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**The existence of three positive solutions to integral type BVPs for second order ODEs with one-dimensional \(p\)-Laplacian.**
*(English)*
Zbl 1243.34036

Summary: This paper is concerned with integral type boundary value problems of second order differential equations with one-dimensional \(p\)-Laplacian
\[
\begin{cases} [\rho (t)\Phi (x'(t))]' + f (t, x(t), x'(t)) = 0, t \in (0, 1), \\ \phi_1(x(0)) = \int^1_0g(s) \phi_1(x(s))ds, \\ \phi_2(x'(1)) = \int_0^1 h(s) \phi_2(x'(s))ds. \end{cases}
\]
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 one-dimensional \(p\)-Laplacian term \([\rho (t)\Phi (x'(t))]'\) involved with the function \(\rho \), which makes the solutions un-concave.

### MSC:

34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |

34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

47N20 | Applications of operator theory to differential and integral equations |