Table of Contents
Table of Contents
Preface
Introduction
I – NONLINEAR BOUNDARY VALUE PROBLEMS: EXISTENCE AND MULTIPLICITY RESULTS
Chapter 1. High Order Periodic Problems
Chapter 2. New Trends on Lidstone Problems
Chapter 3. Multiplicity of Solutions
Chapter 4. High Order Periodic Impulsive Problems
II – FUNCTIONAL BOUNDARY VALUE PROBLEMS
Chapter 5. High Order Problems with Functional Boundary Conditions
Chapter 6. Generalized f−Laplacian Equation with Functional Boundary Conditions
Chapter 7. Functional Boundary
References
Reviews
“This very interesting book is devoted to the study of higher order boundary value problems. The main tool utilized throughout the volume is the method of upper and lower solutions. Of particular interest is the fact that in many cases the authors give an explicit construction of the upper and lower solution. The authors illustrate how this location tool can be utilized to gain qualitative information regarding the solutions: existence, multiplicity, monotonicity. Another nice feature of the book is that the methodology is applied also to real world phenomena: the London Millenium bridge and the periodic oscillations of the axis of a satellite. Overall the book is fairly easy to read and would be helpful for graduate students and young researchers willing to learn more on this method.” Gennaro Infante, Ph.D., Associate Professor, Department of Mathematics and Computer Science, University of Calabria, Italy
References
[1] P. Amster, P. de N´apoli, An application of the antimaximum principle for a fourth order periodic problem, Electronic Journal of Qualitative Theory of Differential Equations,
[2] A. Ambrosetti, G. Prodi, On the inversion on some differential mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. [3] C. Bereanu, Periodic solutions for some fourth-order nonlinear differential equations, Nonlinear Anal.[4] Z. Bai, The upper and lower solution method for some fourth-order boundary value problems,Nonlinear Anal. (2007), 1704–1709.[5] Z. Bai, Themethod of lower and upper solutions for a bending of an elastic beam equation, J. Math. Anal. Appl. 248, (2000), 195–202.[6] V.V. Beletskii, On the oscillations of a satellite,Iskusst, Sputn. Zemli 3, (1959) 1-3.[7] V.V. Beletskii, Essays on the Motion of Celestial Bodies, Birkh¨auser, 2001.[8] Z. Benbouziane, A. Boucherif, S. M. Bouguima, Existence result for impulsive third order periodic boundary value problems, Appl. Math. Comput. 206,no. 2, (2008), 728–737.[9] A. Cabada, The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems. J. Math. Anal. Appl. 185, 2, (1994), 302–320.[10] A. Cabada, J. Tomeˇcek, Extremal solutions for nonlinear functional Laplacian impulsive equations.Nonlinear Anal. 67, no. 3, (2007) 827–841.[11] A. Cabada, J. Angel Cid, On a class of singular Sturm–Liouville periodic boundary value problems, Nonlinear Anal: Real World App. 12 (4), (2011), 2378-2384.[12] A. Cabada, J. ´Angel Cid, L. Sanchez, Existence of solutions for elliptic systems with nonlocal terms in one dimension, Bound. Value Probl. 2011.[13] A. Cabada, F. Minh´os, Fully nonlinear fourth order equations with functional boundary conditions, J. Math. Anal. Appl., Vol. 340/1 (2008) 239–251.[14] A. Cabada, F. Minh´os, A. I. Santos, Solvability for a third order discontinuous fully equation with functional boundary conditions J. Math. Anal. Appl., Vol. 322 (2006) 735–748.[15] A. Cabada, M. R. Grossinho, F. Minh´os,On the Solvability of some Discontinuous Third Order Nonlinear Differential Equations with Two Point Boundary Conditions, J. Math. Anal. Appl. 285 (2003) 174–190.[16] A. Cabada, R. Pouso, F. Minh´os, Extremal solutions to fourth-order functional boundary value problems including multipoint condition, Nonlinear Anal.: Real World Appl., 10 (2009) 2157–2170.[17] C. de Coster, P. Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions, Mathematics in Science and Engineering, 205, Elsevier, 2006.[18] C. de Coster and L. Sanchez,Upper and lower solutions,Ambrosetti-Prodi problem and positive solutions for a fourth order O.D.E., Riv. Mat. Pura Appl., 14 (1994), 57–82.[19] P. Dallard, A. J. