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An error has been detected (and also corrected) in Theorem 2.8 of the paper entitled “Adjoining an Order Unit to a Matrix Ordered Space” (Positivity, (2005)9: 207–223; DOI 10.1007/s11117-003-2778-5). Accordingly, some of the results of the paper have been modified. Also, a notion of C*-matricially, Riesz normed spaces has been introduced.

In this paper, we have pointed out that the proof of Theorem 11 in the recent paper (Lafhim in Positivity, 2019. https://doi.org/10.1007/s11117-019-00685-1 ) is erroneous. Using techniques from variational analysis, we propose other proofs to detect necessary optimality conditions in terms of Karush–Kuhn–Tucker multipliers. Our main results are given in terms of the limiting subdifferentials and the limiting normal cones. Completely detailed first order necessary optimality conditions are then given in the smooth setting while using the generalized differentiation calculus of Mordukhovich.

Present paper deals with applications of asymptotic equivalence relations on operator nets. These relations are defined via unbounded convergences on vector lattices. Given two convergences $$\mathfrak {c}$$ and $$\mathfrak {d}$$ on a vector lattice, we study $$\mathfrak {d}$$ -asymptotic properties of operator nets formed by $$\mathfrak {c}$$ -continuous operators. Asymptotic equivalences are known to be useful and extremely important tools to study infinite behaviors of strongly convergent operator nets and continuous semigroups. After giving a general theory, paper focuses on $$\mathfrak {d}$$ -martingale and $$\mathfrak {d}$$ -Lotz–Rabiger nets.

Many researchers have been interested in the concept of statistical convergence because of the fact that it is stronger than the classical convergence. Also, the concepts of statistical equal convergence and equi-statistical convergence are more general than the statistical uniform convergence. In this paper we define a new type of statistical convergence by using the notions of equi-statistical convergence and statistical equal convergence to prove a Korovkin type theorem. We show that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems which were demonstrated by earlier authors. After, we present an example in support of our definition and result presented in this paper. Finally, we also compute the rates of statistical equi-equal convergence of sequences of positive linear operators.

This paper deals with the existence and uniqueness of positive mild solutions for a class of semilinear evolution equations with nonlocal initial conditions on infinite interval. The existence and uniqueness of mild solution for the associated linear evolution equation nonlocal problem is established, and the spectral radius of resolvent operator is accurately estimated. With the aid of the estimation, the existence and uniqueness of positive mild solutions for nonlinear evolution equation nonlocal problem are obtained by using the monotone iterative method without the assumption of lower and upper solutions. The theorems proved in this paper improve and extend some related results in ordinary differential equations and partial differential equations. An example is also given to illustrate that our results are valuable.

This paper is to illustrate that the main proofs and results of the paper (Verma in Positivity 13, 771–782, 2009) are incorrect.

The purpose of this paper is to prove a strong convergence theorem for a finite family of uniformly L-Lipschitzian mappings in Banach spaces. The results presented in the paper improve and extend some recent results in Chang [1], Cho et al. [2] Ofoedu [5], Schu [7] and Zeng [8, 9].

We study a class of degenerate elliptic second order differential operators acting on some polynomial weighted function spaces on [0,+∞[. We show that these operators are the generators of C0-semigroups of positive operators which, in turn, are the transition semigroups associated with right-continuous normal Markov processes with state space [0,+∞]. Approximation and qualitative properties of both the semigroups and the Markov processes are investigated as well. Most of the results of the paper depend on a representation of the semigroups we give in terms of powers of particular positive operators of discrete type we introduced and studied in a previous paper.

We consider a continuous-time branching random walk on the integer latticeZ d (d> 1) with a finite number of branching sources, or catalysts. The random walk is assumed to be spatially homogeneous and irreducible. The branching mechanism at each catalyst, being independent of the random walk, is governed by a Markov branching process. The quantities of interest are the local numbers of particles (at each site) and the total population size. In the present paper, we derive and analyze the Kolmogorov type backward equations for the corresponding Laplace generating functions and also for the successive integer moments and the process extinction probability. In particular, existence and uniqueness theorems are proved and the problem of explosion is studied in some detail. We then rewrite these equations in the form of integral equations of renewal type, which may serve as a convenient tool for the study of the process long-time behavior. The paper also provides a technical foundation to some results published before without detailed proofs. AMS subject classifications: Primary 60J80, 60J15; Secondary 39A70, 35R10, 60K05

In this paper, we introduce and study a new class of variational inequalities, which is called the generalized mixed variational inequality. Using essentially the resolvent operator concept, we establish the equivalence between the generalized mixed variational inequalities and the system of resolvent equations. This equivalence is used to suggest a number of new iterative algorithms for solving the variational inequalities. Several special cases are discussed which can be obtained from the main results of this paper.

