We introduce the $Q$-lasso which generalizes the well-known lasso of Tibshirani (1996) with $Q$ a closed convex subset of a Euclidean m-space for some integer $m\ge 1$. This set $Q$ can be interpreted as the set of errors within given tolerance level when linear measurements are taken to recover a signal/image via the lasso. Solutions of the $Q$-lasso depend on a tuning parameter $\gamma$. In this paper, we obtain basic properties of the solutions as a function of $\gamma$. Because of ill posedness, we also apply $l_1\text{-}{l}_2$ regularization to the $Q$-lasso. In addition, we discuss iterative methods for solving the $Q$-lasso which include the proximal-gradient algorithm and the projection-gradient algorithm.

We study holomorphic maps between C${\mathrm{}}^{\mathrm{*}}$-algebras $A$ and $B$, when $f:B_A(\mathrm{0},\varrho )\to B$ is a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball $U=B_A(\mathrm{0},\delta )$. If we assume that $f$ is orthogonality preserving and orthogonally additive on $A_{sa}\cap U$ and $f(U)$ contains an invertible element in $B$, then there exist a sequence $(h_n)$ in $B^{\mathrm{*}\mathrm{*}}$ and Jordan ${\mathrm{}}^{\mathrm{*}}$-homomorphisms $\mathrm{\Theta },\tilde{\mathrm{\Theta }}:M(A)\to B^{\mathrm{*}\mathrm{*}}$ such that $f(x)={\sum }_{n=\mathrm{1}}^{\infty }{h}_n\tilde{\mathrm{\Theta }}(a^n)={\sum }_{n=\mathrm{1}}^{\infty }\mathrm{\Theta }(a^n)h_n$ uniformly in $a\in U$. When $B$ is abelian, the hypothesis of $B$ being unital and $f(U)\cap \mathrm{}\text{i}\text{n}\text{v}\mathrm{}(B)\ne \mathrm{\varnothing }$ can be relaxed to get the same statement.

The aim of the present paper is to give approximate versions of Hua’s theorem and other related results for Banach algebras and C*-algebras. We also study linear maps approximately preserving the conorm between unital C*-algebras.

We introduce Mann-type extragradient methods for a general system of variational inequalities with solutions of a multivalued variational inclusion and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type extragradient methods are based on Korpelevich’s extragradient method and Mann iteration method. We first consider and analyze a Mann-type extragradient algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another Mann-type extragradient algorithm in a smooth and uniformly convex Banach space. Under suitable assumptions, we derive some weak and strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

We introduce hybrid and relaxed Mann iteration methods for a general system of variational inequalities with solutions being also common solutions of a countable family of variational inequalities and common fixed points of a countable family of nonexpansive mappings in real smooth and uniformly convex Banach spaces. Here, the hybrid and relaxed Mann iteration methods are based on Korpelevich’s extragradient method, viscosity approximation method, and Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for hybrid and relaxed Mann iteration algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gateaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

This paper is concerned with the existence of mild and strong solutions on the interval $[0,T]$ for some neutral partial differential equations with nonlocal conditions. The linear part of the equations is assumed to generate a compact analytic semigroup of bounded linear operators, whereas the nonlinear part satisfies the Carathëodory condition and is bounded by some suitable functions. We first employ the Schauder fixed-point theorem to prove the existence of solution on the interval $[\delta ,T]$ for $\delta >0$ that is small enough, and, then, by letting $\delta \to 0$ and using a diagonal argument, we have the existence results on the interval $[0,T]$. This approach allows one to drop the compactness assumption on a nonlocal condition, which generalizes recent conclusions on this topic. The obtained results will be applied to a class of functional partial differential equations with nonlocal conditions.

We study the solvability of the system of the adjointable operator equations $A_{1}X=C_1$, $X{B}_2=C_2$, and $A_{3}X{B}_3=C_3$ over Hilbert $C^{\mathrm{*}}$-modules. We give necessary and sufficient conditions for the existence of a solution and a positive solution of the system. We also derive representations for a general solution and a positive solution to this system. The above results generalize some recent results concerning the equations for operators with closed ranges.