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R. M. Ridsdill, M. Willford, The London Millennium Footbridge, The structural engineer, Vol. 79, no 22 (2001), 17-33.[20] P. Dr´abek, G. Holubov´a, A. Matas, P. Neˇcessal, Nonlinear models of suspension bridges: discussion of results, Applications of Mathematics, 48 (2003) 497-514.[21] J. Ehme, P. Eloe, J. Henderson,Upper and lower solutions method for fully nonlinear boundary value problems, J. Differential Equations, 180(2002), 51-64.[22] P.W. Eloe, J. Henderson, Positive solutions for conjugate boundary value problems. Nonlinear Anal.28 (1997), no. 10, 1669–1680.[23] R. Enguic¸a, L. Sanchez, Existence and localization of solutions for fourth order boundary-value problems.Electron. J. Differential Equations 2007, No. 127, 10 pp.[24] Q. Fan, W. Ang, J. Zhou, Periodic solutions for some fourth-order nonlinear differential equations, J. Comp. Appl. Math. 233 (2009) 121-126.[25] J. Fialho, F. Minh´os, Existence and location results for hinged beams with unbounded nonlinearities, Nonlinear Anal. 71 (2009) e1519-e1525.[26] J. Fialho, F. Minh´os, On higher order fully periodic boundary value problems, J. Math. Anal. Appl.395 (2012) 616–625.[27] J. Fialho, F. Minhos, Higher order functional boundary value problems without monotone assumptions, Boundary Value Problems , 2013:81.[28] C. Fabry, J. Mawhin , M. N. Nkashama, A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations. Bull. London Math. Soc. 18 (1986), 173-180.[29] H. Feng, D. Ji, W. Ge, Existence and uniqueness of solutions for a fourth order boundary value problem, Nonlinear Anal.70 (2009) 3761–3566.[30] D. Franco, R. L. Pouso, Nonressonance conditions and extremal solutions for first-order impulsive problems under weak assumptions, ANZIAM J. 44 (2003) 393-407.[31] D. Franco, D. O’Regan, J. Per´an, Fourth-order problems with nonlinear boundary conditions, J. Comput. Appl. Math. 174 (2005) 315–327.[32] C. Fuzhong, Periodic solutions for 2th order ordinary differential equations with nonresonance,Nonlinear Anal.32 (1998) 787-793.[33] J. Graef, L. Kong, A necessary and sufficient condition for existence of positive solutions of nonlinear boundary value problems, Nonl. Analysis 66 (2007) 2389–2412.[34] J. R. Graef, L. Kong, B. Yang, Existence of solutions for a higher order multi-point boundary value problems, Result. Math. 53 (2009), 77–101.[35] J. R. Graef, L. Kong, Solutions of second order multi-point boundary value problems, Math. Proc. Camb. Phil. Soc. 145 (2008), 489–510.[36] J. R. Graef, L. Kong, Q. Kong, Higher order multi-point boundary value problems, Math. Nachr. 284, no. 1, (2011) 39–52.[37] J. R. Graef, L. Kong, F. Minh´os, Higher order-Laplacian BVP with generalized Sturm-Liouville boundary conditions, Differ. Equ. Dyn. Syst. 18, no. 4, (2010) 373–383.[38] J. R. Graef, L. Kong, F. Minh´os, Higher order functional boundary value problems: existence and location results, Acta Scientiarum Mathematicarum (Szeged), 77 (2011), 87–100.[39] J. R. Graef, L. Kong, F. M. Minh´os, J. Fialho, On the lower and upper solution method for higher order functional boundary value problems. Appl. Anal. Discrete Math. 5, no. 1, (2011) 133–146.[40] M.R. Grossinho, F.M.Minh´os, A.I. Santos, Solvability of some third-order boundary value problems with asymmetric unbounded linearities, Nonlinear Analysis, 62 (2005), 1235-1250.[41] M. R. Grossinho, F. Minh´os, A. I. Santos, A note on a class of problems for a higher order fully nonlinear equation under one sided Nagumo type condition, Nonlinear Anal.70 (2009) 4027–4038.[42] M.R. Grossinho, F. Minh´os, Upper and lower solutions for some higher order boundary value problems, Nonlinear Studies, 12 (2005) 165–176.[43] M. R. Grossinho, S. Tersian, The dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation, Nonlinear Anal. Series A: Theory Methods , 41 (2000), no. 3-4, 417–431.[44] M. R. Grossinho, F. Minh´os, S. Tersian, PositiveHomoclinic Solutions for a Class of Second Order Differential Equations, J. Math. Anal. Appl. 240 (1999) 163-173.