As described in [9, 10], polynomial-like iterative equations are critical not only while in the review of practical equations but also in the research of dynamical programs. For example, such equations are encountered in the discussion on transversal Lumacaftor structure homoclinic intersection for diffeomorphisms , typical kind Alisertib (MLN8237) of dynamical programs , and dynamics of a quadratic mapping . Some issues of invariant curves for dynamical methods also cause this kind of iterative equations .For your case that F is linear, wherever (1) could be written as��nfn(x)+��n?1fn?one(x)+?+��1f(x)+��0x=0,(2)lots of results [15�C17] happen to be given to present all of its constant remedies. Disorders that guarantee the uniqueness of this kind of remedies are also given by [18, 19].
For the case that F is nonlinear, the basic troubles this kind of as existence, uniqueness, and stability can't be solved very easily.
In 1986, Zhang , under the restriction that ��1 �� 0, constructed an interesting operator referred to as ��structural operator�� for (1) and used the fixed stage theory in Banach room to get the answers of (one). Hence, he overcame the troubles encountered from the formers. By means of this strategy, Zhang and Si made a series of performs regarding these qualitative difficulties, this kind of as [21�C24]. Following that, (1) along with other type equations had been discussed extensively by employing this notion (see [25�C31] and references therein).Alternatively, good efforts have been produced to resolve the ��leading coefficient problem�� which was raised byAutophagy signaling [32, 33] as an open difficulty.
The essence of solving this dilemma is to abolish the technical restriction ��1 �� 0 and discuss (one) beneath the much more organic assumption ��n �� 0. As stated in [34, 35], a mapping f is explained for being locally expansive (resp., locally contractive) at its fixed point x0, if |f��(x0)|>1 (resp., 0 < |f��(x0)|<1). In 2004, Zhang  gave positive answers to this problem in local C1 solutions in some cases of coefficients, but this paper only discussed the locally expansive case and the nonhyperbolic case. In 2009, Chen and Zhang  gave positive answers to this problem with more combinations between locally expansive mappings and locally contractive ones and combinations between increasing mappings and decreasing ones. The main tools used in the two papers above are Schr?der transformation and Schauder fixed point theorem.
In 2012, J. M. Chen and L.
Chen  think about the locally contractive C1 solutions from the iterative equation G(x, f(x),��, fn(x)) = F(x), and some success on locally contractive solutions of  were generalized. In 2007, Xu and Zhang  answered this dilemma by constructing C0 solutions of (1). Their method would be to construct the options piece by piece via a recursive formula obtained type (1). Following this plan, worldwide escalating and decreasing answers [38, 39] were also investigated.