The generalized method of quasilinearization and nonlinear boundary value problems with integral boundary conditions, Electron

The generalized method of quasilinearization is applied to obtain a mono- tone sequence of iterates converging uniformly and rapidly to a solution of second order nonlinear boundary value problem with nonlinear integral boundary conditions.


Introduction
In this paper, we shall study the method of quasilinearization for the nonlinear boundary value problem with integral boundary conditions where f : J × R → R and h i : R → R (i = 1, 2) are continuous functions and k i are nonnegative constants.Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems.They include two, three, multipoint and nonlocal boundary value problems as special cases.For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers [13,14,15] and the references therein.Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors, for example [11,12,16,17].
The purpose of this paper is to develop the method of quasilinearization for the boundary value problem (1.1).The main idea of the method of quasilinearization as developed by Bellman and Kalaba [1] and generalized by Lakshmikantham [4,5] has been applied to a variety of problems [3,6,7].Recently, Eloe and Gao [8], Ahmad, Khan and Eloe [9] have developed the quasilinearization method for three point boundary value problems.
More recently, Khan, Ahmad [10] developed the method to treat first order problems In the present paper we extend the method of generalized quasilinearization to the boundary value problem (1.1) and we obtain a sequence of solutions converging uniformly and rapidly to a solution of the problem.

Preliminaries
We know that the homogeneous problem has only the trivial solution.Consequently, for any σ(t), ρ 1 (t), ρ 2 (t) ∈ C[0, 1], the corresponding nonhomogeneous linear problem where is the unique solution of the problem is the Green's function of the problem.We note that G(t, s) < 0 on (0, 1) × (0, 1).
Similarly, β is an upper solution of the BVP (1.1), if β satisfies similar inequalities in the reverse direction.Now, we state and prove the existence and uniqueness of solutions in an ordered interval generated by the lower and upper solutions of the boundary value problem.
Theorem 2.2.Assume that α and β are respectively lower and upper solutions of (1.1) continuous and h i (x) ≥ 0, then there exists a solution x(t) of the boundary value problem (1.1) such that Proof.Define the following modifications of f (t, x) and h i (x)(i = 1, 2) and Consider the modified problem which imply that α and β are respectively lower and upper solutions of (2.1).Also, we note that any solution of (2.1) which lies between α and β, is a solution of (1.1).Thus, we only need to show that any solution x(t) of (2.1) is such that a contradiction.If t 0 = 0, then k(0) > 0 and k (0) ≤ 0, but then the boundary conditions and the nondecreasing property of h i gives ) and again k(0) ≤ 0, another contradiction.Similarly, if t 0 = 1, we get a contradiction.Thus α(t) ≤ x(t), t ∈ J.
Similarly, we can show that Theorem 2.3.Assume that α and β are lower and upper solutions of the boundary value problem Using the increasing property of the function f (t, x) in x, we obtain a contradiction.If t 0 = 0, then m(0) > 0 and m (0) ≤ 0. On the other hand, using the boundary conditions (2.2) and the assumption 0 ≤ h 1 (x) < 1, we have a contradiction.If t 0 = 1, then m(1) > 0 and m (1) ≥ 0. But again, the boundary conditions (2.2) and the assumption 0 a contradiction.Hence As a consequence of the theorem (2.3), we have Then, there exists a monotone sequence {w n } of solutions converging uniformly and quadratically to the unique solution of the problem. Proof.Define, For any t ∈ [0, 1], using Taylor's theorem, (3.1) and (A 3 ), we have and where x, y ∈ R. Again applying Taylor's theorem to φ(t, x), we can find ξ ∈ R with which in view of (A 2 ) implies that where x − y = max t∈[0,1] {|x(t) − y(t)|} denotes the supremum norm in the space of continuous functions on [0, 1].Using (3.6) in (3.2), we obtain Now, set w 0 = α and consider the linear problem (3.12) Using (A 1 ), (3.10) and (3.11), we obtain which imply that w 0 and β are respectively lower and upper solutions of (3.12).It follows by theorems 2.2 and 2.3 that there exists a unique solution w 1 of (3.12) such that In view of (3.10), (3.11) and the fact that w 1 is a solution of (3.12), we note that w 1 is a lower solution of (1.1).EJQTDE, 2003 No. 10, p. 7 Now consider the problem Again we can show that w 1 and β are lower and upper solutions of (3.13) and hence by theorems (2.2, 2.3), there exists a unique solution w 2 of (3.13) such that Continuing this process, we obtain a monotone sequence {w n } of solutions satisfying where, the element w n of the sequence {w n } is a solution of the boundary value problem Employing the standard arguments [2], it follows that the convergence of the sequence is uniform.If x(t) is the limit point of the sequence, passing to the limit as n → ∞, (3.14) gives where we obtain In view of (A 3 ), there exist λ i < 1 and (3.16) Further, using Taylor's theorem, (A 2 ), (3.5) and (3.9), we obtain where From (3.16) and (3.17), it follows that EJQTDE, 2003 No. 10, p. 9 where, r(t) ≥ 0 is the unique solution of the boundary value problem Thus, r(t where l is a bound for where, δ = L 1−λ .

Rapid convergence
Theorem 4.1.Assume that are lower and upper solutions of (1.1) respectively such that α(t) ≤ , and Then, there exists a monotone sequence {w n } of solutions converging uniformly to the unique solution of the problem.Moreover the rate of convergence is of order k ≥ 2.
Using (B 3 ), Taylor's theorem and (4.1), we have Expanding φ(t, x) about (t, y) by Taylor's theorem, we can find y ≤ ξ ≤ x, such that which in view of (B 2 ) implies that Using ( x ∈ [α, β]} and define on Ω the functions (4.7) Then, we note that g * (t, x, y) and H * j (x, y) are continuous, bounded and are such that Now, set α = w 0 and consider the linear problem H * 2 (x(s), w 0 (s))ds. (4.11) The assumption (B 1 ) and the expressions (4.9), (4.10) yields H * 1 (w 0 (s), w 0 (s))ds, H * 2 (w 0 (s), w 0 (s))ds and H * 1 (β(s), w 0 (s))ds, imply that w 0 and β are respectively lower and upper solutions of (4.11).Hence by theorems (2.2, 2.3), there exists a unique solution w 1 of (4.11) such that Continuing this process, we obtain a monotone sequence {w n } of solutions satisfying where the element w n of the sequence {w n } is a solution of the boundary value problem In view of (B 3 ), we have It follows that, we can find λ < 1 such that p j (t) ≤ λ, t ∈ [0, 1], (j = 1, 2) and hence e n (0) − k 1 e n (0) ≤ λ

2000
Mathematics Subject Classification.Primary 34A45, 34B15, secondary .Key words and phrases.Quasilinearization, integral boundary value problem, upper and lower solutions, rapid convergence Financial support by the MoST Govt. of Pakistan is gratefully acknowledged.EJQTDE, 2003 No. 10, p. 1 with integral boundary conditions