Showing papers in "Positivity in 2018"

Unbounded absolute weak convergence in Banach lattices

Abstract: Several recent papers investigated unbounded versions of order and norm convergences in Banach lattices. In this paper, we study the unbounded variant of weak convergence and its relationship with other convergences. In particular, we characterize order continuous Banach lattices and reflexive Banach lattices in terms of this convergence.

Some loose ends on unbounded order convergence

Abstract: The notion of almost everywhere convergence has been generalized to vector lattices as unbounded order convergence, which proves to be a very useful tool in the theory of vector and Banach lattices. In this short note, we establish some new results on unbounded order convergence that tie up some loose ends. In particular, we show that every norm bounded positive increasing net in an order continuous Banach lattice is uo-Cauchy and that every uo-Cauchy net in an order continuous Banach lattice has a uo-limit in the universal completion.

Duality for unbounded order convergence and applications

Abstract: Unbounded order convergence has lately been systematically studied as a generalization of almost everywhere convergence to the abstract setting of vector and Banach lattices This paper presents a duality theory for unbounded order convergence We define the unbounded order dual (or uo-dual) $${X_{uo}^\sim }$$ of a Banach lattice X and identify it as the order continuous part of the order continuous dual $${X_n^\sim }$$ The result allows us to characterize the Banach lattices that have order continuous preduals and to show that an order continuous predual is unique when it exists Applications to the Fenchel–Moreau duality theory of convex functionals are given The applications are of interest in the theory of risk measures in Mathematical Finance

Partial order relations on family of sets and scalarizations for set optimization

Abstract: In this study, some new order relations on family of sets are introduced by using Minkowski difference. The relations between these orders and the ordering cone of the vector space are obtained. It is shown that depending on the corresponding cone, these order relations are partial orders on the family of nonempty bounded sets. Some relationships between these order relations and upper and lower set less order relations are investigated. Also, two scalarizing functions are introduced in order to replace set optimization problems with respect to these partial order relations with scalar optimization problems. Moreover, necessary and sufficient optimality conditions are presented.

Unbounded norm topology beyond normed lattices

Abstract: In this paper, we generalize the concept of unbounded norm (un) convergence: let X be a normed lattice and Y a vector lattice such that X is an order dense ideal in Y; we say that a net $$(y_\alpha )$$ un-converges to y in Y with respect to X if $$\bigl |\bigl ||y_\alpha -y|\wedge x\bigr |\bigr |\rightarrow 0$$ for every $$x\in X_+$$ . We extend several known results about un-convergence and un-topology to this new setting. We consider the special case when Y is the universal completion of X. If $$Y=L_0(\mu )$$ , the space of all $$\mu $$ -measurable functions, and X is an order continuous Banach function space in Y, then the un-convergence on Y agrees with the convergence in measure. If X is atomic and order complete and $$Y=\mathbb R^A$$ then the un-convergence on Y agrees with the coordinate-wise convergence.

A note on Szász–Mirakyan–Kantorovich type operators preserving \(e^{-x}\)

Abstract: In the present article, we study modified form of Szasz–Mirakyan–Kantorovich operators, which reproduce constant and $$e^{-x}$$ functions. We discuss a uniform convergence result along with a quantitative estimate for the modified operators.

On Baskakov–Szász–Mirakyan-type operators preserving exponential type functions

Abstract: The current article deals with the study of Baskakov–Szasz–Mirakyan operators which reproduces constant and exponential functions. We discuss a uniform estimate and establish a quantitative result for the modified operators.

u tau-convergence in locally solid vector lattices

Abstract: Let \((x_\alpha )\) be a net in a locally solid vector lattice \((X,\tau )\); we say that \((x_\alpha )\) is unbounded \(\tau \)-convergent to a vector \(x\in X\) if \(|x_\alpha -x |\wedge w \xrightarrow {\tau } 0\) for all \(w\in X_+\). In this paper, we study general properties of unbounded \(\tau \)-convergence (shortly \(u\tau \)-convergence). \(u\tau \)-convergence generalizes unbounded norm convergence and unbounded absolute weak convergence in normed lattices that have been investigated recently. We introduce \(u\tau \)-topology and briefly study metrizability and completeness of this topology.

Weighted iterated Hardy-type inequalities

Abstract: In this paper the inequality $$\begin{aligned} \bigg ( \int _0^{\infty } \bigg ( \int _x^{\infty } \bigg ( \int _t^{\infty } h \bigg )^q w(t)\,dt \bigg )^{r / q} u(x)\,{ ds} \bigg )^{1/r}\le C \,\int _0^{\infty } h v, \quad h \in {\mathfrak {M}}^+(0,\infty ) \end{aligned}$$ is characterized. Here $$0< q ,\, r < \infty $$ and $$u,\,v,\,w$$ are weight functions on $$(0,\infty )$$ .

