Closed Convex Subset Research Articles

VISCOSITY APPROXIMATION METHODS FOR NONEXPANSIVE SEMINGROUPS AND MONOTONE MAPPPINGS

Abstract. Let C be a nonempty closed convex subset of real Hilbertspace Hand F = fS(t) : t0ga nonexpansive self-mapping semigroupof C, and f: C!Cis a xed contractive mapping. Consider the processfx n g:(x n+1 = n x n + (1 n )z n z n = n f(x n ) + (1 n )S(t n )P C (x n r n Ax n ):It is shown that fx n gconverges strongly to a common element of the setof xed points of nonexpansive semigroups and the set of solutions of thevariational inequality for an inverse strongly-monotone mapping whichsolves some variational inequality. 1. IntroductionLet H be a real Hilbert space with inner product h;iand induced normkk, Cbe a nonempty closed convex subset of H. S: C!Cis nonexpansiveif kS(x) S(y)kkx yk;for all x;y2C. The set of xed points of SisF(S) = fx2C: x= Sxg. We assume that F(S) 6= ˚, it is well known thatF(S) is closed convex.A nonexpansive semigroup is a family F = fS(t) : t0gof self-mapping ofCif the following conditions are satis ed:(a) S(0)x= xfor all x2C;(b) S(s+ t) = S(s)S(t) for all s;t0;(c) For each t>0, kS(t)x S(t)ykkx yk;x;y2C:(d) For each x2C, the mapping S()xis continuous.In this paper, we use Fto denote the set of common xed points of F; that is,F= fx2C: S(t)x= x;t0g=

The convergence theorems of Ishikawa iterative process with errors for Φ-hemi-contractive mappings in uniformly smooth Banach spaces

Let D be a nonempty closed convex subset of an arbitrary uniformly smooth real Banach space E, and be a generalized Lipschitz Φ-hemi-contractive mapping with . Let , , , be four real sequences in and satisfy the conditions (i) as and ; (ii) . For some , let , be any bounded sequences in D, and be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the fixed point q of T. A related result deals with the operator equations for a generalized Lipschitz and Φ-quasi-accretive mapping. MSC:47H10.

Simple projection algorithm for a countable family of weak relatively nonexpansive mappings and applications

Abstract Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, let { T n } : C → C be a countable family of weak relatively nonexpansive mappings such that F = ⋂ n = 1 ∞ F ( T n ) ≠ ∅ . For any given gauss x 0 ∈ C , define a sequence { x n } in C by the following algorithm: { C 0 = C , C n + 1 = { z ∈ C n : ϕ ( z , T n x n ) = ϕ ( z , x n ) } , n = 0 , 1 , 2 , 3 , … , x n + 1 = Π C n + 1 x 0 . Then { x n } converges strongly to q = Π F x 0 . MSC:47H05, 47H09, 47H10.

Iterative common solutions for monotone inclusion problems, fixed point problems and equilibrium problems

Abstract Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let α > 0 , and let A be an α-inverse strongly-monotone mapping of C into H. Let T be a generalized hybrid mapping of C into H. Let B and W be maximal monotone operators on H such that the domains of B and W are included in C. Let 0 < k < 1 , and let g be a k-contraction of H into itself. Let V be a γ ¯ -strongly monotone and L-Lipschitzian continuous operator with γ ¯ > 0 and L > 0 . Take μ , γ ∈ R as follows: 0 < μ < 2 γ ¯ L 2 , 0 < γ < γ ¯ − L 2 μ 2 k . Suppose that F ( T ) ∩ ( A + B ) − 1 0 ∩ W − 1 0 ≠ ∅ , where F ( T ) and ( A + B ) − 1 0 , W − 1 0 are the set of fixed points of T and the sets of zero points of A + B and W, respectively. In this paper, we prove a strong convergence theorem for finding a point z 0 of F ( T ) ∩ ( A + B ) − 1 0 ∩ W − 1 0 , where z 0 is a unique fixed point of P F ( T ) ∩ ( A + B ) − 1 0 ∩ W − 1 0 ( I − V + γ g ) . This point z 0 ∈ F ( T ) ∩ ( A + B ) − 1 0 ∩ W − 1 0 is also a unique solution of the variational inequality 〈 ( V − γ g ) z 0 , q − z 0 〉 ≥ 0 , ∀ q ∈ F ( T ) ∩ ( A + B ) − 1 0 ∩ W − 1 0 . Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space. In particular, we solve a problem posed by Kurokawa and Takahashi (Nonlinear Anal. 73:1562-1568, 2010). MSC:47H05, 47H10, 58E35.

