Successively iterative technique of a classical elastic beam equation with Carathéodory nonlinearity.(English)Zbl 1188.34013

The author considers the following nonlinear fourth-order boundary-value problem with non-homogeneous boundary conditions:
\begin{aligned} &u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)),\quad\text{a. e. }t\in[0,1],\\ & u(0)=a,\;u'(1)=b,\;u''(0)=c,\;u'''(1)=d, \end{aligned}\tag{*}
where $$a, b, c, d$$ are given constants and $$f:[0,1]\times(-\infty,\infty)^4\to(-\infty,\infty)$$ is a given continuous function. In mechanics, the problem ($$*$$) describes the equilibrium state of an elastic beam simply supported at left and clamped at right by sliding clamps. An improved iterative sequence is constructed by the help of monotonic technique which approximates successively the solution of the problem ($$*$$) under suitable assumptions. The function $$f$$ is said to be Carathéodory if 8mm
(i)
for a. e. $$t\in[0,1]$$, $$f(t,\cdot,\cdot,\cdot,\cdot):(-\infty,\infty)^4\to(-\infty,\infty)$$ is continuous,
(ii)
for all $$(u,v,w,z)\in(-\infty,\infty)^4$$, $$f(\cdot,u,v,w,z):[0,1]\to(-\infty,\infty)$$ is measurable and
(iii)
for every $$r > 0$$, there exists a non-negative function $$j_r\in L^1[0,1]$$ such that $$|f(t,u,v,w,z)|\leq j_r(t)$$, $$(t,u,v,w,z)\in[0,1]\times[-r,r]^4$$.
An example is presented to illustrate the result obtained.

MSC:

 34A45 Theoretical approximation of solutions to ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text:

References:

 [1] Gupta, P.C.: Existence and uniqueness theorems for the bending of an elastic beam equation. Appl. Anal. 26, 289–304 (1988) · Zbl 0611.34015 [2] 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, 257–262 (1997) · Zbl 0913.34020 [3] Grae, R., Yan, B.: Existence and nonexistence of positive solutions of fourth order nonlinear boundary value problems. Appl. Anal. 74, 201–214 (2000) · Zbl 1031.34025 [4] Yao, Q.: An existence theorem for a nonlinear elastic beam equation with all order derivatives. J. Math. Study 38, 24–28 (2005) · Zbl 1092.34513 [5] Bai, Z.: The upper and lower solution method for some fourth-order boundary value problems. Nonlinear Anal. 67, 1704–1709 (2007) · Zbl 1122.34010 [6] Agarwal, R.P.: On fourth order boundary value problems arising in beam analysis. Differ. Integral Equ. 2, 91–110 (1989) · Zbl 0715.34032 [7] Wong, P.J.Y., Agarwal, R.P.: Multiple solutions for a system of (n i ,p i ) boundary value problems. J. Anal. Appl. 19, 511–528 (2000) · Zbl 1160.34313 [8] Yao, Q.: Monotone iterative technique and positive solutions of Lidstone boundary value problems. Appl. Math. Comput. 138, 1–9 (2003) · Zbl 1049.34028 [9] Yao, Q.: Successive iteration and positive solution of nonlinear second-order three-point boundary value problems. Comput. Math. Appl. 50, 433–444 (2005) · Zbl 1096.34015 [10] Yao, Q.: Successive iteration and positive solution for a discontinuous third-order boundary value problem. Comput. Math. Appl. 53, 741–749 (2007) · Zbl 1149.34316 [11] Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) · Zbl 0582.49001 [12] Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, Berlin (1978) · Zbl 0137.03202
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.