Mann type iterative methods for finding a common solution of split feasibility and fixed point problems

The purpose of this paper is to study and analyze three different kinds of Mann type iterative methods for finding a common element of the solution set ?? of the split feasibility problem and the set Fix(S) of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces. By combining Mann??s iterative method and the extragradient method, we first propose Mann type extragradient-like algorithm for finding an element of the set ${{{\rm Fix}}(S) \cap \Gamma}$ ; moreover, we derive the weak convergence of the proposed algorithm under appropriate conditions. Second, we combine Mann??s iterative method and the viscosity approximation method to introduce Mann type viscosity algorithm for finding an element of the ${{{\rm Fix}}(S)\cap \Gamma}$ ; moreover, we derive the strong convergence of the sequences generated by the proposed algorithm to an element of set ${{{\rm Fix}}(S) \cap \Gamma}$ under mild conditions. Finally, by combining Mann??s iterative method and the relaxed CQ method, we introduce Mann type relaxed CQ algorithm for finding an element of the set ${{{\rm Fix}}(S)\cap \Gamma}$ . We also establish a weak convergence result for the sequences generated by the proposed Mann type relaxed CQ algorithm under appropriate assumptions.     

Weak convergence of the sequential empirical processes of residuals in TAR models

This paper studies the weak convergence of the sequential empirical process K n of the residuals in the threshold autoregressive(TAR)model of order p.Under some mild conditions,it is shown that K n converges weakly to a Kiefer process plus a random variable which converges to a multivariate normal.This differs from that given by Bai(1994)for a stationary autoregressive and moving average(ARMA)model.     

Weak convergence theory for strong materials with p(x)-growth

Let L: Ω × R ^m × R ^{m × n} → R be a Caratheodory integrand with $c_1 |\nu |^{p(x)} + c_2 \leqslant L(x,u,\nu ) \leqslant c_3 |\nu |^{p(x)} + c_4 ,c_3 \geqslant c_1 > 0,n + \varepsilon \leqslant p( \cdot ) \leqslant p < \infty ,\varepsilon > 0.$ Under these assumptions the weak convergence theory holds for the integral functional $J(u): = \int\limits_\Omega {L(x,u(x),Du(x))dx} $ without further requirements. Weak convergence theory includes lower seraicontinuity with respect to the weak convergence of Sobolev functions, the convergence in energy property (weak convergence of Sobolev functions and convergence in energy imply the strong convergence of the functions), the integral representation for the relaxed energy and related questions. The results of the weak convergence theory follows from a characterization of gradient Young measures associated with these functionals.     

On the bilinearity rank of a proper cone and Lyapunov-like transformations

A real square matrix \(Q\) is a bilinear complementarity relation on a proper cone \(K\) in \(\mathbb{R }^n\) if $$\begin{aligned} x\in K, s\in K^*,\,\,\text{ and }\,\,\langle x,s\rangle =0\Rightarrow x^{T}Qs=0, \end{aligned}$$ where \(K^*\) is the dual of \(K\) . The bilinearity rank of \(K\) is the dimension of the linear space of all bilinear complementarity relations on \(K\) . In this article, we continue the study initiated by Rudolf et al. (Math Prog Ser B 129:5–31, 2011). We show that bilinear complementarity relations are related to Lyapunov-like transformations that appear in dynamical systems and in complementarity theory and further show that the bilinearity rank of \(K\) is the dimension of the Lie algebra of the automorphism group of \(K\) . In addition, we correct a result of Rudolf et al., compute the bilinearity ranks of symmetric and completely positive cones, and state some Schur-type results for Lyapunov-like transformations.     

