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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 640183, 7 pages
Existence and Uniqueness of Solution to Nonlinear Boundary Value Problems with Sign-Changing Green’s Function
1Department of Mathematics, Heze University, Heze, Shandong 274000, China
2School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
3Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia
Received 20 July 2013; Accepted 23 August 2013
Academic Editor: Shaoyong Lai
Copyright © 2013 Peiguo Zhang et al. 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.
By using the cone theory and the Banach contraction mapping principle, the existence and uniqueness results are established for nonlinear higher-order differential equation boundary value problems with sign-changing Green’s function. The theorems obtained are very general and complement previous known results.
Boundary value problems (BVPs for short) for nonlinear differential equations arise in a variety of areas of applied mathematics, physics, and variational problems of control theory. The study of multipoint BVPs for second-order differential equations was initiated by Bicadze and Samarskiĭ  and later continued by II'in and Moiseev [2, 3] and Gupta . Since then, great efforts have been devoted to nonlinear multipoint BVPs due to their theoretical challenge and great application potential. Many results on the existence of solutions for multipoint BVPs have been obtained; the methods used therein mainly depend on the fixed point theorems, degree theory, upper and lower techniques, and monotone iteration. The existence results are available in the literature [5–25] and the references therein.
Recently, by applying the fixed point theorems on cones, the authors of papers [5–7] established the existence and multiplicity of positive solutions for the th-order three-point BVP: where and . The th-order -point BVP has been studied in [8–10], where , and with . The existence and multiplicity results of solutions were shown by using various fixed point theorems and fixed point index theory.
By using the cone theory and the Banach contraction mapping principle, the author  established the existence and uniqueness for singular third-order three-point boundary value problems.
The purpose of this paper is to investigate the existence and uniqueness of solution of the following higher-order differential equation boundary value problem: where , ,
Here, we give the unique solution of BVP (3) under the conditions that is mixed nonmonotone. The methods used in this paper are motivated by , and the arguments are based upon the cone theory and the Banach contraction mapping principle.
2. The Preliminary Lemmas
Lemma 1. For any , the BVP has a unique solution , where
Proof. First, suppose that is a solution to problem (4) and (5). It is easy to see by integration of (4) that Substituting (7) into (5), we obtain and so Substituting (9) into (7), we have Conversely, suppose that ; then it is easy to verify that (4) and (5) are satisfied. The lemma is proved.
For any , let
By Lemma 1, the proof follows by routine calculations.
It is easy to see that .
3. Main Results
This section discusses the solution of nonlinear higher-order differential equation BVP (3).
Let . Obviously, is a normal solid cone of Banach space , by Lemma 2.1.2 in , and we have that is a generating cone in .
Theorem 4. Suppose that , , and there exist positive constants with such that for any , , with ,, one has and there exist , such that converges. Then, BVP (3) has a unique solution in , and moreover, for any , the iterative sequence converges to in .
Remark 5. Recently, in the study of BVP (3), almost all the papers have supposed that Green’s function is nonnegative. However, the scope of is not limited to in Theorem 4, so, we do not need to suppose that is nonnegative.
Remark 6. The function in Theorem 4 is not monotone or convex; the conclusions and the proof used in this paper are different from the known papers in essence.
Proof of Theorem 4. It is easy to see that, for any can be divided into finite partitioned monotone and bounded function on , and then, by (15), we have that
converges. Let ; then
For any , let and then . By (14), we have Hence, Following the former inequality, we can easily have that converges, thus, is converged.
Similarly, by , is converged, and we have that converges.
Define the operator by Let By (14) and (25), for any , we have So we can choose , which satisfies , and so there exists a positive integer such that
Since is a generating cone in , from Lemma 3, there exists such that every element can be represented in this implies Let By (31), we know that is well defined for any . It is easy to verify that is a norm in . By (30)–(32), we get
On the other hand, for any which satisfies , we have ; thus, , where denotes the normal constant of . Since is arbitrary, we have It follows from (33) and (34) that the norms and are equivalent. Now, for any and which satisfies , let then .
It follows from (27) that subtracting (37) from (36) + (38), we obtain Let ; then we have
As and are both positive linear bounded operators, so is a positive linear bounded operator, and therefore, . Hence, by mathematical induction, it is easy to know that for natural number in (29), we have since , we see that which implies by virtue of the arbitrariness of that By , we have . Thus, the Banach contraction mapping principle implies that has a unique fixed point in , and so has a unique fixed point in ; by the definition of has a unique fixed point in ; then, by Lemma 2, is the unique solution of (3). And, for any , let ; we have . By the equivalence of and again, we get . This completes the proof.
In this paper, the results apply to a very wide range of functions, and we are following only one example to illustrate.
Consider the following th-order three-point boundary value problem: where , .
To see that, let then is Green’s function of (44). It is easy to verify that , and so .
Let where ; then it is easy to verify that all conditions in Theorem 4 are satisfied.
