Article
MSC:
34B08,
34B09,
34B15,
34B16,
34B18,
34B27,
34L15,
47N20 | MR 3022770 | Zbl 1274.34076 | DOI: 10.1007/s10492-013-0004-8
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Keywords:
nonlinear ordinary differential equation; singular nonlinearity; positive solution; eigenvalue interval
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.
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