Generalized Quasilinearization Method for Nonlinear Boundary Value Problems with Integral Boundary Conditions ∗

The quasilinearization method coupled with the method of upper and lower solutions is used for a class of nonlinear boundary value problems with integral boundary conditions. We obtain some less restrictive sufficient conditions under which corresponding monotone sequences converge uniformly and quadratically to the unique solution of the problem. An example is also included to illustrate the main result.


Introduction
In this paper, we shall consider the following boundary value problem where f : I × R → R, g i , h i : R → R are continuous and k i are nonnegative constants, i = 1, 2.
The purpose of this paper is to continue the recent ideas for problems of type (1).Concretely, we apply the quasilinearization method coupled with the method of upper and lower solutions to obtain approximate solutions to nonlinear BVP (1) assuming some appropriate properties on f, g i and h i (i = 1, 2).Then, we can show that some monotone sequences converge monotonically and quadratically to the unique solution of BVP (1) in the closed set generated by lower and upper solutions.In this work, we define the less restrictive assumptions to make it applicable to a large class of initial and boundary value problems.
As far as we know, the problem has not been studied in the available reference materials.Because of our nonlinear and integral boundary conditions, we generalize and extend some existing results.Boundary value problems with nonlinear boundary conditions have been studied by some authors, for example [2,5,6,10] and the references therein.For example, in [10], the authors studied a class of boundary value problems with the following boundary conditions and presented a quasilinearization method of the problem under a very smart assumption (see Theorem 5 of [10]).For boundary value problems with integral boundary conditions and comments on their importance, we refer the readers to the papers [3,4,7,13,15] and the references therein.Especially, in [4], Ahmad, Alsaedi and Alghamdi considered the following forced equation with integral boundary conditions It should be pointed out that in this paper, we not only quasilinearize the function f but also quasilinearize the nonlinear boundary conditions, while in [10] the nonlinear boundary conditions are not quasilinearized.Furthermore, in this paper, the convexity assumption of f is relaxed and even f ∈ C 2 is not necessary in our framework.
The paper is organized as follows.In section 2, we give some basic concepts and some preparative theorems.Then we present and prove the main result about the quasilinearization method.This is the content of Section 3.

Preliminaries
In this section, we will present some basic concepts and some preparative results for later use.
Proof.It is easy to see that a solution of BVP (2) is where ϕ(t) ≡ t 0 s 0 σ(v)dvds, and (c 1 , c 2 ) is determined by From the assumptions and using standard arguments, we may see that (c 1 , c 2 ) exists uniquely.In fact, if k 1 = 0, the strict monotonicity of the function g 1 implies that there is a unique c 1 such that g 1 (c 1 ) = 1 0 ρ 1 (s)ds, and then the strict monotonicity of the function g 2 implies that there is a unique c 2 such that g 2 (c Using the strict monotonicity of g 1 , g 2 , the left is an strictly increasing function in c 1 which implies that c 1 exists uniquely.And then the existence and uniqueness of c 2 can be obtained.Thus the proof is completed. In BVP (2), if taking g 1 (s) = g 2 (s) = s, then the condition (2) in this lemma is satisfied.The boundary conditions considered here are general.But for this general boundary value problem, we will need the existence and uniqueness of solutions in the next parts of this paper.The role of condition ( 2) is just to ensure that the unique solution exists.
Proof.Clearly, it follows from g ′ 1 > 0, g ′ 2 > 0 that the homogenous problem has only the solution y ≡ 0. Then by the Green's functions method (see for instance Theorem 3.2.1 in [19]), the associate nonhomogeneous problem (obviously, it is an equivalent form of BVP ( 2)) has a unique solution given by and Similarly, β is called an upper solution of the BVP (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 (1).
Corollary 2.1 Assume that the conditions of Theorem 2.1 and Theorem 2.2 hold.Then BVP (1) has a unique solution.

