POSITIVE SOLUTIONS FOR A SINGULAR FOURTH ORDER NONLOCAL BOUNDARY VALUE PROBLEM

Positive solutions are obtained for the fourth order nonlocal boundary value problem, u = f(x, u), 0 < x < 1, u(0) = u′′(0) = u′(1) = u′′(1) − u′′(2/3) = 0, where f(x, y) is singular at x = 0, x = 1, y = 0, and may be singular at y = ∞. The solutions are shown to exist at fixed points for an operator that is decreasing with respect to a cone. AMS Subject Classification: 34B16, 34B18, 34B10, 47H10


Introduction
We obtain positive solutions to the singular fourth order nonlocal boundary value problem, where f (x, y) is singular at x = 0, x = 1, y = 0, and may be singular at y = ∞.Throughout, we assume the following conditions on f : (A1) f (x, y) : (0, 1) × (0, ∞) → (0, ∞) is continuous, and f (x, y) is decreasing in y, for every x.
Equation (1), which is often referred to as the beam equation, has been studied under a variety of boundary conditions.Physical interpretations of some of the boundary conditions for the linear beam equation can be found in Zill and Wright [40].Contributions to the literature for the beam equation involving boundary conditions different from the boundary conditions (2) include the papers [14,16,17,18,26,27,35,38]. The beam equation, with the nonlocal boundary conditions like (2) has been studied by Graef et al. [13,15].Some of the results from these latter two papers will play major roles in this work.
Singular boundary value problems for ordinary differential equations have used to model glacial advance and transport of coal slurries down conveyor belts as examples of non-Newtonian fluid theory in studies of pseudoplastic fluids [9], for problems involving draining flows [1,5], and semipositone and positone problems [2], and as models in boundary layer applications, Emden-Fowler boundary value problems, and reaction-diffusion applications [6,7,8,25].
There has been substantial theoretical interest in singular boundary value problems; we suggest the studies in [4,21,22,31,32,33,36,37,39].In this work, we will convert the problem (1)-( 2) into an integral equation problem, from which we define a sequence of decreasing integral operators associated with a sequence of perturbed integral equations.Applications of a Gatica, Oliker, and Waltman [12] fixed point theorem, for operators that are decreasing with respect to a cone, yield a sequence of fixed points of the integral operators.A solution of (1)-( 2) is then obtained from a subsequence of the fixed points.

Definitions, Cone Properties and the Gatica, Oliker and Waltman Fixed Point Theorem
In this section, we state some definitions and properties of Banach space cones, and we state the fixed point theorem on which the paper's main result depends.Let (B, || • ||) be a real Banach space.A nonempty closed K ⊂ B is called a cone if the following hold: (i) αu + βv ∈ K, for all u, v ∈ K, and for all α, β ∈ [0, ∞).
Given a cone K, a partial order, ≤, is induced on B by x ≤ y, for x, y ∈ B if, and only if, y − x ∈ K. (We sometimes will write x ≤ y (w.r.t.K).)If x, y ∈ B with x ≤ y, let x, y denote the closed order interval between x and y and be defined by, x, y := {z ∈ B | x ≤ z ≤ y}.A cone K is normal in B provided there exists a δ > 0 such that ||e 1 + e 2 || ≥ δ, for all e 1 , e 2 ∈ K with Remark 2.1.If K is a normal cone in B, then closed order intervals are norm bounded.
We now state the Gatica, Oliker, and Waltman [12] fixed point theorem on which the main result of this paper depends.Theorem 2.2.Let B be a Banach space, K a normal cone, J a subset of K such that, if x, y ∈ J, x ≤ y, then x, y ⊆ J, and let T : J → K be a continuous decreasing mapping which is compact on any closed order interval contained in J. Suppose there exists x 0 ∈ J such that T 2 x 0 is defined, and furthermore, T x 0 and T 2 x 0 are order comparable to x 0 .
Then T has a fixed point in J provided that, either (I) T x 0 ≤ x 0 and T 2 x 0 ≤ x 0 , or T x 0 ≥ x 0 and T 2 x 0 ≥ x 0 , or (II) The complete sequence of iterates {T n x 0 } ∞ n=0 is defined, and there exists y 0 ∈ J such that y 0 ≤ T n x 0 , for every n.
We shall also make extensive use of the following theorem due to Graef, Qian and Yang [15].
We will assume hereafter: (A3) It follows from Theorem 2.3 that, for each positive solution u(x) of ( 1)-( 2), there exists a θ > 0 such that Next, we define a subset D ⊂ K by We observe that, for each v ∈ D and and for each positive solution u(x) of ( 1)-( 2), There is a Green's function, G(x, s), for y (4) = 0 satisfying (2) which will play the role of a kernel for certain compact operators meeting the requirements of Theorem 2.2.
First, the Green's function G 1 (x, s) for is given by and second, the Green's function G 2 (x, s) for is given by Both G 1 and G 2 are positive valued on (0, 1] × (0, 1).
Remark 3.1.Graef, Kong, and Yang [13] by direct computation have also given the closed form expression where H(•) denotes the Heaviside function.