The space clcv(ℝ n ) with the Hausdorff-Bebutov metric and statistically invariant sets of control systems

We obtain conditions that allow one to evaluate the relative frequency of occurrence of the reachable set of a control system in a given set. If the relative frequency of occurrence in this set is 1, then the set is said to be statistically invariant. It is assumed that the images of the right-hand side of the differential inclusion corresponding to the given control system are convex, closed, but not necessarily compact. We also study the basic properties of the space clcv(ℝ n ) of nonempty closed convex subsets of ℝ n with the Hausdorff-Bebutov metric.

On invariant convex subsets in algebras defined on a locally compact group G

Suppose that A is either the Banach algebra L1(G) of a locally compact group G, or measure algebra M(G), or other algebras (usually larger than L1(G) and M(G)) such as the second dual, L1(G)**, of L1(G) with an Arens product, or LUC(G)* with an Arenstype product. The left translation invariant closed convex subsets of A are studied. Finally, we obtain necessary and sufficient conditions for LUC(G)* to have 1-dimensional left ideals.

On Olaleru’s open problem on Gregus fixed point theorem

Let (X, d) be a complete metric space and $${TX \longrightarrow X }$$ be a mapping with the property d(Tx, Ty) ? ad(x, y) + bd(x, Tx) + cd(y, Ty) + ed(y, Tx) + fd(x, Ty) for all $${x, y \in X}$$ , where 0 0. We show that if e + f > 0 then T has a unique fixed point and also if e + f ? 0 and X is a closed convex subset of a complete metrizable topological vector space (Y, d), then T has a unique fixed point. These results extend the corresponding results which recently obtained in this field. Finally by using our main results we give an answer to the Olaleru's open problem.

Fixed point theorems of block operator matrices on Banach algebras and an application to functional integral equations

In this paper, we study some fixed point theorems of a 2 × 2 block operator matrix defined on nonempty bounded closed convex subsets of Banach algebras, where the entries are nonlinear operators. Furthermore, we apply the obtained results to a coupled system of nonlinear equations. Copyright © 2012 John Wiley & Sons, Ltd.

Solving the split feasibility problem without prior knowledge of matrix norms

The split feasibility problem (SFP) consists in finding a point in a given closed convex subset of a Hilbert space such that its image under a bounded linear operator belongs to a given closed convex subset of another Hilbert space. Iterative methods can be employed to solve the SFP. The most popular iterative method is Byrne’s CQ algorithm. However, to employ Byrne’s CQ algorithm, one needs to know a priori the norm (or at least an estimate of the norm) of the bounded linear operator (matrix in the finite-dimensional framework). It is the purpose of this paper to introduce a way of selecting the stepsizes such that the implementation of the CQ algorithm does not need any prior information about the operator norm. We also practise this way of selecting stepsizes for variants of the CQ algorithm, including a relaxed CQ algorithm where the two closed convex sets are both level sets of convex functions, and a Halpern-type algorithm. Both weak and strong convergence are investigated. Numerical experiments are included to illustrate the applications in signal processing of the CQ algorithm with stepsizes selected in an adaptive way.

HYBRID METHOD FOR DESIGNING EXPLICIT HIERARCHICAL FIXED POINT APPROACH TO MONOTONE VARIATIONAL INEQUALITIES

Let $C$ be a nonempty closed convex subset of a real Hilbert space $H$. Assume that $F: C \to H$ is a $\kappa$-Lipschitzian and $\eta$-strongly monotone operator with constants $\kappa,\eta \gt 0$, $f: C \to H$ is $L$-Lipschitzian with constant $L \geq 0$ and $T,V: C \to C$ are nonexpansive mappings with ${\rm Fix}(T) \neq \emptyset$. Let $0 \lt \mu \lt 2 \eta/\kappa^2$ and $0 \leq \gamma L \lt \tau$, where $\tau = 1 - \sqrt{1-\mu(2\eta-\mu\kappa^2)}$. Consider the hierarchical monotone variational inequality problem (in short, HMVIP): VI (a): finding $z^* \in {\rm Fix}(T)$ such that $\langle(I-V)z^*, z-z^*\rangle \geq 0$, $\forall z \in {\rm Fix}(T)$; VI (b): finding $x^* \in S$ such that $\langle(\mu F - \gamma f) x^*, x-x^*\rangle \geq 0$, $\forall z \in S$. Here $S$ denotes the nonempty solution set of the VI (a). This paper combines hybrid steepest-descent method, viscosity method and projection method to design an explicit algorithm, that can be used to find the unique solution of the HMVIP. Strong convergence of the algorithm is proved under very mild conditions. Applications in hierarchical minimization problems are also included.

