Abstract: In the paper, we develop sum and chain rules of the generalized contingent derivative for set-valued mappings. Then, their applications to sensitivity analysis and optimality conditions for some particular optimization problems are given. Our results extend some recent existing ones in the literature. PubDate: 2019-03-14

Abstract: We prove a Korovkin-type approximation theorem using the relative uniform convergence of a sequence of functions at a point, which is a method stronger than the classical ones. We give some examples on this new convergence method and we study also rates of convergence. PubDate: 2019-03-07

Abstract: The notion of the mixed brightness was first introduced by Lutwak. Recently, Abardia and Bernig presented complex projection bodies. Based on this notion, we define the mixed complex brightness integrals and establish related Aleksandrov–Fenchel inequality, cyclic inequality and monotonicity inequality, respectively. PubDate: 2019-03-06

Abstract: In the paper it is shown that there exists a function \(U\in L^1[0,1)^2\) , which is universal for all class \(L^{p}[0,1)^2\) , \(p\in (0,1)\) , by rectangles and by spheres with respect to the double Walsh system in the sense of signs of Fourier coefficients. PubDate: 2019-03-06

Abstract: We construct a linear operator R, between two Banach lattices, such that R has a modulus which does not satisfy the Riesz–Kantorovich formula, thereby answering a long-standing question in the theory of regular operators. PubDate: 2019-02-28

Abstract: In this paper we consider Loos symmetric spaces on an open cone \(\Omega \) in the Banach space setting and develop the foundations of a geometric theory based on the (modified) Loos axioms for such cones. In particular we establish exponential and log functions that exhibit many desirable features reminiscent of those of the exponential function from the space of self-adjoint elements to the cone of positive elements in a unital \(C^*\) -algebra. We also show that the Thompson metric arises as the distance function for a natural Finsler structure on \(\Omega \) and its minimal geodesics agree with the geodesics of the spray arising from the Loos structure. We close by showing that some familiar operator inequalities can be derived in this very general setting using the differential and metric geometry of the cone. PubDate: 2019-02-28

Abstract: The purpose of this article is to study the concepts Arens regularity for semigroup algebras \(M_a(S)\) , \(\ell ^1(S)\) , and \({ LUC}(S)^*\) . We give a necessary and sufficient condition for \(M_a(S)\) to be pointwise Arens regular. We then give some characterizations for Arens regularity of \(M_a(S)\) and \(\ell ^1(S)\) . Also, we investigate weakly compact multipliers on \(M_a(S)\) and then for a compactly cancellative foundation semigroup S with identity, we show that S is compact if \({ LUC}(S)^*\) is Arens regular. PubDate: 2019-02-28

Abstract: Let f be a real or complex-valued function on \([1,\infty )\) which is continuous over every finite interval [1, x) for \(1<x<\infty \) . We set \(s(x):=\int _{1}^{x}f(t)dt\) and define \(\sigma _{k}(s(x))\) by $$\begin{aligned} \sigma _{k}(s(x))=\left\{ \begin{array}{ll} \displaystyle {\frac{1}{x}\int _{1}^{x}} \sigma _{k-1}(s(t))dt,&{}\quad k\ge 1\\ s(x),&{}\quad k=0 \end{array} \right. \end{aligned}$$ for each nonnegative integer k. An improper integral $$\begin{aligned} \int _{1}^{\infty } f(x)dx \end{aligned}$$ is said to be integrable to a finite number \(\mu \) by the k-th iteration of Hölder or Cesàro mean method of order one, or for short, the (H, k) integrable to \(\mu \) if $$\begin{aligned} \lim _{x\rightarrow \infty }\sigma _{k}(s(x))=\mu . \end{aligned}$$ In this case, we write \(s(x)\rightarrow \mu \,\,(H,k)\) . It is clear that the (H, k) integrability method reduces to the ordinary convergence for \(k=0\) and the (H, 1) integrability method is (C, 1) integrability method. It is known that \(\lim _{x \rightarrow \infty } s(x) =\mu \) implies \(\lim _{x \rightarrow \infty }\sigma _{k}(s(x)) =\mu \) . But the converse of this implication is not true in general. In this paper, we obtain some Tauberian conditions for the iterations of Hölder integrability method under which the converse implication holds. PubDate: 2019-02-22

