Let ϕ be an arbitrary bijection of ${ℝ}_{+}$. We prove that if the two-place function ${\varphi }^{-1}[\varphi \left(s\right)+\varphi \left(t\right)]$ is subadditive in ${ℝ}_{+}^2$ then $\varphi$ must be a convex homeomorphism of ${ℝ}_{+}$. This is a partial converse of Mulholland’s inequality. Some new properties of subadditive bijections of ${ℝ}_{+}$ are also given. We apply the above results to obtain several converses of Minkowski’s inequality.

We study the asymptotic behavior of the solutions of a scalar convolution sum-difference equation. The rate of convergence of the solution is found by determining the asymptotic behavior of the solution of the transient renewal equation.

Let T: C¹(ℝ) → C(ℝ) be an operator satisfying the “chain rule inequality” T(f∘g) ≤ (Tf)∘g⋅Tg, f,g ∈ C¹(ℝ). Imposing a weak continuity and a non-degeneracy condition on T, we determine the form of all maps T satisfying this inequality together with T(-Id)(0) < 0. They have the form Tf = ⎧ $(H\circ f/H)f^{\text{'}p}$, f’ ≥ 0, ⎨ ⎩ $-A(H\circ f/H)|f^{\text{'}}{|}^p$, f’ < 0, with p > 0, H ∈ C(ℝ), A ≥ 1. For A = 1, these are just the solutions of the chain rule operator equation. To prove this, we characterize the submultiplicative, measurable functions...

We consider the dynamical system (𝒜, Tf), where 𝒜 is a class of differential real functions defined on some interval and Tf : 𝒜 → 𝒜 is an operator Tfφ := fοφ, where f is a differentiable m-modal map. If we consider functions in 𝒜 whose critical values are periodic points for f then, we show how to define and characterize a substitution system associated with (𝒜, Tf). For these substitution systems, we compute the growth rate of the...

A functional characterization of Sugeno's negations is presented and as a consequence, we study a family of non strict Archimedean t-norms whose (vertical-horizontal) sections are straight lines.

We consider a functional equation of the formG(x, phi(f1(x)), ..., phi(fr(x))) = cin the unknown function phi.We present a method to construct the general solution of this equation under suitable hypotheses on the functions Inf i fi and Supi fi.

Let X be an arbitrary Abelian group and E a Banach space. We consider the difference-operators ∆n defined by induction:(∆f)(x;y) = f(x+y) - f(x), (∆nf)(x;y1,...,yn) = (∆n-1(∆f)(.;y1)) (x;y2,...,yn)(n = 2,3,4,..., ∆1=∆, x,yi belonging to X, i = 1,2,...,n; f: X --&gt; E).Considering the difference equation (∆nf)(x;y1,y2,...,yn) = d(x;y1,y2,...,yn) with independent variable increments, the most general solution is given explicitly if d: X x Xn --&gt; E is a given bounded function. Also the...

We consider the summation equation, for $t\in {[\mu -2,\mu +b]}_{{\mathbb{N}}_{\mu -2}}$, $$\begin{array}{ccc}\hfill y\left(t\right)={\gamma }_1\left(t\right)H_1\left(\sum _{i=1}^n{a}_{i}y\left({\xi }_i\right)\right)& +{\gamma }_2\left(t\right)H_2\left(\sum _{i=1}^m{b}_{i}y\left({\zeta }_i\right)\right)\hfill & \hfill +\lambda \sum _{s=0}^{b}G(t,s)f(s+\mu -1,y(s+\mu -1))\end{array}$$ in the case where the map $(t,s)\mapsto G(t,s)$ may change sign; here $\mu \in (1,2]$ is a parameter, which may be understood as the order of an associated discrete fractional boundary value problem. In spite of the fact that $G$ is allowed to change sign, by introducing a new cone we are able to establish the existence of at least one positive solution to this problem by imposing some growth conditions on the functions $H_1$ and $H_2$. Finally, as an application of the abstract existence result,...

We consider the functional equation f(z+σ) - f(z) = g(z) where σ is a complex number, f and g are entire functions of a complex variable z, with growth conditions. We prove the existence of certain types of solutions of this equation by an a priori estimate method in certain weighted L2-spaces.