Levitin–Polyak well-posedness for strong bilevel vector equilibrium problems and applications to traffic network problems with equilibrium constraints

Abstract: In this paper we consider strong bilevel vector equilibrium problems and introduce the concepts of Levitin–Polyak well-posedness and Levitin–Polyak well-posedness in the generalized sense for such problems. The notions of upper/lower semicontinuity involving variable cones for vector-valued mappings and their properties are proposed and studied. Using these generalized semicontinuity notions, we investigate sufficient and/or necessary conditions of the Levitin–Polyak well-posedness for the reference problems. Some metric characterizations of these Levitin–Polyak well-posedness concepts in the behavior of approximate solution sets are also discussed. As an application, we consider the special case of traffic network problems with equilibrium constraints.

On Cesàro summability of vector valued multiplier spaces and operator valued series

Abstract: In this paper, we introduce and study vector valued multiplier spaces with the help of the sequence of continuous linear operators between normed spaces and Cesaro convergence. Also, we obtain a new version of the Orlicz–Pettis Theorem by means of Cesaro summability.

On approximate solutions for robust convex semidefinite optimization problems

Abstract: In this paper, we consider approximate solutions ( $$\epsilon $$ -solutions) for a convex semidefinite programming problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we prove an approximate optimality theorem and approximate duality theorems for $$\epsilon $$ -solutions in robust convex semidefinite programming problem under the robust characteristic cone constraint qualification. Moreover, an example is given to illustrate the obtained results.

On the class of almost L-weakly and almost M-weakly compact operators

Abstract: In this paper, we introduce and study new concepts of almost L-weakly and almost M-weakly compact operators.

Measures under the flat norm as ordered normed vector space

Abstract: The space of real Borel measures $$\mathcal {M}(S)$$ on a metric space S under the flat norm (dual bounded Lipschitz norm), ordered by the cone $$\mathcal {M}_+(S)$$ of nonnegative measures, is considered from an ordered normed vector space perspective in order to apply the well-developed theory of this area. The flat norm is considered in place of the variation norm because subsets of $$\mathcal {M}_+(S)$$ are compact and semiflows on $$\mathcal {M}_+(S)$$ are continuous under much weaker conditions. In turn, the flat norm offers new challenges because $$\mathcal {M}(S)$$ is rarely complete and $$\mathcal {M}_+(S)$$ is only complete if S is complete. As illustrations serve the eigenvalue problem for bounded additive and order-preserving homogeneous maps on $$\mathcal {M}_+(S)$$ and continuous semiflows. Both topics prepare for a dynamical systems theory on $$\mathcal {M}_+(S)$$ .

Set-valued loss-based risk measures

Abstract: In this paper, we introduce a new class of set-valued risk measures, named set-valued convex loss-based risk measures. Representation results are provided. This new class can be considered as a set-valued extension of those introduced by Cont et al. (Stat Risk Model Appl Finance Insur 30(2):133–167, 2013) and Chen et al. (Positivity, 2017). Finally, examples are also given to illustrate the set-valued convex loss-based risk measures.

$um$-Topology in multi-normed vector lattices

Abstract: Let \({\mathcal {M}}=\{m_\lambda \}_{\lambda \in \Lambda }\) be a separating family of lattice seminorms on a vector lattice X, then \((X,{\mathcal {M}})\) is called a multi-normed vector lattice (or MNVL). We write \(x_\alpha \xrightarrow {\mathrm {m}} x\) if \(m_\lambda (x_\alpha -x)\rightarrow 0\) for all \(\lambda \in \Lambda \). A net \(x_\alpha \) in an MNVL \(X=(X,{\mathcal {M}})\) is said to be unbounded m-convergent (or um-convergent) to x if \(|x_\alpha -x |\wedge u \xrightarrow {\mathrm {m}} 0\) for all \(u\in X_+\). um-Convergence generalizes un-convergence (Deng et al. in Positivity 21:963–974, 2017; Kandic et al. in J Math Anal Appl 451:259–279, 2017) and uaw-convergence (Zabeti in Positivity, 2017. doi: 10.1007/s11117-017-0524-7), and specializes up-convergence (Aydin et al. in Unbounded p-convergence in lattice-normed vector lattices. arXiv:1609.05301) and \(u\tau \)-convergence (Dabboorasad et al. in \(u\tau \)-Convergence in locally solid vector lattices. arXiv:1706.02006v3). um-Convergence is always topological, whose corresponding topology is called unbounded m-topology (or um-topology). We show that, for an m-complete metrizable MNVL \((X,{\mathcal {M}})\), the um-topology is metrizable iff X has a countable topological orthogonal system. In terms of um-completeness, we present a characterization of MNVLs possessing both Lebesgue’s and Levi’s properties. Then, we characterize MNVLs possessing simultaneously the \(\sigma \)-Lebesgue and \(\sigma \)-Levi properties in terms of sequential um-completeness. Finally, we prove that every m-bounded and um-closed set is um-compact iff the space is atomic and has Lebesgue’s and Levi’s properties.

