© Ahmed Alsaedi. 2011
Received: 11 May 2010
Accepted: 2 October 2010
Published: 4 October 2010
The monotone iterative technique coupled with the method of upper and lower solutions [1–7] manifests itself as an effective and flexible mechanism that offers theoretical as well as constructive existence results in a closed set, generated by the lower and upper solutions. In general, the convergence of the sequence of approximate solutions given by the monotone iterative technique is at most linear [8, 9]. To obtain a sequence of approximate solutions converging quadratically, we use the method of quasilinearization . This method has been developed for a variety of problems [11–20]. In view of its diverse applications, this approach is quite an elegant and easier for application algorithms.
The subject of multipoint nonlocal boundary conditions, initiated by Bicadze and Samarskiĭ , has been addressed by many authors, for instance, [22–32]. The multipoint boundary conditions appear in certain problems of thermodynamics, elasticity and wave propagation, see  and the references therein. The multipoint boundary conditions may be understood in the sense that the controllers at the endpoints dissipate or add energy according to censors located at intermediate positions.
We need the following result  to establish the main result.
Let be a continuous function satisfying the Nagumo condition on where are continuous functions such that for all Then there exists a constant (depending only on the Nagumo function ) such that every solution of (1.1)-(1.2) with , satisfies
3. Main Results
Thus, are lower and upper solutions of (3.14), respectively. Since satisfies a Nagumo condition, there exists a constant (depending on and a Nagumo function) such that any solution of (3.14) with satisfies on
Let and be, respectively, lower and upper solutions of (3.60). Clearly and are not the solutions of (3.60) and Also, the assumptions of Theorem 3.1 are satisfied. Thus, the conclusion of Theorem 3.1 applies to the problem (3.60).
The author is grateful to the referees and professor G. Infante for their valuable suggestions and comments that led to the improvement of the original paper.
- Ladde GS, Lakshmikantham V, Vatsala AS: Monotone Iterative Techniques for Nonlinear Differential Equations, Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics, 27. Pitman, Boston, Mass, USA; 1985:x+236.Google Scholar
- Nieto JJ, Jiang Y, Jurang Y: Monotone iterative method for functional-differential equations. Nonlinear Analysis: Theory, Methods & Applications 1998, 32(6):741–747. 10.1016/S0362-546X(97)00524-5View ArticleMathSciNetMATHGoogle Scholar
- Vatsala AS, Yang J: Monotone iterative technique for semilinear elliptic systems. Boundary Value Problems 2005, 2005(2):93–106. 10.1155/BVP.2005.93View ArticleMathSciNetMATHGoogle Scholar
- Drici Z, McRae FA, Devi JV: Monotone iterative technique for periodic boundary value problems with causal operators. Nonlinear Analysis: Theory, Methods & Applications 2006, 64(6):1271–1277. 10.1016/j.na.2005.06.033View ArticleMathSciNetMATHGoogle Scholar
- Jiang D, Nieto JJ, Zuo W: On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations. Journal of Mathematical Analysis and Applications 2004, 289(2):691–699. 10.1016/j.jmaa.2003.09.020View ArticleMathSciNetMATHGoogle Scholar
- Nieto JJ, Rodríguez-López R: Monotone method for first-order functional differential equations. Computers & Mathematics with Applications 2006, 52(3–4):471–484. 10.1016/j.camwa.2006.01.012View ArticleMathSciNetMATHGoogle Scholar
- Ahmad B, Sivasundaram S: The monotone iterative technique for impulsive hybrid set valued integro-differential equations. Nonlinear Analysis: Theory, Methods & Applications 2006, 65(12):2260–2276. 10.1016/j.na.2006.01.033View ArticleMathSciNetMATHGoogle Scholar
- Cabada A, Nieto JJ: Rapid convergence of the iterative technique for first order initial value problems. Applied Mathematics and Computation 1997, 87(2–3):217–226. 10.1016/S0096-3003(96)00285-8View ArticleMathSciNetMATHGoogle Scholar
- Lakshmikantham V, Nieto JJ: Generalized quasilinearization for nonlinear first order ordinary differential equations. Nonlinear Times and Digest 1995, 2(1):1–9.MathSciNetMATHGoogle Scholar
- Bellman RE, Kalaba RE: Quasilinearization and Nonlinear Boundary-Value Problems, Modern Analytic and Computional Methods in Science and Mathematics. Volume 3. American Elsevier, New York, NY, USA; 1965:ix+206.Google Scholar
- Lakshmikantham V, Vatsala AS: Generalized Quasilinearization for Nonlinear Problems, Mathematics and Its Applications. Volume 440. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1998:x+276.View ArticleGoogle Scholar
