# On positive solutions of a class of second order nonlinear differential equations on the halfline

• Volume: 62, Issue: 2, page 123-142
• ISSN: 0066-2216

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## Abstract

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The differential equation of the form ${\left(q\left(t\right)k\left(u\right){\left({u}^{\text{'}}\right)}^{a}\right)}^{\text{'}}=f\left(t\right)h\left(u\right){u}^{\text{'}}$, a ∈ (0,∞), is considered and solutions u with u(0) = 0 and (u(t))² + (u’(t))² > 0 on (0,∞) are studied. Theorems about existence, uniqueness, boundedness and dependence of solutions on a parameter are given.

## How to cite

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Svatoslav Staněk. "On positive solutions of a class of second order nonlinear differential equations on the halfline." Annales Polonici Mathematici 62.2 (1995): 123-142. <http://eudml.org/doc/262632>.

@article{SvatoslavStaněk1995,
abstract = {The differential equation of the form $(q(t)k(u)(u^\{\prime \})^a)^\{\prime \} = f(t)h(u)u^\{\prime \}$, a ∈ (0,∞), is considered and solutions u with u(0) = 0 and (u(t))² + (u’(t))² > 0 on (0,∞) are studied. Theorems about existence, uniqueness, boundedness and dependence of solutions on a parameter are given.},
author = {Svatoslav Staněk},
journal = {Annales Polonici Mathematici},
keywords = {nonlinear second order differential equation; nonnegative solution; existence and uniqueness of solutions; bounded solution; dependence of solutions on the parameter; boundary value problem on a noncompact interval; Tikhonov-Schauder fixed point theorem; water percolation; existence; uniqueness; boundedness; continuous dependence; Tikhonov-Schauder fixed point theorems},
language = {eng},
number = {2},
pages = {123-142},
title = {On positive solutions of a class of second order nonlinear differential equations on the halfline},
url = {http://eudml.org/doc/262632},
volume = {62},
year = {1995},
}

TY - JOUR
AU - Svatoslav Staněk
TI - On positive solutions of a class of second order nonlinear differential equations on the halfline
JO - Annales Polonici Mathematici
PY - 1995
VL - 62
IS - 2
SP - 123
EP - 142
AB - The differential equation of the form $(q(t)k(u)(u^{\prime })^a)^{\prime } = f(t)h(u)u^{\prime }$, a ∈ (0,∞), is considered and solutions u with u(0) = 0 and (u(t))² + (u’(t))² > 0 on (0,∞) are studied. Theorems about existence, uniqueness, boundedness and dependence of solutions on a parameter are given.
LA - eng
KW - nonlinear second order differential equation; nonnegative solution; existence and uniqueness of solutions; bounded solution; dependence of solutions on the parameter; boundary value problem on a noncompact interval; Tikhonov-Schauder fixed point theorem; water percolation; existence; uniqueness; boundedness; continuous dependence; Tikhonov-Schauder fixed point theorems
UR - http://eudml.org/doc/262632
ER -

## References

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1. [1] F. A. Atkinson and L. A. Peletier, Similarity profiles of flows through porous media, Arch. Rational Mech. Anal. 42 (1971), 369-379. Zbl0249.35043
2. [2] F. A. Atkinson and L. A. Peletier, Similarity solutions of the nonlinear diffusion equation, Arch. Rational Mech. Anal. 54 (1974), 373-392. Zbl0293.35039
3. [3] J. Bear, D. Zaslavsky and S. Irmay, Physical Principles of Water Percolation and Seepage, UNESCO, 1968.
4. [4] J. Goncerzewicz, H. Marcinkowska, W. Okrasiński and K. Tabisz, On the percolation of water from a cylindrical reservoir into the surrounding soil, Zastos. Mat. 16 (1978), 249-261. Zbl0403.76078
5. [5] P. Natanson, Theorie der Funktionen einer reellen Veränderlichen, Akademie-Verlag, Berlin, 1969.
6. [6] W. Okrasiński, Integral equations methods in the theory of the water percolation, in: Mathematical Methods in Fluid Mechanics, Proc. Conf. Oberwolfach, 1981, Band 24, P. Lang, Frankfurt/M, 1982, 167-176.
7. [7] W. Okrasiński, On a nonlinear ordinary differential equation, Ann. Polon. Math. 49 (1989), 237-245. Zbl0685.34038
8. [8] S. Staněk, Nonnegative solutions of a class of second order nonlinear differential equations, Ann. Polon. Math. 57 (1992), 71-82. Zbl0774.34017
9. [9] S. Staněk, Qualitative behavior of a class of second order nonlinear differential equations on the halfline, Ann. Polon. Math. 58 (1993), 65-83. Zbl0777.34027

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