Báo cáo hóa học: "OSCILLATION AND NONOSCILLATION THEOREMS FOR A CLASS OF EVEN-ORDER QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: OSCILLATION AND NONOSCILLATION THEOREMS FOR A CLASS OF EVEN-ORDER QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS | OSCILLATION AND NONOSCILLATION THEOREMS FOR A CLASS OF EVEN-ORDER QUASILINEAR FUNCTIONAL DIFFERENTIAL eQuATIONS JELENA MANOJLOVIC AND TOMOYUKI TANIGAWA Received 13 November 2005 Accepted 30 January 2006 We are concerned with the oscillatory and nonoscillatory behavior of solutions of evenorder quasilinear functional differential equations of the type I y n t Ia sgn y n t n q t Iy g t P sgny g t 0 where a and Ỉ are positive constants g t and q t are positive continuous functions on 0 to and g t is a continuously differentiable function such that g t 0 limt TOg t to. We first give criteria for the existence of nonoscilla-tory solutions with specific asymptotic behavior and then derive conditions sufficient as well as necessary and sufficient for all solutions to be oscillatory by comparing the above equation with the related differential equation without deviating argument. Copyright 2006 J. Manojlovic and T. Tanigawa. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction We consider even-order quasilinear functional differential equations of the form I y n t I a sgn y n t n q t I y g t Ip sgn y g t 0 A where a a and Ỉ are positive constants b q 0 to 0 to is a continuous function c g 0 to 0 to is a continuously differentiable function such that g t 0 t 0 and limt TOg t to. By a solution of A we mean a function y Ty to R which is n times continuously differentiable together with I y n Ia sgn y n and satisfies A at all sufficiently large t. Those solutions which vanish in a neighborhood of infinity will be excluded from our consideration. A solution is said to be oscillatory if it has a sequence of zeros clustering around to and nonoscillatory otherwise. Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 42120 Pages 1-22 DOI JIA 2006 42120 2 .

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