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.     

On certain nonlinear problems arising in the theory of lubrication

Various nonlinear elliptic boundary value problems motivated by the theory of lubrication are stated and studied. We deal mainly with the aspects of existence and uniqueness of solutions.     

On a multiplicative type sum form functional equation and its role in information theory

In this paper, we obtain all possible general solutions of the sum form functional equations

A nonautonomous nonlinear functional differential equation arising in the theory of population dynamics

A nonlinear periodic functional differential equation with unbounded delay describing the growth of a single species with depensation is considered. The global bifurcation of positive periodic solutions from the null one is studied and the differences from logistic-type equations are shown, namely the multiplicity of non-trivial solutions and the occurrence of a new bifurcation phenomenon. The biological meaning of the results is discussed.     

Convex solutions of a functional equation arising in information theory

Given a convex function f defined for positive real variables, the so-called Csiszár f-divergence is a function If defined for two n-dimensional probability vectors p=(p₁,…,pn) and q=(q₁,…,qn) as . For this generalized measure of entropy to have distance-like properties, especially symmetry, it is necessary for f to satisfy the following functional equation: for all x>0. In the present paper we determine all the convex solutions of this functional equation by proposing a way of generating all of them. In doing so, existing usual f-divergences are recovered and new ones are proposed.     

On a nonlinear diffusion system arising in reactor dynamics

In this paper the existence, uniqueness and asymptotic behavior of the solutions of a nonlinear diffusion system $\text{?u}\text{?t}\text{? Lu = f(t, x, u) + (H(u))(t, x)}$, occurring in reactor dynamics is studied. The problem is approached on the bases of the monotone method and the notion of upper and lower solutions.     

On a nonlinear nonlocal ODE arising in magnetic recording

In this paper we study the uniqueness of the solution for a nonlinear ODE with nonlocal terms. We consider a limit case of a one-dimensional equation arising in magnetic recording. The equation models the tape deflection where the magnetic head profile, with trenches to control the tape position, is a known function.     

On some properties of elliptic and parabolic functional differential operators arising in nonlinear optics

Quasilinear parabolic functional differential equations containing multiple transformations of spatial variables are considered with the Neumann boundary-value conditions. Sufficient conditions of the Andronov-Hopf bifurcation of periodic solutions are obtained along with expansions of the solutions in powers of a small parameter. Spectral properties of the linearized elliptic operator of this problem are investigated. Necessary and sufficient conditions of normality are obtained for such operators. Examples illustrating their properties are given. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 21, Proceedings of the Seminar on Differential and Functional Differential Equations Supervised by A. Skubachevskii (Peoples’ Friendship University of Russia), 2007.     

On the solvability of a nonlinear boundary value problem arising in the theory of growing cell populations

In this paper we establish some results regarding the existence of solution on L₁ spaces to a nonlinear boundary value problem originally proposed by Lebowitz and Rubinow (J. Math. Biol. 1974; 1 :17–36) to model an age‐structured proliferating cell population. Our approach, based on topological methods, uses essentially the specific properties of weakly compact sets on L₁ spaces. Our results provide positive answers to the questions posed in Jeribi (Nonlinear Anal. Real World Appl. 2002; 3 :85–105). Copyright © 2005 John Wiley & Sons, Ltd.     

On a nonlinear elliptic equation arising in a free boundary problem

 Let p ^* =n/(n−2) and n≥3. In this paper, we first classify all non-constant solutions of

On existence and approximate solutions for a nonlinear integral equation arising in the theory of creep in metals

Existence of a solution to a nonlinear integral equation arising in the theory of creep in symmetric pressure vessels is established by means of a recursively defined sequence of functions. It is shown that this sequence converges monotonically and uniformly to the solution.

---

Zusammenfassung Das Problem des Kriechens in symmetrischen Druckgefässen führt auf eine nichtlineare Integralgleichung. Zum Beweis der Existenz einer Lösung dieser Gleichung wird rekursiv eine Folge von Näherungslösungen konstruiert. Es wird gezeigt, dass diese Folge monoton und gleichmässig nach der Lösung der Integralgleichung konvergiert.

---

---

This research was supported by the National Science Foundation under Grant MPS-75-07450.