Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order
A nonlinear, fourth-order boundary value problem (BVP) \[ u^{(4)}(t)=\lambda h(t) f(t,u(t),u'(t),u''(t)),\quad u(0)=u'(1)=u''(0)=u'''(1)=0 \eqno(P1) \] is considered for \(t\in (0,1)\); the prime stands for the derivative with respect to \(t\). The continuity and nonnegativity of both \(h\) and \(f\) are assumed, but a singularity of \(h(t)f(t,x,y,z)\) is allowed at \(t=0\), \(t=1\), \(x=0\), \(y=0\), and \(z=0\). Further assumptions comprise both \(t\)-dependent bounds put on the value of \(f\) and a condition constraining the values of \(h(t)\). Besides (P1), a simplified problem (P2), where \(f\equiv f(t,u(t))\), is also introduced. By employing the Green function for a simpler BVP, by defining a cone of functions and introducing completely continuous integral operators, and by applying the Guo-Krasnosel'skii fixed point theorem, the author formulates and proves two main results. Roughly speaking, if \(\lambda \) is bounded from below and from above by particular expressions, then (P1) has at least one increasing positive solution. If \(\lambda \) complies with even trickier inequalities, then (P1) has at least two strictly increasing positive solutions. Parallel statements are also formulated for (P2). Moreover, it is shown that the obtained results generalize the existence theorem by \textit{J. R. Graef} and \textit{B. Yang} [Appl. Anal. 74, 201--214 (2000; Zbl 1031.34025)]. Finally, the theory is applied to a nonlinear fourth-order BVP.
nonlinear ordinary differential equation
scientific article; zbMATH DE number 6162814
scientific article