| A more ecumenical second-order nonlinear neutral delay differential equationwhereτ> 0,σ1,σ2,…,σn≥0, P, r∈C([t0,∞), R), F∈C([t0, +∞)×Rn, R), G∈C([t0, +∞), R) and c is a constant, is studied in this paper, and some sufficient conditions for existence of nonoscillatory solutions for this equation are established and expatiated through five theorems according as the range of value of function P(t). These theorems are mainly proofed as follows. At first, construct a contraction mapping T : S→X on a nonempty closed convex subset S of the Banach space X and illuminate the mapping T is a self-mapping on subset S by the method of classified discussion. Afterwards, make use of Banach contraction mapping theory to gain a unique fixed point x∈S of mapping T, and then the fixed point x is a solution of the above equation. In the last section, five examples are presented to illustrated that our works are proper generalizations of the corresponding results of Kulenovic and Hadziomerspahic [8], Cheng and Annie [3], Yu and Wang [16]. Furthermore, our results omit the restriction of Q1(t) dominating Q2(t) (See condition C in the text). In addition, because of the limit of ability and knowledge, for some range of the value of function P(t), the existence of nonoscillatory solutions for the above equation is inextricability in this paper. Therefore, in order to solve the existence of nonoscillatory solutions for the above equation completely early, some open questions are shown in the text.
|
PDF Full Text Request