Weak automorphisms of dihedral groups

Let ${\mathcal{A} = (A; F)}$ be an algebra with T the set of all its term operations. For any permutation τ of A, the induced mapping ${f \to \tau\circ f\circ\tau^{-1}}$ defines a permutation ${\tau^{\star}}$ of the set of all finitary operations on the set A. We say that τ is a weak automorphism of ${\mathcal{A}}$ if and only if τ*(T) = T. Of course any automorphism α of ${\mathcal{A}}$ is a weak automorphism, because α*(t) = t for all ${t \in T}$ . The set of all weak automorphisms of ${\mathcal{A}}$ forms a subgroup of the symmetric group on A. In this paper, we describe weak automorphisms of the dihedral groups ${\mathcal{D}_n}$ for n ≥ 3. We show that the weak automorphism group of ${\mathcal{D}_n}$ is a semidirect product of the group of automorphisms of ${\mathcal{D}_n}$ and some group related to the group of invertible elements of the ring ${\mathbb{Z}_n}$ .     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

Weak solutions of the stationary Navier–Stokes equations for a viscous incompressible fluid past an obstacle

Consider the stationary Navier–Stokes equations in an exterior domain $\varOmega \subset \mathbb{R }^3 $ with smooth boundary. For every prescribed constant vector $u_{\infty } \ne 0$ and every external force $f \in \dot{H}_2^{-1} (\varOmega )$ , Leray (J. Math. Pures. Appl., 9:1–82, 1933) constructed a weak solution $u $ with $\nabla u \in L_2 (\varOmega )$ and $u - u_{\infty } \in L_6(\varOmega )$ . Here $\dot{H}^{-1}_2 (\varOmega )$ denotes the dual space of the homogeneous Sobolev space $\dot{H}^1_{2}(\varOmega ) $ . We prove that the weak solution $u$ fulfills the additional regularity property $u- u_{\infty } \in L_4(\varOmega )$ and $u_\infty \cdot \nabla u \in \dot{H}_2^{-1} (\varOmega )$ without any restriction on $f$ except for $f \in \dot{H}_2^{-1} (\varOmega )$ . As a consequence, it turns out that every weak solution necessarily satisfies the generalized energy equality. Moreover, we obtain a sharp a priori estimate and uniqueness result for weak solutions assuming only that $\Vert f\Vert _{\dot{H}^{-1}_2(\varOmega )}$ and $|u_{\infty }|$ are suitably small. Our results give final affirmative answers to open questions left by Leray (J. Math. Pures. Appl., 9:1–82, 1933) about energy equality and uniqueness of weak solutions. Finally we investigate the convergence of weak solutions as $u_{\infty } \rightarrow 0$ in the strong norm topology, while the limiting weak solution exhibits a completely different behavior from that in the case $u_{\infty } \ne 0$ .     

Uniform Continuity, Uniform Convergence, and Shields

Let $\mathcal{B}$ be a bornology in a metric space $\langle X,d \rangle$ , that is, a cover of X by nonempty subsets that also forms an ideal. In Beer and Levi (J Math Anal Appl 350:568–589, 2009), the authors introduced the variational notions of strong uniform continuity of a function on $\mathcal{B}$ as an alternative to uniform continuity of the restriction of the function to each member of $\mathcal{B}$ , and the topology of strong uniform convergence on $\mathcal{B}$ as an alternative to the classical topology of uniform convergence on $\mathcal{B}$ . Here we continue this study, showing that shields as introduced in Beer, Costantini and Levi (Bornological Convergence and Shields, Mediterranean J. Math, submitted) play a pivotal role. For example, restricted to continuous functions, the topology of strong uniform convergence on $\mathcal{B}$ reduces to the classical topology if and only if the natural closure of the bornology is shielded from closed sets. The paper also further develops the theory of shields and their applications.     

Domain perturbation and invariant manifolds

We consider a reaction-diffusion equation defined on a sequence of bounded open sets ${(\Omega_n)_n \in \mathbb{N}}$ , converging to ${\Omega}$ in the sense of Mosco, and we prove stability of invariant manifolds of the flux with respect to domain perturbation.     

