# Article

 Title: Solvability of a class of elastic beam equations with strong Carathéodory nonlinearity (English) Author: Yao, Qingliu Language: English Journal: Applications of Mathematics ISSN: 0862-7940 (print) ISSN: 1572-9109 (online) Volume: 56 Issue: 6 Year: 2011 Pages: 543-555 Summary lang: English . Category: math . 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. (English) Keyword: nonlinear ordinary differential equation Keyword: boundary value problem Keyword: existence Keyword: fixed point theorem MSC: 34B15 MSC: 34B16 MSC: 47N20 MSC: 74K10 idZBL: Zbl 1249.34064 idMR: MR2886237 DOI: 10.1007/s10492-011-0032-1 . Date available: 2011-12-16T15:05:13Z Last updated: 2020-07-02 Stable URL: http://hdl.handle.net/10338.dmlcz/141766 . Reference: [1] Agarwal, R. P.: On fourth order boundary value problems arising in beam analysis.Differ. Integral Equ. 2 (1989), 91-110. Zbl 0715.34032, MR 0960017 Reference: [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. MR 1773979, 10.1016/S0362-546X(98)00341-1 Reference: [3] Clarke, F. H.: Optimization and Nonsmooth Analysis.John Wiley & Sons New York (1983). Zbl 0582.49001, MR 0709590 Reference: [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. Zbl 0913.34020, MR 1444725, 10.1155/S0161171297000343 Reference: [5] Gupta, C. P.: Existence and uniqueness theorems for the bending of an elastic beam equation.Appl. Anal. 26 (1988), 289-304. Zbl 0611.34015, MR 0922976, 10.1080/00036818808839715 Reference: [6] Hewitt, E., Stromberg, K.: Real and Abstract Analysis.Springer Berlin-Heidelberg-New York (1975). Zbl 0307.28001, MR 0367121 Reference: [7] Ma, R.: Existence and uniqueness theorems for some fourth-order nonlinear boundary value problems.Int. J. Math. Math. Sci. 23 (2000), 783-788. Zbl 0959.34015, MR 1764120, 10.1155/S0161171200003057 Reference: [8] O'Regan, D.: Singular Dirichlet boundary value problems. I: Superlinear and nonresonant case.Nonlinear Anal., Theory Methods Appl. 29 (1997), 221-245. Zbl 0884.34028, MR 1446226, 10.1016/S0362-546X(96)00026-0 Reference: [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. Zbl 1127.34308, MR 2247273 Reference: [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. Zbl 1199.34085, MR 2458444 Reference: [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 Reference: [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. MR 2091853, 10.1016/j.aml.2003.09.011 Reference: [13] Yao, Q.: Solvability of a fourth-order beam equation with all-order derivatives.Southeast Asian Bull. Math. 32 (2008), 563-571. Zbl 1174.34365, MR 2416172 Reference: [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. Zbl 1174.34354, MR 2416279 Reference: [15] Yao, Q.: Successive iteration and positive solution for a discontinuous third-order boundary value problem.Comput. Math. Appl. 53 (2007), 741-749. MR 2327630, 10.1016/j.camwa.2006.12.007 Reference: [16] Wang, J.: Solvability of singular nonlinear two-point boundary value problems.Nonlinear Anal., Theory Methods Appl. 24 (1995), 555-561. Zbl 0876.34017, MR 1315694, 10.1016/0362-546X(95)93091-H .

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