Let $H$ be a weak Hopf algebra with bijective antipode. In this paper we follow Woronowicz's fundamental method to characterize bicovariant differential calculi on $H$. We show that there exists a 1-1 correspondence between bicovariant differential calculi and some right ideals of $H$ contained in $ ker \varepsilon_s $ such that these ideals are right $H$-comodules with coadjoint maps, where $\varepsilon_s$ is the source map of $H$. This is a generalization of well-known Woronowicz's theorem about bicovariant differential calculi on quantum groups.

Let $G$ be a finite abelian group, and $k_1,k_2$ be two integers. For any subset $A\subset G$, let $r_{k_1,k_2}(A,n)$ denote the number of solutions of $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. In this paper, we generalize a result of Q.-H. Yang and Y.-G. Chen to finite abelian groups. More precisely, we characterize all subsets $A\subset G$ such that $r_{k_1,k_2}(A,n)=r_{k_1,k_2}(G\backslash A,n)$ for all $n\in G$.

This paper proposes the sufficient conditions of approximate controllability for a class of fractional order stochastic variational inequalities driven by Poisson jumps. The possibilities of finding the approximate controllability of a given problem of this type introduce the smoothing system corresponding to the fractional order stochastic variational inequalities driven by Poisson jumps. The results are achieved upon the Moreau-Yosida approximation of subdifferential operator. Sufficient conditions for the approximate controllability of smoothing system are discussed under the boundedness condition on control operator. The results are formulated and proved by using the fractional calculus, semigroup theory, stochastic analysis techniques. An example is provided to illustrate the obtained theory.

A set of improvements of a well-balanced scheme for the model of fluid flows in a nozzle with variable cross-section is presented. Relying on the well-balanced method introduced in our earlier work, we use the steady state solutions to absorb the nonconservative term. The underlying numerical fluxes operating on these steady states are obtained as convex combinations of the numerical fluxes of a first-order and a second-order schemes. The improvements are still well-balanced schemes. Then, we present many numerical tests, which establishes the efficiency of these numerical schemes. These schemes can provide us with very desirable approximations for any initial data: data in supersonic or subsonic regions, and data in both of these two kinds of regions. All the tests also show that the accuracy of the method by the improvements is improved.

General linear time-varying differential systems with delay are considered. Several explicit criteria for exponential stability are presented. Furthermore, an explicit robust stability bound for systems subject to time-varying perturbations is given. Two examples are given to illustrate the obtained results. To the best of our knowledge, the results of this paper are new.

In this paper we introduce the weighted Poincaré distance and the induced distance by the weighted Bloch type space. We prove that the weighted Poincaré distance is identical to the inner distance generated by the induced distance.

In this article, a hybrid of the parametric method and discretization approach is proposed for a class of continuous-time quadratic fractional programming problems (CQFP). This approach leads to an approximation algorithm that solves the problem (CQFP) to any required accuracy. The analysis also shows that we can predetermine the size of discretization such that the accuracy of the corresponding approximate solution can be controlled within the predefined error tolerance. Hence, the trade-off between the quality of the results and the simplification of the problem can be controlled by the decision maker. Moreover, we prove the convergence of the searched sequence of approximate solutions.

We introduce a new family of nonlinear operators called $k$-set-pseudo-contractions where several well-known mappings, such as, the condensing mappings (for $k=1$) and the compact perturbations of $k$-pseudo-contractive mappings are embraced in the class of $k$-set-pseudo-contractions. We prove an invariance of domain theorem and (as a consequence) a fixed point theorem for a $k$-set-pseudo-contraction ($0 \lt k \lt 1$) which is also an $L$-set-contraction ($L \geq 0$). Several well known results can be deduced from our theorems.

In this paper, we obtain Chen's inequalities for submanifolds of a Riemannian manifold of quasi-constant curvature endowed with a semi-symmetric metric connection. Also, some results of A. Mihai and C. Özgur's paper have been modified.

In this paper, some sufficient conditions are obtained for theexistence and multiplicity of sign-changing solutions for the dampedvibration problem with impulsive effects \begin{eqnarray*} \left\{ \begin{array}{ll} -u''(t)+g(t)u'(t)=f(t,u(t)), & \hbox{a.e. $t\in [0,T]$;} \\ u(0)=u(T)=0, & \hbox{} \\ \Delta u'(t_{j})=u'(t_{j}^{+})-u'(t_{j}^{-})=I_{j}(u(t_{j})), & \hbox{$j=1,2,\ldots,p,$} \end{array} \right. \end{eqnarray*} where $t_{0}=0\lt t_{1}\lt t_{2}\lt \ldots\lt t_{p}\lt t_{p+1}=T,g\in L^{1}(0,T;\mathbb{R}),I_{j}:\mathbb{R}\rightarrow\mathbb{R},j=1,2,\ldots,p$ are continuous, $f:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function with subcritical growth condition:

$(A) |f(t, u)| ≤ C(1 + |u|^{s-1}), \forall t \in[0,T], u\in \mathbb{R}, s\in [2,+\infty)$.