This paper deals with (1, 1; r)-convexity of sequences. First, we prove several results on the sets of (1, 1; r)-convex sequences for various values of r. Then we show that the Cesaro matrix does not preserve (1, 1; r)-convexity of sequences for any $$r\ge 2$$ .

In this paper, we study sufficient conditions for the (strict) inverse-positivity of the linear fourth order operator corresponding to the one-dimensional beam equation with a spatially variable coefficient. We use a modification of results obtained by the operator reduction technique introduced by Schroder and show that the extrema of the coefficient can go beyond the originally derived bounds significantly.

In this paper, we introduce polynomial versions of the weak Dunford–Pettis property and the weak Dunford–Pettis $$^{*}$$ property for Banach lattices. By using Fremlin projective Banach lattice tensor products, we obtain several characterizations of the weak Dunford–Pettis property and the weak Dunford–Pettis $$^{*}$$ property in terms of regular homogeneous polynomials on Banach lattices.

In the paper, we introduce higher-order tangent epiderivatives for set-valued maps. Then, we study some basic properties of these concepts. Finally, we establish some results on duality in set-valued optimization. Several examples are given to illustrate the obtained results.

In this paper, we study the boundedness of multilinear commutators of Calderon–Zygmund operators $$T_{\mathbf {b}}$$ on generalized variable exponent Morrey spaces $$M^{p(\cdot ), \varphi }$$ . Let $$\mathbf {b}=(b_1,\ldots ,b_m)$$ and $$b_i \in BMO$$ for $$i=1,\ldots ,m$$ . Then the sufficient conditions on the pair $$(\varphi _1,\varphi _2)$$ , which ensure the boundedness of the operator $$T_{\mathbf {b}}$$ from $$M^{p(\cdot ), \varphi _1}$$ to $$M^{p(\cdot ), \varphi _2}$$ , are found.

Let $$\mathfrak {A}$$ be a unital $$C^*$$ -algebra and $$\mathfrak {A}'$$ be its topological dual space. Let a be a positive element in $$\mathfrak {A}$$ , and set $$\mathscr {S}_a(\mathfrak {A}):= \left\{ f\in \mathfrak {A}': f\ge 0, f(a)=1\right\} .$$ The a-numerical range and a-numerical radius of any element $$x\in \mathfrak {A}$$ are defined by $$\begin{aligned} V_a(x):= \left\{ f(ax): f\in \mathscr {S}_a(\mathfrak {A})\right\} , \end{aligned}$$ and $$\begin{aligned} v_a(x):=\sup \left\{ \left| z\right| :z\in V_a(x)\right\} , \end{aligned}$$ respectively. In this paper, we establish some permanence properties of the a-numerical range and a-numerical radius of elements in $$\mathfrak {A}$$ . In particular, we investigate when the a-numerical range of an element of $$\mathfrak {A}$$ is closed, and provide explicit formulas for the a-numerical radius of the so-called a-hermitian elements of $$\mathfrak {A}$$ . Furthermore, given a positive operator A on a complex Hilbert space $${\mathscr {H}}$$ , we study and investigate the relationship between the algebraic and spatial A-numerical ranges of bounded linear operators on $${\mathscr {H}}$$ .

In this paper, we give some sufficient and necessary conditions for discrete cosine operator functions on solid Banach function spaces to be chaotic or topologically transitive.

In this paper we establish some Positivstellensatze for polynomial matrices, applying the Scherer–Hol theorem. Firstly, we give a representation for polynomial matrices positive definite on subsets of compact polyhedra. Then we establish a Putinar-Vasilescu Positivstellensatz for polynomial matrices. Next we propose a matrix version of the Dickinson–Povh Positivstellensatz. Finally, we establish a version of Marshall’s theorem for polynomial matrices, approximating positive semi-definite polynomial matrices using sums of squares.