[45] M. R. Grossinho, F. Minh´os, S. Tersian, Homoclinic and Periodic Solutions for some Class of Second Order Differential Equations, Progress in Nonlinear Differential Equations and their Applications, vol. 43, Birkhauser, Boston, 289-298, 2001.[46] C. P. Gupta, Existence and Uniqueness Theorems for the Bending of an Elastic Beam Equation, Appl. Anal., 26(1988), 289-304.[47] C. P. Gupta, Existence and Uniqueness Theorems for a Fourth Order Boundary Value Problem of Sturm-Liouville Type, Differential and Integral Equations, vol. 4, Number 2, March 1991, 397-410.[48] T. Gyulov, S. Tersian, Existence of Trivial and Nontrivial Solutions of a Fourth-Order Ordinary Differential Equation, Electron. J. Diff. Eqns, Vol. 2004 (2004), no. 41, 1-14.[49] D.D. Hai, Note on a differential equation describing the periodic motion of a satellite in its elliptical orbit,Nonlinear Anal.12 (12) (1988) 1337–1338.[50] S. Heikkil¨a, V. Lashmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker Inc., New York, 1994.[51] J. Henderson, Solutions of multipoint boundary value problems for second order equations, Dynam. Systems Appl. 15 (2006), 111–117.[52] J. Henderson, S. K. Ntouyas, Positive solutions for systems of order three-point nonlocal boundary value problems, Electron. J. Qual. Theory Differ. Equ.(2007), No. 18, 1–12.[53] I. Kiguradze, T. Ksunanno, Periodic solutions for nonautonomous ordinary differential equations of higher order, Differential Equations, 35 (1), (1999) 70-77.[54] A.C. Lazer, P.J. Mckenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Review 32 (1990) 537-578.[55] V. Lakshmikantham, D. D. Ba˘ınov, P. S. Simeonov, Theory of impulsive differential equations. Series in Modern Applied Mathematics, 6. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. xii+273 pp.[56] Y. Li, On the existence and uniqueness for higher order periodic boundary value problems, Nonlinear Anal. 70 (2009) 711-718.[57] S. Liang, J. Zhanga, The method of lower and upper solutions for 2nth order multi-point boundary value problems, Nonlinear Anal. 71 (2009) 4581 4587.[58] Y. Liu, An existence result for solutions of nonlinear Sturm-Liouville boundary value problems for higher order p-Laplacian differential equations, Rocky Mountain J. Math, 39 (2009) 147-163.[59] Y. Liu, Solvability of periodic value problems for -order ordinary differential equations, Comp. Math. Appl.52 (2006) 1165-1182.[60] Y. Liu, G.Weigao, Solutions of a generalized multi-point conjugate BVPs for higher order impulsive differential equations. Dynam. Systems Appl. 14, no. 2, (2005) 265–279.[61] E. Liz, J. J. Nieto, Periodic solutions of discontinuous impulsive differential systems. J. Math. Anal. Appl.161 (1991), no. 2, 388–394.[62] A. Love, A Treatise on the Mathematical Theory of Elasticity, Dover Books on Physics and Chemistry, 1944.[63] X.L. Lu, W.T. Li, Positive solutions of the non-linear fourth-order beam equation with three parameters, J. Math. Anal. Appl. 303 (2005) 150-163.[64] D. Ma, X. Yang, Upper and lower solution method for fourth-order four point boundary value problems, J. Comput. Appl. Math., 223 (2009), 543–551.[65] H. Ma, Positive solutions for m-point boundary value problems of fourth order, J. Math. Anal. Appl. 321 (2006) 37–49. [66] T.F. Ma, J. da Silva, Iterative solutions for a beam equation with nonlinear boundary conditions of third order, Appl.Math. Comp. 159 (2004) 11–18.[67] J.Mawhin, Topological degree methods in nonlinear boundary value problems, Regional Conference Series in Mathematics, AMS, 40 , 1979.[68] J.Mawhin, On a differential equation for the periodicmotions of a satellite around its center of mass, Asymptotic Methods in Mathematical Physics, 302, Naukova Dumka, Kiev, 1988, pp. 150–157.[69] F. Minh´os, On some third order nonlinear boundary value problems: existence, location and multiplicity results, J. Math. Anal. Appl., Vol. 339/2(2008) 1342-1353.[70] F. Minh´os, J. Fialho, On the solvability of some fourth-order equations with functional boundary conditions, Discrete Contin. Dyn. Syst., 2009, suppl., 564–573.[71] F. Minh´os, T. Gyulov, A. I. Santos, Lower and upper solutions for a fully nonlinear beam equations, Nonlinear Anal., 71 (2009) 281–292.