A Korovkin-type theorem for double sequences of positive linear operators via power series method

Abstract: In this paper, using power series method we obtain a Korovkin type theorem for double sequences of real valued functions defined on a compact subset of $$\mathbb {R}^{2}$$ (the real two-dimensional space). We also present an example that satisfies our theorem. Finally, we calculate the rate of convergence.

Monotonicity and non-monotonicity results for sequential fractional delta differences of mixed order

Abstract: We consider the discrete fractional sequential difference $$\Delta _{1+a-\mu }^{ u }\Delta _a^{\mu }f(t)$$ , where $$t\in \mathbb {N}_{3-\mu - u +a}$$ , in two separate cases, where in each case we require that $$\mu + u \in (1,2)$$ . In the first case, we show that when $$\mu \in (0,1)$$ and $$ u \in (1,2)$$ it follows that the condition $$\Delta _{1+a-\mu }^{ u }\Delta _a^{\mu }f(t)\ge 0$$ implies that f is an increasing map when we impose that $$f(a)\ge 0$$ , $$\Delta f(a)\ge 0$$ , and $$\Delta f(a+1)\ge 0$$ . On the other hand, when $$\mu \in (1,2)$$ and $$ u \in (0,1)$$ we demonstrate that the situation is very different and that this type of monotonicity result only holds when restricted to a proper subregion of the $$(\mu , u )$$ -parameter space coupled with some additional auxiliary conditions.

Some applications of the regularity principle in sequence spaces

Abstract: The Hardy–Littlewood inequalities for m-linear forms have their origin with the seminal paper of Hardy and Littlewood (Q J Math 5:241–254, 1934) Nowadays it has been extensively investigated and many authors are looking for the optimal estimates of the constants involved For \(m

Fatou closedness under model uncertainty

Abstract: We provide a characterization in terms of Fatou closedness for weakly closed monotone convex sets in the space of $${\mathcal P}$$ -quasisure bounded random variables, where $${\mathcal P}$$ is a (possibly non-dominated) class of probability measures. Applications of our results lie within robust versions the Fundamental Theorem of Asset Pricing or dual representation of convex risk measures.

On surjective second order non-linear Markov operators and associated nonlinear integral equations

Abstract: It was known that orthogonality preserving property and surjectivity of nonlinear Markov operators, acting on finite dimensional simpleces, are equivalent. It turns out that these notions are no longer equivalent when such kind of operators are considered over on infinite dimensional spaces. In the present paper, we find necessary and sufficient condition to be equivalent of these notions, for the second order nonlinear Markov operators. To do this, we fully describe all surjective second order nonlinear Markov operators acting on infinite dimensional simplex. As an application of this result, we provided some sufficient conditions for the existence of positive solutions of nonlinear integral equations whose domain are not compact.

Commutators of fractional maximal operator on generalized Orlicz–Morrey spaces

Abstract: In the present paper, we shall give necessary and sufficient conditions for the Spanne and Adams type boundedness of the commutators of fractional maximal operator on generalized Orlicz–Morrey spaces, respectively. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators.

Extrapolation results in grand Lebesgue spaces defined on product sets

Abstract: Extrapolation results in weighted grand Lebesgue spaces defined with respect to product measure $$\mu \times u $$ on $$X\times Y$$ , where $$(X, d, \mu )$$ and $$(Y, \rho , u )$$ are spaces of homogeneous type, are obtained. As applications of the derived results we prove new one-weight estimates for multiple integral operators such as strong maximal, Calderon–Zygmund and fractional integral operators with product kernels in these spaces.

