# Some properties of solutions of the homogeneous nonlinear second order differential equations

## 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

## PDF

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

Mathematica

1222-9016

2601-744X

MR3611700

##### ZBL

[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)