ABSTRACT
Existence and/or nonexistence of various kinds of solutions to (2) have been investigated. Troy [15] made a thorough research of (2) in the case λ=1/2; he imposes the initial condition and pursues the trajectory in detail regarding β as a parameter. Main results of [15] state that (2) with λ=1/2 has at least two odd periodic solutions and that there further exist at least two globally bounded solutions obeying respectively. In [16], the same method is applied to derive the nonexistence of monotonic global solutions; that is, there is no solution to (2) with such that y′(x)>0 for all and
respectively. The nonexistence of monotonic global solutions is later extended in [6] to include all λ>0. See also [12]. The method of proof in [6] should be compared with the one [1] given for a similar ODE
which arises from dendritic crystal growth. After a pioneering observation by [7], it is now also concluded that there is no solution y of (4) which satisfies y′(x)>0 for all and respectively. We remark that Toland [14] already gave an elementary approach to proving the nonexistence of monotonic solutions of (2) if λ≥ 2/9. Our first result [5], on the other hand, presents a simple unified argument toward the nonexistence of monotonic global solutions for certain class of ODEs including (2) and (4); however, [5] restricts the range of parameters λ and ε depending on nonlinearities. Here is our theorem.
