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Existence and multiplicity of solutions to \(2m\)th-order ordinary differential equations. (English) Zbl 1119.34014

Summary: The existence and multiplicity of solutions are obtained for the \(2m\)th-order ordinary differential equation two-point boundary value problems \[ (-1)^mu^{(2 m)}(t)+\sum^m_{i=1}(-1)^{m-i}a_i u^{(2(m-i))}(t)=f(t,u(t))\text{ for all }t\in [0,1] \] subject to Dirichlet, Neumann, mixed and periodic boundary value conditions, respectively, where \(f\) is continuous, \(a_i\in\mathbb{R}\) for all \(i=1,2,\dots,m\). Since these four boundary value problems have some common properties and they can be transformed into the integral equation of form \[ u+ \sum^m_{i=1}a_iT^iu=T^m{\mathbf f}u, \] we firstly deal with this nonlinear integral equation. By using the strongly monotone operator principle and the critical point theory, we establish some conditions on \(f\) which are able to guarantee that the integral equation has a unique solution, at least one nonzero solution, and infinitely many solutions. Furthermore, we apply the abstract results on the integral equation to the above four \(2m\)th-order two-point boundary problems and successfully resolve the existence and multiplicity of their solutions.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] Bai, Z.; Wang, H., On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270, 357-368 (2002) · Zbl 1006.34023
[2] Conti, M.; Terracini, S.; Verzini, G., Infinitely many solutions to fourth order superlinear periodic problems, Trans. Amer. Math. Soc., 356, 3283-3300 (2004) · Zbl 1074.34047
[3] Davis, J. M.; Eloe, P. W.; Henderson, J., Triple positive solutions and dependence on higher order derivatives, J. Math. Anal. Appl., 237, 710-720 (1999) · Zbl 0935.34020
[4] Davis, J. M.; Henderson, J.; Wong, P. J.Y., General Lidstone problems: Multiplicity and symmetry of solutions, J. Math. Anal. Appl., 251, 527-548 (2000) · Zbl 0966.34023
[5] Erbe, L. H.; Hu, S.; Wang, H., Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl., 184, 640-648 (1994) · Zbl 0805.34021
[6] Graef, J. R.; Qian, C.; Yang, B., Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations, Proc. Amer. Math. Soc., 131, 577-585 (2003) · Zbl 1046.34037
[7] Li, F.; Liang, Z., Existence of positive periodic solutions to nonlinear second order differential equations, Appl. Math. Lett., 11, 1256-1264 (2005) · Zbl 1088.34038
[8] Li, F.; Liang, Z.; Zhang, Q., Existence of solutions to a class of nonlinear second order two-point boundary value problems, J. Math. Anal. Appl., 312, 357-373 (2005) · Zbl 1088.34012
[9] Li, F.; Zhang, Q.; Liang, Z., Existence and multiplicity of solutions of a kind of fourth-order boundary value problem, Nonlinear Anal., 62, 803-816 (2005) · Zbl 1076.34015
[10] Li, F.; Li, Y.; Liang, Z., Existence of solutions to nonlinear Hammerstein integral equations and applications, J. Math. Anal. Appl., 323, 209-227 (2006) · Zbl 1104.45003
[11] Li, Y., Positive solutions of fourth-order boundary value problems with two parameters, J. Math. Anal. Appl., 281, 477-484 (2003) · Zbl 1030.34016
[12] Li, Y., Positive solutions of fourth-order periodic boundary value problems, Nonlinear Anal., 54, 1069-1078 (2003) · Zbl 1030.34025
[13] Li, Y., Abstract existence theorems of positive solutions for nonlinear boundary value problems, Nonlinear Anal., 57, 211-227 (2004) · Zbl 1064.47058
[14] Ma, Y., Existence of positive solutions of Lidstone boundary value problems, J. Math. Anal. Appl., 314, 97-108 (2006) · Zbl 1085.34021
[15] Pang, C.; Dong, W.; Wei, Z., Multiple solutions for fourth-order boundary value problem, J. Math. Anal. Appl., 314, 464-476 (2006) · Zbl 1094.34012
[16] Sun, J.; Li, W.; Cheng, S., Three positive solutions for second-order Neumann boundary value problems, Appl. Math. Lett., 17, 1079-1084 (2004) · Zbl 1061.34014
[17] Taylor, A. E.; Lay, D. C., Introduction to Functional Analysis (1980), Wiley · Zbl 0501.46003
[18] Yao, Q., Existence, multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem, Nonlinear Anal., 63, 237-246 (2005) · Zbl 1082.34025
[19] Zhang, B.; Liu, X., Existence of multiple symmetric positive solutions of higher order Lidstone problems, J. Math. Anal. Appl., 284, 672-689 (2003) · Zbl 1048.34054
[20] Zhang, Z.; Wang, J., On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations, J. Math. Anal. Appl., 281, 99-107 (2003) · Zbl 1030.34024
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