Now we define an integral operator
We shall show that T is well-defined on D, is decreasing, and T : D → D. To that end, let v, u ∈ D be given, with v(x) ≤ u(x).Then, there exists θ > 0 such that g θ (x) ≤ v(x).By Assumptions (A1) and (A3), and the positivity of G, Therefore, T is well-defined on D and T is a decreasing operator.

A priori Bounds on Norms of Solutions
In this section, we exhibit that solutions of (1)-( 2) have positive a priori upper and lower bounds on their norms.From (4) or (5), Therefore, Next, let M > 0 be defined by (A2) implies there exists m 0 ∈ N such that, for each m ≥ m 0 and Then, for m ≥ m 0 , So, for m ≥ m 0 and 0 ≤ x ≤ 1, we have which, in view of (A3), contradicts lim m→∞ ||u m || = ∞.Therefore, there exists an S > 0 such that ||u|| ≤ S, for any solution u ∈ D of (1)- (2).
Following that, we now exhibit positive a priori lower bounds on the solution norms.

Existence of Positive Solutions
In this section, we will construct a sequence of operators, {T m } ∞ m=1 , each of which is defined on all of K. Applications of Theorem 2.2 yield that, for each m ∈ N, T m has a fixed point φ m ∈ K.Then, we will extract a subsequence from the fixed points {φ m } ∞ m=1 that converges to a fixed point of the operator T .Since f is decreasing with respect to its second component, we have 0 < u m+1 (x) < u m (x), for 0 < x < 1, and by (A2), lim m→∞ u m (x) = 0 uniformly on [0, 1].
Next, we define f m (x, y) Then, f m is continuous and f m does not have the singularity at y = 0 possessed by f .Moreover, for (x, y) ∈ (0, 1) × (0, ∞), Next, we define a sequence of operators, T m : K → K, for φ ∈ K and 0 ≤ x ≤ 1, by Then standard arguments yield that each T m is a compact operator on K. Furthermore, and Theorem 2.2 implies, with J = K and x 0 = 0, that T m has a fixed point in K, for each m.That is, for each m, there exists φ m ∈ K such that So, for each m ≥ 1, φ m satisfies the boundary conditions (2), and also, That is, for each 0 ≤ x ≤ 1 and for each m, Using arguments similar to those in the proofs of Lemmas 4.1 and 4.2, there exist R > 0 and S > 0 such that R ≤ ||φ m || ≤ S, for every m.Now, let θ := R. Since φ m belongs to K and is a fixed point of T m , the conditions of Theorem 2.3 hold.So, for every m and 0 ≤ x ≤ 1, So, the sequence {φ m } ∞ m=1 is contained in the closed order interval g θ , S , and therefore, the sequence is contained in D. Since T is a compact mapping, we may assume without loss of generality that lim m→∞ T φ m exists; let us call the limit φ * .
To complete the proof, it suffices to show that lim m→∞ uniformly on [0, 1], from which it will follow that φ * ∈ g θ , S .

Dependence on Higher Order Derivatives
The techniques of proof of Theorem 5.1 can be extended to a boundary value problem of the form with boundary conditions (2) using methods developed by Henderson and Yin [24] if one extends Theorem 2.3 in the following way.
The Banach space for this section is