On L² and limit-point type solutions of fourth order differential equations

Abstract

We study differential expressions of the fourth order and find sutficient conditions to ensure that they are not of the limit-circle type. In particular, we show that the differential expression y⁽⁴⁾ + ry is never of the limit-circle type as long as r is not an unbounded oscillatory function; this partially answers an open question. Some of the results are deduced as consequences of new results on the nonlinear limit-point/limit-circle problem.

Keywords:

• differential expressions of fourth order
• nonlinear (linear) differential equations of fourth order
• Weyl's classification of limit-pointilirnit-circle
• nonlinear limit-point
• nonlinear limit-circle
• spectral theory

¹Research supported by grant 201/93/0452 of Czech Grant Agency (Prague).

¹Research supported by grant 201/93/0452 of Czech Grant Agency (Prague).

²Research supported by the Mississippi State University Biological and Physical Sciences Research Institute.

¹Research supported by grant 201/93/0452 of Czech Grant Agency (Prague).

¹Research supported by grant 201/93/0452 of Czech Grant Agency (Prague).

²Research supported by the Mississippi State University Biological and Physical Sciences Research Institute.

Notes

¹Research supported by grant 201/93/0452 of Czech Grant Agency (Prague).

¹Research supported by grant 201/93/0452 of Czech Grant Agency (Prague).

²Research supported by the Mississippi State University Biological and Physical Sciences Research Institute.