- Cabada A, Nieto JJ: Quasilinearization and rate of convergence for higher-order nonlinear periodic boundary-value problems. Journal of Optimization Theory and Applications 2001, 108(1):97–107. 10.1023/A:1026413921997View ArticleMathSciNetMATHGoogle Scholar
- Ahmad B, Nieto JJ, Shahzad N: The Bellman-Kalaba-Lakshmikantham quasilinearization method for Neumann problems. Journal of Mathematical Analysis and Applications 2001, 257(2):356–363. 10.1006/jmaa.2000.7352View ArticleMathSciNetMATHGoogle Scholar
- Eloe PW, Gao Y: The method of quasilinearization and a three-point boundary value problem. Journal of the Korean Mathematical Society 2002, 39(2):319–330.View ArticleMathSciNetMATHGoogle Scholar
- Akyildiz FT, Vajravelu K: Existence, uniqueness, and quasilinearization results for nonlinear differential equations arising in viscoelastic fluid flow. Differential Equations & Nonlinear Mechanics 2006, 2006:-9.Google Scholar
- Ahmad B: A quasilinearization method for a class of integro-differential equations with mixed nonlinearities. Nonlinear Analysis: Real World Applications 2006, 7(5):997–1004. 10.1016/j.nonrwa.2005.09.003View ArticleMathSciNetMATHGoogle Scholar
- Krivec R, Mandelzweig VB: Quasilinearization method and WKB. Computer Physics Communications 2006, 174(2):119–126. 10.1016/j.cpc.2004.12.017View ArticleMathSciNetMATHGoogle Scholar
- Amster P, De Nápoli P: A quasilinearization method for elliptic problems with a nonlinear boundary condition. Nonlinear Analysis: Theory, Methods & Applications 2007, 66(10):2255–2263. 10.1016/j.na.2006.03.016View ArticleMathSciNetMATHGoogle Scholar
- Ahmad B, Alsaedi A, Alghamdi BS: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Analysis: Real World Applications 2008, 9(4):1727–1740. 10.1016/j.nonrwa.2007.05.005View ArticleMathSciNetMATHGoogle Scholar
- Ahmad B, Nieto JJ: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(10):3291–3298. 10.1016/j.na.2007.09.018View ArticleMathSciNetMATHGoogle Scholar
- Bicadze AV, Samarskiĭ AA: Some elementary generalizations of linear elliptic boundary value problems. Doklady Akademii Nauk SSSR 1969, 185: 739–740.MathSciNetGoogle Scholar
- Gupta CP: A second order -point boundary value problem at resonance. Nonlinear Analysis: Theory, Methods & Applications 1995, 24(10):1483–1489. 10.1016/0362-546X(94)00204-UView ArticleMathSciNetMATHGoogle Scholar
- Ma R: Multiple positive solutions for nonlinear -point boundary value problems. Applied Mathematics and Computation 2004, 148(1):249–262. 10.1016/S0096-3003(02)00843-3View ArticleMathSciNetMATHGoogle Scholar
- Eloe PW, Ahmad B: Positive solutions of a nonlinear th order boundary value problem with nonlocal conditions. Applied Mathematics Letters 2005, 18(5):521–527. 10.1016/j.aml.2004.05.009View ArticleMathSciNetMATHGoogle Scholar
- Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach. Journal of the London Mathematical Society 2006, 74(3):673–693. 10.1112/S0024610706023179View ArticleMathSciNetMATHGoogle Scholar
- Khan RA: Generalized approximations and rapid convergence of solutions of -point boundary value problems. Applied Mathematics and Computation 2007, 188(2):1878–1890. 10.1016/j.amc.2006.11.138View ArticleMathSciNetMATHGoogle Scholar
- Pei M, Chang SK: The generalized quasilinearization method for second-order three-point boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(9):2779–2790. 10.1016/j.na.2007.02.025View ArticleMathSciNetMATHGoogle Scholar
- Pei M, Chang SK: A quasilinearization method for second-order four-point boundary value problems. Applied Mathematics and Computation 2008, 202(1):54–66. 10.1016/j.amc.2008.01.026View ArticleMathSciNetMATHGoogle Scholar
- Ahmad B: Approximation of solutions of the forced Duffing equation with -point boundary conditions. Communications in Applied Analysis 2009, 13(1):11–20.MathSciNetMATHGoogle Scholar
- Wang L, Pei M, Ge W: Existence and approximation of solutions for nonlinear second-order four-point boundary value problems. Mathematical and Computer Modelling 2009, 50(9–10):1348–1359. 10.1016/j.mcm.2008.11.018View ArticleMathSciNetMATHGoogle Scholar
- Chang Y-K, Nieto JJ, Li W-S: On impulsive hyperbolic differential inclusions with nonlocal initial conditions. Journal of Optimization Theory and Applications 2009, 140(3):431–442. 10.1007/s10957-008-9468-1View ArticleMathSciNetMATHGoogle Scholar
- Graef JR, Webb JRL: Third order boundary value problems with nonlocal boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(5–6):1542–1551. 10.1016/j.na.2008.12.047View ArticleMathSciNetMATHGoogle Scholar
- Grossinho M, Minhós FM: Upper and lower solutions for higher order boundary value problems. Nonlinear Studies 2005, 12(2):165–176.MathSciNetMATHGoogle Scholar
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