Systems-disconjugacy of a Fourth-order Differential Equation1
نویسندگان
چکیده
where r(x) >0 and p(x) are both continuous on [a, oo) and p(x) does not change sign, and related conjugate point properties to oscillation. They gave extensive evidence indicating that the oscillatory behavior when p(x) is positive is essentially different from that when p(x) is negative. Relatively little is known in general when p(x) changes sign or when derivative terms of order less than four are present—except those facts which can be obtained by a simple piecing together of intervals or by the relatively few transformations into the type (1). Earlier study of this subject was done by W. M. Whyburn [8] and others listed in the bibliography of [5]. This paper is primarily concerned with the fourth-order equation (1) where r(x) and p(x) are both positive and continuous on [a, oo)3 and the designation "(1)" will denote the equation (1), together with these restrictions on the coefficients except in one or two cases in which departure from this convention will be explicitly given. For example, in §3 there is a theorem insuring the disconjugacy of (1) without regard of the sign, or changes of sign, of p(x). Leighton and Nehari [5] introduced the double-zero conjugate point concept of which the first conjugate point r]i(a) of a is defined as follows and is called the first LN'-conjugate point: Definition 1. The number 171(a) is the smallest number ¿>G(a, <*>) for which the two point boundary conditions4
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تاریخ انتشار 2010