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Differential operators; linear differential equation of third order; canonical forms; adjoint equation; cyclic permutation; oscillatory solution; Kneser solution; property $\mathrm A$
Consider the third order differential operator $L$ given by \[L(\cdot )\equiv \,\frac {1}{a_3(t)}\frac {\mbox{d}}{\mbox{d} t}\frac {1}{a_2(t)}\frac {\mbox{d}}{\mbox{d} t} \frac {1}{a_1(t)}\frac {\mbox{d}}{\mbox{d} t}\,(\cdot ) \] and the related linear differential equation $L(x)(t)+x(t)=0$. We study the relations between $L$, its adjoint operator, the canonical representation of $L$, the operator obtained by a cyclic permutation of coefficients $a_i$, $ i=1,2,3$, in $L$ and the relations between the corresponding equations. We give the commutative diagrams for such equations and show some applications (oscillation, property A).
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