Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order. (English) Zbl 1274.34076

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 J. R. Graef and B. Yang [Appl. Anal. 74, 201–214 (2000; Zbl 1031.34025)]. Finally, the theory is applied to a nonlinear fourth-order BVP.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34B09 Boundary eigenvalue problems for ordinary differential equations


Zbl 1031.34025
Full Text: DOI


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