Assume that X is a reflexive Banach space with an unconditional basis. In this paper, we show that the Banach space K(X) of compact operators on X has Pelczynski’s property V.

In this paper, we study Sobolev type inequalities for fractional maximal functions in the half space. We also discuss Sobolev type inequalities for Green potentials in the half space.

In the present paper we give some criteria for statistical convergence with respect to power series method. The main tool for providing this result is the inverse Fourier transformation. We will also deal with sequences x that admit a factorization $$x=yz$$ in which y is strongly convergent and z is statistically convergent with respect to a power series method.

In this paper, we define the $$L_p$$ -mixed affine surface area probability measure and obtain the log-Minkowski inequalities for nonsymmetric convex bodies of $$L_p$$ -mixed affine surface area. Besides, we also establish the functional inequalities for $$L_p$$ -mixed affine surface area, which are more general form than the log-Minkowski inequalities of $$L_p$$ -mixed affine surface area.

In the present paper, we introduce a new concept of positive p-majorizing operators as a dual notion of positive p-summing operators and generalize the concept of majorizing operators introduced by Schaefer (Isr J Math 13:400–415, 1972). We introduce the concept of positive (p, q)-dominated operators and prove a positive version of the famous Kwapien’s factorization theorem for (p, q)-dominated operators via positive p-majorizing operators. We also introduce the notion of disjoint p-summing operators which is a new larger class of operators than positive p-summing operators and use it to characterize the Radon–Nikodým property. Finally, we investigate the maximal properties of these four classes of operators and prove that they are maximal in corresponding sense.

In this paper, we establish a robust version of the S-lemma that we use to characterize robust solutions for classes of homogeneous and non-homogeneous quadratic problems with a quadratic inequality constraint under interval uncertainty and a linear equality constraint. Necessary and sufficient conditions of global optimality of robust solution of these problems are given.

This paper deals with second-order optimality conditions and regularity of Lagrange multipliers for a class of optimal control problems governed by semilinear elliptic equations with mixed pointwise constraints in which controls act both in the domain and on the boundary. We give some criteria under which the optimality conditions are of KKT type and the multipliers are of $$L^p$$ -spaces. Moreover, we show that the multipliers are Lipschitz continuous functions.

In this paper we prove that the $$\ell $$ -group tensor product of two Archimedean f-rings is again an f-ring. We will use this result to characterize multiplicative $$\ell $$ -bimorphisms between unital f-rings.

The main purpose of this paper is to study higher-order sensitivity analysis in parametric vector optimization problems. Firstly, the higher-order proto-differentiability/the higher-order asymptotic proto-differentiability of the feasible map of a parametric vector optimization problem are considered. Then, we verify that the weak efficient solution map and the weak perturbation map of a parameterized vector optimization problem are higher-order proto-differentiable/higher-order asymptotic proto-differentiable under some suitable qualification conditions.

In this paper, under several hypotheses and using a dual gap functional, weak sharp solutions are studied for a multidimensional variational-type inequality governed by $$(\rho , \mathbf {b}, \mathbf {d})$$ -convex multiple integral functional. Moreover, a relation between the minimum principle sufficiency property and weak sharpness of solutions for the considered multidimensional variational-type inequality is established.

In this paper, we introduce some strong and weak bilevel vector equilibrium problems in locally convex Hausdorff topological vector spaces and present some conditions for the existence of solutions to these problems by using the Kakutani–Fan–Glicksberg fixed-point theorem. Furthermore, as a real-world application, we obtain the existence of solutions to traffic network problems with equilibrium constraints.

In the paper, we mention a parametrized vector equilibrium problem via sum of two given set-valued maps, considered as a unified model of some problems in variational analysis and optimization. Then, we study sensitivity analysis for this problem in terms of the second-order contingent derivatives. Finally, corresponding results for special cases are implied.

The purpose of this paper is to study the pessimistic version of bilevel programming problems in finite-dimensional spaces. Problems of this type are intrinsically nonsmooth (even for smooth initial data). By using optimal value function, we transform the initial problem into a generalized minimax optimization problem. Using convexificators, first-order necessary optimality conditions are then established. An example that illustrates our findings is also given.

