On certain comparison theorems for half-linear dynamic equations on time scales.
In this paper, recent results on the existence and uniqueness of (continuous and homeomorphic) solutions φ of the equation φ ∘ f = g ∘ φ (f and g are given self-maps of an interval or the circle) are surveyed. Some applications of these results as well as the outcomes concerning systems of such equations are also presented.
Let I ⊂ ℝ be an interval, Y be a normed linear space and Z be a Banach space. We investigate the Banach space Lip₂(I,Z) of all functions ψ: I → Z such that , where [r,s,t;ψ]:= ((s-r)ψ(t)+(t-s)ψ(r)-(t-r)ψ(s))/((t-r)(t-s)(s-r)). We show that ψ ∈ Lip₂(I,Z) if and only if ψ is differentiable and its derivative ψ’ is Lipschitzian. Suppose the composition operator N generated by h: I × Y → Z, (Nφ)(t):= h(t,φ(t)), maps the set (I,Y) of all affine functions φ: I → Y into Lip₂(I,Z). We prove that if N is...
This paper discusses continuous solutions of the functional equation φ[f(x)] = g(x,φ(x)) in topological spaces.
A continuous multifunction F:[a,b] → clb(Y) is *-concave if and only if the inclusion holds for every s,t ∈ [a,b], s < t.