EXISTENCE OF POSITIVE SOLUTIONS FOR MULTI-POINT BOUNDARY VALUE PROBLEMS

The existence of positive solutions are established for the multi-point bound- ary value problems � ( 1) n u (2n) (x) = �p(x)f(u(x)), 0 < x < 1 u (2i) (0) = P m=1 aju (2i) (�j), u (2i+1) (1) = P m=1 bju (2i+1) (�j), i = 0,1,...,n 1 where aj,bj 2 (0,1), j = 1,2,...,m, with 0 < P m j=1 aj < 1, 0 < Pm j=1 bj < 1, and �j 2 (0,1) with 0 < �1 < �2 < ... < �m < 1, under certain conditions on f and p using the Krasnosel'skii fixed point theorem for certain values of �. We use the positivity of the Green's function and cone theory to prove our results.

Our methods will involve those utilized in dealing with elliptic problems in annular regions.These methods have been effectively adapted for the cases when f is superlinear (i.e., f 0 = 0 and f ∞ = ∞) and when f is sublinear (i.e., f 0 = ∞ and f ∞ = 0).EJQTDE, 2007 No. 26, p. 1 A large part of the literature on multiple solutions to boundary value problems seems to be traced back to Kransnosel'skii's work on nonlinear operator equations [9], especially the part dealing with the theory of cones in Banach Spaces.In 1994, Erbe and Wang [6] applied Krasnoselskii's work to eigenvalue problems such as the one above to establish intervals of the parameter λ for which there is at least one positive solution.Many authors have used this approach or a variation thereof to obtain eigenvalue intervals.For a small sample of such work, we refer the reader to works by Davis, Henderson, Prasad and Yin [2], Eloe and Henderson [4], Eloe, Henderson and Wong [5], Erbe and Wang [6], and references therein.
The paper is organized as follow.In section 2, we are going to define the appropriate Green's function which is, later, used to define the operator of which the fixed points are the solutions of our boundary value problem ( 1) and ( 2).The Krasnosel'skii fixed point theorem stated in that section 2 will be applied in section 3 to yield positive solutions for certain intervals of eigenvalues.

Preliminary Results
In this section, we construct the Green's function and state it's properties.We also state the fixed point theorem.The following lemma helps us in constructing the Green's function for the problem (1) and (2).
So, for 0 < s ≤ 1, we have Note that, for 0 < s ≤ 1, If we continue this way, we would get that for 0 Now we are going to define the Banach Space and the cone where we will produce the solution of our boundary value problems.
Consider the Banach space The following straightforward results will be used in proving our main theorems.
Note that the operator T preserves P; i.e., T : P → P and is completely continuous.
To see this, let 0 < λ < ∞ be given and u ∈ P.
which implies that (T u) is increasing on [0, 1] and so, T u = T u(1).

EJQTDE, 2007
No. 26, p. 5 Also, Hence, T u ∈ P, that is, T : P → P.Moreover, the standard argument shows that T is completely continuous.
The fixed point theorem by Krasnosel'skii which can be found in [9] and used to obtain our results in next section is as follows.
Krasnosel'skii Fixed Point Theorem Let X be a Banach space and let K be a cone in X. Assume that Ω 1 and Ω 2 are two bounded open subsets of X with 0 ∈ Ω 1 , Ω 1 ⊂ Ω 2 , and let Φ : K ∩ (Ω 2 \ Ω 1 ) → K be a completely continuous operator such that either Then, Φ has a fixed point in K ∩ (Ω 2 \ Ω 1 ).

Main Results
Now we are going to prove the existence of at least one positive solution of the above mentioned boundary value problems.
Theorem 3. Let 0 < λ < ∞, and the function f satisfies superlinearity conditions, that is, f 0 = 0, f ∞ = ∞ and the conditions (A), (B), and (C) hold.Then the boundary value problem (1) and (2) has at least one solution belonging to the cone P.
So, by part (i) of Krasnoselskii's Fixed Point Theorem, there is a fixed point of the operator T , that belongs to P ∩ (Ω 2 − Ω 1 ).The fixed point u(x) is the desired solution of ( 1) and ( 2) for the given λ.
The next result proves the existence of at least one positive solution when f satisfies sublinearity conditions.Theorem 4. Let 0 < λ < ∞, and the function f satisfies sublinearity conditions, that is, f 0 = ∞, f ∞ = 0 and the conditions (A), (B), and (C) hold.Then the boundary value problem (1) and (2) has at least one solution belonging to the cone P.
Proof.Let u ∈ P with u = J 1 .Then, Then, for u ∈ P with u = J 2 , and 0 ≤ x ≤ 1, we have Case(b) f is unbounded.Then there exists Therefore, by part (ii) of Krasnoselskii's Fixed Point Theorem, there is a fixed point of T belonging to P ∩ (Ω 2 \ Ω 1 ) and is a solution of the problem (1) and ( 2) for given λ.
The following two results give us existence of at least one positive solution when f 0 , f ∞ exist as positive real numbers and λ satisfies certain inequality.
Theorem 5. Let f 0 , f ∞ exist as positive real numbers.Assume (A), (B) hold.Then, for each λ satisfying, 1 there is at least one solution of (1) and (2) belonging to P.
Proof.Let ε > 0 be given such that is a finite positive real number, there exists an x is a finite positive real number, there exists an Ĥ2 > 0 such that So, by part (i) of Krasnoselskii's Fixed Point Theorem, there is a fixed point of the operator T , that belongs to P ∩ (Ω 2 − Ω 1 ).The fixed point u(x) is the desired solution of (1) and (2) for the given λ.Theorem 6.Let f 0 , f ∞ exist as positive real numbers.Assume (A), (B) hold.Then, for each λ satisfying, there is at least one solution of (1) and (2) belonging to P.
Proof.Let ε > 0 be given such that x is a finite positive real number, there exists a x is finite positive real number, there exists an Ĵ2 > 0 such that f (x) ≤ (f ∞ + ε)x for x ≥ Ĵ2 .
Case(a) The function f is bounded.Let N > 0 such that f (x) ≤ N for all x ≥ 0 and J 2 = max 2J 1 , N λ So, by part (ii) of Krasnoselskii's Fixed Point Theorem, there is a fixed point of the operator T , that belongs to P ∩ (Ω 2 − Ω 1 ), say u(x) which is the desired solution of ( 1) and (2) for the given λ.

1 0
s)f (s)ds + Cx + D where C and D are arbitrary constants and D(x, s) is defined above.Then u (x) = 1 x f (s)ds + C = H(s − x)f (s)ds + C, and u (x) = −f (x).