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Existence of three solutions for perturbed nonlinear difference equations

Heidarkhani S., Moghadam M. K.

Opuscula Mathematica

2014

Vol. 34, no. 4

747--761

Using critical point theory, we study the existence of at least three solutions for perturbed nonlinear difference equations with discrete boundary-value condition depending on two positive parameters.

Reduction and continuation theorems for Brouwer degree and applications to nonlinear difference equations

Mawhin J.

Opuscula Mathematica

2008

Vol. 28, no. 4

541-560

The aim of this note is to describe the continuation theorem of [39,40] directly in the context of Brouwer degree, providing in this way a simple frame for multiple applications to nonlinear difference equations, and to show how the corresponding reduction property can be seen as an extension of the well-known reduction formula of Leray and Schauder [24], which is fundamental for their construction of Leray-Schauder's degree in normed vector spaces.

Oscillatory and asymptotic behavior of fourth order nonlinear delay difference equations

Graef J. R., Thandapani E.

Fasciculi Mathematici

2001

Nr 31

23-36

The authors consider the nonlinear difference equation (E) delta2 ((delta(bn delta yn))+f(n,yn-t)=0, n należy N(no)={no,no+1,...}, here {an} and {bn} are positive real sequences, I is a nonnegative integer, f: N(no) x R R is a continuous function with uf(n, u) > 0 for all u nierówne 0. They obtain necessary and sufficient conditions for the existence of nonoscillatory solutions with a specified asymptotic behavior. They also obtain sufficient conditions for all solutions to be oscillatory if/ is either strongly sublinear or strongly superlinear. Examples of their results are also included.

On the asymptotic behaviour of solutions of nonlinear difference equations

Migda M., Schmeidel E.

Fasciculi Mathematici

2001

Nr 31

63-69

This paper consists of three theorems. For the nonlinear difference equation (E) wzór sufficient conditions for the existence of the asymptotically constant solutions are given in Th. 1. In Th. 2 conditions under which there exists a solution (xn) of Eq. (E) such that xn = cn + o(1), are given. In Th. 3 conditions under which every solution (xn) of Eq. (E) possesses property: the sequence (xn/n is convergent in R, are presented.

Asymptotic behaviour of solutions of nonlinear delay difference equations

Migda M.

Fasciculi Mathematici

2001

Nr 31

57-62

Asymptoic properties of the solutions of the difference equation of the form ^(r(n-1)^x(n-1))+anf(x(n-k)=bn are studied.