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Existence of solutions for a class of nonlinear boundary value problems on half-line

Türker Ertem* and Ağacık Zafer

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Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey

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Citation and License

Boundary Value Problems 2012, 2012:43  doi:10.1186/1687-2770-2012-43

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/43

Received:28 January 2012
Accepted:16 April 2012
Published:16 April 2012

© 2012 Ertem and Zafer; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Consider the infinite interval nonlinear boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M1">View MathML</a>

where u and v are principal and nonprincipal solutions of (p(t)x')' + q(t)x = 0, r1(t) = o(u(t)(v(t))μ) and r2(t) = o(v(t)(u(t))μ) for some μ ∈ (0, 1), and a and b are arbitrary but fixed real numbers.

Sufficient conditions are given for the existence of a unique solution of the above problem for i = 1, 2. An example is given to illustrate one of the main results.

Mathematics Subject Classication 2011: 34D05.

Boundary value problem; singular; half-line; principal; nonprincipal

1. Introduction

Boundary value problems on half-line occur in various applications such as in the study of the unsteady flow of a gas through semi-infinite porous medium, in analyzing the heat transfer in radial flow between circular disks, in the study of plasma physics, in an analysis of the mass transfer on a rotating disk in a non-Newtonian fluid, etc. More examples and a collection of works on the existence of solutions of boundary value problems on half-line for differential, difference and integral equations may be found in the monographs [1,2] For some works and various techniques dealing with such boundary value problems (we may refer to [3-6] and the references cited therein).

In this article by employing principal and nonprincipal solutions we introduce a new approach to study nonlinear boundary problems on half-line of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M2">View MathML</a>


<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M3">View MathML</a>


<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M4">View MathML</a>


where a and b are any given real numbers, u and v are principal and nonprincipal solutions of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M5">View MathML</a>


and p C([0, ∞), (0, ∞)), q C([0, ∞), ℝ) and f C([0, ∞) × ℝ, ℝ).

We will show that the problem (1.1)-(1.3) has a unique solution in the case when

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M8">View MathML</a>



<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M9">View MathML</a>


where μ ∈ (0, 1) is arbitrary but fixed real numbers.

The nonlinear boundary value problem (1.1)-(1.3) is also closely related to asymptotic integration of second order differential equations. Indeed, there are several important works in the literature, see [7-16], dealing with mostly the asymptotic integration of solutions of second order nonlinear equations of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M10">View MathML</a>

The authors are usually interested in finding conditions on the function f(t, x) which guarantee the existence of a solution asymptotic to linear function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M11">View MathML</a>


We should point out that u(t) = 1 and v(t) = t are principal and nonprincipal solutions of the corresponding unperturbed equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M12">View MathML</a>

and the function x(t) in (1.7) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M13">View MathML</a>

Note that v(t) → ∞ as t → ∞ but u(t) is bounded in this special case. It turns out such information is crucial in investigating the general case. Our results will be applicable whether or not u(t) → ∞ (v(t) → ∞) as t → ∞.

2. Main results

It is well-known that [17,18] if the second order linear Equation (1.4) has a positive solution or nonoscillatory at , then there exist two linearly independent solutions u(t) and v(t), called principal and nonprincipal solutions of the equation. The principal solution u is unique up to a constant multiple. Moreover, the following useful properties are satisfied:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M14">View MathML</a>

where t* ≥ 0 is a sufficiently large real number.

Let v be a principal solution of (1.4). Without loss of generality we may assume that v (t) > 0 if t t1 for some t1 ≥ 0. It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M15">View MathML</a>


is a nonprincipal solution of (1.4), which is strictly positive for t > t1.

Theorem 2.1. Let t0 > t1. Assume that the function f satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M16">View MathML</a>



<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M17">View MathML</a>


where g C([0, ∞), [0, ∞)) is bounded; h1, h2, k C([t0, ∞), [0, ∞)). Suppose further that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M18">View MathML</a>



<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M19">View MathML</a>


for some β C([t0, ∞), [0, ∞)) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M20">View MathML</a>


If either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M21">View MathML</a>


or else

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M22">View MathML</a>


then there is a unique solution x(t) of (1.1)-(1.3), where r is given by (1.5).

Proof. Denote by M the supremum of the function g over [0, ∞). Let X be a space of functions defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M23">View MathML</a>


<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M24">View MathML</a>


<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M25">View MathML</a>

Note that X is a complete metric space with the metric d defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M26">View MathML</a>

Define an operator F on X by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M27">View MathML</a>

In view of conditions (2.2) and (2.5) we see that F is well defined. Next we show that F X X. Indeed, let x X, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M28">View MathML</a>

which means that F x X.

Using (2.1), (2.3) and (2.4) we also see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M29">View MathML</a>

where x1, x2 X arbitrary. This implies that F is a contracting mapping.

Thus according to Banach contraction principle F has a unique fixed point x. It is not difficult to see that the fixed point solves (1.1) and (1.2). It remains to show that x(t) satisfies (1.3) as well. It is not difficult to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M30">View MathML</a>


<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M31">View MathML</a>

If (2.7) is satisfied, then in view (2.6) and the above inequality we easily obtain (1.3). In case (2.8) holds, then c = 0 and hence we still have (1.3).

From Theorem 2.1 we deduce the following Corollary.

Corollary 2.2. Assume that the function f satisfies (2.2) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M32">View MathML</a>

where k C([t0, ∞), [0, ∞)). Suppose further that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M33">View MathML</a>

for some μ ∈ (0, 1) and β C([t0, ∞), [0, ∞)), where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M34">View MathML</a>

Then for each a, b ∈ ℝ the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M36">View MathML</a>

has a unique solution.