In this paper, we study the strong ergodicity of non-homogeneous $$\mathbf {p}$$ -majorizing quadratic stochastic operators acting on the finite dimension simplex. We first provide a criterion for the strong ergodicity of such operators. We then establish the strong ergodicity of non-homogeneous $$\mathbf {p}$$ -majorizing quadratic stochastic operators associated with scrambling, Sarymsakov, and Wolfowitz cubic $$\mathbf {p}$$ -stochastic matrices.

This paper is devoted to the study of efficient elements for set-valued maps. We propose two new notions of relative weak $$\epsilon $$ -efficient element and strict relative weak $$\epsilon $$ -efficient element of set-valued maps and provide new necessary optimality conditions for the proposed concepts. We provide existence results for efficient elements. The critical ingredients for the existence results for efficient elements are the well-known separation arguments and Fan’s lemma. As an application of the existence results, we derive relationships between the efficiency concepts and the local optimizers of certain optimization problems.

In this paper, we study the close relationship between multivariate coherent and convex risk measures. Namely, starting from a multivariate convex risk measure, we propose a family of multivariate coherent risk measures induced by it. In return, the convex risk measure can be represented by its induced coherent risk measures. The representation result for the induced coherent risk measures is given in terms of the minimal penalty function of the convex risk measure. Finally, an example is also given.

In this paper, we study \(\sigma \)-subdifferentials of \(\sigma \)-convex functions. Two equivalent conditions for \(\sigma \)-convexity are given. The formula for the \(\sigma \)-subdifferential of a sum of two functions is established. In terms of \(\sigma \)-subdifferential and Clarke’s normal cone, some Fermat’s rules for minimization problems are obtained.

In this paper, we are concerned with the optimistic formulation of a semivectorial bilevel optimization problem. Introducing a new scalarization technique for multiobjective programs, we transform our problem into a scalar-objective optimization problem by means of the optimal value reformulation and establish its theoretical properties. Detailed necessary conditions, to characterize local optimal solutions of the problem, were then provided, while using the weak basic CQ together with the generalized differentiation calculus of Mordukhovich. Our approach is applicable to nonconvex problems and is different from the classical scalarization techniques previously used in the literature and the conditions obtained are new.

In this paper, we establish, in the setting of infinite dimensional Hilbert space, a new existence result for nonconvex sweeping process with right uniformly lower semicontinuous sets. This class of sets is more general than the classical assumption of absolutely continuous sets, and contains hyperplanes and half-spaces. Further, we apply on the problem an unbounded set-valued perturbation and we state the existence of solution.

In this paper, we deal with nonlinear resistive networks in the framework of modular sequence spaces, introduced by De Michele and Soardi. We consider ideal boundaries of a network and investigate Dirichlet boundary value problems for solutions of Poisson equations.

The aim of this paper is to prove that the cosine Fourier coefficients $$a_{mn}$$ (with $$m,n\ge 1)$$ of a Popoviciu convex function of two variables are nonnegative.

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.

EgeUn###, In this paper, we obtain some Tauberian conditions in terms of slow oscillation and slow decreasing in certain senses, under which convergence of a double sequence in Pringsheim’s sense follows from its statistical (C, 1, 1) summability. © 2019, Springer Nature Switzerland AG., Firat University Scientific Research Projects Management Unit, FÃ?BAP: 513, This study is supported by Ege University Scientific Research Projects Coordination Unit. Project Number 513.

In this paper, we investigate the existence of a positive periodic solution for the following Lienard equation with a indefinite singularity $$\begin{aligned} x''+f(x)x'+\frac{b(t)}{x}=p(t), \end{aligned}$$ where $$b\in C({\mathbb {R}},{\mathbb {R}})$$ is a T-periodic sign-changing function. The novelty of the present article is that for the first time we show that a indefinite singularity enables the achievement of a new existence criterion of positive periodic solutions through a application of a topological degree theorem by Mawhin. Recent results in the literature are generalized and significantly improved, and we give the existence interval of a positive periodic solution of this equation. At last, an example is given to show applications of the theorems.