Fixed points, eigenvalues and surjectivity for (ws)-compact operators on unbounded convex sets

The paper studies the existence of fixed points for some nonlinear (ws)-compact, weakly condensing and strictly quasibounded operators defined on an unbounded closed convex subset of a Banach space. Applications of the newly developed fixed point theorems are also discussed for proving the existence of positive eigenvalues and surjectivity of quasibounded operators in similar situations. The main condition in our results is formulated in terms of axiomatic measures of weak noncompactness.

A two-step iterative process for common fixed points in Banach spaces

Suppose that K is a nonempty closed convex subset of a uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with two sequences {kn(i)}⊂[1,∞) satisfying ∑n=1∞(kn(i)−1)<∞(i=1,2) and F≔F(T1)∩F(T2)={x∈K:T1x=T2x=x}≠0̸, respectively. For any given x1∈K, suppose that {xn} is a sequence generated iteratively by {xn+1=(1−αn)(PT1)nxn+αn(PT2)nyn,yn=(1−βn)xn+βn(PT1)nxn,n∈N, where {αn} and {βn} are sequences in [a,1−a] for some a∈(0,1). Strong convergence of {xn} to a common fixed point of T1 and T2 is obtained by assuming the complete continuity of only one of the two mappings and then under the condition (A′) which is weaker than compactness of the domain. Moreover, weak convergence is obtained under Opial’s condition. Our results generalize a number of corresponding results in the contemporary literature. Another interesting fact is that our results obtained even for only one mapping are new.

TWO GENERALIZED STRONG CONVERGENCE THEOREMS OF HALPERN’S TYPE IN HILBERT SPACES AND APPLICATIONS

Let $C$ be a closed convex subset of a real Hilbert space $H$. Let $A$ be an inverse-strongly monotone mapping of $C$ into $H$ and let $B$ be a maximal monotone operator on $H$ such that the domain of $B$ is included in $C$. We introduce two iteration schemes of finding a point of $(A+B)^{-1}0$, where $(A+B)^{-1}0$ is the set of zero points of $A+B$. Then, we prove two strong convergence theorems of Halpern's type in a Hilbert space. Using these results, we get new and well-known strong convergence theorems in a Hilbert space.

Efficient implementation of a modified and relaxed hybrid steepest-descent method for a type of variational inequality

To reduce the difficulty and complexity in computing the projection from a real Hilbert space onto a nonempty closed convex subset, researchers have provided a hybrid steepest-descent method for solving VI(F, K) and a subsequent three-step relaxed version of this method. In a previous study, the latter was used to develop a modified and relaxed hybrid steepest-descent (MRHSD) method. However, choosing an efficient and implementable nonexpansive mapping is still a difficult problem. We first establish the strong convergence of the MRHSD method for variational inequalities under different conditions that simplify the proof, which differs from previous studies. Second, we design an efficient implementation of the MRHSD method for a type of variational inequality problem based on the approximate projection contraction method. Finally, we design a set of practical numerical experiments. The results demonstrate that this is an efficient implementation of the MRHSD method.

Lagrange multiplier characterizations of robust best approximations under constraint data uncertainty

In this paper we explain how to characterize the best approximation to any x in a Hilbert space X from the set C∩{x∈X:gi(x)≤0,i=1,2,…,m} in the face of data uncertainty in the convex constraints, gi(x)≤0,i=1,2,…,m, where C is a closed convex subset of X. Following the robust optimization approach, we establish Lagrange multiplier characterizations of the robust constrained best approximation that is immunized against data uncertainty. This is done by characterizing the best approximation to any x from the robust counterpart of the constraints where the constraints are satisfied for all possible uncertainties within the prescribed uncertainty sets. Unlike the traditional Lagrange multiplier characterizations without data uncertainty, for constrained best approximation problems in the face uncertainty, we show that the strong conical hull intersection property (strong CHIP) alone is not sufficient to guarantee the Lagrange multiplier characterizations. We present conditions which guarantee that the strong CHIP is necessary and sufficient for the multiplier characterization. We also establish that the strong CHIP is automatically satisfied for the cases of polyhedral constraints with polytope uncertainty, and linear constraints with interval uncertainty. As an application, we show how robust solutions of shape preserving interpolation problems under ellipsoidal and box uncertainty cases can be obtained in terms of Lagrange multipliers under strict robust feasibility conditions.

