Comparioson Theory and Asymptotic Theory for Differential Equations

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Sufficient conditions for every solution to oscillate were obtained by establishing a Picone identity for half-linear elliptic equation. In the case where nonlinear term is the sum of superlinear term and sublinear term oscillation results were obtained via Picone-type inequalities. We established Sturmian comparison theorems for fourth order differential operators by deriving a Picone identity. For quasilinear elliptic equations where principal parts are more general than p-Laplacin we obtained oscillation results by establishing Picone identity for half-linear case and Picone-type inequality for the case where nonlinear term is the sum of superlinear term and sublinear term. Moreover, oscillation theorems were obtained for superlinear elliptic equations with forcing terms. Using Picone-type inequality applicable to quasilinear parabolic equations, we derived unboundedness of solutions and sufficient conditions for bounded solution to oscillate. All of above results are very significa … More nt because importance of Picone identity or Picone-type inequality was realized again. Sufficient conditions for solution of boundary value problems of parabolic equations with functional arguments and hyperbolic equations with functional arguments to oscillate were established by reducing to functional differential equations. We obtained sufficient conditions for every solution of second order neutral differential equations with positive and negative coefficients to oscillate. These results seem to contribute greatly to oscillation theory for partial differential equations and oscillation theory for functional differential equations with one variable. I gave lectures on international conferences "EQUADIFF 11", "Conference on Differential and Difference Equations and Applications", "Colloquium on Differential and Difference Equations", "EQUADIFF 2007" and it was very fruitful for me to be able to communicate with foreign researchers and get newest informations. Also I could participate in various workshops in Japan, and presented research results, and moreover I could do research arrangements. I believe that comparison theory and asymptotic theory for differential equations could be developed greatly. Less