Existence theorems of positive solutions for a fourth-order three-point boundary value problem (Q2372399)

The paper deals with the solvability of the fourth-order three-point BVP with \(p\)-Laplacian \[ (\phi_p(u''(t))''-a(t)f(u(t))=0, \leqno(1) \] \[ u(0)=\xi u(1),\quad u'(1)=\eta u'(0),\quad u''(0)=\alpha_1u''(\delta),\quad u''(1)=\beta_1u''(\delta), \leqno(2) \] where \(f:\mathbb R\to[0,\infty)\) and \(a:(0,1)\to [0,\infty)\) are continuous functions, \(\alpha_1,\beta_1\geq 0\), \(\xi\not=1\), \(\eta\not=1\), \(0<\delta<1\) and \(\phi_p(z)=| z| ^{p-2}z\) for \(p>1\). The paper establishes the existence of at least three positive solutions of the problem (1),(2). The arguments involve the use of the concavity and integral representation of solutions and a fixed point theorem which is a generalization of the Leggett-Williams fixed point theorem. No illustrative examples are shown.