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Ma, Ruyun; Chen, Tianlan

Existence of positive solutions of fourth-order problems with integral boundary conditions. (English) Zbl 1208.34016

Bound. Value Probl. 2011, Article ID 297578, 17 p. (2011).
Summary: We study the existence of positive solutions of the following fourth-order boundary value problem with integral boundary conditions
\[ u^{(4)}(t)=f(t,u(t),u''(t)),\quad t\in (0,1), \]
\[ u(0)=\int_0^1g(s)u(s)\,ds,\quad u(1)=0,\quad u''(0)=\int^1_0 h(s)u''(s)\,ds,\;u''(1)=0, \]
where \(f:[0,1]\times [0,+\infty)\times (-\infty,0]\to [0,+\infty)\) is continuous, \(g,h\in L^1[0,1]\) are nonnegative. The proof of our main result is based upon the Krein-Rutman theorem and global bifurcation techniques.

Cited in 6 Documents

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34C23 Bifurcation theory for ordinary differential equations

Keywords:

Krein-Rutman theorem; bifurcation techniques
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\textit{R. Ma} and \textit{T. Chen}, Bound. Value Probl. 2011, Article ID 297578, 17 p. (2011; Zbl 1208.34016)
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References:

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