The sign-changing solutions are sought by means of some sign-changing critical point theorems and two examples are presented to illustrate the feasibility and effectiveness of our results.

Recently, Wardowski [Fixed Point Theory Appl. 2012:94, 2012] introduced and studied a new contraction called F-contraction to prove a fixed point result as a generalization of the Banach contraction principle. Abbas et al. [2] further generalized the concept of F-contraction and proved certain fixed and common fixed point results. In this paper, we introduce an $\alpha$-GF-contraction with respect to a general family of functions $G$ and establish Wardowski type fixed point results in metric and ordered metric spaces. As an application of our results we deduce Suzuki type fixed point results for GF-contractions. We also derive certain fixed and periodic point results for orbitally continuous generalized F-contractions. Moreover, we discuss some illustrative examples to highlight the realized improvements.

Well-posedness for vector optimization problems has been extensively studied. More recently, some attempts to extend thee results to set-valued optimization have been proposed, mainly applying some scalarization. In this paper we propose a new definition of global well-posedness for set-optimization problems. Using an embedding technique proposed by Kuroiwa and Nuriya (2006), we prove well-posedness property of a class of generalized convex set-valued maps.

As a generalization of semi-slant submersions, h-slant submersions, and h-semi-invariant submersions, we introduce the notions of h-semi-slant submersions and almost h-semi-slant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We obtain characterizations and investigate the integrability of distributions, the geometry of fibers, and the harmonicity of such maps. We also find a condition for such maps to be totally geodesic. Moreover, we give some examples of such maps.

Let $\varphi$ be an analytic self-map of the unit disk $\mathbb{D}$, $H(\mathbb{D})$ the space of analytic functions on $\mathbb{D}$ and $\psi_{1},\psi_{2}\in H(\mathbb{D})$. Recently Stevi$\acute{\hbox{c}}$ and co-workers defined the following operator $$T_{\psi_{1},\psi_{2},\varphi}f(z)=\psi_{1}(z)f(\varphi(z))+\psi_{2}(z)f'(\varphi(z)),\ \ \ f\in H(\mathbb{D}).$$ The boundedness and compactness of the operators $T_{\psi_{1},\psi_{2},\varphi}$ from mixed-norm spaces to weighted-type spaces are investigated in this paper.

In this paper, we obtain some stability results for a class of parametric generalized system. Under weaker conditions, a density theorem of positive proper efficient solutions in the set of strong solutions is obtained. Then, by using the density result, we establish the upper semicontinuity of strong solution mappings to parametric generalized system without monotonicity. Our results are different from the corresponding ones in the literature. Some examples are given to illustrate the results.

We consider the boundary value problem \begin{equation} \tag{0.1} \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(x, u), \ \ \ \ & x\in \Omega,\\ u=0, \ \ \ \ & x\in \partial\Omega, \end{array} \right. \end{equation} where $ \Omega \subset \mathbb R^N$ be a bounded domain, $\inf_{\Omega}V(x)\gt -\infty$, $f$ is a superlinear, subcritical nonlinearity. Inspired by previous work of Szulkin and Weth (2009)) [21] and (2010) [22], we develop a more direct and simpler approach on the basis of one used in [21], to deduce weaker conditions under which problem (0.1) has a ground state solution of generalized Nehari type or infinity many nontrivial solutions. Unlike the Nehari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the generalized Nehari manifold by using the diagonal method.

A point set $X$ in the plane is called a $k$-distance set if there are exactly $k$ different distances between two distinct points in $X$. Let $D=D(X) $ be the diameter of a finite set $X$, and let $X_{D} = \{x\in X : d(x, y) = D$ for some $y \in X\}$, the diameter graph $DG(X_{D})$ of $X_{D}$ is the graph with $X_{D}$ as its vertices and where two vertices $x, y \in X_{D} $ are adjacent if $d(x, y) = D$. We prove the set $X$ having at most five distances with $DG(X_{D})=C_{7}$ has the unique $X_{D}=R_{7}$, and the set $X$ having at most six distances with $DG(X_{D})=C_{9}$ has the unique $X_{D}=R_{9}$, and give a conjecture for $k$-distance set with $DG(X_{D})=C_{2k-3}$.

In this paper the idea of studying real hypersurfaces in non-flat complex space forms, whose *-Ricci tensor satisfies geometric conditions is presented. More precisely, three dimensional real hypersurfaces in non-flat complex space forms with parallel *-Ricci tensor are studied. At the end of the paper ideas for further research on $^{*}$-Ricci tensor are given.

In this paper, we introduce the concept of a weak $q$-distance and for this distance we derive a set-valued version of Ekeland's variational principle in the setting of uniform spaces. By using this principle, we prove the existence of solutions to a vector optimization problem with a set-valued map. Moreover, we define the $(p,\varepsilon)$-condition of Takahashi and the $(p,\varepsilon)$-condition of Hamel for a set-valued map. It is shown that these two conditions are equivalent. As an application, we discuss the relationship between an $\varepsilon$-approximate solution and a solution of a vector optimization problem with a set-valued map. Also, a well-posedness result for a vector optimization problem with a set-valued map is given.