We prove fixed point theorems in a space with a distance function that takes values in a partially ordered monoid. On the one hand, such an approach allows one to generalize some fixed point theorems in a broad class of spaces, including metric and uniform spaces. On the other hand, compared to the so-called cone metric spaces and K-metric spaces, we do not require that the distance function range has a linear structure. We also consider several applications of the obtained fixed point theorems. In particular, we consider the questions of the existence of solutions of the Fredholm integral equation in L-spaces.

We consider the problems of optimal recovery of an operator A (generally speaking, nonlinear) defined on a unit ball BH in a Hilbert space H based on the information about elements of this unit ball BH given by a bounded linear operator T : H → Y, where Y is a Banach space. For a fixed information operator T, it is shown that the optimal method of recovery is offered by the so-called T-interpolating splines. For fixed Y, we also solve the problem of finding the optimal information operator. Moreover, for a bounded linear self-adjoint operator A, it is shown that if T is the optimal information operator for the recovery of A on BH, then any other operator TAn, n ∈ ℕ, is also the optimal information operator.

In this paper we solve the problem of optimal recovery of the operator $A_\alpha x= (\alpha_1x_1,\alpha_2x_2,\ldots)$ on the class $W^T_q = \{(t_1h_1,t_2h_2,\ldots)\,:\,\|h\|_{\ell_q}\le 1\}$, where $1\le q < \infty$ and $t_1\ge t_2\ge \ldots \ge 0$, and $\alpha_1t_1\ge\alpha_2t_2\ge\ldots\ge 0$ are given, in the space $\ell_q$. We solve this problem under assumption that $\lim_{n\to\infty}t_n = \lim_{n\to\infty}\alpha_nt_n = 0$. Information available about a sequence $x\in W^T_q$ is provided either (i) by an element $y\in\mathbb{R}^n$, $n\in\mathbb{N}$, whose distance to the first $n$ coordinates $\left(x_1,\ldots,x_n\right)$ of $x$ in the space $\ell_p^n$, $0 < p \le \infty$, does not exceed given $\varepsilon\ge 0$, or (ii) by a sequence $y\in\ell_p$ whose distance to $x$ in the space $\ell_r$ does not exceed $\varepsilon$. We show that the optimal method of recovery in this problem is either operator $\Phi^*_m$ with some $m\in\mathbb{Z}_+$ ($m\le n$ in case $y\in\ell^n_p$), defined by $$\Phi^*_m(y) = \left\{\alpha_1y_1\left(1 - \frac{\alpha_{m+1}^qt_{m+1}^q}{\alpha_1^qt_{1}^q}\right),\ldots,\alpha_my_m\left(1 - \frac{\alpha_{m+1}^qt_{m+1}^q}{\alpha_m^qt_{m}^q}\right),0,\ldots\right\},$$where $y\in\mathbb{R}^n$ or $y\in\ell_p$ or convex combination $(1-\lambda) \Phi^*_{m+1} + \lambda\Phi^*_{m}$, or the operator $A_\alpha$ itself.

This work is dedicated to solving problems of optimal recovery of operator $A$ (not necessarily linear), defined on a subset $\mathfrak{M}$ of a Banach space $H$ using information about elements of the $\mathfrak{M}$, given by a linear bounded operator $T\colon H\to Y$ where $Y$ is some Banach space. We show that under certain condition the optimal method of recovery is given by abstract interpolation splines in $H$ generated by $T$ ($T$-interpolating splines).

In this article we obtain sharp Kolmogorov-type inequalities that estimate the uniform norm of a hypersingular integral operator $$D^{w,\Omega}_K f(x): = \int_{C} w(|t|_K) (f(x+t) - f(x))\Omega(t)dt, x\in C, $$using the uniform norm of the function $f$ and either the norm $\|f\|_{H^\omega_K(C)}$ determined by a modulus of continuity $\omega$, or the weighted integral norm $\| \Omega^{\frac 1p} \cdot |\nabla f|_{K^\circ}\|_{L_p(C)}$ of the gradient $\nabla f$. Here $C$ is a convex cone in ${\mathbb R}^d$, $d\geq 2$, $\Omega\colon C\to\mathbb R$ is a non-negative homogeneous of degree 0 locally integrable function, $w\colon (0,\infty)\to [0,\infty)$ is some weight function, $|\cdot|_K$ is an arbitrary norm in ${\mathbb R}^d$, $|\cdot|_{K^\circ}$ is its polar norm, and $p\in (d,\infty]$.

For non-empty sets X we define notions of distance and pseudo metric with values in a partially ordered set that has a smallest element $\theta $. If $h_X$ is a distance in $X$ (respectively, a pseudo metric in $X$), then the pair $(X,h_X)$ is called a distance (respectively, a pseudo metric) space. If $(T,h_T)$ and $(X,h_X)$ are pseudo metric spaces, $(Y,h_Y)$ is a distance space, and $H(T,X)$ is a class of Lipschitz mappings $f\colon T\to X$, for a broad family of mappings $\Lambda\colon H (T,X)\to Y$, we obtain a sharp inequality that estimates the deviation $h_Y(\Lambda f(\cdot),\Lambda f(t))$ in terms of the function $h_T(\cdot, t)$. We also show that many known estimates of such kind are contained in our general result.

