




Volume 35 • Number 2 • 2012 

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The Existence of Three Positive Solutions to Integral Type BVPs for Second Order ODEs with OneDimensional $p$Laplacian
Yuji Liu
Abstract.
This paper is concerned with the integral type boundary value problems of the second order differential equations with onedimensional $p$Laplacian
$$ \left\{\begin{array}{l}[\rho(t)\Phi(x'(t))]'+f(t,x(t),x'(t))=0,\quad t\in (0,1),\\
\phi_1(x(0))=\int_0^1g(s)\phi_1(x(s))ds,\\
\phi_2( x'(1))=\int_0^1h(s)\phi_2(x'(s))ds.\end{array}\right.$$
Sufficient conditions to guarantee the existence of at least three positive solutions of this BVP are established. An example is presented to illustrate the main results. The emphasis is put on the onedimensional $p$Laplacian term $[\rho(t)\Phi(x'(t))]'$ involved with the function $\rho$, which makes the solutions unconcave.
2010 Mathematics Subject Classification: 34B10, 34B15, 35B10, 65L10.
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