Let $C$ be a closed bounded convex subset of a real Banach space $X$ with $0$ as its interior and $p_C$ the Minkowski functional generated by the set $C$. For a nonempty set $G$ in $X$ and $x\in X$, $g_0\in G$ is called the generalized best approximation to $x$ from $G$ if $p_C(g_0-x)\le p_C(g-x)$ for all $g\in G$. In this paper, we will give a distance formula under $p_C$ from a point to a closed hyperplane $H(x^{\ast },\alpha )$ in $X$ determined by a nonzero continuous linear functional $x^{\ast }$ in $X$ and a real number α, a representation of the generalized metric projection onto $H(x^{\ast },\alpha )$, and investigate the continuity of this generalized metric projection, extending corresponding results for the case of norm.

Making use of a linear operator, which is defined here by means of the Hadamard product (or convolution), we consider two subclasses $F_{p,n}(a,c,\lambda ,A,B)$ and $G_{p,n}(a,c,\lambda ,A,B)$ of multivalent analytic functions with negative coefficients in the open unit disk. Some modified Hadamard products, integral transforms, and the partial sums of functions belonging to these classes are studied.

An estimation of uniqueness ball of a zero point of a mapping on Lie group is established. Furthermore, we obtain a unified estimation of radius of convergence ball of Newton’s method on Lie groups under a generalized L-average Lipschitz condition. As applications, we get estimations of radius of convergence ball under the Kantorovich condition and the $\gamma$-condition, respectively. In particular, under the $\gamma$-condition, our results improve the corresponding results in (Li et al. 2009, Corollary 4.1) as showed in Remark 17. Finally, applications to analytical mappings are also given.

The present paper is concerned with the semilocal as well as the local convergence problems of Newton-Steffensen’s method to solve nonlinear operator equations in Banach spaces. Under the assumption that the second derivative of the operator satisfies $\gamma$-condition, the convergence criterion and convergence ball for Newton-Steffensen’s method are established.

We study the well-posedness for generalized set equilibrium problems (GSEP) and propose two types of the well-posed concepts for these problems in topological vector space settings. These kinds of well-posedness arise from some well-posedness in the vector settings. We also study the relationship between these well-posedness concepts and present several criteria for the well-posedness of GSEP. Our results are new or include as special cases recent existing results.

We consider a generalized equilibrium problem involving DC functions. By using the properties of the epigraph of the conjugate functions, some sufficient and/or necessary conditions for the weak and strong duality results and optimality conditions for generalized equilibrium problems are provided.

The aim of this paper is to define modified weak $\alpha$-$\psi$-contractive mappings and to establish fixed point results for such mappings defined on partial metric spaces using the notion of triangular $\alpha$-admissibility. As an application, we prove new fixed point results for graphic weak $\psi$-contractive mappings. Moreover, some examples and an application to integral equation are given here to illustrate the usability of the obtained results.

We consider an implicit algorithm for the split fixed point and convex feasibility problems. Strong convergence theorem is obtained.

We consider a class of nonsmooth generalized semi-infinite programming problems. We apply results from parametric optimization to the lower level problems of generalized semi-infinite programming problems to get estimates for the value functions of the lower level problems and thus derive necessary optimality conditions for generalized semi-infinite programming problems. We also derive some new estimates for the value functions of the lower level problems in terms of generalized differentiation and further obtain the necessary optimality conditions.

We introduce nonsmooth vector quasi-variational-like inequalities (NVQVLI) by means of a bifunction. We establish some existence results for solutions of these inequalities by using Fan-KKM theorem and a maximal element theorem. By using the technique and methodology adopted in Al-Homidan et al. (2012), one can easily derive the relations among these inequalities and a vector quasi-optimization problem. Hence, the existence results for a solution of a vector quasi-optimization problem can be derived by using our results. The results of this paper extend several known results in the literature.

We introduce and analyze hybrid implicit and explicit extragradient methods for finding a zero of an accretive operator and solving a general system of variational inequalities and a fixed point problem of an infinite family of nonexpansive self-mappings in a uniformly convex Banach space $X$ which has a uniformly Gateaux differentiable norm. We establish some strong convergence theorems for hybrid implicit and explicit extra-gradient algorithms under suitable assumptions. Furthermore, we derive the strong convergence of hybrid implicit and explicit extragradient algorithms for finding a common element of the set of zeros of an accretive operator and the common fixed point set of an infinite family of nonexpansive self-mappings and a self-mapping whose complement is strictly pseudocontractive and strongly accretive in $X$. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.