approximate smooth solutions to an integral equation of the second kind with a degenerate kernel. It is assumed that the inverse operator is unbounded.     

Power series solutions to Volterra integral equations

Power series type solutions are given for a wide class of linear and q-dimensional nonlinear Volterra equations on R^p. The basic assumption on the kernel K(x, y) is that K(x, xt) has a power series in x. For example, this holds for any analytic kernel.The kernel may be strongly singular, provided certain constants are finite. One and only one such power series solution exists. Its coefficients are given by a simple iterative formula. In many cases this may be solved explicitly. In particular an explicit formula for the resolvent is given.     

Representations of solutions to linear integral equations with difference kernels

For linear integral equations with difference kernels, we give a formula for the inverse in terms of the solutions to two specific problems. If the equation is self-adjoint, these solutions are simply related.     

Asymptotic behaviour of solutions to some pseudoparabolic equations

The aim of this paper is to investigate the behaviour as t→∞ of solutions to the Cauchy problem u_t−△u_t−v△u−(b,∇u)=∇⋅F(u),u(x,0)=u₀(x), where v>0 is a fixed constant, t≥0, x∈Ω, Ω is a bounded domain in Rⁿ. We will first establish an a priori estimate. Then, we establish the global existence, uniqueness and continuous dependence of the weak solution for the Sobolev-Galpern type equation with the Dirichlet boundary.     

Asymptotic behavior of solutions to polynomial renewal equations

Many special functions arise as “renormalized” limits of sequences of polynomials that satisfy a polynomial renewal equation. We determine the asymptotic behavior of these sequences of polynomials for both ordinary and “coefficientwise” convergence, and illustrate it with specific examples.     

Asymptotic behaviour of linear evolutionary integral equations

The asymptotic behaviour of bounded solutions of evolutionary integral equations in a Banach spaceX