Peiguo Zhang and Lishan Liu were supported financially by the National Natural Science Foundation of China (11071141, 11371221), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province, and the Project of Shandong Province Higher Educational Science and Technology Program (J11LA06, J13LI02). Yonghong Wu was supported financially by the Australian Research Council through an ARC Discovery Project grant.
- A. V. Bicadze and A. A. Samarskiĭ, “Some elementary generalizations of linear elliptic boundary value problems,” Doklady Akademii Nauk SSSR, vol. 185, pp. 739–740, 1969.
- V. A. II’in and E. I. Moiseev, “Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects,” Differential Equations, vol. 23, pp. 803–810, 1987.
- V. A. II’in and E. I. Moiseev, “Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator,” Differential Equations, vol. 23, pp. 979–987, 1987.
- C. P. Gupta, “Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation,” Journal of Mathematical Analysis and Applications, vol. 168, no. 2, pp. 540–551, 1992.
- P. W. Eloe and B. Ahmad, “Positive solutions of a nonlinear th order boundary value problem with nonlocal conditions,” Applied Mathematics Letters, vol. 18, no. 5, pp. 521–527, 2005.
- X. Hao, L. Liu, and Y. Wu, “Positive solutions for nonlinear th-order singular nonlocal boundary value problems,” Boundary Value Problems, vol. 2007, Article ID 74517, 2007.
- J. R. Graef and T. Moussaoui, “A class of th-order BVPs with nonlocal conditions,” Computers & Mathematics with Applications, vol. 58, no. 8, pp. 1662–1671, 2009.
- C. Pang, W. Dong, and Z. Wei, “Green's function and positive solutions of th order -point boundary value problem,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1231–1239, 2006.
- J. Yang and Z. Wei, “Positive solutions of th order -point boundary value problem,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 715–720, 2008.
- Y. Guo, Y. Ji, and J. Zhang, “Three positive solutions for a nonlinear th-order -point boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 11, pp. 3485–3492, 2008.
- M. ur Rehman and R. A. Khan, “Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1038–1044, 2010.
- M. El-Shahed and J. J. Nieto, “Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3438–3443, 2010.
- G. Zhang and J. Sun, “Positive solutions of -point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 291, no. 2, pp. 406–418, 2004.
- M. Feng and W. Ge, “Existence results for a class of th order -point boundary value problems in Banach spaces,” Applied Mathematics Letters, vol. 22, no. 8, pp. 1303–1308, 2009.
- X. Hao, L. Liu, and Y. Wu, “On positive solutions of an -point nonhomogeneous singular boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 8, pp. 2532–2540, 2010.
- W. Jiang, “Multiple positive solutions for th-order -point boundary value problems with all derivatives,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 5, pp. 1064–1072, 2008.
- J. R. Graef and B. Yang, “Positive solutions to a multi-point higher order boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 409–421, 2006.
- M. Zhang, Y. Yin, and Z. Wei, “Positive solution of singular higher-order -point boundary value problem with nonlinearity that changes sign,” Applied Mathematics and Computation, vol. 201, no. 1-2, pp. 678–687, 2008.
- J. R. Graef, L. Kong, and B. Yang, “Existence of solutions for a higher order multi-point boundary value problem,” Results in Mathematics, vol. 53, no. 1-2, pp. 77–101, 2009.
- Y. Ji and Y. Guo, “The existence of countably many positive solutions for some nonlinear th order -point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 187–200, 2009.
- X. Zhang, M. Feng, and W. Ge, “Multiple positive solutions for a class of -point boundary value problems,” Applied Mathematics Letters, vol. 22, no. 1, pp. 12–18, 2009.
- J. Zhao and W. Ge, “Existence results of -point boundary value problem of Sturm-Liouville type with sign changing nonlinearity,” Mathematical and Computer Modelling, vol. 49, no. 5-6, pp. 946–954, 2009.
- S. Liang and J. Zhang, “Existence of countably many positive solutions of th-order -point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 527–537, 2009.
- H. Su and X. Wang, “Positive solutions to singular semipositone -point -order boundary value problems,” Journal of Applied Mathematics and Computing, vol. 36, no. 1-2, pp. 187–200, 2011.
- J. Henderson and R. Luca, “Existence and multiplicity for positive solutions of a multi-point boundary value problem,” Applied Mathematics and Computation, vol. 218, no. 21, pp. 10572–10585, 2012.
- P. Zhang, “Iterative solutions of singular boundary value problems of third-order differential equation,” Boundary Value Problems, vol. 2011, Article ID 483057, 10 pages, 2011.
- D. Guo, Semi-Ordered Method in Nonlinear Analysis, Shandong Scientific Technical Press, Jinan, China, 2000, Chinese.
- D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, NY, USA, 1988.
- D. Guo, V. Lakshmikantham, and X. Liu, Nonlinear Integral Equations in Abstract Spaces, vol. 373 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.