Class Of Nonlinear Differential Equations Research Articles

Homoclinic Solutions for a Class of Nonlinear Difference Equations

We prove the existence of homoclinic solutions of a class of nonlinear difference equations with superlinear nonlinearity by using the generalized Nehari manifold approach. For the case where the nonlinearity is odd, we obtain infinitely many homoclinic solutions of the equations. Recent results in the literature are generalized and improved.

Approximate Periodic Solutions for Oscillatory Phenomena Modelled by Nonlinear Differential Equations

We apply the Fourier-least squares method (FLSM) which allows us to find approximate periodic solutions for a very general class of nonlinear differential equations modelling oscillatory phenomena. We illustrate the accuracy of the method by using several significant examples of nonlinear problems including the cubic Duffing oscillator, the Van der Pol oscillator, and the Jerk equations. The results are compared to those obtained by other methods.

Approximate Solutions by Truncated Taylor Series Expansions of Nonlinear Differential Equations and Related Shadowing Property with Applications

This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial condition. The class of differential equations is assumed to be approximated by well-posed truncated Taylor series expansions up to a certain order obtained about certain, in general nonperiodic, sampling pointsti∈[t0,tJ]fori=0,1,…,Jof the solution. Two examples are provided.

Turnpike Theorem for an Infinite Horizon Optimal Control Problem with Time Delay

An optimal control problem for systems described by a special class of nonlinear differential equations with time delay is considered. The cost functional adopted could be considered as an analogue of the terminal functional defined over an infinite time horizon. The existence of optimal solutions as well as the asymptotic stability of optimal trajectories (that is, the turnpike property) are established under some quite mild restrictions on the nonlinearities of the functions involved in the description of the problem. Such mild restrictions on the nonlinearities allowed us to apply these results to a blood cell production model.

Stability of solutions of pseudolinear differential equations with impulse action

In this paper, we develop a new approach to constructing piecewise differentiable Lyapunov functions for certain classes of nonlinear differential equations with impulse action. This approach is based on the method of “frozen” coefficients, and the required function is constructed as a pseudoquadratic form. For the case under consideration, stability conditions in the sense of Lyapunov are obtained. The proposed approach can be used to study the stability of the critical equilibrium states of systems of differential equations with impulse action.

Polynomial solutions of a nonlinear difference equation

A class of nonlinear difference equations is considered. We show how their polynomial solutions can be computed in a systematic manner.

Stability of nonlinear time-varying perturbed differential equations

The goal of this paper is twofold. The first part presents a converse Lyapunov theorem for the notion of uniform practical exponential stability of nonlinear differential equations in presence of small perturbation. This class of nonlinear differential equations can be viewed as parametric differential equations. The second part provides the classical perturbation method of seeking an approximate solution as a finite Taylor expansion of the exact solution. The practical asymptotic validity on the approximate is established on infinite-time interval. Finally, we give a numerical example to prove the validity of our methods.

Wirtinger inequality and nonlinear differential systems

Picone identity for a class of nonlinear differential equations is established and various qualitative results (such as Wirtinger-type inequality and the existence of zeros of first components of solutions) are obtained with the help of this new formula.

Existence and precise asymptotics of positive solutions for a class of nonlinear differential equations of the third order

Existence and precise asymptotics of positive solutions for a class of nonlinear differential equations of the third order

Uniformly boundedness of a class of non-linear differential equations of third order with multiple deviating arguments

This paper deals with a certain third-order non-linear differential equation with multiple deviating arguments. Some sufficient conditions are set up for all solutions and their derivatives to be uniformly bounded.

A limit boundary value problem for a nonlinear difference equation

We give an asymptotic description of the monotone increasing solutions of a limit boundary value problem for a class of nonlinear difference equations with continuous arguments and rationally non-related shifts.

On a class of non-linear differential equations with exact solution

The present paper deals with a class of non-linear ordinary second-order differential equations with exact solutions. A procedure for finding the general exact solution based on a known particular one is derived. For illustration solutions of some non-linear equations occurred in many problems of solid mechanics are considered.

Asymptotic representations of solutions of one class of nonlinear differential equations of the second order

We establish asymptotic properties of some types of solutions of one class of essentially nonlinear nonautonomous differential equations of the second order and find necessary and sufficient conditions for the existence of these solutions.

Infinitely many homoclinic solutions for a class of nonlinear difference equations

Infinitely many homoclinic solutions for a class of nonlinear difference equations

UNIQUENESS OF POSITIVE RADIAL-SYMMETRICAL SOLUTION OF DIRICHLET PROBLEM IN ANNULAR DOMAINS FOR ONE CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

The existence and uniqueness of positive radially symmetric solution of Dirichlet problem in annular domain for one class of nonlinear differential equations of the second order is proved.

The global stability conditions for solutions of nonstationary differential equations with instant impulsive effects in pseudolinear form

We obtained the sufficient conditions for the stability of solutions of a class of nonlinear differential equations with fixed instant impulsive effects in the Banach space. With the use of the Slyusarchuk’s condition and methods of the theory of operators in a partially ordered Banach space, the problem is reduced to the study of the stability of a linear system of second-order impulsive differential equations.

Solitary wave solutions for a certain class of nonlinear differential equations

Generalized forms of exact solitary wave solutions of the class (1.1) are investigated. The analysis rests mainly on the standard a direct algebraic method. The most general solutions are obtained, possibly having a constant term in their expansion into real exponentials. These solutions of the class (1.1) are performed under certain conditions for the relationship between the coefficients of the nonlinear, dispersive and dissipative terms. The analytical solutions of this class are of pulse-type and of kink-type solitary wave solutions and they are obtained with an arbitrary constant phase shift.

On the solvability of a boundary value problem on the real line

Abstract We investigate the existence of heteroclinic solutions to a class of nonlinear differential equations ( a ( x ) Φ ( x ′ ( t ) ) ) ′ = f ( t , x ( t ) , x ′ ( t ) ) , a . e . t ∈ ℝ governed by a nonlinear differential operator Φ extending the classical p- Laplacian, with right-hand side f having the critical rate of decay -1 as |t| → +∞, that is f ( t , ⋅ , ⋅ ) ≈ 1 t . We prove general existence and non-existence results, as well as some simple criteria useful for right-hand side having the product structure f(t, x, x') = b(t, x)c(x, x'). Mathematical subject classification: Primary: 34B40; 34C37; Secondary: 34B15; 34L30.

On the stability and boundedness of solutions of a class of nonautonomous differential equations of second order with multiple deviating arguments

This paper is devoted to the mathematical analysis of the stability, boundedness and uniform boundedness of solutions of a class of nonlinear differential equations of second order with multiple constant deviating arguments. We use Lyapunov functionals to verify the stability and boundedness of the solutions and some examples are given to illustrate the theoretical analysis in this work.

Positive almost periodic type solutions to a class of nonlinear difference equation

This paper is concerned with positive almost periodic type solutions to a class of nonlinear difference equation with delay. By using a fixed point theorem in partially ordered Banach spaces, we establish several theorems about the existence and uniqueness of positive almost periodic type solutions to the addressed difference equation. In addition, in order to prove our main results, some basic and important properties