Smoothing neural network for constrained non-Lipschitz optimization with applications.

In this paper, a smoothing neural network (SNN) is proposed for a class of constrained non-Lipschitz optimization problems, where the objective function is the sum of a nonsmooth, nonconvex function, and a non-Lipschitz function, and the feasible set is a closed convex subset of . Using the smoothing approximate techniques, the proposed neural network is modeled by a differential equation, which can be implemented easily. Under the level bounded condition on the objective function in the feasible set, we prove the global existence and uniform boundedness of the solutions of the SNN with any initial point in the feasible set. The uniqueness of the solution of the SNN is provided under the Lipschitz property of smoothing functions. We show that any accumulation point of the solutions of the SNN is a stationary point of the optimization problem. Numerical results including image restoration, blind source separation, variable selection, and minimizing condition number are presented to illustrate the theoretical results and show the efficiency of the SNN. Comparisons with some existing algorithms show the advantages of the SNN.

Invariance for stochastic reaction-diffusion equations

Given a stochastic reaction-diffusion equation on a bounded open subset $\mathcal O$ of $\mathbb{R}^n$, we discuss conditions for the invariance of a nonempty closed convex subset $K$ of $L^2(\mathcal O)$ under the corresponding flow. We obtain two general results under the assumption that the fourth power of the distance from $K$ is of class $C^2$, providing, respectively, a necessary and a sufficient condition for invariance. We also study the example where $K$ is the cone of all nonnegative functions in $L^2(\mathcal O)$.

A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let α > 0 and let A be an α-inverse-strongly monotone mapping of C into H and let B be a maximal monotone operator on H. Let F be a maximal monotone operator on H such that the domain of F is included in C. Let 0 0 }\) and L > 0. Take \({\mu, \gamma \in \mathbb R}\) as follows: $${0 < \mu < \frac{2\overline{\gamma}}{L^2}, \quad 0 < \gamma < \frac{\overline{\gamma}-\frac{L^2 \mu}{2}}{k}.}$$ In this paper, under the assumption \({(A+B)^{-1}0 \cap F^{-1}0 \neq \emptyset}\), we prove a strong convergence theorem for finding a point \({z_0\in (A+B)^{-1}0\cap F^{-1}0}\) which is a unique solution of the hierarchical variational inequality $${\langle (V-\gamma g)z_0, q-z_0 \rangle \geq 0, \quad \forall q\in (A+B)^{-1}0 \cap F^{-1}0.}$$ Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space which are useful in nonlinear analysis and optimization.

Hybrid iteration method for common fixed points of an infinite family of nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space, and let K be a nonempty closed convex subset of E. Let be a sequence of nonexpansive mappings from K to itself with F :={x ∈ K :T i x = x, ∀i ≥ 1}≠ ∅. For an arbitrary initial point x1 ∈ K, the modified hybrid iteration scheme {x n } is defined as follows: , where A: K → K is an L-Lipschitzian mapping, with i satisfying: n = [(k-i+1)(i+k)/2]+[1+(i-1)(i+2)/2],k ≥ i-1(i = 1,2,...),{λ n } ⊂ [0,1), and {α n } is a sequence in [a, 1 - a] for some a ∈ (0,1). Under some suitable conditions, the strong and weak convergence theorems of {x n } to a common fixed point of the nonexpansive mappings are obtained. The results in this article extend those of the authors whose related researches are restricted to the situation of finite families of nonexpansive mappings. Mathematics Subject Classifications 2000: 47H09; 47J25.

Solving a Generalized Heron Problem by Means of Convex Analysis

The classical Heron problem states: on a given straight line in the plane, find a point C such that the sum of the distances from C to the given points A and B is minimal. This problem can be solved using standard geometry or differential calculus. In the light of modern convex analysis, we are able to investigate more general versions of this problem. In this paper we propose and solve the following problem: on a given nonempty closed convex subset of ℝs, find a point such that the sum of the distances from that point to n given nonempty closed convex subsets of ℝs is minimal.