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

Abstract

Consider the infinite interval nonlinear boundary value problem

( p ( t ) x ) + q ( t ) x = f ( t , x ) , t t 0 0 , x ( t 0 ) = x 0 , x ( t ) = a v ( t ) + b u ( t ) + o ( r i ( t ) ) , t ,

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.

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 [36] 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

( p ( t ) x ) + q ( t ) x = f ( t , x ) , t t 0 ,
(1.1)
x ( t 0 ) = x 0 ,
(1.2)
x ( t ) = a v ( t ) + b u ( t ) + o ( r ( t ) ) , t ,
(1.3)

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

( p ( t ) x ) + q ( t ) x = 0 , t 0
(1.4)

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

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

r ( t ) = o ( u ( t ) ( v ( t ) ) μ )
(1.5)

and

r ( t ) = o ( v ( t ) ( u ( t ) ) μ ) ,
(1.6)

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 [716], dealing with mostly the asymptotic integration of solutions of second order nonlinear equations of the form

x = f ( t , x ) .

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

x ( t ) = a t + b , t .
(1.7)

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

x = 0 ,

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

x = a v ( t ) + b u ( t ) .

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:

lim t u ( t ) v ( t ) = 0 , t * 1 p ( t ) u 2 ( t ) d t = , t * 1 p ( t ) v 2 ( t ) d t < ,

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 tt1 for some t1 ≥ 0. It is easy to see that

v ( t ) = u ( t ) t 1 t 1 p ( s ) u 2 ( s ) d s
(2.1)

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

f ( t , x ) h 1 ( t ) g ( x ) + h 2 ( t ) , t t 0
(2.2)

and

f ( t , x 1 ) - f ( t , x 2 ) k ( t ) v ( t ) x 1 - x 2 , t t 0 ,
(2.3)

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

t 0 u ( s ) k ( s ) d s μ
(2.4)

and

1 p ( t ) u 2 ( t ) t u ( s ) h i ( s ) d s β ( t ) , t t 0 , i = 1 , 2
(2.5)

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

t 0 t β ( s ) d s = o ( ( v ( t ) ) μ ) , t .
(2.6)

If either

v ( t ) , t
(2.7)

or else

b = x 0 u ( t 0 ) - a t 1 t 0 1 p ( s ) u 2 ( s ) d s ,
(2.8)

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

X = x C ( t 0 , , ) | x ( t ) l 1 v ( t ) + l 2 u ( t ) , t t 0 ,

where

l 1 = ( M + 1 ) p ( t 0 ) u 2 ( t 0 ) β ( t 0 ) + a

and

l 2 = x 0 u ( t 0 ) + a t 1 t 0 1 p ( s ) u 2 ( s ) d s .

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

d ( x 1 , x 2 ) = sup t t 0 1 v ( t ) x 1 ( t ) - x 2 ( t ) , x 1 , x 2 X .

Define an operator F on X by

( F x ) ( t ) = - u ( t ) t 0 t 1 p ( s ) u 2 ( s ) s u ( τ ) f ( τ , x ( τ ) ) d τ d s + a v ( t ) + x 0 u ( t 0 ) - a t 1 t 0 1 p ( s ) u 2 ( s ) d s u ( t ) .

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

( F x ) ( t ) u ( t ) t 0 t 1 p ( s ) u 2 ( s ) s u ( τ ) f ( τ , x ( τ ) ) d τ d s + a v ( t ) + l 2 u ( t ) u ( t ) t 0 t 1 p ( s ) u 2 ( s ) t 0 u ( τ ) f ( τ , x ( τ ) ) d τ d s + a υ ( t ) + l 2 u ( t ) u ( t ) t 0 t 1 p ( s ) u 2 ( s ) t 0 u ( τ ) ( h 1 ( τ ) g ( x ( τ ) ) + h 2 ( τ ) ) d τ d s + a v ( t ) + l 2 u ( t ) u ( t ) t 0 t 1 p ( s ) u 2 ( s ) t 0 u ( τ ) ( M h 1 ( τ ) + h 2 ( τ ) ) d τ d s + a υ ( t ) + l 2 u ( t ) ( M + 1 ) p ( t 0 ) u 2 ( t 0 ) β ( t 0 ) u ( t ) t 0 t 1 p ( s ) u 2 ( s ) d s + a v ( t ) + l 2 u ( t ) l 1 v ( t ) + l 2 u ( t ) ,

which means that F x X.

