Distribution of zeros and disconjugacy of fourth order differential equations (Q2786728)

The authors consider the fourth-order linear differential equation NEWLINE\[NEWLINE (r(x)y''')'-(p(x)y')'+q(x)y=0, \eqno{(*)} NEWLINE\]NEWLINE where \(r\), \(p\), \(q\) are continuous functions and \(r(x)>0\), together with various boundary conditions at the endpoints \(x=\alpha\), \(x=\beta\) of the interval \([\alpha,\beta]\), typical ones are NEWLINE\[NEWLINE y(\alpha)=y'(\alpha)=0=y(\beta)=y'(\beta). \eqno{(**)} NEWLINE\]NEWLINE One of the main results given in the paper is the following statement.NEWLINENEWLINETheorem. Suppose that \(r(x)\equiv1\) in \((*)\). If there exists a nontrivial solution of \((*)\) satisfying \((**)\), then we have the following lower bounds for the distance of \(\alpha\) and \(\beta\): NEWLINE\[NEWLINE \sqrt{\beta-\alpha}\left[\int_\alpha^\beta |P(x)|^2\,dx\right]^{\frac{1}{2}}+ \frac{(\beta-\alpha)^{5/2}}{6}\left[\int_\alpha^\beta |Q(x)|^2\,dx\right]^{\frac{1}{2}} \geq 1NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \sqrt{\beta-\alpha}\left[\int_\alpha^\beta [P(x)+2Q_2(x)]^2\,dx\right]^{\frac{1}{2}} + \frac{(\beta-\alpha)^{3/2}}{\sqrt{12}}\left[\int_\alpha^\beta Q_1^2(x)\,dx\right]^{\frac{1}{2}} \geq 1 NEWLINE\]NEWLINE where \(P\), \(Q\) are primitive functions of \(p\), \(q\) respectively, \(Q_1'=Q\), \(Q_2'=Q_1\).NEWLINENEWLINEThe results of the paper are proved using new inequalities of Hardy's type and Opial and Wirtinger inequalities with weight functions.