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Existence of solutions for a class of nonlinear boundary value problems on half-line
© Ertem and Zafer; licensee Springer. 2012
Received: 28 January 2012
Accepted: 16 April 2012
Published: 16 April 2012
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, 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 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).
and p ∈ C([0, ∞), (0, ∞)), q ∈ C([0, ∞), ℝ) and f ∈ C([0, ∞) × ℝ, ℝ).
where μ ∈ (0, 1) is arbitrary but fixed real numbers.
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
where t* ≥ 0 is a sufficiently large real number.
is a nonprincipal solution of (1.4), which is strictly positive for t > t1.
then there is a unique solution x(t) of (1.1)-(1.3), where r is given by (1.5).
which means that F x ∈ X.
where x1, x2 ∈ X arbitrary. This implies that F is a contracting mapping.
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.
has a unique solution.
Then from (2.9), we have v (t) ≥ u(t) for t ≥ t2, which is needed in the proof of the next theorem.
then there is a unique solution x(t) of (1.1) - (1.3), where r is given by (1.6).
Again, X is a complete metric space with the metric d defined in the proof of the previous theorem.
The remainder of the proof proceeds similarly as in that of Theorem 2.1 by using (2.2), (2.3), (2.9)-(2.14).
has a unique solution.
3. An example
for any given μ ∈ (0, 1) there is a t0 such that (3.4) holds.
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.
by taking (a, b) = (0, 1) and (a, b) = (1, 0), respectively.
This work was supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under project number 108T688.
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