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

( F x 1 ) ( t ) - ( F x 2 ) ( t ) u ( t ) t 0 t 1 p ( s ) u 2 ( s ) s u ( τ ) f ( τ , x 1 ( τ ) ) - f ( τ , x 2 ( τ ) ) d τ d s u ( t ) t 0 t 1 p ( s ) u 2 ( s ) s u ( τ ) k ( τ ) v ( τ ) x 1 ( τ ) - x 2 ( τ ) d τ d s d ( x 1 , x 2 ) u ( t ) t 0 t 1 p ( s ) u 2 ( s ) s u ( τ ) k ( τ ) d τ d s d ( x 1 , x 2 ) u ( t ) t 0 t 1 p ( s ) u 2 ( s ) t 0 u ( τ ) k ( τ ) d τ d s d ( x 1 , x 2 ) v ( t ) t 0 u ( τ ) k ( τ ) d τ μ d ( x 1 , x 2 ) v ( t ) ,

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

x ( t ) - a v ( t ) - b u ( t ) u ( t ) t 0 t 1 p ( s ) u 2 ( s ) s u ( τ ) f ( τ , x ( τ ) ) d τ d s + c u ( t ) u ( t ) t 0 t 1 p ( s ) u 2 ( s ) s u ( τ ) ( M h 1 ( τ ) + h 2 ( τ ) ) d τ d s + c u ( t ) ( M + 1 ) u ( t ) t 0 t β ( s ) d s + c u ( t ) ,

where

c = x 0 u ( t 0 ) - a t 1 t 0 1 p ( s ) u 2 ( s ) d s - b .

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

f ( t , x 1 ) - f ( t , x 2 ) k ( t ) t x 1 - x 2 , t t 0 ,

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

t 0 k ( s ) d s μ ; t h i ( s ) d s β ( t ) , t t 0 , i = 1 , 2

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

t 0 t β ( s ) d s = o ( t μ ) , t .

Then for each a, b the boundary value problem

x = f ( t , x ) , t t 0 , x ( t 0 ) = x 0 , x ( t ) = a t + b + o ( t μ ) , t

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 tt2 for some t2 ≥ 0. It is easy to see that [17, 18]

u ( t ) = v ( t ) t 1 p ( s ) v 2 ( s ) d s
(2.9)

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

t 1 p ( s ) v 2 ( s ) d s 1 .

Then from (2.9), we have v (t) ≥ u(t) for tt2, which is needed in the proof of the next theorem.

Theorem 2.3. Let t0t2. Assume that the function f satisfies (2.2) and (2.3). Suppose further that

t 0 v ( s ) k ( s ) d s μ
(2.10)

and

1 p ( t ) v 2 ( t ) t v ( s ) h i ( s ) d s β ( t ) , t t 0 , i = 1 , 2
(2.11)

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

t 0 t β ( s ) d s = o ( ( u ( t ) ) μ ) , t .
(2.12)

If either

u ( t ) , t
(2.13)

or else

a = x 0 v ( t 0 ) - b t 0 1 p ( s ) v 2 ( s ) d s ,
(2.14)

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

X = x C ( t 0 , , ) | x ( t ) l 1 v ( t ) + l 2 u ( t ) , t t 0 ,

where

l 1 = ( M + 1 ) p ( t 0 ) u ( t 0 ) v ( t 0 ) β ( t 0 ) + x 0 v ( t 0 ) + b t 0 1 p ( s ) v 2 ( s ) d s and l 2 = b .

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

( F x ) ( t ) = - v ( t ) t 0 t 1 p ( s ) v 2 ( s ) s v ( τ ) f ( τ , x ( τ ) ) d τ d s + x 0 v ( t 0 ) - b t 0 1 p ( s ) v 2 ( s ) d s v ( t ) + b u ( t ) .

