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Single-valley-extended continuous solutions for the Feigenbaum's functional equation f (φ(x)) = φ2(f(x))

Zhang M.

Demonstratio Mathematica

2014

Vol. 47, nr 3

615--626

This work deals with the Feigenbaum's functional equation in the broad sense (…), where φ2 is the 2-fold iteration of φ, f(x) is a strictly increasing continuous function on [0, 1] and satisfies (...). Using constructive method, we discuss the existence of single-valley-extended continuous solutions of the above equation.

Eigenvalue-eigenfunction problem for Steklov's smoothing operator and differential-difference equations of mixed type

Iakovlev S. I., Iakovleva V.

Opuscula Mathematica

2013

Vol. 33, no. 1

81-98

It is shown that any μ ∈ C is an infinite multiplicity eigenvalue of the Steklov smoothing operator Sh acting on the space [formula]. For μ ≠ 0 the eigenvalue-eigenfunction problem leads to studying a differential-difference equation of mixed type. An existence and uniqueness theorem is proved for this equation. Further a transformation group is defined on a countably normed space of initial functions and the spectrum of the generator of this group is studied. Some possible generalizations are pointed out.