Abstract: The aim of this paper is to establish scalarizations for minimal and weak minimal solutions of a set optimization problem using generalized oriented distance function introduced by Crespi et al. (Math Methods Oper Res 63:87–106, 2006). The solution concepts are based on a partial set order relation on the family of nonempty bounded sets proposed by Karaman et al. (Positivity 22:783–802, 2018). Finally, we also provide existence results for minimal solutions and sufficient conditions for the solution sets to be closed. PubDate: 2019-02-22

Abstract: We show that the free Banach lattice \(\mathrm {FBL}(A)\) may be constructed as the completion of \(\mathrm {FVL}(A)\) with respect to the maximal lattice seminorm \(\nu \) on \(\mathrm {FVL}(A)\) with \(\nu (a)\leqslant 1\) for all \(a\in A\) . We present a similar construction for the free Banach lattice \(\mathrm {FBL}[E]\) generated by a Banach space E. PubDate: 2019-02-21

Abstract: In this paper we characterize the closed subspaces of \(L^2({\mathcal {F}})\) that reduce the operators of the form \(E^{{\mathcal {A}}}M_u\) , in which \({\mathcal {A}}\) is a \(\sigma \) - subalgebra of \({\mathcal {F}}\) . We show that \(L^2(A)\) reduces \(E^{{\mathcal {A}}}M_u\) if and only if \(E^{{\mathcal {A}}}(\chi _A)=\chi _A\) on the support of \(E^{{\mathcal {A}}}( u ^2)\) , where \(A\in {\mathcal {F}}\) . Also, some necessary and sufficient conditions are provided for \(L^2({\mathcal {B}})\) to reduces \(E^{{\mathcal {A}}}M_u\) , for the \(\sigma \) -subalgebra \({\mathcal {B}}\) of \({\mathcal {F}}\) . PubDate: 2019-02-21

Abstract: We consider positive operator semigroups on ordered Banach spaces and study the relation of their long time behaviour to two different domination properties. First, we analyse under which conditions almost periodicity and mean ergodicity of a semigroup \(\mathcal {T}\) are inherited by other semigroups which are asymptotically dominated by \(\mathcal {T}\) . Then, we consider semigroups whose orbits asymptotically dominate a positive vector and show that this assumption is often sufficient to conclude strong convergence of the semigroup as time tends to infinity. Our theorems are applicable to time-discrete as well as time-continuous semigroups. They generalise several results from the literature to considerably larger classes of ordered Banach spaces. PubDate: 2019-02-19

Abstract: We study a nonlinear eigenvalue problem on the exterior to a simply connected bounded domain in \(\mathbb {R}^N\) containing the origin. We consider positive weak solutions satisfying Dirichlet boundary conditions on the compact boundary and decaying to zero at infinity. We discuss multiplicity and uniqueness results of solutions with respect to a bifurcation parameter and conjecture an S-shaped bifurcation diagram for positive reaction terms which are singular at the origin and sublinear at infinity. As a by-product, on regions exterior to a ball with radially symmetric weight functions, we obtain radial symmetry of solutions when uniqueness holds. PubDate: 2019-02-19

Abstract: A new convergence criteria using the concept of semi-interior points has been defined in E-metric spaces with non-solid and non-normal set of positive elements \(E^{+}\) of a real normed space E, also known as a positive cone. Many examples are provided to insure the existence of semi-interior points of \(E^{+}\) with empty interior. New generalizations of Banach, Kannan and Chatterjea fixed point theorems are proved. PubDate: 2019-02-08