For a function $f$ from the Sobolev space $W^{1,p}(C)$ ($C\subset\mathbb{R}^d$ is an open convex cone), a sharp inequality that estimates $\| f\|_{L_{\infty}}$ via the $L_{p}$-norm of its gradient and a seminorm of the function is obtained. With the help of this inequality, a sharp inequality is proved, which estimates the ${L_{\infty}}$-norm of the Radon--Nikodym derivative of a charge defined on Lebesgue measurable subsets of $C$ via the $L_p$-norm of the gradient of this derivative and a seminorm of the charge. In the case, when $C=\mathbb{R}_+^m\times \mathbb{R}^{d-m}$, $0\le m\le d$, we obtain inequalities that estimate the ${L_{\infty}}$-norm of a mixed derivative of a function $f\colon C\to \mathbb{R}$ using its ${L_{\infty}}$-norm and the $L_p$-norm of the gradient of the function's mixed derivative.

Abstract We prove an analogue of the Korneichuk–Stechkin lemma for functions with values in L-spaces. As applications, we obtain sharp Ostrowski type inequalities and solve problems of optimal recovery of identity and convexifying operators, as well as the problem of integral recovery on the classes of L-space valued functions with given majorant of modulus of continuity. The recovery is done based on n mean values of the functions over intervals. Moreover, on the classes of functions with given majorant of modulus of continuity of their Hukuhara type derivative, we solve the problem of optimal recovery of the function and the Hukuhara type derivative. The recovery is done based on n values of the function. We obtain sharp Landau type inequalities and solve an analogue of the Stechkin problem about approximation of unbounded operators by bounded ones and the problem of optimal recovery of an unbounded operator on a class of elements, known with error. Consideration of L-space valued functions gives a unified approach to solution of the mentioned above extremal problems for the classes of multi- and fuzzy-valued functions, and for the classes of functions with values in Banach spaces, in particular random processes, and many other classes of functions.

In this article we prove sharp Landau-Kolmogorov type inequalities on a class of charges defined on Lebesgue measurable subsets of a cone in $\mathbb{R}^d$, $d\geqslant 1$, that are absolutely continuous with respect to the Lebesgue measure. In addition we solve the Stechkin problem of approximation of the Radon-Nikodym derivative of such charges by bounded operators and two related problems. As an application, we also solve these extremal problems on classes of essentially bounded functions $f$ such that their distributional partial derivative $\frac{\partial ^d f}{\partial x_1\ldots\partial x_d}$ belongs to the Sobolev space $W^{1,\infty}$.

We solve the problem of the best approximation of closed operators by linear bounded operators in Hilbert spaces under assumption that the operator transforms orthogonal basis in Hilbert space into an orthogonal system. As a consequence, sharp additive Hardy-Littlewood-Pólya type inequality for multiple closed operators is established. We also demonstrate application of these results in concrete situations: for the best approximation of powers of the Laplace-Beltrami operator on classes of functions defined on closed Riemannian manifolds, for the best approximation of differentiation operators on classes of functions defined on the period and on the real line with the weight $e^{-x^2}$, and for the best approximation of functions of self-adjoint operators in Hilbert spaces.

We solve the Stechkin problem about approximation of generally speaking unbounded hypersingular integral operators by bounded ones. As a part of the proof, we also solve several related and interesting on their own problems. In particular, we obtain sharp Landau-Kolmogorov type inequalities in both additive and multiplicative forms for hypersingular integral operators and prove a sharp Ostrowski type inequality for multivatiate Sobolev classes. We also give some applications of the obtained results, in particular study the modulus of continuity of the hypersingular integral operators, and solve the problem of optimal recovery of the value of a hypersingular integral operator based on the argument known with an error.

In this paper we solve the problem of approximating functionals (phi(A)x, f) (where phi(A) is some function of self-adjoint operator A) on the class of elements of a Hilbert space that is defined using another function psi(A) of the operator A. In addition, we obtain a series of sharp Taikov-type additive inequalities that estimate vertical bar(phi(A)x, f)vertical bar with the help of parallel to psi(A)x parallel to and parallel to x parallel to. We also present several applications of the obtained results. First, we find sharp constants in inequalities of the type used in Hormander theorem on comparison of operators in the case when operators are acting in a Hilbert space and are functions of a self-adjoint operator. Second, we obtain Taikov-type inequalities for functions of the operator 1/i d/dt in the spaces L-2(R) and L-2(T), as well as for integrals with respect to spectral measures, defined with the help of classical orthogonal polynomials.

The goal of the article is to characterize continuous $(\lambda,\varphi)$-additive operators acting on measurable bounded functions with values in $L$-spaces. As an application, we prove a sharp Ostrowski type inequality for such operators.