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International Journal of Differential Equations
Volume 2013 (2013), Article ID 728149, 6 pages
Picard Type Iterative Scheme with Initial Iterates in Reverse Order for a Class of Nonlinear Three Point BVPs
Department of Mathematics, BITS Pilani, Pilani, Rajasthan 333031, India
Received 30 April 2013; Accepted 23 June 2013
Academic Editor: Yuji Liu
Copyright © 2013 Mandeep Singh and Amit K. Verma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the following class of three point boundary value problem , , where , , the source term is Lipschitz and continuous. We use monotone iterative technique in the presence of upper and lower solutions for both well-order and reverse order cases. Under some sufficient conditions, we prove some new existence results. We use examples and figures to demonstrate that monotone iterative method can efficiently be used for computation of solutions of nonlinear BVPs.
In recent years, multipoint boundary value problems have been extensively studied by many authors ([1–5] and the references there in). Multipoint BVPs have lots of applications in various branches of science and engineering; for example, Webb  studied a second-order nonlinear boundary value problem subject to some nonlocal boundary conditions, which models a thermostat, and Zou et al.  studied the design of a large size bridge with multipoint supports.
It is well known that one of the most important tools for dealing with existence results for nonlinear problems is the method of upper and lower solutions. The method of upper and lower solutions has a long history and some of its ideas can be traced back to Picard . Later, it was extensively studied by Dragoni .
Recently, there have been numerous results in the presence of an upper solution and a lower solution with . But, in many cases, the upper and lower solutions may occur in the reversed order also, that is, . Cabada et al.  considered the monotone iterative method for the following BVP: with reversed ordered upper and lower solutions. So far, there have been some results in the presence of reverse ordered upper and lower solutions [10–13]. Xian et al.  considered the following second-order three point BVP: where , , , . They used the fixed point index theory with non-well-ordered upper and lower solutions.
Recently, Li et al.  studied the existence and uniqueness of solutions of second-order three point BVP with upper and lower solutions in the reversed order via the monotone iterative method in Banach space.
The present work proves some new existing results for three point BVPs. Our technique is based on Picard-type iterative scheme and is quite simple and efficient from computational point of view. We believe that it can be very well adapted for this type of problem. In this paper we consider the following three point BVP: where , , , . We have allowed to take both negative and positive values.
The paper is divided into 4 sections. In Section 2, we construct Green's function and establish maximum and antimaximum principle. In Section 3, we generate monotone sequences by using results of Section 2 with upper and lower solutions as initial iterates ordered in one way or the other. We prove our final result of existence. In Section 4, we show that the monotone iterative scheme is a powerful technique. For that by using iterative scheme proposed in this paper we have computed the members of sequences in both cases (well-ordered and non-well-ordered case).
2.1. Construction of Green's Function
To investigate (4), we consider the following linear three point BVP: where and is any constant. In this section, we construct the Green's function. We divide it into two cases.
Case I (). Let us assume that, , .
It is easy to see that can be satisfied.
Lemma 1. Green's function for the following linear three point BVP is given by and if holds then .
Proof. See proof of Lemma 2.1 in .
Lemma 2. When , is a solution of boundary value problem (5) and is given by
Proof. See proof of Lemma 2.3 in .
Remark 3. Particulary is a solution of the boundary value problem (5) if and only if is a solution of the integral equation
Case II (). Assume that, , .
It is easy to see that can be satisfied.
Lemma 4. Green's function for the following three point BVP for is given by where and if holds then .
Proof. Proof is same as given in Lemma 1.
Lemma 5. When , is a solution of boundary value problem (5) and is given by
Proof. Proof is same as given in Lemma 2.
2.2. Maximum and Antimaximum Principle
Proposition 6 (antimaximum principle). Let , , , , and is such that ; then is nonpositive on .
Proposition 7 (maximum principle). Let , , , and is such that ; then is nonnegative on .
3. Three Point Nonlinear BVP
Based on maximum and antimaximum Principle we develop theory to solve the three point nonlinear BVP and divide it into the following two subsections.
3.1. Reverse Ordered Lower and Upper Solutions
Theorem 8. Let there exist , in such that and satisfies If is continuous on and there exists such that for all , then the boundary value problem (5) has at least one solution in the region . Further, if a constant such that and is satisfied, then the sequences generated by where , with initial iterate converge monotonically (non-decreasing) and uniformly towards a solution of (5). Similarly, using as an initial iterate leads to a nonincreasing sequences converging to a solution . Any solution in must satisfy
Proof. From (13) and (16) (for )
Since and , by using Proposition 6, we have .
In view of , from (16) we get and if , then Since , then from (20) (for ) and (16) (for ), we get From Proposition 6 we have .
Now from (14) and (16) (for ) Thus, follows from Proposition 6.
Now assuming that , , we show that and for all . From (16) (for ) and (20), we get and hence from Proposition 6 we have .
From (16) (for ) and (14) we get Then, from Proposition 6, and hence we have and starting with it is easy to get Finally, we show that for all . For this, by assuming , we show that . From (16) it is easy to get Hence, from Proposition 6, . Thus we have So, the sequences and are monotonically nondecreasing and nonincreasing, respectively, and are bounded by and . Hence by Dini's theorem they converge uniformly. Let and .
Using Lemma 2, the solution of (16) is given by Then, by Lebesgue's dominated convergence theorem, taking the limit as approaches to , we get which is the solution of boundary value problem (5).
Any solution in can play the role of ; hence, and similarly one concludes that .
3.2. Well-Ordered Lower and Upper Solutions
Theorem 9. Let there exist , in such that and satisfies If is continuous on and there exists such that for all then the boundary value problem (5) has at least one solution in the region . If a constant , such that and is satisfied, then the sequences generated by where , with initial iterate , converge monotonically (nonincreasing) and uniformly towards a solution of (5). Similarly, using as an initial iterate leads to a nondecreasing sequences converging to a solution . Any solution in must satisfy
Proof. Proof follows from the analysis of Theorem 8.
4. Numerical Illustration
To verify our results, we consider examples and show that there exists at least one value of such that iterative scheme generates monotone sequences which converge to solutions of nonlinear problem. Thus, these examples validate sufficient conditions derived in this paper.
Example 10 (reverse order). Consider the boundary value problem Here, , , . This problem has and as lower and upper solutions; that is, it is non-well-ordered case. The nonlinear term is Lipschitz in and continuous for all value of , and Lipschitz constants are . For we can see that will be true. To verify , in Figure 1 we plot inequalities assumed in . From Figures 2, 3, and 4, we plot members of monotone sequences , for different values of .
Example 11 (well order). Consider the boundary value problem Here, , , . This problem has and as lower and upper solutions; that is, it is well-ordered case. The nonlinear term is Lipschitz in and continuous for all value of , and Lipschitz constants are . For , we can see that will be true. To verify , in Figure 5 we plot inequalities assumed in . From Figures 6, 7, and 8, we plot members of monotone sequences , for different values of .
The monotone iterative technique coupled with upper and lower solutions is a powerful tool for computation of solutions of nonlinear three point boundary value problems. It proves the existence of solutions analytically and gives us a tool so that numerical solutions can also be computed and then some real-life problems, for example, bridge design problem, thermostat problem, and so forth, can be solved. We have plotted sequences for both and . The plots are quite encouraging and will motivate researchers to explore further possibilities. Employing this technique, Mathematica/Maple/MATLAB user-friendly packages can be developed (see ).
This work is partially supported by Grant provided by UGC, New Delhi, India, File no. F.4-1/2006 (BSR)/7-203/2009(BSR).
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