Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4633974 | Applied Mathematics and Computation | 2008 | 9 Pages |
Abstract
The half-linear differential equation (ϕp(x′))′+a(t)ϕp(x′)+b(t)ϕp(x)=0(ϕp(x′))′+a(t)ϕp(x′)+b(t)ϕp(x)=0 is considered under the assumption that the coefficient a(t) and an indefinite integral B(t) of b(t) are periodic functions with period T > 0. It is proved that {(p-1)ϕp∗(B(t))-a(t)}B(t)⩽0(0⩽t⩽T) is sufficient for all nontrivial solutions to be nonoscillatory. Here, p > 1 and ϕq(y)=|y|q-2yϕq(y)=|y|q-2y for q = p or q = p∗ = p/(p − 1). The proof is given by means of Riccati technique. The condition is shown to be sharp. Sufficient conditions are also presented for all nontrivial solutions are oscillatory in the linear case p = 2. Some examples and simulations are included to illustrate our results.
Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
Jitsuro Sugie, Kouhei Matsumura,