QUASILINEARIZATION METHOD AND NONLOCAL SINGULAR THREE POINT BOUNDARY VALUE PROBLEMS

The method of upper and lower solutions and quasilinearization for nonlinear singular equations of the type −x ′′ (t) + �x ′ (t) = f(t,x(t)), t ∈ (0, 1),


Introduction
Nonlocal singular boundary value problems (BVPs) have various applications in chemical engineering, underground water flow and population dynamics.These problems arise in many areas of applied mathematics such as gas dynamics, Newtonian fluid mechanics, the theory of shallow membrane caps, the theory of boundary layer and so on; see for example, [2,7,12,13,16] and the references therein.An excellent resource with an extensive bibliography was produced by Agarwal and O'Regan [1].Existence theory for nonlinear multi-point singular boundary value problems has attracted the attention of many researchers; see for example, [3,4,5,14,15,17,18] and the references therein.
For the existence theory, we develop the method of upper and lower solutions and to approximate the solution of the BVP (1), we develop the quasilinearization technique [5,8,9,10,11].We obtain a monotone sequence of solutions of linear problems and show that, under suitable conditions on f , the sequence converges uniformly and quadratically to a solution of the original nonlinear problem (1).
We recall the concept of upper and lower solutions for the BVP (2).
Theorem 2.1.Assume that there exist lower and upper solutions α, β ∈ C[0, 1] ∩ C 2 (0, 1) of the BVP (2) such that α(1) = β(1), and 0 < α ≤ β on [0, 1), and α(0) − δα(η) < β(0) − δβ(η).Assume that f : (0, 1) × R \ {0} → (0, ∞) is continuous and there exists h(t) such that e −λt h(t) ∈ L 1 [0, 1] and where ᾱ = min{α(t Proof.Let {a n }, {b n } be two monotone sequences satisfying for sufficiently large n, and choose two null sequences {τ n } and Define a partial order in Define a modification F of f with respect to α, β as follows: Clearly, F is continuous and bounded on (0, 1) × C[0, 1].For each n ∈ N, consider the nonsingular modified problems We write the BVP ( 9) as an equivalent integral equation Clearly, G n (t, s) → G(t, s) as n → ∞.By a solution of (10), we mean a solution of the operator equation where I is the identity and for each Since F is continuous and bounded on [ Firstly, we show that α Hence, z has a negative minimum at some point However, in view of the definition of F and that of lower solution, we obtain a contradiction.Hence z has no negative local minimum.

Approximation of solution
We develop the approximation technique (quasilinearization) and show that under suitable conditions on f , there exists a bounded monotone sequence of solutions of linear problems that converges uniformly and quadratically to a solution of the nonlinear original problem.Choose a function Φ(t, x) such that Φ, Φ x , Here, we do not require the condition that where β = max{β(t) : t ∈ [0, 1]}.(A 2 ) f, f x , f xx ∈ C((0, 1) × R) and there exist h 1 , h 2 , h 3 such that e −λt h i ∈ L 1 [0, 1] and for |x| ≤ β, t ∈ (0, 1), i = 0, 1, 2.Moreover, f is non-increasing in x for each t ∈ (0, 1).
Then, there exists a monotone sequence {w n } of solutions of linear problems converging uniformly and quadratically to a unique solution of the BVP (2).
Proof.The conditions (A 1 ) and (A 2 ) ensure the existence of a C 1 positive solution x of the BVP (2) such that For t ∈ (0, 1), using (16), we obtain where x, y ∈ (0, β].The mean value theorem and the fact that Φ x is increasing in x on [0, β] for each t ∈ [0, 1], yields where x, y ∈ [0, β] such that y ≤ c ≤ x.Substituting in (17), we have We note that g(t, x, y) is continuous on (0, 1) × R × R \ {0}.Moreover, for every t ∈ (0, 1) and x, y ∈ (0, β], g satisfies the following relations Moreover, for every t ∈ (0, 1) and x, y ∈ (0, β], using mean value theorem, we have where y < c < β.Consequently, in view of (A 2 ), we obtain , for every t ∈ (0, 1) and x, y ∈ (0, β], where Now, we develop the iterative scheme to approximate the solution.As an initial approximation, we choose w 0 = α and consider the linear problem Using ( 21) and the definition of lower and upper solutions, we get which imply that w 0 and β are lower and upper solutions of (23) respectively.Hence by Theorem 2.1, there exists a C 1 positive solution Using (21) and the fact that w 1 is a solution of (23), we obtain which implies that w 1 is a lower solution of (2).Similarly, in view of (A 1 ), ( 21) and (24), we can show that w 1 and β are lower and upper solutions of Hence by Theorem 2.1, there exists a C 1 positive solution Continuing in the above fashion, we obtain a bounded monotone sequence {w n } of C 1 [0, 1] positive solutions of the linear problems satisfying where the element w n of the sequence is a solution of the linear problem −x ′′ (t) + λx ′ (t) = g(t, x(t), w n−1 (t)), t ∈ (0, 1) x(0) = δx(η), x(1) = 0 and for each t ∈ (0, 1), is given by Hence w satisfy the boundary conditions.Moreover, from (22), the sequence {g(t, w n , w n−1 )} is uniformly bounded by h 3 (t) ∈ L 1 [0, 1] on (0, 1).Hence, the continuity of the function g on (0, 1) × (0, β] × (0, β] and the uniform boundedness of the sequence EJQTDE Spec.Ed.I, 2009 No. 17 {g(t, w n , w n−1 )} implies that the sequence {g(t, w n , w n−1 )} converges pointwise to the function g(t, w, w) = f (t, w).By Lebesgue dominated convergence theorem, for any t ∈ (0, 1), G(η, s)f (s, w(s))ds, t ∈ (0, 1); that is, w is a solution of (2).Now, we show that the convergence is quadratic.Set v n (t) = w(t)−w n (t), t ∈ [0, 1], where w is a solution of (2).Then, v n (t) ≥ 0 on [0, 1] and the boundary conditions imply that v n (0) = δv n (η) and v n (1) = 0. Now, in view of the definitions of F and g, we obtain Using the mean value theorem repeatedly and the fact that   (32) gives quadratic convergence.