On cell modules of symmetric cellular algebras

Let A be a symmetric cellular algebra with cell datum (??, M, C, i) and let ${\Lambda_1=\{\lambda \in \Lambda_0 \mid W(\lambda) \, {\rm is \, simple}\}}$ . We prove that ??₁ consists of two parts: one gives a lower bound for the cardinality of the set of cell modules with zero bilinear forms and the other parametrizes all the projective cell modules. Moreover, it is proved in Li (arxiv: math0911.3524, 2009) that the dual basis of ${\{C_{S, T}^{\lambda} \mid \lambda \in \Lambda, S,T \in M(\lambda)\}}$ is again cellular. In this paper, we will study the cell modules defined by dual basis. In particular, we study the dual basis of the Murphy basis.     

Convergence of symmetric Markov chains on {\mathbb{Z}^d}

For each n let ${Y^{(n)}_t}$ be a continuous time symmetric Markov chain with state space ${n^{-1} \mathbb{Z}^d}$ . Conditions in terms of the conductances are given for the convergence of the ${Y^{(n)}_t}$ to a symmetric Markov process Y _t on ${\mathbb{R}^d}$ . We have weak convergence of $\{{Y^{(n)}_t: t \leq t_0\}}$ for every t ₀ and every starting point. The limit process Y has a continuous part and may also have jumps.     

Passing from bulk to bulk-surface evolution in the Allen–Cahn equation

In this paper we formulate a boundary layer approximation for an Allen–Cahn-type equation involving a small parameter ${\varepsilon}$ . Here, ${\varepsilon}$ is related to the thickness of the boundary layer and we are interested in the limit ${\varepsilon \to 0}$ in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L²-gradient flow with the corresponding Allen–Cahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Γ- and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions.     

Pseudo PCF

We continue our investigation on pcf with weak forms of the axiom of choice. Characteristically, we assume DC+P(Y) when looking at \(\prod\nolimits_{s \in Y} {{\delta _s}} \) . We get more parallels of pcf theorems.     

Superconvergence of finite volume element method for elliptic problems

We study the superconvergence of finite volume element (FVE) method for elliptic problems by using linear trial functions. Under the condition of C-uniform meshes, we first establish a superclose weak estimate for the bilinear form of FVE method. Then, we prove that all interior mesh points are the optimal stress points of interpolation function and further we give the superconvergence result of gradient approximation: $\displaystyle {\max _{P\in S}}\left |\left (\nabla u-\overline {\nabla }u_{h}\right )(P)\right |=O\left (h^{2}\right )\left |\ln h\right |$ , where S is the set of mesh points and $\overline {\nabla }$ denotes the average gradient on elements containing vertex P.     

Lower semicontinuity for functionals with (p, q)-growth in Carnot groups

Let ?? be a bounded open subset of ${\mathbb{G}}$ , where ${\mathbb{G}}$ is a Carnot group, and let ${u: \Omega \rightarrow \mathbb{R}^d}$ be a vector valued function. We prove a lower semicontinuity result in the weak topology of the horizontal Sobolev space ${W^{1,p}_X(\Omega,\mathbb{R}^d)}$ , with p?>?1, of the integral functional of the calculus of variations of the type $$F(u)=\int\limits_\Omega f(Xu)\,dx$$ where f is a X-quasiconvex function satisfying a non-standard growth conditions and Xu is the horizontal gradient of u.     

The structures of Hausdorff metric in non-Archimedean spaces

For non-Archimedean spaces X and Y, let $\mathcal{M}_\flat \left( X \right)$ , $\mathfrak{M}\left( {V \to W} \right)$ and $\mathfrak{D}_\flat \left( {X,Y} \right)$ be the ballean of X (the family of the balls in X), the space of mappings from X to Y, and the space of mappings from the ballean of X to Y, respectively. By studying explicitly the Hausdorff metric structures related to these spaces, we construct several families of new metric structures (e.g., $\hat \rho _u$ , $\hat \beta _{X,Y}^\lambda$ , $\hat \beta _{X,Y}^{ * \lambda }$ ) on the corresponding spaces, and study their convergence, structural relation, law of variation in the variable λ, including some normed algebra structure. To some extent, the class $\hat \beta _{X,Y}^\lambda$ is a counterpart of the usual Levy-Prohorov metric in the probability measure spaces, but it behaves very differently, and is interesting in itself. Moreover, when X is compact and Y = K is a complete non-Archimedean field, we construct and study a Dudly type metric of the space of K-valued measures on X.     

