Geodesic
Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem
Applications of Mathematics, Tome 58 (2013) no. 1, pp. 93-110

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We consider the classical nonlinear fourth-order two-point boundary value problem $$ \begin {cases} u^{(4)}(t)=\lambda h(t)f(t,u(t),u'(t),u''(t)),\quad 01,\\ u(0)=u'(1)=u''(0)=u'''(1)=0. \end {cases} $$ 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.
DOI : 10.1007/s10492-013-0004-8
Classification : 34B08, 34B09, 34B15, 34B16, 34B18, 34B27, 34L15, 47N20
Keywords: nonlinear ordinary differential equation; singular nonlinearity; positive solution; eigenvalue interval 
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Yao, Qingliu. Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem. Applications of Mathematics, Tome 58 (2013) no. 1, pp. 93-110. doi: 10.1007/s10492-013-0004-8

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