Abstract
A multi-valued iterative functional equation of order n is considered. A result on the existence and uniqueness of K-convex solutions in some class of multifunctions is presented. MSC:39B12, 37E05, 54C60.
Highlights
Where S is a subset of a linear space over R, F : S → S is a given function, λis (i =, . . . , n) are real constants, f : S → S is the unknown function, and f i is the ith iterate of f, i.e., f i(x) = f (f i– (x)) and f (x) = x for all x ∈ S, is one of the important forms of a functional equation since the problem of iterative roots and the problem of invariant curves can be reduced to the kind of equations
A multifunction is an important class of mappings often used in control theory [ ], stochastics [ ], artificial intelligence [ ], and economics [ ]
Discussing Eq ( . ) for n ≥ evokes great interest, but the greatest difficulty is that the multifunction has no Lipschitz condition
Summary
) for n = with an increasing upper semi-continuous (USC) multifunction G on I = [a, b] and proved the existence and uniqueness of USC solutions under the assumption that G has fixed points a and b and λ , λ are both constants such that λ > λ ≥ and λ + λ = . In , this difficulty was overcome by introducing the class of unblended multifunctions, the existence of USC multi-valued solutions for a modified form of the equation λ F(x) = G(x) – λ F (x) – · · · – λnFn(x), x ∈ I, Up to now, there are no results on convexity of multi-valued solutions for the iterative equation
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