Positive Solutions for Second Order Singular Boundary Value Problems with Derivative Dependence on Infinite Intervals
- PDF / 675,060 Bytes
- 39 Pages / 439.37 x 666.142 pts Page_size
- 93 Downloads / 241 Views
Positive Solutions for Second Order Singular Boundary Value Problems with Derivative Dependence on Infinite Intervals Baoqiang Yan · Donal O’Regan · Ravi P. Agarwal
Received: 24 January 2008 / Accepted: 24 January 2008 / Published online: 14 February 2008 © Springer Science+Business Media B.V. 2008
Abstract The existence of at least one positive solution and the existence of multiple positive solutions are established for the singular second-order boundary value problem
1 (p(t)x (t)) p(t)
x(0) = 0,
+ (t)f (t, x, px ) = 0, limt→+∞ p(t)x (t) = 0
0 < t < +∞,
using the fixed point index, where f may be singular at x = 0 and px = 0. Keywords Boundary value problems · Singularity · Fixed point index
1 Introduction In this paper, we consider the singular boundary value problem
1 (p(t)x (t)) p(t)
x(0) = 0,
+ (t)f (t, x(t), p(t)x (t)) = 0,
t ∈ (0, +∞),
limt→+∞ p(t)x (t) = 0,
(1.1)
where f (t, x, z) ∈ C(R + × R0+ × R0+ , R + ) may be singular at x = 0 and z = 0; here R + = [0, +∞), R0+ = (0, +∞).
The project is supported by the fund of National Natural Science (10571111) and the fund of Natural Science of Shandong Province. B. Yan Department of Mathematics, Shandong Normal University, Ji-nan, 250014, China D. O’Regan Department of Mathematics, National University of Ireland, Galway, Ireland R.P. Agarwal () Department of Mathematical Science, Florida Institute of Technology, Melbourne, FL 32901, USA e-mail: [email protected]
20
B. Yan et al.
In [6], Bobisud, using a diagonalization procedure, established the existence of bounded solutions when p(t) ≡ 1 and f is singular at x = 0 but not singular at z = 0. In [2], the authors proved the existence of at least one bounded solution of the above problem and obtained an exponential asymptotic estimate in the case when p(t) ≡ 1, f (t, x, z) = f (t, x) and f is singular at x = 0 under conditions different from [6]. Recently Yang in [18] considered the existence of at least one positive solution to (1.1) when p(t) ≡ 1 and f (t, x, z) is singular at x = 0 and z = 0. We refer the reader also to [4, 5, 7, 8, 10–12, 16, 17]. In this paper with conditions different from [2, 6] (f has no singularity at z = 0) and [18] (f is bounded at x = +∞) we establish the existence of at least one bounded positive solutions for (1.1) when f is singular at x = 0 and z = 0. Moreover, motivated by many results on the existence of multiple positive solutions for singular second-order boundary value problems on finite intervals (see [1, 3, 14, 19]) we establish the existence of multiple positive solutions for (1.1) when f is singular at x = 0 and z = 0. There are five main sections in our paper. In Sect. 2, we define a special norm for our space, construct a special cone and give its properties and list some conditions that will be needed in Sect. 3 to Sect. 6. In Sect. 3, using the theory of fixed point index on a cone, we present the existence of at least one solution and multiple solutions to (1.1) when the nonlinearity has no singularities. The result is new since o
Data Loading...
