## Abstract

In this paper we consider the following nonlinear homogeneous second order differential equations, \(F(x,y,y^{\prime},y^{\prime\prime})=0.\) We present for the solutions, \(y\in C^2[a,b]\), of this equation, extremal principle, Sturm-type, Nicolescu-type and Butlewski-type separation theorems.

Some applications and examples are given. Open problems are also presented.

## Authors

V. **Ilea
**(Babes Bolyai Univ)

D. **Otrocol
**(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

I.A. **Rus
**(Babes Bolyai Univ.)

## Keywords

Homogeneous nonlinear second order differential equation, zeros of solutions, Sturm-type theorem, Nicolescu-type theorem, Butlewski-type theorem, bilocal problem, Cauchy problem, open problem, extremal principle.

## Cite this paper as:

V. Ilea, D. Otrocol, I. A. Rus, *Some properties of solutions of the homogeneous nonlinear second order differential equations*, Mathematica, 57 (80) (2015), no 1-2, pp. 38-43

2016-Ilea-Otrocol-Rus-Some properties of solutions.pdf ??

## About this paper

##### Journal

Mathematica

##### Publisher Name

Romanian Academy Publishing House

##### DOI

Paper on journal website

http://math.ubbcluj.ro/~mathjour/articles/2015/ilea-otrocol-rus.pdf

##### Print ISSN

1222-9016

##### Online ISSN

2601-744X

##### MR

MR3611700

##### ZBL

## Google Scholar

[1] Agratini, O*., Approximation by linear operators*, Cluj University Press, 2001 (in Romanian)

[2] Agratini, O. and Rus, I. A., *Iterates of a class of discrete linear operators via contraction principle*, Comment. Math. Univ. Caroline, 44 (2003), 555–563

[3] Agratini, O. and Rus, I. A., *Iterates of some bivariate approximation process via weakly Picard operators*, Nonlinear Analysis Forum, 8 (2003), No. 2, 159–168

[4] Altomare, F. and Campiti, M., *Korovkin-type Approximation Theory and its Applications*, de Gruyter Studies in Mathematics, 17, Walter de Gruyter & Co., Berlin, 1994

[5] Andras, S. and Rus, I. A., *Iterates of Cesaro operators, via fixed point principle*, Fixed Point Theory 11 (2010), No. 2, 171–178

[6] Bohl, E., *Linear operator equations on a partially ordered vector space*, Aeq. Math., 4 (1970), fas. 1/2, 89–98

[7] Cristescu, R., *Ordered vector spaces and linear operators*, Abacus Press, 1976

[8] Gavrea, I. and Ivan, M., *The iterates of positive linear operators preserving the constants*, Appl. Math. Lett., 24 (2011), No. 12, 2068–2071

[9] Gavrea, I. and Ivan, M., *On the iterates of positive linear operators*, J. Approx. Theory, 163 (2011), No. 9, 1076–1079

[10] Gavrea, I. and Ivan, M., *Asymptotic behaviour of the iterates of positive linear operators*, Abstr. Appl. Anal., 2011, Art. ID 670509, 11 pp.

[11] Gavrea, I. and Ivan, M., *On the iterates of positive linear operators preserving the affine functions*, J. Math. Anal. Appl, 372 (2010), No. 2, 366–368

[12] Gonska, H., Pitul, P. and Rasa, I., *Over-iterates of Bernstein-Stancu operators*, Calcolo, 44 (2007), 117–125

[13] Heikkila, S. and Roach, G. F., *On equivalent norms and the contraction mapping principle*, Nonlinear Anal., 8 (1984), No. 10, 1241–1252

[14] Rasa, I., *Asymptotic behaviour of iterates of positive linear operators*, Jaen J. Approx., 1 (2009), No. 2, 195–204

[15] Rus, I. A*., Generalized contractions and applications*, Cluj Univ. Press, 2001

[16] Rus, I. A., *Iterates of Stancu operators, via contraction principle*, Studia Univ. Babes¸-Bolyai Math., 47 (2002), No. 4, 101–104

[17] Rus, I. A., *Iterates of Bernstein operators, via contraction principle*, J. Math. Anal. Appl., 292 (2004), No. 1, 259–2 172 Teodora Catinas¸, Diana Otrocol and Ioan A. Rus ˘

[18] Rus, I. A., *Iterates of Stancu operators (via contraction principle) revisited*, Fixed Point Theory, 11 (2010), No. 2, 369–374

[19] Rus, I. A., *Fixed points and interpolation point set of a positive linear operator on C*(D), Studia Univ. Babes-Bolyai, Math., 55 (2010), No. 4, 243–248

[20] Rus, I. A., *Heuristic introduction to weakly Picard operators theory*, Creative Math. Inf., 23 (2014), No. 2, 243–252

[21] Rus, I. A., *Iterates of increasing linear operators, via Maia’s fixed point theorem*, Stud. Univ. Babes¸-Bolyai Math., 60 (2015), No. 1, 91–98

[22] Stancu, D. D., *On some generalization of Bernstein polynomials*, Studia Univ. Babes-Bolyai Math., 14 (1969), No. 2, 31-45 (in Romanian)