On a functional equation arising in information theory

We derive the representation of generalized additive measures of information, depending on two probability distributions and having the sum property with a regular generating function. This family of measures includes the Shannon entropy, the entropy of degree (,), the generalized information energy, the sine entropy and the generalized directed divergence.     

Convex solutions of polynomial-like iterative equations

In this paper convex solutions and concave solutions of polynomial-like iterative equations are investigated. A result for non-monotonic solutions is given first and applied then to prove the existence of convex continuous solutions and concave ones. Furthermore, another condition for convex solutions, which is weaker in some aspects, is also given. The uniqueness and stability of those solutions are also discussed.     

Convex solutions of boundary value problems

We establish two criteria for the existence of convex solutions for a boundary value problem arising from the study of the existence of convex radial solutions for the Monge-Ampère equations. We shall use fixed point theorems in a cone.     

Converses of the Jensen inequality derived from the Green functions with applications in information theory

Jensen integral inequality has got much importance regarding their applications in different fields of mathematics. In this paper, two converses of Jensen integral inequality for convex function are obtained. The results are applied to establish converses of Hölder and Hermite-Hadamard inequalities as well. At the end, some useful applications in information theory of the obtained results are given. The idea used in this paper may inculcate further research.     

On a functional equation arising from comparison of utility representations

We solve the functional equation F₁(t)−F₁(t+s)=F₂[F₃(t)+F₄(s)] for real functions defined on intervals, assuming that F₂ is positive valued and strictly monotonic and that F₃ is continuous. The equation arose from the equivalence problem of utility representations under assumptions of separability, homogeneity and segregation (e-distributivity).     

Convex solutions of a general similarity boundary layer equation for power-law fluids

This paper is devoted to a general similarity boundary layer equation for power-law fluids, which includes many important similarity boundary layer problems such as the Falker-Skan equation and the magnetohydrodynamic boundary layer equation which arises in the study of self-similar solutions of the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluids along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. By a rigorous mathematical analysis, the uniqueness, existence and nonexistence results for convex solutions, normal convex solutions and generalized convex solutions to the general similarity boundary layer equation are established. Also the asymptotic behavior of the normal convex solutions at the infinity are displayed.     

Local analytic solutions of a functional differential equation

In this paper, we are concerned with the existence of analytic solution of a functional differential equation αz+βx^′(z)=x(az+bx^″(z)), where are four complex numbers. We first discuss the existence of analytic solutions for some special cases of the above equation. Then, by reducing the equation with the Schröder transformation to the another functional equation with proportional delay, an existence theorem is established for analytic solutions of the original equation. For the constant λ given in the Schröder transformation, we discuss the case 0<|λ|<1 and λ on the unit circle S¹, i.e., |λ|=1. We study λ is at resonance, i.e., at a root of the unity and λ is near resonance under the Brjuno condition.     

Solution of a functional equation arising in an axiomatization of the utility of binary gambles

For a new axiomatization, with fewer and weaker assumptions, of binary rank-dependent expected utility of gambles the solution of the functional equation

is needed under some monotonicity and surjectivity conditions. We furnish the general such solution and also the solutions under weaker suppositions. In the course of the solution we also determine all sign preserving solutions of the related general equation

     

Differentiability of solutions of second-order functional differential equations with unbounded delay

In this paper we study the differentiability of solutions of the second-order semilinear abstract retarded functional differential equation with unbounded delay, specially when the underlying space is reflexive or at least has the Radon–Nikodym property. We apply our results to characterize the infinitesimal generators of several strongly continuous semigroups of linear operators that arise in the theory of linear abstract retarded functional differential equations with unbounded delay on a phase space defined axiomatically.     

Three periodic solutions for a nonlinear first order functional differential equation

This paper is concerned with the existence of three positive T-periodic solutions of the first order functional differential equations of the form

x^′(t)=a(t)x(t)-λb(t)f(t,x(h(t))),     

Meromorphic solutions of equations over non-Archimedean fields

In this paper, we give some conditions to assure that the equation P(X)=Q(Y) has no meromorphic solutions in all K, where P and Q are polynomials over an algebraically closed field K of characteristic zero, complete with respect to a non-Archimedean valuation. In particular, if P and Q satisfy the hypothesis (F) introduced by H. Fujimoto, a necessary and sufficient condition is obtained when deg P=deg Q. The results are presented in terms of parametrization of a projective curve by three entire functions. In this way we also obtain similar results for unbounded analytic functions inside an open disk.