A function F is said to have a generalized Peano derivative at x if F is continuous in a neighborhood of x and if there exists a positive integer q such that a qth primitive of F in the neighborhood has the (q+n)th Peano derivative at x; in this case the latter is called the generalized nth Peano derivative of F at x and denoted by $F_{\left[n\right]}\left(x\right)$. We show that generalized Peano derivatives belong to the class [Δ’]. Also we show that they are path derivatives with a nonporous system of paths satisfying the I.I.C....

Mathematics Subject Classification: 33D60, 33D90, 26A33Fractional q-integral operators of generalized Weyl type, involving generalized basic hypergeometric functions and a basic analogue of Fox’s H-function have been investigated. A number of integrals involving various q-functions have been evaluated as applications of the main results.

We use the general Riemann approach to define the Stratonovich integral with respect to Brownian motion. Our new definition of Stratonovich integral encompass the classical Stratonovich integral and more importantly, satisfies the ideal Itô formula without the “tail” term, that is, $$f\left(W_t\right)=f\left(W_0\right)+{\int }_0^t{f}^{\text{'}}\left(W_s\right)\circ \mathrm{d}{W}_s.$$ Further, the condition on the integrands in this paper is weaker than the classical one.

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 $F_i=1,...,N$ be affine mappings of ${ℝ}^n$. It is well known that if there exists j ≤ 1 such that for every ${\sigma }_1,...,{\sigma }_j\in 1,...,N$ the composition (1) $F_{\sigma 1}\circ ...\circ F_{{\sigma }_j}$ is a contraction, then for any infinite sequence ${\sigma }_1,{\sigma }_2,...\in 1,...,N$ and any $z\in {ℝ}^n$, the sequence (2)$F_{\sigma 1}\circ ...\circ F_{{\sigma }_n}\left(z\right)$ is convergent and the limit is independent of z. We prove the following converse result: If (2) is convergent for any $z\in {ℝ}^n$ and any $\sigma ={\sigma }_1,{\sigma }_2,...$ belonging to some subshift Σ of N symbols (and the limit is independent of z), then there exists j ≥ 1 such that for every $\sigma ={\sigma }_1,{\sigma }_2,...\in \Sigma$ 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 $l_p$-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...