Multiple positive solutions for nonlinear third order general two-point boundary value problems

We consider the existence of positive solutions and multiple positive solutions for the third order nonlinear differential equation subject to the general two-point boundary conditions using different fixed point theorems.


Introduction
In this paper we consider the existence of positive solutions to the third order nonlinear differential equation, where the coefficients α 11 , α 12 , α 21 , α 22 , α 31 , α 32 are positive real constants.The BVPs of this form arise in the modeling of nonlinear diffusions generated by nonlinear sources, in thermal ignition of gases, and in concentration in chemical or biological problems.In these applied settings, only positive solutions are meaningful.
There is much current attention focussed on existence of positive solutions to the boundary value problems for ordinary differential equations, as well as for the finite difference equations; see [5,6,8,9] to name a few.The book by Agarwal, Wong and O'Regan [1] gives a good overview for much of the work which has been done and the methods used.Shuhong and Li [22] obtained the existence of single and multiple positive solutions to the nonlinear singular third-order two-point boundary value problem y (t)+λa(t)f (y(t)) = 0, 0 < t < 1 y(0) =y (0) = y (1) = 0 by using Krasnosel'skii fixed point theorem [16].In [23], Sun and Wen considered the existence of multiple positive solutions to third order equation, y (t) = a(t)f (y(t)), 0 < t < 1 under the boundary conditions αy (0) − βy (0) = 0, y(1) = y (1) = 0.
We extend these results to general two-point boundary value problems in the interval [a, b], where b > a ≥ 0. We use the following notation for simplicity, We assume that throughout the paper: We define the nonnegative extended real numbers f 0 , f 0 , f ∞ and f ∞ by and assume that they will exist.We note that f 0 = 0 and f ∞ = ∞ correspond to the superlinear case, and f 0 = ∞ and f ∞ = 0 correspond to the sublinear case.By the positive solution of ( 1)-( 2) we mean that y(t) is positive on [a, b] and satisfies the differential equation ( 1) and the boundary conditions (2).This paper is organized as follows.In Section 2, as a fundamental importance, we estimate the bounds for Green's function corresponding to the BVP (1)- (2).In Section 3, we present a lemma which is needed in our main result and establish a criterion for the existence of at least one positive solution for the BVP (1)-(2) by using Krasnosel'skii fixed point theorem [16].In Section 4, some existence criteria for at least three positive solutions to the BVP (1)-( 2) are established by using the well-known Leggett-Williams fixed point theorem [18].And then, for arbitrary positive integer m, existence results for at least 2m − 1 positive solutions are obtained.Finally as an application, we give two examples to demonstrate our results.

Green's Function and bounds
In this section, we estimate the bounds of the Green's function for the homogeneous two-point BVP corresponding to (1)- (2).The Green's function for the homogeneous problem −y = 0, satisfying the boundary conditions (2) can be constructed after computation and is given by .
And also, for s < t, G(t, s) attains minimum value at EJQTDE, 2009, No. 9, p. 3 The above two Lemmas can be proved easily by considering minimum and maximum of function of two variables.
Theorem 2.3 Let G(t, s) be the Green's function for the homogeneous BVP corresponding to (1)-( 2), then where Proof: The Green's function G(t, s) for the homogeneous problem of the BVP (1)-( 2) is given in (3).Clearly First we establish the right side inequality by assuming the conditions given by (A2)-(A3).For t < s, and for s < t, Hence, G(t, s) ≤ G(s, s).
By assuming the conditions given by (A2)-(A4), we establish the other inequality.For t < s, from Lemma 2.1 and Lemma 2.2, we have and for s < t, we have Therefore, where 0 In this section, first we prove a lemma which is needed in our main result and establish criteria for the existence of at least one positive solution of the BVP (1)- (2).
Let y(t) be the solution of the BVP (1)-( 2), and be given by Define with norm Then (X, .) is a Banach space.Define a set κ by then it is easy to see that κ is a positive cone in X.
Define the operator T : κ → X by If y ∈ κ is a fixed point of T , then y satisfies (6) and hence y is a positive solution of the BVP (1)- (2).We seek the fixed points of the operator T in the cone κ.
Lemma 3.1 The operator T defined by ( 8) is a self map on κ.
Proof: If y ∈ κ, then by ( 4) EJQTDE, 2009, No. 9, p. 5 Moreover, if y ∈ κ, Therefore, min Also, from the positivity of G(t, s), it clear that for y ∈ κ, that (T y)(t) ≥ 0, a ≤ t ≤ b, and so T y ∈ κ; thus T : κ → κ.Further arguments yields that T is completely continuous. 2 The existence of at least one positive solution of ( 1)-( 2) is based on an application of the following fixed point theorem [16].
Theorem 3.2 (Krasnosel'skii) Let X be a Banach space, K ⊆ X be a cone, and suppose that Ω 1 , Ω 2 are open subsets of X with 0 ∈ Ω 1 and Suppose further that T : K ∩ (Ω 2 \Ω 1 ) → K is completely continuous operator such that either holds.Then T has a fixed point in K ∩ (Ω 2 \Ω 1 ).
Proof: Let T be the cone preserving, completely continuous operator defined as in (8).Since f 0 = 0, we may choose where η 1 > 0 satisfies Thus, if y ∈ κ and y = H 1 , then we have Therefore, T y ≤ y .
Proof: Let T be the cone preserving, completely continuous operator defined as in (8).Since f 0 = ∞, we choose where η 1 γ 2 b a G(s, s)ds ≥ 1.Then for y ∈ κ and y = J 1 , we have Now, since f ∞ = 0, there exists where η 2 > 0 satisfies It follows that We consider two cases: In this case, we may choose G(s, s)ds , so that y ∈ κ with y = J 2 , we have and therefore T y ≤ y .

