# Article

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Keywords:
nonlinear ordinary differential equation; boundary value problem; existence; fixed point theorem
Summary:
We study the existence of a solution to the nonlinear fourth-order elastic beam equation with nonhomogeneous boundary conditions $\begin {cases} u^{(4)}(t)=f\bigl (t,u(t),u'(t),u''(t),u'''(t)\bigr ),\quad \text {a.e.} \ t\in [0,1],\\ u(0)=a, \ u'(0)=b, \ u(1)=c, \ u''(1)=d, \end {cases}$ where the nonlinear term $f(t,u_{0},u_{1},u_{2},u_{3})$ is a strong Carathéodory function. By constructing suitable height functions of the nonlinear term $f(t,u_{0},u_{1},u_{2},u_{3})$ on bounded sets and applying the Leray-Schauder fixed point theorem, we prove that the equation has a solution provided that the integration of some height function has an appropriate value.
References:
[1] Agarwal, R. P.: On fourth order boundary value problems arising in beam analysis. Differ. Integral Equ. 2 (1989), 91-110. MR 0960017 | Zbl 0715.34032
[2] Agarwal, R. P., O'Regan, D., Lakshmikantham, V.: Singular $(p,n-p)$ focal $(n,p)$ higher order boundary value problems. Nonlinear Anal., Theory Methods Appl. 42 (2000), 215-228. DOI 10.1016/S0362-546X(98)00341-1 | MR 1773979
[3] Clarke, F. H.: Optimization and Nonsmooth Analysis. John Wiley & Sons New York (1983). MR 0709590 | Zbl 0582.49001
[4] 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), 257-262. DOI 10.1155/S0161171297000343 | MR 1444725 | Zbl 0913.34020
[5] 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
[6] Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer Berlin-Heidelberg-New York (1975). MR 0367121 | Zbl 0307.28001
[7] Ma, R.: Existence and uniqueness theorems for some fourth-order nonlinear boundary value problems. Int. J. Math. Math. Sci. 23 (2000), 783-788. DOI 10.1155/S0161171200003057 | MR 1764120 | Zbl 0959.34015
[8] O'Regan, D.: Singular Dirichlet boundary value problems. I: Superlinear and nonresonant case. Nonlinear Anal., Theory Methods Appl. 29 (1997), 221-245. DOI 10.1016/S0362-546X(96)00026-0 | MR 1446226 | Zbl 0884.34028
[9] Yao, Q.: A local existence theorem for nonlinear elastic beam equations fixed at left and simply supported at right. J. Nat. Sci. Nanjing Norm. Univ. 8 (2006), 1-4. MR 2247273 | Zbl 1127.34308
[10] Yao, Q.: Existence of $n$ positive solutions to a singular beam equation rigidly fixed at left and simply supported at right. J. Zhengzhou Univ., Nat. Sci. Ed. 40 (2008), 1-5. MR 2458444 | Zbl 1199.34085
[11] 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. 69 (2008), 1570-1580. MR 2424530
[12] Yao, Q.: Solution and positive solution to a class of semilinear third-order two-point boundary value problem. Appl. Math. Lett. 17 (2004), 1171-1175. DOI 10.1016/j.aml.2003.09.011 | MR 2091853
[13] Yao, Q.: Solvability of a fourth-order beam equation with all-order derivatives. Southeast Asian Bull. Math. 32 (2008), 563-571. MR 2416172 | Zbl 1174.34365
[14] Yao, Q.: Solvability of singular beam equations fixed at left and simply supported at right. J. Lanzhou Univ., Nat. Sci. 44 (2008), 115-118 Chinese. MR 2416279 | Zbl 1174.34354
[15] Yao, Q.: Successive iteration and positive solution for a discontinuous third-order boundary value problem. Comput. Math. Appl. 53 (2007), 741-749. DOI 10.1016/j.camwa.2006.12.007 | MR 2327630
[16] Wang, J.: Solvability of singular nonlinear two-point boundary value problems. Nonlinear Anal., Theory Methods Appl. 24 (1995), 555-561. DOI 10.1016/0362-546X(95)93091-H | MR 1315694 | Zbl 0876.34017

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