In the present paper, we will characterize the boundedness of the generalized fractional integral operators I? and the generalized fractional maximal operators M? on Orlicz spaces, respectively. Moreover, we will give a characterization for the Spanne-type boundedness and the Adams-type boundedness of the operators M? and I? on generalized Orlicz–Morrey spaces, respectively. Also we give criteria for the weak versions of the Spanne-type boundedness and the Adams-type boundedness of the operators M? and I? on generalized Orlicz–Morrey spaces. © 2018, Springer Nature Switzerland AG., Ministry of Education and Science of the Russian Federation: 15H03621, 02. FEF.A4.18.019, EIF-BGM-4-RFTF-1/2017-21/01/1 Japan Society for the Promotion of Science: 16K05209, The research of F. Deringoz was partially supported by the Grant of Ahi Evran University Scientific Research Project (FEF.A4.18.019). The research of V. Guliyev was partially supported by the Grant of 1st Azerbaijan–Russia Joint Grant Competition (Agreement Number No. EIF-BGM-4-RFTF-1/2017-21/01/1) and by the Ministry of Education and Science of the Russian Federation (the Agreement No. 02.a03.21.0008). Eiichi Nakai was supported by Grant-in-Aid for Scientific Research (B), No. 15H03621, Japan Society for the Promotion of Science. Yoshihiro Sawano was supported by Grant-in-Aid for Scientific Research (C) (16K05209), the Japan Society for the Promotion of Science and by Peoples Friendship University of Russia. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

On the set of positive invertible elements in a finite von Neumann algebra carrying a faithful normalized trace $$\tau $$ the numerical quantity $$\begin{aligned} d_{\tau }(A,B)=\tau (A + B)/2 - \tau \left( A^{1/2}\left( A^{-1/2}BA^{-1/2}\right) ^{1/2}A^{1/2}\right) \end{aligned}$$can be viewed as a measure of the difference of the arithmetic and the geometric mean. In this paper, we study maps between the positive definite cones of operator algebras which respect the above distance measure. We obtain the interesting fact that any such map originates from a trace-preserving Jordan $${}^*$$-isomorphisms (either algebra $${}^*$$-isomorphism or algebra $${}^*$$-antiisomorphism in the more restrictive case of factors) between the underlying von Neumann algebras.

In this paper we obtain the Voronovskaya formula for the sequence of positive linear operators constructed using the squared Bernstein polynomials.

In this paper, we consider the following quasilinear Lienard equation with a singularity $$\begin{aligned} (\phi _p(x'(t)))'+f(x(t))x'(t)+g(t,x(t))=e(t), \end{aligned}$$where g has a attractive singularity at the origin and satisfies superlinear condition at $$x=+\infty $$. By using Manasevich–Mawhin continuous theorem, we prove that this equation has at least one positive T-periodic solution. We solve a difficulty to estimate it a priori bounds of a periodic solution for quasilinear Lienard equation in the case that superlinear condition. At last, example and numerical solution (phase portrait and time series portrait of the positive periodic solution of example) are given to show applications of the theorem.

In this paper we prove results on logarithmic convexity of fixed points of stochastic kernel operators. These results are expected to play a key role in the economic application to strategic market games.

In this paper we obtain quantitative results for the convergence of the iterates of some positive linear operators which preserve certain functions. Some examples involving q-operators are given. We show that the considered q-operators, $$0

In this paper we complete the analysis of a statistical mechanics model on Cayley trees of any degree, started in Botirov (Positivity 21(3):955–961, 2017), Eshkabilov et al. (J Stat Phys 147(4):779–794, 2012), Eshkabilov and Rozikov (Math Phys Anal Geom 13:275–286, 2010), Botirov et al. (Lobachevskii J Math 34(3):256–263 2013) and Jahnel et al. (Math Phys Anal Geom 17:323–331 2014). The potential is of nearest-neighbor type and the local state space is compact but uncountable. Based on the system parameters we prove existence of a critical value $$\theta _{\mathrm{c}}$$ such that for $$\theta \le \theta _{\mathrm{c}}$$ there is a unique translation-invariant splitting Gibbs measure. For $$\theta _{\mathrm{c}}

In the paper, we first extend calculus rules of variational sets, known as a kind of generalized derivatives for set-valued maps, from the first order to the second order. Then, we study sensitivity analysis of parametric set-valued equilibrium problems under the weak efficiency in terms of these sets.