Abstract: In this paper we formulate and proof Girsanov’s theorem in vector lattices. To reach this goal, we develop the theory of cross-variation processes, derive the cross-variation formula and the Kunita–Watanabe inequality. Also needed and derived are properties of exponential processes, Itô’s rule for multi-dimensional processes and the integration by parts formula for martingales. After proving Girsanov’s theorem for the one-dimensional case, we also discuss the multi-dimensional case. PubDate: 2019-02-07

Abstract: In the dual \(L_{\varPhi ^*}\) of a \(\varDelta _2\) -Orlicz space \(L_\varPhi \) , that we call a dual Orlicz space, we show that a proper (resp. finite) convex function is lower semicontinuous (resp. continuous) for the Mackey topology \(\tau (L_{\varPhi ^*},L_\varPhi )\) if and only if on each order interval \([-\zeta ,\zeta ]=\{\xi : -\zeta \le \xi \le \zeta \}\) ( \(\zeta \in L_{\varPhi ^*}\) ), it is lower semicontinuous (resp. continuous) for the topology of convergence in probability. For this purpose, we provide the following Komlós type result: every norm bounded sequence \((\xi _n)_n\) in \(L_{\varPhi ^*}\) admits a sequence of forward convex combinations \({{\bar{\xi }}}_n\in \text {conv}(\xi _n,\xi _{n+1},\ldots )\) such that \(\sup _n {\bar{\xi }}_n \in L_{\varPhi ^*}\) and \({\bar{\xi }}_n\) converges a.s. PubDate: 2019-02-07

Abstract: In this paper we define Lipschitz and Triebel–Lizorkin spaces associated with the differential-difference Dunkl operators on \(\mathbb {R}^d\) . We study inclusion relations among them. Next, some interpolation results and continuity results of some important operators (the Dunkl–Poisson semigroup and Dunkl–Flett potentials) on them are established. Also, we prove certain inclusion relations between Dunkl–Sobolev classes \(\mathcal{L}^p_{\alpha ,k}(\mathbb {R}^d)\) of positive fractional order \(\alpha \) , Dunkl–Lipschitz spaces \(\wedge ^k_{\alpha ,p,q}(\mathbb {R}^d)\) and Dunkl–Triebel–Lizorkin spaces \(F^k_{\alpha ,p,q}(\mathbb {R}^d)\) . PubDate: 2019-02-04

Abstract: The Sandwich theorem (König in Archiv der Mathematik 23:500–1972, 1972) yields the existence of a linear functional sandwiched in between a given superlinear functional and a given sublinear functional. By combining an argument implicit in Fuchssteiner and Maitland Wright (Q J Math 28:155–162, 1977) with a theorem of Pataraia (in: Presented at the 65th peripatetic seminar on sheaves and logic, Aarhus, Denmark, November, 1997), we show that such a linear functional obtains as a common fixed point of a certain family of mappings. PubDate: 2019-02-01

Abstract: Let X and Y be locally compact Hausdorff spaces. In this paper we study surjections \(T: A \longrightarrow B\) between certain subsets A and B of \(C_0(X)\) and \(C_0(Y)\) , respectively, satisfying the norm condition \(\Vert \varphi (Tf, Tg)\Vert _Y=\Vert \varphi (f,g)\Vert _X\) , \(f,g \in A\) , for some continuous function \(\varphi : {\mathbb {C}}\times {\mathbb {C}}\longrightarrow {\mathbb {R}}^+\) . Here \(\Vert \cdot \Vert _X\) and \(\Vert \cdot \Vert _Y\) denote the supremum norms on \(C_0(X)\) and \(C_0(Y)\) , respectively. We show that if A and B are (positive parts of) subspaces or multiplicative subsets, then T is a composition operator (in modulus) inducing a homeomorphism between strong boundary points of A and B. Our results generalize the recent results concerning multiplicatively norm preserving maps, as well as, norm additive in modulus maps between function algebras to more general cases. PubDate: 2019-02-01