Related papers: Mathematical Constraint on Functions with Continuo…
We prove that functions defined on a lattice in a finite dimensional torus with bounded finite differences can be smoothly extended to the whole torus, and relate the bounds on the extension's derivatives with bounds on the original…
The mathematics of K-conserving functional differentiation, with K being the integral of some invertible function of the functional variable, is clarified. The most general form for constrained functional derivatives is derived from the…
In this paper, we establish several new inequalities for twice differantiable mappings that are connected with the celebrated Hermite-Hadamard integral inequality. Some applications for special means of real numbers are also provided.
We show that a form for the second partial derivative of $1/r$ proposed by Frahm and subsequently used by other workers applies only when averaged over smooth functions. We use dyadic notation to derive a more general form without that…
We consider a one-parameter family of functions $\{F(t,x)\}_{t}$ on $[0,1]$ and partial derivatives $\partial_{t}^{k} F(t, x)$ with respect to the parameter $t$. Each function of the class is defined by a certain pair of two square matrices…
The function spaces of continuously differentiable functions are extensively studied and appear in various mathematical settings. In this context, we investigate the spaces of continuously fractional differentiable functions of order…
Transport phenomena play a vital role in various fields of science and engineering. In this work, exact solutions are derived for advection equations with integer- and fractional-order time derivatives and a constant time-delay in the…
We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary value $\partial_t^{\alpha} u(x,t) = -Au(x,t)$, where $-A = \sum}{i,j=1}^d \partial_i(a_{ij}(x)\partial_j) +…
Generalised Ito formulae are proved for time dependent functions of continuous real valued semi-martingales. The conditions involve left space and time first derivatives, with the left space derivative required to have locally bounded…
In this chapter, we mainly review theoretical results on inverse source problems for diffusion equations with the Caputo time-fractional derivatives of order $\alpha\in(0,1)$. Our survey covers the following types of inverse problems: 1.…
Replacing operators with continuous operator-valued functions, we prove time-dependent versions of well-known results on compressions and diagonals of bounded operators. The setting of smooth functions is also addressed. Our results have no…
Variational convexity, together with ist strong counterpart, of extended-real-valued functions has been recently introduced by Rockafellar. In this paper we present second-order characterizations of these properties, i.e., conditions using…
In this work, approximations for real two variables function $f$ which has continuous partial $(n-1)$-derivatives $(n \ge 1)$ and has the $n$--th partial derivative of bounded bivariation or absolutely continuous are established. Explicit…
We comment on a Mazur problem from "Scottish Book" concerning second partial derivatives. It is proved that, if a function $f(x,y)$ of real variables defined on a rectangle has continuous derivative with respect to $y$ and for almost all…
In this paper, a general integral identity for convex functions is derived. Then, we establish new some inequalities of the Simpson and the Hermite-Hadamard's type for functions whose absolute values of derivatives are convex. Some…
We investigate necessary and sufficient conditions under which entire functions in de Branges spaces can be recovered from function values and values of derivatives. Our main focus is on spaces with a structure function whose logarithmic…
In this article we prove a general result which in particular suggests that, on a simply connected domain in C, all the derivatives and anti-derivatives of the generic holomorphic function are unbounded. A similar result holds for the…
It is well known that in $R^n$ , G{\^a}teaux (hence Fr{\'e}chet) differ-entiability of a convex continuous function at some point is equivalent to the existence of the partial derivatives at this point. We prove that this result extends…
We prove that if $f:I\subset \Bbb R\to \Bbb R$ is of bounded variation, then the noncentered maximal function $Mf$ is absolutely continuous, and its derivative satisfies the sharp inequality $\|DMf\|_1\le |Df|(I)$. This allows us obtain,…
We introduce two kinds of fractional integral operators; the one is defined via the exponential-integral function $$ E_1(x)=\int_x^\infty \frac{e^{-t}}{t}\,dt,\quad x>0, $$ and the other is defined via the special function $$…