Study On The Oscillation And Asymptotic Behavior Of Solutions Of Several Classes Generalized Elastic-Rod Equations(Systems)

Rod and rod groups have been drawing interest in the research realm of nonlinear vibration mechanics. Based on the two sidedness of oscillation, it is really instructively important for the modern engineering study to make clear the oscillation state of rod. In view of the difficulty to acquire the exact or approximate solutions, with the aid of oscillatory theory of differential equation. the oscillation of several complicated nonlinear oscillation system of elastic rod can be achieved and their oscillation state on mechanics as well as physics can also be analyzed.In this paper, using Lebesgue dominated convergence theorem, Schauder-Tychonoff theorem, Banach contraction principle, differential inequality theory etc tools, the forced oscillation of a class generalized elastic-rod equations with fixed boundary in solid me-chanics differential equations with forced type, neutral differential equations with positive and negative coefficients, neutral differential equations with distributed delay, and the ex-istence and asymptotic of oscillatory and non-oscillatory solutions for these equations were correspondingly obtained. The acquired results were applied to the generalized elastic-rod equations and the corresponding results were achieved. Finally, the boundary questions of two classes systems were discussed and the oscillation of equations systems was also gained.1. The Schauder-Tychonoff theorem was adopted to obtain new sufficient condition for global existence and asymptotic behavior of oscillatory solution for a forced second order nonlinear delay differential equations. The resultant conclusions were applied to a class generalized rod equations with forced type, under the condition of fixed boundary, the sufficient condition for global existence and asymptotic:behavior of oscillatory solution was obtained. The conclusion present that the oscillation state of the rod-the forced oscillation happened with the amplitude being more and more small. When tâ†'∞, it was a tiny oscillation.2. Higher-order differential equations with positive and negative coefficients and the system of higher-order neutral differential equations with distributed delays were consid-ered respectively. Using the Banach contraction principle, new sufficient conditions for the existence of nonoscillatory solutions were obtained. The above conclusion were adopted to a class generalized rod Equations, Under the condition of fixed boundary, obtained the existence of nonoscillatory solutions. The result propose the oscillation state of the rod in this state, that is, the oscillation won’t happen.3. The existence of nonoscillatory solutions of variable coefficients higher-order d-ifferential equations with distributed delays was considered.. The Lebesgue dominated convergence theorem and comparison theorem were used to obtain new necessary and suf-ficient condition for the existence and asymptotic behavior of bounded positive solutions. With the above conclusion, asymptotic behavior of bounded positive solutions for another class generalized rod Equations with distributed delays characteristics was also obtained.4. Two classes systems of even order nonlinear partial functional differential equa-tions with distributed delay characteristics were considered, with the aid of mathematical methods, The new sufficient condition for oscillation of equations systems were obtained. The result reflected the oscillation state of the rod in this case that the oscillation kept happening.