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
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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