Main Result
Now, we develop the approximation scheme and show that under suitable conditions on f , g and h, there exists a monotone sequence of solutions of linear problems that converges uniformly and quadratically to a solution of the original nonlinear problem.Theorem 3.1.Assume that the conditions of Theorem 2.1 and Theorem 2.2 hold.And assume that g i , h i ∈ C 2 (R) satisfy g ′′ i (s) 0, h ′′ i (s) 0, s ∈ R.Then, there exists a monotone sequence {α n } which converges uniformly to the unique solution x of BVP (1) and the convergence is in a quadratic manner.EJQTDE, 2010 No. 66, p. 7 Proof.In view of the assumptions, by Corollary 2.1, BVP (1) has a unique solution where function F : [0, 1] × R → R is selected to be such that F (t, x), F x (t, x), F xx (t, x) are continuous on [0, 1] × R and Obviously, the function satisfying the above conditions is very easily found.F and Φ are two auxiliary functions in this proof.Using the mean value theorem and the assumptions, we obtain In particular, we consider the proof only on the set We divide the proof into two steps.
Step 1. Construction of a convergent sequence Now, set α 0 = α and consider the following BVP Then H2 (β(s); α 0 (s))ds, which implies that α 0 and β are lower and upper solutions of BVP (4), respectively.Also, it is easy to see that F , Ḡi and Hi (i = 1, 2) are such that the assumptions of Corollary 2.1.Hence, by Corollary 2.1, BVP (4) has a unique solution Furthermore, we note that which implies that α 1 is a lower solution of BVP (1).Now, consider the following BVP Again, we find that α 1 and β are lower and upper solutions of BVP (5), respectively.Also, it is easy to see that F , Ḡi and Hi (i = 1, 2) are such that the assumptions of Corollary 2.1.Hence, by Corollary 2.1, BVP (5) has a unique solution Employing the same arguments successively, we conclude that for all n and t ∈ [0, 1], where the elements of the monotone sequence {α n } are the unique solutions of the BVP EJQTDE, 2010 No. 66, p. 9 Consider the following Robin type BVP From Lemma 2.1, BVP (6) has a unique solution.It is easy to see that α n is the unique solution.Thus, we may conclude that where By similar arguments to some references, see for instance [4], employing the fact that [0,1] is compact and the monotone convergence is pointwise, it follows by the Ascoli-Arzela Theorem and Dini's Theorem that the convergence of the sequence is uniform.If x is the limit point of the sequence α n , then passing to the limit n → ∞, (7) gives Thus, x(t) is the solution of the BVP (1). Step where α n−1 (t) ξ 1 ξ 2 x(t) and α n (t) ξ 3 x(t).Since F xx 0 and f x > 0, it follows that there exists γ > 0 and an integer N such that where where α n−1 (0) ξ 4 x(0) and α n−1 (s) ξ 5 x(s).On the other hand, noticing that where α n−1 (1) ξ 6 x(1) and α n−1 (s) ξ 7 x(s).Let Now, we consider the following BVP ( From ( 8) and ( 9), it follows that e n (t) is a lower solution of BVP (10).Let then it is clear that r ′′ (t) = γr(t) − M e n−1 2 ≡ 0.
Also, if we let γ > 0 be sufficiently small, we have From ( 11) and ( 12), it follows that r(t) is an upper solution of BVP (10).Hence, by Theorem 2.2, we obtain This establishes the quadratic convergence of the iterates.Now we will illustrate the main result by the following example (which is a modified version of the example in [4]): where 0 k 1 (3/2 − c/4), 0 k 2 , 0 c < 1.It can easily be verified that α(t) = −1 and β(t) = t are the lower and super solutions of BVP (13), respectively.Also the assumptions of Theorem 3.1 are satisfied.Hence we can obtain a monotone sequence of approximate solutions converging uniformly and quadratically to the unique solution of BVP (13).
By a direct calculation, one can see that in the foregoing example, f xx does not exist.However, in many references (see for example [3,4,7,12]), the existence of f xx is an important condition.