A function class of strictly positive definite and logarithmically completely monotonic functions related to the modified Bessel functions

Abstract: In this paper, we give some conditions for a class of functions related to Bessel functions to be positive definite or strictly positive definite. We present some properties and relationships involving logarithmically completely monotonic functions and strictly positive definite functions. In particular, we are interested with the modified Bessel functions of the second kind. As applications, we prove the logarithmically monotonicity for a class of functions involving the modified Bessel functions of second kind and we established new inequalities for this function.

Quadratic stochastic operators on Banach lattices

Abstract: We study the convergence of iterates of quadratic stochastic operators that are mean monotonic. They are defined on the convex set of probability measures concentrated on a weakly compact order interval $$S = [0, f]$$ of a fixed Banach lattice F. We study their regularity and identify the limits of trajectories either as the “infimum” or “supremum” of the support of initial distributions.

Unitarily invariant strictly positive definite kernels on spheres

Abstract: We present a Fourier characterization for the continuous and unitarily invariant strictly positive definite kernels on the unit sphere in $${\mathbb {C}}^{q}$$ , thus adding to a celebrated work of I J Schoenberg on positive definite functions on real spheres

Closed range and nonclosed range adjointable operators on Hilbert $$C^*$$ C ∗ -modules

Abstract: In this paper, we show that for a positive operator A on a Hilbert $$C^*$$ -module $$ \mathscr {E} $$ , the range $$ \mathscr {R}(A) $$ of A is closed if and only if $$ \mathscr {R}(A^\alpha ) $$ is closed for all $$\alpha \in (0,1)\cup (1,+\,\infty )$$ , and this occurs if and only if $$ \mathscr {R}(A)=\mathscr {R}(A^\alpha ) $$ for all $$\alpha \in (0,1)\cup (1,+\,\infty )$$ . As an application, we prove that for an adjontable operator A if $$\mathscr {R}(A)$$ is nonclosed, then $$\dim \left( \overline{\mathscr {R}(A)}/\mathscr {R}(A)\right) =+\,\infty $$ . Finally, we show that for an adjointable operator A if $$ \overline{\mathscr {R}(A^*) } $$ is orthogonally complemented in $$ \mathscr {E} $$ , then under certain coditions there exists an idempotent C and a unique operator X such that $$ XAX=X, AXA=CA, AX=C $$ and $$ XA=P_{A^*} $$ , where $$ P_{A^*} $$ is the orthogonal projection of $$ \mathscr {E} $$ onto $$ \overline{\mathscr {R}(A^*)}$$ .

A Sawyer-type sufficient condition for the weighted Poincaré inequality

Abstract: In this paper we prove the Poincare-type weighted inequality $$\begin{aligned} \Vert v^{1/q} f \Vert _{L^q(\Omega )} \le C \Vert \omega ^{1/p} abla f \Vert _{L^p(\Omega )}, \quad q\ge p>1, \end{aligned}$$ for a locally Lipschitz function f with a weighted mean equal to zero over a convex bounded domain $$\Omega $$ ; here the weights v, $$\omega $$ are positive measurable functions which satisfy a certain compatibility condition. This result is a generalization of the well-known weighted Poincare inequality to the case of more general weights in the sense that we do not use the traditional conditions of high summability $$v,\, \omega ^{-\frac{1}{p-1}}\in L^{r,loc}$$ with $$r>1$$ for $$q=p$$ or the reverse doubling condition on the function v for $$q>p$$ . In other words, a Sawyer type sufficient condition on weight functions is established.

Coherent and convex loss-based risk measures for portfolio vectors

Abstract: In this paper, we introduce two new classes of risk measures, named coherent and convex loss-based risk measures for portfolio vectors. These new risk measures can be considered as a multivariate extension of univariate loss-based risk measures introduced by Cont et al. (Stat Risk Model 30:133–167, 2013). Representation results for these new introduced risk measures are provided. The links between convex loss-based risk measures for portfolios and convex risk measures for portfolios introduced by Burgert and Ruschendorf (Insur Math Econ 38:289–297, 2006) or Wei and Hu (Stat Probab Lett 90:114–120, 2014) are stated. Finally, applications to the multi-period coherent and convex loss-based risk measures are addressed.

Power type asymptotically uniformly smooth and asymptotically uniformly flat norms

Abstract: We provide a short characterization of p-asymptotic uniform smoothability and asymptotic uniform flatenability of operators and of Banach spaces. We use these characterizations to show that many asymptotic uniform smoothness properties pass to injective tensor products of operators and of Banach spaces. In particular, we prove that the injective tensor product of two asymptotically uniformly smooth Banach spaces is asymptotically uniformly smooth. We prove that for $$1