In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10],...

We prove that the study of the Łojasiewicz exponent at infinity of overdetermined polynomial mappings $ℂⁿ\to {ℂ}^m$, m > n, can be reduced to the one when m = n.

The Lukacs theorem is one of the most brilliant results in the area of characterizations of probability distributions. First, because it gives a deep insight into the nature of independence properties of the gamma distribution; second, because it uses beautiful and non-trivial mathematics. Originally it was proved for probability distributions concentrated on (0,∞). In 1962 Olkin and Rubin extended it to matrix variate distributions. Since that time it has been believed that the fundamental reason...

In this paper we prove that every collection of measurable functions fα , |α| = m, coincides a.e. withmth order derivatives of a function g ∈ Cm−1 whose derivatives of order m − 1 may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin, Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal a.e.

In this paper, we define the $M_{\alpha }$-integral of real-valued functions defined on an interval $[a,b]$ and investigate important properties of the $M_{\alpha }$-integral. In particular, we show that a function $f:[a,b]\to R$ is $M_{\alpha }$-integrable on $[a,b]$ if and only if there exists an $AC{G}_{\alpha }$ function $F$ such that $F^{\text{'}}=f$ almost everywhere on $[a,b]$. It can be seen easily that every McShane integrable function on $[a,b]$ is $M_{\alpha }$-integrable and every $M_{\alpha }$-integrable function on $[a,b]$ is Henstock integrable. In addition, we show that the $M_{\alpha }$-integral is equivalent to the $C$-integral....

We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality ${\left|\right|Mf\left|\right|}_{p),w}{\le c\left|\right|f\left|\right|}_{p),w}$ holds with some c independent of f iff w belongs to the well known Muckenhoupt class $A_p$, and therefore iff ${\left|\right|Mf\left|\right|}_{p,w}{\le c\left|\right|f\left|\right|}_{p,w}$ for some c independent of f. Some results of similar type are discussed for the case of small Lebesgue spaces....

We prove that the topographic map structure of upper semicontinuous functions, defined in terms of classical connected components of its level sets, and of functions of bounded variation (or a generalization, the WBV functions), defined in terms of M-connected components of its level sets, coincides when the function is a continuous function in WBV. Both function spaces are frequently used as models for images. Thus, if the domain Ω' of the image is Jordan domain, a rectangle, for instance, and...

Some relationships between the vector valued Henstock and McShane integrals are investigated. An integral for vector valued functions, defined by means of partitions of the unity (the PU-integral) is studied. In particular it is shown that a vector valued function is McShane integrable if and only if it is both Pettis and PU-integrable. Convergence theorems for the Henstock variational and the PU integrals are stated. The families of multipliers for the Henstock and the Henstock variational integrals...

In this article, we formalized the measurability of complex-valued functional sequences. First, we proved the measurability of the limits of real-valued functional sequences. Next, we defined complex-valued functional sequences dividing real part into imaginary part. Then using the former theorems, we proved the measurability of each part. Lastly, we proved the measurability of the limits of complex-valued functional sequences. We also showed several properties of complex-valued measurable functions....

In this paper we use the upper and lower solutions method to investigate the existence of solutions of a class of impulsive partial hyperbolic differential inclusions at fixed moments of impulse involving the Caputo fractional derivative. These results are obtained upon suitable fixed point theorems.

We prove two versions of the Monotone Convergence Theorem for the vector integral of Kurzweil, ${\int }_R\mathrm{d}\alpha \left(t\right)f\left(t\right)$, where $R$ is a compact interval of ${ℝ}^n$, $\alpha$ and $f$ are functions with values on $L(Z,W)$ and $Z$ respectively, and $Z$ and $W$ are monotone ordered normed spaces. Analogous results can be obtained for the Kurzweil vector integral, ${\int }_R\alpha \left(t\right)\mathrm{d}f\left(t\right)$, as well as to unbounded intervals $R$.