Let υ be a nonprincipal solution of (1.4). Without loss of generality we may assume that v(t) > 0, if t t2 for some t2 ≥ 0. It is easy to see that [17,18]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M37">View MathML</a>


is a principal solution of (1.4) which is strictly positive. Take t2 large enough so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M38">View MathML</a>

Then from (2.9), we have v (t) ≥ u(t) for t t2, which is needed in the proof of the next theorem.

Theorem 2.3. Let t0 t2. Assume that the function f satisfies (2.2) and (2.3). Suppose further that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M39">View MathML</a>



<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M40">View MathML</a>


for some β C([t0, ∞), [0, ∞)) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M41">View MathML</a>


If either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M42">View MathML</a>


or else

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M43">View MathML</a>


then there is a unique solution x(t) of (1.1) - (1.3), where r is given by (1.6).

Proof. Let X be a space of functions defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M23">View MathML</a>


<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M44">View MathML</a>

Again, X is a complete metric space with the metric d defined in the proof of the previous theorem.

We define an operator F on X by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M45">View MathML</a>

The remainder of the proof proceeds similarly as in that of Theorem 2.1 by using (2.2), (2.3), (2.9)-(2.14).

Corollary 2.4. Assume that the function f satisfies (2.2) and (2.3). Suppose further that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M46">View MathML</a>

for some μ ∈ (0, 1) and β C([t0, ∞), [0, ∞)), where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M47">View MathML</a>

If for any given a, b ∈ ℝ the condition (2.14) holds then the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M48">View MathML</a>

has a unique solution.

3. An example

Consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M49">View MathML</a>


<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M50">View MathML</a>


<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M51">View MathML</a>


where t0 > t1 = 1 and μ ∈ (0, 1) are chosen to satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M52">View MathML</a>


Note that since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M53">View MathML</a>

for any given μ ∈ (0, 1) there is a t0 such that (3.4) holds.

Comparing with the boundary value problem (1.1)-(1.3) we see that p(t) = t, q(t) = 0, and f(t, x) = (1/t2) arctan x + tυ. The corresponding linear equation becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M54">View MathML</a>

Clearly, we may take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M55">View MathML</a>


<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M56">View MathML</a>

then it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M57">View MathML</a>


<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M35">View MathML</a>

i.e., all the conditions of Theorem 2.1 are satisfied. Therefore we may conclude that if (3.4) holds, then the boundary value problem (3.1)-(3.3) has a unique solution.

Furthermore, we may also deduce that there exist solutions x1(t) and x2(t) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M7">View MathML</a>


<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/43/mathml/M6">View MathML</a>

by taking (a, b) = (0, 1) and (a, b) = (1, 0), respectively.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

Both authors contributed to this work equally, read and approved the final version of the manuscript.


This work was supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under project number 108T688.


  1. Agarwal, RP, O'Regan, D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic Publisher, Netherlands (2001)

  2. O'Regan, D: Theory of Singular Boundary Value Problems, World Scientific, River Edge, New Jersey, USA (1994)

  3. Agarwal, RP, O'Regan, D: Boundary value problems on the half line in the theory of colloids. Math Probl Eng. 8(2), 143–150 (2002). Publisher Full Text OpenURL

  4. Chen, SZ, Zhang, Y: Singular boundary value problems on a half-line. J Math Anal Appl. 195, 449–468 (1995). Publisher Full Text OpenURL

  5. Rachůnková, I, Tomeček, J: Superlinear singular problems on the half line. Boundary Value Probl. 2010, 18 (2010) pages (Article ID 429813

  6. Li, J, Nieto, JJ: Existence of positive solutions for multipoint boundary value problem on the half-line with impulses. Boundary Value Probl. 2009, 12 (2009) pages (Article ID 834158

  7. Caligo, D: Comportamento asintotico degli integrali dell'equazione y+ a(t)y = 0, nell'ipotesi limt→∞.a(t) = 0. Boll Un Mat Ital. 3, 286–295 (1941)

  8. Boas, ML, Boas, RP Jr., Levinson, N: The growth of solutions of a differential equation. Duke Math J. 9, 847–853 (1942). Publisher Full Text OpenURL

  9. Haupt, O: Über das asymptotische verhalten der lösungen gewisser linearer gewöhnlicher differentialgleichungen. Math Z. 48, 289–292 (1942). Publisher Full Text OpenURL

  10. Bihari, I: Researches of the boundedness and stability of the solutions of non-linear differential equations. Acta Math Acad Sci Hung. 8, 261–278 (1957)

  11. Hale, JK, Onuchic, N: On the asymptotic behavior of solutions of a class of differential equations. Contrib Diff Equ. 2, 61–75 (1963)

  12. Lipovan, O: On the asymptotic behavior of the solutions to a class of second order nonlinear differential equations. Glasgow Math J. 45, 179–187 (2003). Publisher Full Text OpenURL

  13. Mustafa, OG, Rogovchenko, YV: Asymptotic integration of a class of nonlinear differential equations. Appl Math Lett. 19, 849–853 (2006). Publisher Full Text OpenURL

  14. Agarwal, RP, Djebali, S, Moussaoui, T, Mustafa, OG: On the asymptotic integration of nonlinear differential equations. J Comput Appl Math. 202, 352–376 (2007). Publisher Full Text OpenURL

  15. Eastham, MSP: The Asymptotic Solution of Linear Differential Systems. Applications of the Levinson Theorem, Clarendon Press, Oxford (1989)

  16. Kiguradze, IT, Chanturia, TA: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Academic Publishers, Dordrecht (1993)

  17. Hartman, P: Ordinary Differential Equations, Wiley, New York (1964)

  18. Kelley, W, Peterson, A: The Theory of Differential Equations: Classical and Qualitative, Pearson Education, Inc., Upper Saddle River, New Jersey (2004)