# Article

Full entry | PDF   (0.2 MB)
Keywords:
nonlinear ordinary differential equation; singular nonlinearity; positive solution; eigenvalue interval
Summary:
We consider the classical nonlinear fourth-order two-point boundary value problem $$\begin {cases} u^{(4)}(t)=\lambda h(t)f(t,u(t),u'(t),u''(t)),\quad 0<t<1,\\ u(0)=u'(1)=u''(0)=u'''(1)=0. \end {cases}$$ In this problem, the nonlinear term $h(t)f(t,u(t),u'(t),u''(t))$ contains the first and second derivatives of the unknown function, and the function $h(t)f(t,x,y,z)$ may be singular at $t=0$, $t=1$ and at $x=0$, $y=0$, $z=0$. By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.
References:
[1] Agarwal, R. P., O'Regan, D.: Positive solutions to superlinear singular boundary value problems. J. Comput. Appl. Math. 88 (1998), 129-147. DOI 10.1016/S0377-0427(97)00205-7 | MR 1609070 | Zbl 0902.34017
[2] Agarwal, R. P., O'Regan, D.: Multiplicity results for singular conjugate, focal, and $(N,P)$ problems. J. Differ. Equations 170 (2001), 142-156. DOI 10.1006/jdeq.2000.3808 | MR 1813103 | Zbl 0978.34018
[3] Atanackovic, T. M.: Stability Theory of Elastic Rods. Series on Stability, Vibration and Control of Systems, Series A. World Scientific Singapore (1997).
[4] Bai, Z., Wang, H.: On positive solutions of some nonlinear fourth-order beam equations. J. Math. Anal. Appl. 270 (2002), 357-368. DOI 10.1016/S0022-247X(02)00071-9 | MR 1915704 | Zbl 1006.34023
[5] Bai, Z.: The upper and lower solution method for some fourth-order boundary value problems. Nonlinear Anal., Theory Methods Appl. 67 (2007), 1704-1709. DOI 10.1016/j.na.2006.08.009 | MR 2326022 | Zbl 1122.34010
[6] Dunninger, D. R.: Multiplicity of positive solutions for a nonlinear fourth order equation. Ann. Pol. Math. 77 (2001), 161-168. DOI 10.4064/ap77-2-3 | MR 1869312 | Zbl 0989.34014
[7] Elgindi, M. B. M., Guan, Z.: On the global solvability of a class of fourth-order nonlinear boundary value problems. Int. J. Math. Math. Sci. 20 (1997), 257-261. DOI 10.1155/S0161171297000343 | MR 1444725 | Zbl 0913.34020
[8] Feng, H., Ji, D., Ge, W.: Existence and uniqueness of solutions for a fourth-order boundary value problem. Nonlinear Anal., Theory Methods Appl. 70 (2009), 3561-3566. DOI 10.1016/j.na.2008.07.013 | MR 2502764 | Zbl 1169.34308
[9] Graef, J. R., Yang, B.: Existence and nonexistence of positive solutions of fourth order nonlinear boundary value problems. Appl. Anal. 74 (2000), 201-214. DOI 10.1080/00036810008840810 | MR 1742276 | Zbl 1031.34025
[10] Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Notes and Reports in Mathematics in Science and Engineering, 5. Academic Press Boston (1988). MR 0959889
[11] Gupta, C. P.: Existence and uniqueness theorems for the bending of an elastic beam equation. Appl. Anal. 26 (1988), 289-304. DOI 10.1080/00036818808839715 | MR 0922976 | Zbl 0611.34015
[12] Henderson, J., Wang, H.: Positive solutions for nonlinear eigenvalue problems. J. Math. Anal. Appl. 208 (1997), 252-259. DOI 10.1006/jmaa.1997.5334 | MR 1440355 | Zbl 0876.34023
[13] Hewitt, E., Stromberg, K.: Real and Abstract Analysis. A Modern Treatment of The Theory of Functions of a Real Variable. 3rd printing. Springer Berlin, Heidelberg, New York (1975). MR 0367121 | Zbl 0307.28001
[14] Liu, B.: Positive solutions of fourth-order two point boundary value problems. Appl. Math. Comput. 148 (2004), 407-420. DOI 10.1016/S0096-3003(02)00857-3 | MR 2015382 | Zbl 1039.34018
[15] Ma, R., Ma, Q.: Multiplicity results for a fourth-order boundary value problem. Appl. Math. Mech., Engl. Ed. 16 (1995), 961-969. DOI 10.1007/BF02538837 | MR 1365596 | Zbl 0836.73032
[16] Marcos, J., Lorca, S., Ubilla, P.: Multiplicity of solutions for a class of non-homogeneous fourth-order boundary value problems. Appl. Math. Lett. 21 (2008), 279-286. DOI 10.1016/j.aml.2007.02.025 | MR 2433742 | Zbl 1169.34317
[17] Meehan, M., O'Regan, D.: Positive solutions of singular integral equations. J. Integral Equations Appl. 12 (2000), 271-280. DOI 10.1216/jiea/1020282208 | MR 1810743 | Zbl 0988.45004
[18] Staněk, S.: Positive solutions of singular Dirichlet boundary value problems with time and space singularities. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 4893-4905. DOI 10.1016/j.na.2009.03.043 | MR 2548721 | Zbl 1192.34027
[19] Wei, Z.: A class of fourth order singular boundary value problems. Appl. Math. Comput. 153 (2004), 865-884. DOI 10.1016/S0096-3003(03)00683-0 | MR 2066157 | Zbl 1057.34006
[20] Yao, Q.: Positive solutions of nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69 (2008), 1570-1580. DOI 10.1016/j.na.2007.07.002 | MR 2424530
[21] Yao, Q.: Successively iterative technique of a classical elastic beam equation with Carathéodory nonlinearity. Acta Appl. Math. 108 (2009), 385-394. DOI 10.1007/s10440-008-9317-0 | MR 2551480 | Zbl 1188.34013
[22] Yao, Q.: Local existence of multiple positive solutions to a singular cantilever beam equation. J. Math. Anal. Appl. 363 (2010), 138-154. DOI 10.1016/j.jmaa.2009.07.043 | MR 2559048 | Zbl 1191.34031

Partner of