Project/Area Number  09640237 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
解析学

Research Institution  Fukuoka University 
Principal Investigator 
KUSANO Takashi Fukuoka Univ., Fac.of Science, Professor, 理学部, 教授 (70033868)

CoInvestigator(Kenkyūbuntansha) 
YOSHIDA Norio Toyama Univ.Fac.of Science, Professor, 理学部, 教授 (80033934)
NAITO Manabu Ehime Univ., Fac.of Science, Professor, 理学部, 教授 (00106791)
KUROKIBA Masaki Fukuoka Univ., Fac.of Science, Assistant, 理学部, 助手 (60291837)
TANAKA Naoto Fukuoka Univ., Fac.of Science, Assoc.Prof., 理学部, 助教授 (00247222)
SAIGO Megumi Fukuoka Univ., Fac.of Science, Professor, 理学部, 教授 (10040403)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 1998 : ¥1,300,000 (Direct Cost : ¥1,300,000)
Fiscal Year 1997 : ¥1,700,000 (Direct Cost : ¥1,700,000)

Keywords  oscillation theory / nonlinear differential equation / nonlinear SturmLiouville operator / oscillatory / zero / singular solution / 振動理論 / 非線形微分方程式 / 非線形SturmLiouville作用素 / 振動解、非振動解 / 零点分布 / 特異解 / 振動解.非振動解 / 定性的理論 / 定量解析 
Research Abstract 
The present research project is devoted to the investigation of the oscillatory behavior of various types of differential equations involving the nonlinear SturmLiouville differential operators. The main results obtained are as follows. (1)We have established Picorietype identities for the nonlinear SturmLiouville operators and have applied them to the study of comparison and oscillation of solutions to halflinear ordinary and partial differential equations. We observe that it has long been unknown whether there is a class of nonlinear differential operators for which a Piconetype identity can be established. (2)We have developed a theory of generalized trigonometric functions, on the basis of which one can successfully construct an exact nonlinear analogue of the wellknown Sturmian theory for linear differential equations. The generalized Prufer transformations thus defined has made it possible to count the number of zeros of nonoscillatory solutions to a certain class of halflinear ordinary differential equations. (3)We have made a detailed analysis of the asymptotic behavior of positive solutions to some nonlinear SturmLiouville equations with singularities. A similar analysis has also been made of differential equations involving singular SturmLiouville operators. As an unexpected byproduct we have discovered a new type of singular solution which has never appeared in the literature. (4)Regarding the oscillation of functional differential equations, we have established (i) a new comparison theorem holding for a special class of nonneutral equations and (ii) an effective criterion for oscillation of neutral equations involving the nonlinear SturmLiouville differential operators.
