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 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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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     author = {Yao, Qingliu},
     title = {Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem},
     journal = {Applications of Mathematics},
     pages = {93--110},
     year = {2013},
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     number = {1},
     doi = {10.1007/s10492-013-0004-8},
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     zbl = {1274.34076},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10492-013-0004-8/}
}
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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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