A Maschke type theorem for weak group entwined modules and applications

Let π be a discrete group. Given a weak π-entwining structure \({(A,C)_{\pi - \psi }}\) and α ∈ π, we give the necessary and sufficient conditions for the forgetful functor \({F^{(\alpha )}}\) from the category \(U_A^{\pi - C}(\psi )\) of right \({(A,C)_{\pi - \psi }}\) -modules to the category \({M_{{A_\alpha }}}\) of right \({A_\alpha }\) -modules to be separable. This leads to a generalized notion of integrals. The results are applied to weak Doi-Hopf π-modules and to weak entwining modules.     

An Empirical Process Central Limit Theorem for Multidimensional Dependent Data

Let $(U_{n}(t))_{t\in\mathbb{R}^{d}}$ be the empirical process associated to an ? ^d -valued stationary process (X _i ) _i≥0. In the present paper, we introduce very general conditions for weak convergence of $(U_{n}(t))_{t\in\mathbb{R}^{d}}$ , which only involve properties of processes (f(X _i )) _i≥0 for a restricted class of functions $f\in\mathcal{G}$ . Our results significantly improve those of Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011) and provide new applications. The central interest in our approach is that it does not need the indicator functions which define the empirical process $(U_{n}(t))_{t\in\mathbb{R}^{d}}$ to belong to the class  $\mathcal{G}$ . This is particularly useful when dealing with data arising from dynamical systems or functionals of Markov chains. In the proofs we make use of a new application of a chaining argument and generalize ideas first introduced in Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011). Finally we will show how our general conditions apply in the case of multiple mixing processes of polynomial decrease and causal functions of independent and identically distributed processes, which could not be treated by the preceding results in Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011).     

Decay Rates to Equilibrium for Nonlinear Plate Equations with Degenerate, Geometrically-Constrained Damping

We analyze the convergence to equilibrium of solutions to the nonlinear Berger plate evolution equation in the presence of localized interior damping (also referred to as geometrically constrained damping). Utilizing the results in (Geredeli et al. in J. Differ. Equ. 254:1193–1229, 2013), we have that any trajectory converges to the set of stationary points $\mathcal{N}$ . Employing standard assumptions from the theory of nonlinear unstable dynamics on the set $\mathcal{N}$ , we obtain the rate of convergence to an equilibrium. The critical issue in the proof of convergence to equilibria is a unique continuation property (which we prove for the Berger evolution) that provides a gradient structure for the dynamics. We also consider the more involved von Karman evolution, and show that the same results hold assuming a unique continuation property for solutions, which is presently a challenging open problem.     

{\mathcal{C}^{\alpha}}-regularity for a class of non-diagonal elliptic systems with p-growth

We consider weak solutions to nonlinear elliptic systems in a W ^1,p -setting which arise as Euler equations to certain variational problems. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the dependent and independent variables. We impose new structure conditions on the coefficients which yield everywhere ${\mathcal{C}^{\alpha}}$ -regularity and global ${\mathcal{C}^{\alpha}}$ -estimates for the solutions. These structure conditions cover variational integrals like ${\int F(\nabla u)\; dx}$ with potential ${F(\nabla u):=\tilde F (Q_1(\nabla u),\ldots, Q_N(\nabla u))}$ and positively definite quadratic forms in ${\nabla u}$ defined as ${Q_i(\nabla u)=\sum_{\alpha \beta} a_i^{\alpha \beta} \nabla u^\alpha \cdot \nabla u^\beta}$ . A simple example consists in ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{2}} + |\xi_2|^{\frac{p}{2}}}$ or ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{4}}|\xi_2|^{\frac{p}{4}}}$ . Since the Q _i need not to be linearly dependent our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. The proof uses a new weighted norm technique with singular weights in an L ^p -setting.