We consider the Abel equation φ[f(x)] = φ(x) + a on the plane ℝ², where f is a free mapping (i.e. f is an orientation preserving homeomorphism of the plane onto itself with no fixed points). We find all its homeomorphic and diffeomorphic solutions φ having positive Jacobian. Moreover, we give some conditions which are equivalent to f being conjugate to a translation.
We consider ideal equal convergence of a sequence of functions. This is a generalization of equal convergence introduced by Császár and Laczkovich [Császár Á., Laczkovich M., Discrete and equal convergence, Studia Sci. Math. Hungar., 1975, 10(3–4), 463–472]. Our definition of ideal equal convergence encompasses two different kinds of ideal equal convergence introduced in [Das P., Dutta S., Pal S.K., On and *-equal convergence and an Egoroff-type theorem, Mat. Vesnik, 2014, 66(2), 165–177]_and [Filipów...
We present an example of a locally BV-integrable function in the real line whose indefinite integral is not the sum of a locally absolutely continuous function and a function that is Lipschitz at all but countably many points.
Let be affine mappings of . It is well known that if there exists j ≤ 1 such that for every the composition (1) is a contraction, then for any infinite sequence and any , the sequence (2) is convergent and the limit is independent of z. We prove the following converse result: If (2) is convergent for any and any belonging to some subshift Σ of N symbols (and the limit is independent of z), then there exists j ≥ 1 such that for every the composition (1) is a contraction. This result...
Some usual and unusual properties of the Riemann integral for functions x : [a,b] → X where X is an F-space are investigated. In particular, a continuous integrable -valued function (0 < p < 1) with non-differentiable integral function is constructed. For some class of quasi-Banach spaces X it is proved that the set of all X-valued functions with zero derivative is dense in the space of all continuous functions, and for any two continuous functions x and y there is a sequence of differentiable...
In this paper we derive the Integration-by-Parts Formula using the generalized Riemann approach to stochastic integrals, which is called the Itô-Kurzweil-Henstock integral.