The Oscillation of Fourth Order Linear Differential Operators

1975 ◽  
Vol 27 (1) ◽  
pp. 138-145 ◽  
Author(s):  
Roger T. Lewis
Keyword(s):  
Fourth Order ◽  
Differential Operators ◽  
Adjoint Operator ◽  
Oscillatory Behavior ◽  
The Self ◽  
Order Operator ◽  
Order Linear ◽  
Linear Differential Operators ◽  
Even Order ◽  
Linear Differential

Define the self-adjoint operatorwhere r(x) > 0 on (0, ∞) and q and p are real-valued. The coefficient q is assumed to be differentiate on (0, ∞) and r is assumed to be twice differentia t e on (0, ∞).The oscillatory behavior of L4 as well as the general even order operator has been considered by Leigh ton and Nehari [5], Glazman [2], Reid [7], Hinton [3], Barrett [1], Hunt and Namb∞diri [4], Schneider [8], and Lewis [6].

scholarly journals II. On the interchange of the variables in certain linear differential operators

1890 ◽  
Vol 181 ◽  
pp. 19-51
Keyword(s):  
Differential Operators ◽  
The Self ◽  
Differential Invariants ◽  
Linear Differential Operators ◽  
Independent Variable ◽  
The Subject ◽  
Functional Differential ◽  
Linear Differential ◽  
Derivatives Of ◽  
Single Relation

The operators to be considered, include or involve all those which have presented themselves as annihilators and generators in recent theories of functional differential invariants, reciprocants, cyclicants, &c. The general form of the binary operators, operators whose arguments are the derivatives of one dependent with regard to one independent variable, which I propose first to consider, is adopted in accordance with that used in two remarkable papers by Major MacMahon. They are his operators in four elements. The analogous ternary operators to which I subsequently devote attention, are distinct from his operators of six elements. Their arguments are the partial derivatives of one of three variables, supposed connected by a single relation, with regard to the two others. The only'previous contribution, of which I am aware, to the subject of the reversion of MacMahon operators, is a paper by Professor L. J. Rogers, in which he obtains the operator reciprocal to { μ, v ; 1, 1}, and alludes to the self reciprocal property of V which is expressed with more precision in (38) below.

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