Electronic Journal of Qualitative Theory of Differential Equations

In this paper, some nonexistence, existence and multiplicity of positive solutions are established for a class of singular boundary value problem. The authors also obtain the relation between the existence of the solutions and the parameter λ. The arguments are based upon the fixed point index theory and the upper and lower solutions method.

In the special cases i) f (t, x) = q(t)x −λ 1 , λ 1 > 0, and ii) f (t, x) = q(t)x λ 1 , 0 < λ 1 < 1, where q > 0 for t ∈ (0, 1), the existence and uniqueness of positive solutions for the BVP (1.2) as λ = 1 have been studied completely by Taliaferro in [1] with the shooting method and by Zhang [2] with the sub-super solutions method, respectively.In the special case iii)f (t, x) = q(t)g(x), q(t) is singular only at t = 0 and g(x) ≥ e x , the existence of multiple positive solutions for the BVP (1.2) have been studied by Ha and Lee in [3] with the sub-super solutions method.In the special case iv) f (t, x) = q(t)g(x), q(t) is singular only at t = 0 and g(x) ∈ C(−∞, +∞), [0.+ ∞), the existence of multiple positive solutions for the BVP (1.2) have been studied by Wong in [4] with the shooting method.
Motivated by the results mentioned above, in this paper we study the existence, multiplicity, and nonexistence of positive solutions for the BVP (1.1) by new technique(different from [3,5,11,12,13,14]) to overcome difficulties arising from the appearances of p(t) and p(t) is singular at t = 0 and t = 1.On the other hand, to the best of our knowledge, there are very few literatures considering the existence, multiplicity, and nonexistence of positive solutions for the case when p(t) is singular at t = 0 and t = 1.The arguments are based upon the fixed point index theory and the upper and lower solutions method.
Fixed point index theorems have been applied to various boundary value problems to show the existence of multiple positive solutions.An overview of such results can be found in Guo and Lakshmikantham V., [16] and in Guo and Lakshmikantham V., Liu X.Z., [17] and in Guo, [18] and in K. Deimling, [19] and in M. Krasnoselskii, [20].
The paper is organized in the following fashion.In Section 2, we provide some necessary background.In particular, we state some properties of the Green , s function associated with the BVP (1.1) λ .In Section 3, the main results will be stated and proved.Finally some examples are worked out to demonstrate our main results in this section.
Let G(t, s) be Green , s function of the following BVP Then G(t, s) is defined by where ∆ = ad + ac 1 0 dr p(r) + bc.It is easy to prove that G(t, s) has the following properties.Property 2.1.For all t, s ∈ [0, 1] we have where , It is easy to see that 0 < σ 0 < 1. Definition 2.1.Letting x(t) ∈ C[0, 1] ∩ C 1 (0, 1), we say x(t) is a lower solution for the BVP (1.1) λ if x(t) satisfies : 1), we say y(t) is an upper solution for the BVP (1.1) λ if y(t) satisfies : Firstly, we consider the following BVP: In order to prove the following results we define a cone by Where σ 0 is given by (2.5), θ ∈ (0, 1 2 ).It is easy to see that Q is a closed convex cone of E and Q ⊂ P .Lemma 2.2.Let (H 1 ) − (H 3 ) be satisfied.Then T 0 λ (Q) ⊂ Q and T 0 λ : Q → Q is completely continuous and nondecreasing.Proof.For any x ∈ P , we have by (2.2) and (2.7) On the other hand, for any t ∈ J θ , we have by (2.4) and (2.7) Proof.For any λ ∈ S, let x λ is a solution of the BVP (1.1) λ .Then we have the result is easily obtained.On the other hand, if ||x λ || ≥ 1, then we have by (H 3 )  [3]) Suppose f : [0, +∞) → (0, +∞) is continuous and increasing.
EJQTDE, 2006 No. 13, p. 5 Let λ * = supS.Now we prove λ * < +∞.If not, then we must have N ⊂ S, where N denotes natural number set.Therefore, for any n ∈ N, by Lemma 2.1, there exists x n ∈ Q satisfying G(s, s)p(s)g(s)ds] −1 and x n ≥ 1.Then, by Lemma 2.1 and (H 3 ), we have Let xλ be a solution of the BVP (1.1)λ.Suppose xλ (t) = xλ + ρ, ρ ∈ (0, ρ 0 ).Then Hence for any µ ≥ 1, we have T 0 λ y = µy, y ∈ ∂Ω.Therefore by Lemma 1.1 we have It remains to prove that the conditions of Lemma 1.2 are held.Firstly, we check the condition (1) of Lemma 1.2 is satisfied.In fact, for any x ∈ Q, by (H 4 ) and (2.5) we have Taking R > 0, such that Rm−1 1−θ θ G( 1 2 , s)λp(s)g(s) δσ m 0 ds > 1.Therefore, for any R > R and B R ⊂ Q, we have by (3.3) where B R = {x ∈ Q|||x|| < R}.Hence the condition (1) of Lemma 1.2 is held.Now we prove the condition (2) of Lemma 1.2 is satisfied.In fact, if the condition (2) of Lemma 1.2 is not held, then there exist x 1 ∈ Q ∂B R , 0 < µ 1 ≤ 1, such that T 0 λ x 1 = µ 1 x 1 .Therefore ||T 0 λ x 1 || ≤ ||x 1 ||.This conflicts with (3.4).Hence the condition (2) of Lemma 1.2 is satisfied.By Lemma 1.2 we have Consequently, by the additivity of the fixed point index, Therefore, by the solution property of the fixed point index, there is a fixed point of T 0 λ in Ω and a fixed point of T 0 λ in B R \ Ω, respectively.Therefore the BVP (1.1) λ by Lemma 2.1 has at least two solutions.