Abstract
Consider the infinite interval nonlinear boundary value problem
where u and v are principal and nonprincipal solutions of (p(t)x')' + q(t)x = 0, r_{1}(t) = o(u(t)(v(t))^{μ}) and r_{2}(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.
Keywords:
Boundary value problem; singular; halfline; principal; nonprincipal1. Introduction
Boundary value problems on halfline occur in various applications such as in the study of the unsteady flow of a gas through semiinfinite 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 nonNewtonian fluid, etc. More examples and a collection of works on the existence of solutions of boundary value problems on halfline 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 halfline of the form
where a and b are any given real numbers, u and v are principal and nonprincipal solutions of
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
and
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
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
We should point out that u(t) = 1 and v(t) = t are principal and nonprincipal solutions of the corresponding unperturbed equation
and the function x(t) in (1.7) can be written as
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 wellknown 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:
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 ≥ t_{1 }for some t_{1 }≥ 0. It is easy to see that
is a nonprincipal solution of (1.4), which is strictly positive for t > t_{1}.
Theorem 2.1. Let t_{0 }> t_{1}. Assume that the function f satisfies
and
where g ∈ C([0, ∞), [0, ∞)) is bounded; h_{1}, h_{2}, k ∈ C([t_{0}, ∞), [0, ∞)). Suppose further that
and
for some β ∈ C([t_{0}, ∞), [0, ∞)) such that
If either
or else
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
where
and
Note that X is a complete metric space with the metric d defined by
Define an operator F on X by
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
which means that F x ∈ X.
Using (2.1), (2.3) and (2.4) we also see that
where x_{1}, x_{2 }∈ 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
where
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
where k ∈ C([t_{0}, ∞), [0, ∞)). Suppose further that
for some μ ∈ (0, 1) and β ∈ C([t_{0}, ∞), [0, ∞)), where
Then for each a, b ∈ ℝ the boundary value problem
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 ≥ t_{2 }for some t_{2 }≥ 0. It is easy to see that [17,18]
is a principal solution of (1.4) which is strictly positive. Take t_{2 }large enough so that
Then from (2.9), we have v (t) ≥ u(t) for t ≥ t_{2}, which is needed in the proof of the next theorem.
Theorem 2.3. Let t_{0 }≥ t_{2}. Assume that the function f satisfies (2.2) and (2.3). Suppose further that
and
for some β ∈ C([t_{0}, ∞), [0, ∞)) such that
If either
or else
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
where
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
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
for some μ ∈ (0, 1) and β ∈ C([t_{0}, ∞), [0, ∞)), where
If for any given a, b ∈ ℝ the condition (2.14) holds then the boundary value problem
has a unique solution.
3. An example
Consider the boundary value problem
where t_{0 }> t_{1 }= 1 and μ ∈ (0, 1) are chosen to satisfy
Note that since
for any given μ ∈ (0, 1) there is a t_{0 }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/t^{2}) arctan x + t^{υ}. The corresponding linear equation becomes
Clearly, we may take
Let
then it is easy to see that
and
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 x_{1}(t) and x_{2}(t) such that
and
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.
Acknowledgements
This work was supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under project number 108T688.
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