Existence of Multiple Positive Solutions
In this section, we establish the existence of at least three positive solutions to the BVP (1)-( 2).And then, for an arbitrary positive integer m, existence of at least 2m − 1 positive solutions are obtained.
Let E be a real Banach space with cone P .A map S : P → [0, ∞) is said to be a nonnegative continuous concave functional on P , if S is continuous and S(λx for all x, y ∈ P and λ ∈ [0, 1].Let α and β be two real numbers such that 0 < α < β and S be a nonnegative continuous concave functional on P .We define the following convex sets P α = {y ∈ P : y < α}, EJQTDE, 2009, No. 9, p. 10 and P (S, α, β) = {y ∈ P : α ≤ S(y), y ≤ β}.
We now state the famous Leggett-Williams fixed point theorem [18].
Theorem 4.1 Let T : P a 3 → P a 3 be completely continuous and S be a nonnegative continuous concave functional on P such that S(y) ≤ y for all y ∈ P a 3 .Suppose that there exist 0 < d < a 1 < a 2 ≤ a 3 such that the following conditions hold.
Then, T has at least three fixed points y 1 , y 2 , y 3 in P a 3 satisfying convenience, we let Theorem 4.2 Assume that the conditions (A1) − (A4) are satisfied and also there exist real numbers d 0 , d 1 and c with 0 Then the BVP (1)-( 2 It is easy to check that S is a nonnegative continuous concave functional on P with S(y) ≤ y forall y ∈ P .Further the operator T : P → P is a completely continuous by an application of the Ascoli-Arzela theorem [12] and the fixed points of T are the solutions of the BVP (1)-( 2).First, we prove that if there exists a positive number r such that f (t, y(t)) < r D for y ∈ [0, r], then T : P r → P r .Indeed, if y ∈ P r , then for t ∈ [a, b], Thus, T y < r, that is, T y ∈ P r .Hence, we have shown that if ( 13) and ( 15 Then, the BVP (1)-( 2) has at least 2m − 1 positive solutions in P dm .
EJQTDE, 2009, No. 9, p. 13 Proof: We use induction on m.First, for m = 1, we know from ( 16) that T : P d 1 → P d 1 , then, it follows from Schauder fixed point theorem that the BVP (1)-( 2) has at least one positive solution in P d 1 .
Next, we assume that this conclusion holds for m = k.In order to prove that this conclusion holds for m = k + 1, we suppose that there exist numbers d i (1 ≤ i ≤ k + 1) and a j (1 ≤ j ≤ k) with 0 < d 1 < a w(t) < a k .

Examples
Now, we give some examples to illustrate the main results.

y + y 2 ( 1
to see that all the conditions of Theorem 3.3 hold.It follows from Theorem 3.3, the BVP (20) has at least one positive solution.