×

A monotone method for fourth order value problems involving a factorizable linear operator. (English) Zbl 1137.34315

Summary: We consider the nonlinear fourth order beam equation
\[ u^{\text{iv}}=f(t,u,u''), \]
with boundary conditions corresponding to the periodic or the hinged beam problem. In presence of upper and lower solutions, we consider a monotone method to obtain solutions. The main idea is to write the equation in the form
\[ u^{\text{iv}}-cu''+du=g(t,u,u''), \]
where \(c\), \(d\) are adequate constants, and use maximum principles and a suitable decomposition of the operator appearing in the left-hand side.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34A55 Inverse problems involving ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bai, Zhanbing - The method of lower and upper solutions for a bending of an elastic beam equation, J. Math. Anal. Appl., 248 (2000), 195-202. · Zbl 1016.34010 · doi:10.1006/jmaa.2000.6887
[2] Bai, Zhanbing; Ge, Weigao and Wang, Yifu - The method of lower and upper solutions for some fourth order equations, J. Inequalities in Pure Appl. Math., 5(1) (2004), article 13. · Zbl 1063.34010
[3] Bai, Zhanbing and Wang, Haiyan - On positive solutions of some nonlinear fourth order beam equations, J. Math. Anal. Appl., 270 (2002), 357-368. · Zbl 1006.34023 · doi:10.1016/S0022-247X(02)00071-9
[4] Bellen, A. - Monotone methods for periodic solutions of second order scalar functional differential equations, Numer. Math., 42 (1983), 15-30. · Zbl 0536.65065 · doi:10.1007/BF01400915
[5] van den Berg, J.B. - Uniqueness of solutions for the extended F-K equation, C. R. Acad. Sci. Paris Sér. I Math., 326(4) (1998), 447-452. · Zbl 0913.34052 · doi:10.1016/S0764-4442(97)89790-X
[6] Cabada, A. - The method of lower and upper solutions for n-th order periodic boundary value problems, J. Appl. Math. Stochastic Anal., 7(1) (1994), 33-47. · Zbl 0801.34026 · doi:10.1155/S1048953394000043
[7] Cabada, A.; Cid, J. Á. and Sanchez, L. - Positivity and lower and upper solu- tions for fourth order boundary value problems, Nonlinear Anal., Theory Methods Appl., 67(5)(A) (2007), 1599-1612. · Zbl 1125.34010 · doi:10.1016/j.na.2006.08.002
[8] Chaparova, J.; Peletier, L.A. and Tersian, S. - Existence and nonexistence of nontrivial solutions of semilinear fourth and sixth order differential equations, Adv. Diff. Eq., 8 (2003), 1237-1258. · Zbl 1126.34316
[9] Cherpion, M.; De Coster, C. and Habets, P. - A constructive monotone iterative method for second-order BVP in the presence of lower and upper solutions, Appl. Maths. Computation, 123 (2001), 75-91. · Zbl 1024.65063 · doi:10.1016/S0096-3003(00)00058-8
[10] Conti, M.; Terracini, S. and Verzini, G. - Infinitely many solutions to fourth order superlinear periodic problems, Trans. Am. Math. Soc., 356(8) (2004), 3283-3300. · Zbl 1074.34047 · doi:10.1090/S0002-9947-03-03514-1
[11] Coppel, W.A. - Disconjugacy, Lect. Notes Math., 220 (1971).
