Many-point boundary-value problem for some differential-operational equations

The existence and uniqueness of strong solution of many-point boundary-value problems for some differential-operational equations and its continuous dependence on the coefficients are investigated. The tool employed in our analysis is the method of an energy inequality.     

Holomorphic complex differential equations: An elementary proof that the time-p map is holomorphic

We give an elementary proof that, in its domain of definition, the time-p map of a scalar, autonomous holomorphic, complex differential equation, is itself holomorphic. This result is used by Sverdlove [1] when considering limit cycles in complex holomorphic differential equations. However no proof or reference for the result is given in [1]. Although this result must be well established, a proof does not appear to be readily accessible in the reference literature.     

Asymptotic behavior of the unbounded solutions of some boundary layer equations

We give an asymptotic equivalent at infinity of the unbounded solutions of some boundary layer equations arising in fluid mechanics.Received: 24 May 2004     

Asymptotic forms of positive solutions of quasilinear ordinary differential equations with singular nonlinearities

In this paper we consider positive solutions of second order quasilinear ordinary differential equations with singular nonlinearities. We obtain asymptotic equivalence theorems for asymptotically superlinear solutions and decaying solutions. By using these theorems, exact asymptotic forms of such solutions are determined. Furthermore, we can establish the uniqueness of decaying solutions as an application of our results.     

Basic theory of fractional differential equations

In this paper, the basic theory for the initial value problem of fractional differential equations involving Riemann–Liouville differential operators is discussed employing the classical approach. The theory of inequalities, local existence, extremal solutions, comparison result and global existence of solutions are considered.     

Homoclinic and heteroclinic orbits for a class of Hamiltonian systems

The existence of a rich structure of homoclinic and heteroclinic solutions is established for a family of Hamiltonian systems that serve as a simpler model for the multiple pendulum system. The proof is based on recently developed arguments from the calculus of variations that have proved useful in finding actual solutions of an equation near approximate solution.This research was sponsorted in part by the National Science Foundation under grant #MCS-8110556 and the U. S. Army Research Office under Contract #DAAL03-87-K-0043. Any reproduction for the purpose of the United States Government is permitted.     

On the behavior of the Lorenz equation backward in time

The sets of solutions to the Lorenz equations that exist backward in time and are bounded at an exponential rate determined by the eigenvalues of the linear part of the equation are examined. The set associated with the middle eigenvalue is shown to project surjectively onto a plane, thereby providing a lower estimate for its dimension. Specific bounds are also found for a cone containing this set.     

Oscillation of second-order half-linear delay dynamic equations with damping on time scales

By using the generalized Riccati transformation and the inequality technique, we establish a few new oscillation criterions for certain second-order half-linear delay dynamic equations with damping on a time scale. Our results extend and improve some known results, but also unify the oscillation of the second-order half-linear delay differential equation with damping and the second-order half-linear delay difference equation with damping.     

Growth and long-run stability

This paper analyses the implications of persistent growth upon the stability properties of dynamic models. Besides the traditional concept of asymptotic stability, new stability criteria-strong/weak absolute, strong/weak relative, strong/weak logarithmic stability-are introduced, and global stability conditions for satisfying these criteria are stated for general first-order autonomous differential equations. The conflict between rapidity of growth and the degree of stability is demonstrated. Economic applications of the stability theorems are illustrated within the growth models of Harrod and Solow.     

Quasilinearization for functional differential equations with retardation and anticipation

The technique of generalised quasilinearization is developed, using the method of lower and upper solutions and the monotone iterative technique (MIT), to prove the existence of unique solutions for functional differential equations (FDE) with retardation and anticipation.     

Monotone iterative technique for functional differential equations with retardation and anticipation

The method of the monotone iterative technique together with coupled lower and upper solutions is employed to prove existence and uniqueness results for functional differential equations involving retardation and anticipation.     

Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions

The existence, uniqueness and continuous dependence of a mild solution of an impulsive neutral functional differential evolution nonlocal Cauchy problem in general Banach spaces are studied, by using the fixed point technique and semigroup of operators.     

Differential equations with bounded positive Green’s functions and generalized Aizerman’s hypothesis

We derive conditions for the positivity and boundedness of the Green functions of the higher order linear nonautonomous ODE. By virtue of these conditions, the existence of positive solutions for a class of nonlinear equations is proved. In addition, upper and lower estimates for the Green functions are established. Moreover, it is shown that nonlinear equations, having separated nonautonomous linear parts, satisfy the generalized Aizerman hypothesis on absolute stability, if they have the positive Green functions.     

The stability of the equilibrium of the damped oscillator with damping changing sign

In this paper, we prove a sufficient and necessary condition for the stability of the equilibrium x=x^′=0 of the damped oscillator with damping changing sign

     

Asymptotic preconditioning of linear homogeneous systems of differential equations

We consider the asymptotic behavior of solutions of a linear differential system x^′=A(t)x, where A is continuous on an interval ([a,∞). We are interested in the situation where the system may not have a desirable asymptotic property such as stability, strict stability, uniform stability, or linear asymptotic equilibrium, but its solutions can be written as x=Pu, where P is continuously differentiable on [a,∞) and u is a solution of a system u^′=B(t)u that has the property in question. In this case we say that P preconditions the given system for the property in question.     

Optimal regularity of robustness for parameterized perturbations

For exponential dichotomies defined by nonautonomous linear equations, we show that sufficiently small C¹-parameterized perturbations originate a family of exponential dichotomies of class C¹ in the parameter. We consider the general case of nonuniform exponential dichotomies, and also the general case of arbitrary growth rates of the form eλρ(t) where ρ is an arbitrary function. This includes the usual exponential behavior as a very special case when ρ(t)=t.     

Smooth robustness of parameterized perturbations of exponential dichotomies

We establish the robustness of nonuniform exponential dichotomies in Banach spaces, under sufficiently small C¹-parameterized perturbations. Moreover, we show that the stable and unstable subspaces of the exponential dichotomies obtained from the perturbation are also of class C¹ on the parameter, thus yielding an optimal smoothness.