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

Liu, Yuji

Bull. Malays. Math. Sci. Soc. (2) 35, No. 2, 359-372 (2012).

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

Keywords:

second order differential equation with \(p\)-Laplacian; integral type boundary value problem; positive solution; the five functionals fixed point theorem