[72] F. Minh´os, T. Gyulov, A. I. Santos, Existence and location results for the bending of an elastic beam, Proceedings Equadiff 11, Eds.: M. Fila, A. Handlovicova, K. Mikula, M. Medved, P.Quittner and D. Sevcovic, 273-282, 2007.[73] F. Minh´os, T. Gyulov, A. I. Santos, Existence and location result for a fourth order boundary value problem, Discrete Contin. Dyn. Syst., Supp., (2005) 662–671.[74] F. Minh´os, T. Gyulov, A. I. Santos, On an elastic beam fully equation with nonlinear boundary conditions, Proc. of Conference on Differential & Difference Equations and Applications (2005), Eds. R. Agarwal and K.Perera, Hindawi Publishing Corporation, 805-814, 2006.[75] S. Mukhigulashvili, On a periodic boundary value problem for third order linear functional differential equations, Nonlinear Anal., 66 (2007) 527-535.[76] M. Nagumo, ¨Uber die differential gleichung Proc. Phys.-Math. Soc. Japan 19, (1937), 861-866.[77] J. Nieto, Periodic solutions for third order ordinary differential equations,Comment. Math. Univ. Carolinae. 32/3 (1991) 495-499.[78] H. Pang,W. Ge, Existence results for some fourth order multi-point boundary value problem, Math. Comput. Model. 49 (2009) 1319-1325.[79] C. V. Pao, Y.M. Wang, Fourth-order boundary value problems with multipoint boundary conditions, Comm. Appl. Nonlin. Anal., 16 (2009), 1–22.[80] I. Rach°unkov´a, Periodic boundary value problems for third order differential equations, Math. Slovaca, 41/3, (1991) 241-248.[81] I. Rach°unkov´a, M. Tvrd´y, Existence results for impulsive second-order periodic problems. Nonlinear Anal. 59 (2004), no. 1-2, 133–146.[82] W. Rudin, Functional Analysis,McGraw-Hill (1991).[83] M. Ruyun, Z. Jihui, F. Shengmao, The method of lower and upper solutions for fourth-order boundary value problems. J. Math. Anal. Appl. 215, (1997), 415-422.[84] M. ˇSenkyˇr´ık, Existence of multiple solutions for a third order three-point regular boundary value problem, Mathematica Bohemica, 119, n 2 (1994), 113–121.[85] M. ˇSenkyˇr´ık, Fourth order boundary value problems and nonlinear beams, Appl. Analysis, 59 (1995) 15–25.[86] S. Tersian, J. Chaparova, Periodic and homoclinic solutions of some semilinear sixth-order differential equations, J. Math. Anal. Appl. 272 (2002) 223-239.[87] M. X. Wang, A. Cabada, J. J. Nieto, Monotone method for nonlinear second order periodic boundary value problems with Carath´eodory functions, Ann. Polon. Math. 58 (1993), 221–235.[88] W. Wang, J. Shen, Z. Luo, Multi-point boundary value problems for second-order functional differential equations, Comput. Math. Appl. 56 (2008), 2065–2072.[89] X. Wang, F. Zhang, Y. Jiang, Boundary value problems of third-order singularly perturbed impulsive differential equations. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. (2004), Added Volume, 82–88.[90] J. R. L. Webb, Higher order non-local conjugate type boundary value problems. Mathematical models in engineering, biology and medicine, 332–341, AIP Conf. Proc. 1124, Amer. Inst. Phys., Melville,NY, (2009).[91] S.Weng, H. Gao, D. Jiang, X. Hou, Upper and lower solutions method for fourth-order periodic boundary value problems. J. Appl. Anal. 14 (2008), 53–61.[92] B. Yang, Positive solutions of the conjugate boundary value problem, Elect. J. Qualitative Theory of Diff. Eq 53(2010), 1-13.[93] X. Zhang and L. Liu, Nontrivial solutions for higher order multi-point boundary value problems, Comput. Math. Appl. 56 (2008), 861–873.[94] H. Zhao, A note on upper and lower solutions method for fourth-order boundary value problems, Ann. Differential Equations 24 (2008), 117–120.
These contents have of wide scope amongst researchers in various areas like Mathematics, Engineering and Finance as they offer applications to theoretical problems as well as techniques to build and define lower and upper solutions for higher order problems. It is also very useful for students from undergraduate to graduate level in those fields as it provides arguments based on elementary and basic concepts, without “deep mathematics tools” and includes examples, where lower and upper solutions are properly defined and applied.
