Lyapunov-type inequalities for third-order linear differential equations (Q2794822)

This paper is concerned with Lyapunov-type inequalities for third order linear differential equations of the form NEWLINE\[NEWLINEx'''+q(t)x=0,\eqno{(*)}NEWLINE\]NEWLINE where \(q\in C([a,c], \mathbb R) \).NEWLINENEWLINEThe main result of this paper runs as follows: Suppose that \( x(t) \) is a solution of \((\ast)\) satisfying NEWLINE\[NEWLINE x(a)=x(b)=x(c)=0, -\infty <a<b<c<\infty,\text{ and } x(t)\neq 0 \text{ for } t\in (a,b)\cup (b,c) .NEWLINE\]NEWLINE Then one of the following inequalities holds: NEWLINENEWLINENEWLINE (i) \(\int_{a}^{c}(t-a)(c-t)q_{-}(t)dt>2\), NEWLINENEWLINENEWLINE (ii) \(\int_{a}^{c}(t-a)(c-t)q_{+}(t)dt>2\), NEWLINENEWLINENEWLINE (iii) \(\int_{a}^{b} (t-a)(c-t)q_{-}(t)dt + \int_{b}^{c} (t-a)(c-t)q_{+}(t)dt >2\),NEWLINENEWLINEwhere \( q_{+}(t)=\max \{q(t), 0 \} \) and \( q_{-}(t)=\min \{q(t),0\}\).NEWLINENEWLINEThe Green's function technique is used to establish the above results. They generalize their work to third order linear differential equations of the type NEWLINE\[NEWLINE(p(t)x'')'+q(t)x=0,NEWLINE\]NEWLINE where \( p,q \in C([a,c], \mathbb R)\) such that \( p(t)>0, t\in [a,c]\), NEWLINE\[NEWLINEx'''+f(t)x''+h(t)x=0,NEWLINE\]NEWLINE where \( f \) and \( h\in C([a,c], \mathbb R)\) and NEWLINE\[NEWLINEx'''+f(t)x'' + g(t) x' + h(t) x =0,NEWLINE\]NEWLINE where \( f,g \) and \( h\in C([a,b],\mathbb R)\). Some applications of these results are stated.