Recent results on functional equations in a single variable, perspectives and open problems

Abel’s Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions

“…It was probably mentioned for the first time by Abel [1] in his note published posthumously (comp. also [45]) when a relation between f and f 0 as above was given in the case of strictly increasing solutions of the Abel equation with some additional assumptions on g. There is also a broad literature about the equation in various function classes [34,Chapter 7] or [8,Sec. 9].…”

“…2.20] [2][3][4][5][6][7] and a papers of Szekeres [42][43][44]. For iteration roots we also have a broad literature: see [34,Chapter 15], [8,Sec. 2].…”

“…In [8,Section 2] examples of polynomials ϕ without embedding into an iteration semigroup are mentioned. By [34,Th.…”

“…That is why (1) is often called the Schröder equation. There is an impressive bibliography of the subject (see [34,Chapter 6], [35,Chapter 9] and citations therein or a J. Bonet and P. Domański IEOT survey paper [8,Sec. 8], as well papers on the holomorphic case [18,40,41]) but, in spite of extensive literature inquires, we could not find any known complete solution of the problem in our setting.…”

Finding the Optimal Continuous Model for Discrete Data by Neural Network Interpolation of Fractional Iteration

“…In fact, even roots of polynomials are not described. Even worse: we do not know whether every complex cubic polynomial has a square root..." [12] For polynomials of order two at least the question of existence had been solved by Rice et.al. in "When is for all complex ?"…”