[12] De Coster, C. and Habets, P. - Two-Point Boundary Value Problems: Lower and Upper Solutions, Elsevier, Amsterdam, 2006. · Zbl 1330.34009
[13] Elias, U. - Eigenvalue problems for the equations Ly + \lambda p(x)y = 0, J. Differential Equations, 29(1) (1978), 28-57. · Zbl 0351.34014 · doi:10.1016/0022-0396(78)90039-6
[14] Graef, J. and Yang, Bo - Existence and nonexistence of positive solutions of fourth order nonlinear boundary value problems, Applicable Analysis, 74 (2000), 201-214. · Zbl 1031.34025 · doi:10.1080/00036810008840810
[15] Jiang, Daqing; Gao, Wenjie and Wan, Aying - A monotone method for constructing extremal solutions to fourth-order periodic boundary value problems, Appl. Math. Comput., 132 (2002), 411-421. · Zbl 1036.34020 · doi:10.1016/S0096-3003(01)00201-6
[16] Kantorovich, L. - The method of successive approximations for functional equa- tions, Acta Math., 71 (1939), 63-97. · Zbl 0021.13604 · doi:10.1007/BF02547750
[17] Li, Yongxiang - Positive solutions of fourth order boundary value problems with two parameters, J. Math. Anal. Appl., 281 (2003), 477-484. · Zbl 1030.34016 · doi:10.1016/S0022-247X(03)00131-8
[18] Li, Yongxiang - Positive solutions of fourth-order periodic boundary value prob- lems, Nonlinear Analysis, 54 (2003), 1069-1078. · Zbl 1030.34025 · doi:10.1016/S0362-546X(03)00127-5
[19] Liu, B. - Positive solutions of fourth order two point boundary value problems, Appl. Math. Comp., 148 (2004), 407-420. · Zbl 1039.34018 · doi:10.1016/S0096-3003(02)00857-3
[20] Liu, Xi-Lan and Li, Wan-Tong - Positive solutions of the nonlinear fourth order beam equation with three parameters, J. Math. Anal. Appl., 303 (2005), 150-163. · Zbl 1077.34027 · doi:10.1016/j.jmaa.2004.08.026
[21] Mizel, V.J.; Peletier, L.A. and Troy, W.C. - Periodic phases in second-order materials, Arch. Rat. Mech. Anal., 145 (1998), 343-382. · Zbl 0931.74006 · doi:10.1007/s002050050133
[22] Omari, P. and Trombetta, M. - Remarks on the lower and upper solutions method for second- and third-order periodic problems, Applied Math. Comp., 50 (1992), 1-21 and 56 (1993), 101. · Zbl 0760.65078 · doi:10.1016/0096-3003(92)90007-N
[23] Peletier, L.A. and Troy, W.C. - Multibump periodic travelling waves in sus- pension bridges, Proc. Roy. Soc. Edinburgh sect A, 128(3) (1998), 631-659. · Zbl 0909.35143 · doi:10.1017/S0308210500021661
[24] Rynne, B.P. - Infinitely many solutions of superlinear fourth order boundary value problems, Top. Meth. Nonl. Anal., 19 (2002), 303-312. · Zbl 1017.34015
[25] Rynne, B.P. - Bifurcation for 2mth order boundary value problems and infinitely many solutions of superlinear problems, J. Diff. Eq., 188 (2003), 461-472. · Zbl 1029.34015 · doi:10.1016/S0022-0396(02)00146-8
[26] Rynne, B.P. - Solution curves of 2m-th order boundary value problems, Electr. J. Diff. Eq., 2004(32) (2004), 1-16. · Zbl 1060.34011
[27] Senky\check rik, M. - Fourth order boundary value problems and nonlinear beams, Applicable Analysis, 59 (1995), 15-25. · Zbl 0841.34023 · doi:10.1080/00036819508840387
[28] Vandervorst, R.C.A.M. and van den Berg, J.B. - Periodic orbits for fourth order conservative systems and Morse type theory, in “International Conf. on Diff. Eq.”, Vol. 1,2 (Berlin, 1999), 241-245, World Sci. Publishing, River Edge, NJ, 2000. · Zbl 0966.37032
[29] Yao, Qingliu - On the positive solutions of a nonlinear fourth order boundary value problem with two parameters, Applicable Analysis, 83 (2004), 97-107. · Zbl 1051.34018 · doi:10.1080/00036810310001632817
[30] Yuji, Liu and Weigao, Ge - Double positive solutions of fourth order nonlinear boundary value problems, Applicable Analysis, 82 (2003), 369-380. · Zbl 1037.34017 · doi:10.1080/0003681031000063685
[31] Zeidler, E. - Nonlinear Functional Analysis and Its Applications. I: Fixed Point Theorems, Springer-Verlag, New York, 1986. · Zbl 0583.47050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.