Non-oscillation of solutions of difference equations of third order

In this paper, sufficient conditions in terms of coefficient functions are obtained for non-oscillation of all solutions of a class of linear homogeneous third order difference equations of the form

     

Asymptotic and oscillatory behavior of solutions of general nonlinear difference equations of second order

The asymptotic and oscillatory behavior of solutions of some general second-order nonlinear difference equations of the form

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=0 nZ,

is studied. Oscillation criteria for their solutions, when “p_n” is of constant sign, are established. Results are also presented for the damped-forced equation

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=e_n nZ.

Examples are inserted in the text for illustrative purposes.     

Liouvillian solutions of third order differential equations

The Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper we refine parts of the algorithm to find Liouvillian solutions of third order equations. We show that, except for four finite groups and a reduction to the second order case, it is possible to give a formula in the imprimitive case. We also give necessary conditions and several simplifications for the computation of the minimal polynomial for the remaining finite set of finite groups (or any known finite group) by extracting ramification information from the character table. Several examples have been constructed, illustrating the possibilities and limitations.     

A note on oscillation criteria of second order nonlinear neutral delay differential equations

In recent years, many results on oscillation criteria of second order nonlinear neutral delay differential equations have been obtained, but some of these criteria are incorrect due to a mistake in a crucial auxiliary result. This note examines several of these results, provides some counterexamples and points out what causes the problems.     

Oscillatory property of higher order nonlinear difference equations

In this paper, we study the oscillatory and asymptotic behaviour of solutions of higher order nonlinear difference equations of the form

We obtain some necessary and sufficient conditions for all bounded solutions of (*) to be oscillatory and for (*) to have a nonoscillatory solution of a special form.     

Asymptotic behavior of certain Riccati difference equations

Matrix Riccati difference equations are investigated on the infinite index set. Under natural assumptions an existence and uniqueness theorem is proven. The existence of the asymptotic expansion of the solution and computability of its coefficients are shown, provided the coefficients of the equation have such an expansion.     

Asymptotic behaviour of the solutions of second order functional differential equations

This paper presents integral criteria to determine the asymptotic behaviour of the solutions of second order nonlinear differential equations of the type y^″(x)+q(x)f(y(x))=0, with q(x)>0 and f(y) odd and positive for y>0, as x tends to +∞. It also compares them with the results obtained by Chanturia (1975) in [11] for the same problem.     

Asymptotic behavior of solutions of asymptotically constant coefficient systems of linear differential equations

Sufficient conditions are given for the n × n system y'=(A+P(t))y to have a solution such that as t → ∞, where λ is an eigenvalue of the constant matrix A and v is an associated eigenvector. The integrability conditions on P allow conditional convergence and the o(1) terms are specified precisely.     

Iterative solutions to non-linear fourth order differential equations through multi-integral methods

Implicit solution to a certain class of non-linear fourth order differential equation is presented. Bounds on sup norm for the derivative of a certain function f involved in the equation with different boundary values, are computed. These bounds provide the rate of convergence of the iterative sequence of approximate solutions obtained by Picard method, to the exact solution.     

On a third order family of methods for solving nonlinear equations

In this article we present a third-order family of methods for solving nonlinear equations. Some well-known methods belong to our family, for example Halley's method, method (24) from [M. Basto, V. Semiao, and F.L. Calheiros, A new iterative method to compute nonlinear equations, Appl. Math. Comput. 173 (2006), pp. 468–483] and the super-Halley method from [J.M. Gutierrez and M.A. Hernandez, An acceleration of Newton's method: Super-Halley method, Appl. Math. Comput. 117 (2001), pp. 223–239]. The convergence analysis shows the third order of our family. We also give sufficient conditions for the stopping inequality |x _n+1?α|≤|x _n+1?x _n | for this family. Comparison of the family members shows that there are no significant differences between them. Several examples are presented and compared.     

Existence of solutions for a singular system of nonlinear fractional differential equations

In this paper, we establish the existence of positive solutions for a singular system of nonlinear fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Green’s function and its corresponding properties, we transform the derivative systems into equivalent integral systems. The existence is based on a nonlinear alternative of Leray-Schauder type and Krasnoselskii’s fixed point theorem in a cone.     

Integral averages and oscillation of second-order nonlinear differential equations

We present some criteria for the oscillation of the second-order nonlinear differential equation

Full-size image

where a ε C¹([t₀, ∞)) is a nonnegative function, q ε C ([t₀, ∞)) are allowed to change sign on [t₀, ∞), ψ, f ε C¹ , ψ(x) > 0, xf(x) > 0, f′(x) ≥ 0 for x ≠ 0. These criteria are obtained by using a general class of the parameter functions H(t,s) in the averaging techniques and represent extension, as well as improvement of known oscillation criteria of Philos and Purnaras for the generalized Emden-Fowler equation.     

Oscillation of solutions for second-order nonlinear difference equations with nonlinear neutral term

Some Riccati type difference inequalities are given for the second-order nonlinear difference equations with nonlinear neutral term.

and using these inequalities, we obtain some oscillation criteria for the above equation.     

Asymptotic behavior of positive solutions of sublinear differential equations of Emden-Fowler type

This paper is concerned with asymptotic analysis of positive solutions of the second-order nonlinear differential equation

(A)     

On parameter dependence of solutions of algebraic riccati equations

We study the behavior of Hermitian solutions, especially the maximal ones, of algebraic Riccati equations whose coefficients depend on real parameters. The cases of analytic dependence on one parameter andC′ dependence (0≤r≤∞) on many parameters are considered. The basic assumption made is stabilizability. Partially supported by an NSF grant.