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

t 0 s k ( s ) d s μ ; 1 t 2 t s h i ( s ) d s β ( t ) , t t 0 , i = 1 , 2

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

1 t β ( s ) d s = o ( 1 ) , t .

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

x = f ( t , x ) , t t 0 , x ( t 0 ) = x 0 , x ( t ) = a t + b + o ( t ) , t

has a unique solution.

3. An example

Consider the boundary value problem

( t x ) = 1 t 2 arctan x + t ν , t t 0 , ν < - 2 ,
(3.1)
x ( t 0 ) = x 0 ,
(3.2)
x ( t ) = a ln t + b + o ( ( ln t ) μ ) , t .
(3.3)

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

1 + ln t 0 t 0 μ .
(3.4)

Note that since

lim t 0 1 + ln t 0 t 0 = 0

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

( t x ) = 0 , t t 0 .

Clearly, we may take

u ( t ) = 1 and v ( t ) = ln t .

Let

h 1 ( t ) = 1 t 2 , h 2 ( t ) = t ν , g ( x ) = arctan x , k ( t ) = ln t t 2 , β ( t ) = 1 t 2 ,

then it is easy to see that

f ( t , x ) 1 t 2 arctan x + t ν = h 1 ( t ) g x + h 2 ( t ) , f ( t , x 1 ) - f ( t , x 2 ) 1 t 2 x 1 - x 2 = k ( t ) v ( t ) x 1 - x 2 , t 0 k ( s ) d s = t 0 ln s s 2 d s = 1 + ln t 0 t 0 μ by (3 .4) 1 t t h 1 ( s ) d s 1 t t 1 s 2 d s = 1 t 2 = β ( t ) , t t 0 , 1 t t h 2 ( s ) d s = - t ν ν + 1 β ( t ) , t t 0 , t 0 t β ( s ) d s = t 0 t 1 s 2 d s = 1 t 0 - 1 t = o ( ( ln t ) μ ) , t , μ ( 0 , 1 ) ,

and

v ( t ) = ln t , t ,

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

x 1 ( t ) = 1 + o ( ( ln t ) μ ) , t

and

x 2 ( t ) = ln t + o ( ( ln t ) μ ) , t .

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

References

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

    Chapter  Google Scholar 

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

    Book  MATH  Google Scholar 

  3. Agarwal RP, O'Regan D: Boundary value problems on the half line in the theory of colloids. Math Probl Eng 2002, 8(2):143-150. 10.1080/10241230212905

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen SZ, Zhang Y: Singular boundary value problems on a half-line. J Math Anal Appl 1995, 195: 449-468. 10.1006/jmaa.1995.1367

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  8. Boas ML, Boas RP Jr, Levinson N: The growth of solutions of a differential equation. Duke Math J 1942, 9: 847-853. 10.1215/S0012-7094-42-00959-1

    Article  MATH  MathSciNet  Google Scholar 

  9. Haupt O: Über das asymptotische verhalten der lösungen gewisser linearer gewöhnlicher differentialgleichungen. Math Z 1942, 48: 289-292. 10.1007/BF01180019

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  12. Lipovan O: On the asymptotic behavior of the solutions to a class of second order nonlinear differential equations. Glasgow Math J 2003, 45: 179-187. 10.1017/S0017089502001143

    Article  MATH  MathSciNet  Google Scholar 

  13. Mustafa OG, Rogovchenko YV: Asymptotic integration of a class of nonlinear differential equations. Appl Math Lett 2006, 19: 849-853. 10.1016/j.aml.2005.10.013

    Article  MATH  MathSciNet  Google Scholar 

  14. Agarwal RP, Djebali S, Moussaoui T, Mustafa OG: On the asymptotic integration of nonlinear differential equations. J Comput Appl Math 2007, 202: 352-376. 10.1016/j.cam.2005.11.038

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

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Acknowledgements

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

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Ertem, T., Zafer, A. Existence of solutions for a class of nonlinear boundary value problems on half-line. Bound Value Probl 2012, 43 (2012). https://doi.org/10.1186/1687-2770-2012-43

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