HIGHER ORDER BOUNDARY VALUE PROBLEMS WITH TWO POINT SEPARATED NONHOMOGENEOUS BOUNDARY CONDITIONS

John R. Graef; LINGJU KONG; QINGKAI KONG; BO YANG

Title	HIGHER ORDER BOUNDARY VALUE PROBLEMS WITH TWO POINT SEPARATED NONHOMOGENEOUS BOUNDARY CONDITIONS
Publication Type	Journal Article
Year of Publication	2011
Authors	Graef, JR, KONG, LINGJU, KONG, QINGKAI, YANG, BO
Volume	15
Issue	4
Start Page	435
Pagination	22
Date Published	2011
ISSN	1083-2564
AMS	34B15, 34B18
Abstract	The authors study the 2n-th order nonlinear boundary value problems with two point separated nonhomogeneous boundary conditions u (2n) = g(t)f(t, u), t ∈ (0, 1),    αu(2i) (0) − βu(2i+1)(0) = (−1)iλ2i , γu(2i) (1) + δu(2i+1)(1) = (−1)iλ2i+1. i = 0, . . . , n − 1, Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the above problem. Conditions are determined by the relationship between the behavior of the quotient f(t, x)/x for x near 0 and ∞, and the smallest positive characteristic values of some associated linear integral operator. This work improves and extends a number of recent results in the literature on this topic. The results are illustrated with examples.
URL	http://www.acadsol.eu/en/articles/15/4/2.pdf
Refereed Designation	Refereed
Full Text	REFERENCES [1] Z. Bai and H. Wang, On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl. 270 (2002), 357–368. [2] J. V. Baxley and L. J. Haywood, Nonlinear boundary value problems with multiple solutions, Nonlinear Anal. 47 (2001), 1187–1198. [3] J. M. Davis, L. Erbe, and J. Henderson, Multiplicity of positive solutions for higher order Sturm-Liouville problems, Rocky Mountain J. Math. 31 (2001), 169–184. [4] L. Erbe, Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems, Math. Comput. Model. 32 (2000), 529–539. [5] L. Erbe and R. M. Mathsen, Positive solutions for singular nonlinear boundary value problems, Nonlinear Anal. 46 (2001), 979–986. [6] J. R. Graef and L. Kong, Existence results for nonlinear periodic boundary value problems, Proc. Edinburgh Math. Soc. 52 (2009), 79–95. [7] J. R. Graef, L. Kong, and Q. Kong, Symmetric positive solutions of nonlinear boundary value problems, J. Math. Anal. Appl. 326 (2007), 1310–1327. [8] J. R. Graef, C. Qian, and B. Yang, Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations, Proc. Amer. Math. Soc, 131 (2003), 577–585. [9] J. R. Graef and B. Yang, Existence and nonexistence of positive solutions of fourth order nonlinear boundary value problems, Appl. Anal. 74 (2000), 201–214. [10] Y. Guo, W. Shan, and W. Ge, Positive solutions for second-order m-point boundary value problems, J. Comput. Appl. Math. 151 (2003), 415–424. [11] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Orlando, 1988. [12] G. Han and Y. Wu, Nontrivial solutions of singular two-point boundary value problems with sign-changing nonlinear terms, J. Math. Anal. Appl. 325 (2007), 1327–1338. [13] J. Henderson and H. B. Thompson, Multiple symmetric positive solutions for a second order boundary value problems, Proc. Amer. Math. Soc. 128 (2000), 2373–2379. [14] J. Henderson and H. Wang, Positive solutions for nonlinear eigenvalue problems, J. Math. Anal. Appl. 208 (1997), 252–259. [15] L. Kong and Q. Kong, Higher order boundary value problems with nonhomogeneous boundary conditions, Nonlinear Anal. 72 (2010), 240–261. [16] L. Kong and J. S. W. Wong, Positive solutions for multi-point boundary value problems with nonhomogeneous boundary conditions, J. Math. Anal. Appl. 367 (2010), 588–611. [17] M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, New York, 1964. [18] R. Ma, Positive solutions for nonhomogeneous m-point boundary value problems, Comput. Math. Appl. 47 (2004), 689–698. [19] R. Ma and Y. An, Global structure of positive solutions for superlinear 2mth-boundary value problems, Czechoslovak Math. J. 60(135) (2010), 161–172. [20] R. Ma and H. Wang, On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal. 59 (1995), 225–231. [21] Y. Sun, Positive solutions for third-order three-point nonhomogeneous boundary value problems, Appl. Math. Lett. 22 (2009), 45–51. [22] J. Sun and G. Zhang, Nontrivial solutions of singular superlinear Sturm-Liouville problems, J. Math. Anal. Appl. 313 (2006), 518–536. [23] J. Sun and G. Zhang, Nontrivial solutions of singular sublinear Sturm-Liouville problems, J. Math. Anal. Appl. 326 (2007), 242–251. [24] J. R. L. Webb, Positive solutions of some higher order nonlocal boundary value problems, Electron. J. Qual. Theory Differ. Equ. 2009, Special Edition I, No. 29, 15 pp. [25] J. R. L. Webb, Nonlocal conjugate type boundary value problems of higher order, Nonlinear Anal. 71 (2009), 1933–1940. [26] J. R. L. Webb, Optimal constants in a nonlocal boundary value problem, Nonlinear Anal. 63 (2005), 672–685. [27] J. R. L. Webb, Multiple positive solutions of some nonlinear heat flow problems, Discrete Contin. Dyn. Syst. (2005), Suppl., 895–903. [28] J. R. L. Webb and G. Infante, Non-local boundary value problems of arbitrary order, J. Lond. Math. Soc. (2) 79 (2009), 238–258. [29] J. R. L. Webb and G. Infante, Positive solutions of non-local boundary value problems: a unified approach, J. London Math. Soc. (2) 74 (2006), 673–693. [30] J. R. L. Webb and G. Infante, and D. Franco, Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), 427–446. [31] J. R. L. Webb and K. Q. Lan, Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type, Topol. Methods Nonlinear Anal. 27 (2006), 91–115. [32] B. Yang, Positive solutions for a fourth order boundary value problem, Electron. J. Qual. Theory Differ. Equ. 2005, No. 3, 17 pp. [33] B. Yang, Positive solutions to a boundary value problem for the beam equation, Z. Anal. Anwend. 26 (2007), 221–230. [34] E. Zeidler, Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems, Springer-Verlag, New York, 1986. [35] Z. Zhang and J. Wang, Positive solutions to a second-order three point boundary value problem, J. Math. Anal. Appl. 285 (2003), 237–249.