Existence of positive solutions to periodic boundary value problems with sign-changing Green's function
© Zhong and An; licensee Springer. 2011
Received: 27 January 2011
Accepted: 27 July 2011
Published: 27 July 2011
This paper deals with the periodic boundary value problems
where is a constant and in which case the associated Green's function may changes sign. The existence result of positive solutions is established by using the fixed point index theory of cone mapping.
Keywordsperiodic boundary value problem positive solution sign-changing Green's function cone fixed point theorem
where f is a continuous or L1-Caratheodory type function have been extensively studied. A very popular technique to obtain the existence and multiplicity of positive solutions to the problem is Krasnosel'skii's fixed point theorem of cone expansion/compression type, see for example [1–4], and the references contained therein. In those papers, the following condition is an essential assumptions:
(A) The Green function G(t, s) associated with problem (1) is positive for all (t, s) ∈ [0, T] × [0, T].
Under condition (A), Torres get in  some existence results for (1) with jumping nonlinearities as well as (1) with a repulsive or attractive singularity, and the authors in  obtained the multiplicity results to (1) when f(t, u) has a repulsive singularity near x = 0 and f(t, u) is super-linear near x = +∞. In , a special case, a(t) ≡ m2 and , was considered, the multiplicity results to (1) are obtained when the nonlinear term f(t, u) is singular at u = 0 and is super-linear at u = ∞.
Recently, in , the hypothesis (A) is weakened as
(B) The Green function G(t, s) associated with problem (1) is nonnegative for all (t, s) ∈ [0, T] × [0, T] but vanish at some interior points.
By defining a new cone, in order to apply Krasnosel'skii's fixed point theorem, the authors get an existence result when and is sub-linear at u = 0 and u = ∞ or is super-linear at u = 0 and u = ∞ with is convex and nondecreasing.
In , the author study (1) with f(t, u) = λb(t)f(u) under the following condition:
where G+ and G- are the positive and negative parts of G.
Let E denote the Banach space C[0, T] with the norm ||u|| = maxt∈[0,T]|u(t)|.
To prove our result, we need the following fixed point index theorem of cone mapping.
Lemma 1 (Guo and Lakshmikantham ). Let E be a Banach space and let K ⊂ E be a closed convex cone in E. Let L : K → K be a completely continuous operator and let i(L, K r , K) denote the fixed point index of operator L.
3 Existence result
We make the following assumptions: (H 1) f : [0, +∞) → [0, +∞) is continuous;
(H 2) 0 ≤ m = inf u∈[0,+ ∞]f (u) and M = supu∈[0,+ ∞)f (u) ≤ +∞;
and suppose that f0, f∞ ∈ [0, ∞].
It can be easily verified that u ∈ K is a fixed point of L if and only if u is a positive solution of (2).
Lemma 2. Suppose that (H1), (H2) and (H3) hold, then L : E → E is completely continuous and L(K) ⊆ K.
i.e., L(K) ⊆ K. A standard argument can be used to show that L : E → E is completely continuous.
Now we give and prove our existence theorem:
Theorem 3. Assume that (H1), (H2) and (H3) hold. Furthermore, suppose that f0 > ρ2 and f∞ < ρ2 in case of γ = +∞. Then problem (2) has at least one positive solution.
If there exist u0 ∈ K and 0 < μ0 ≤ 1 such that μ0Lu0 = u0, then (5) is valid.
4 An example
The authors are very grateful to the anonymous referee whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript.
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