Nonempty Closed Convex Subset Research Articles

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.

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 $\kappa,\eta>0$, $f:C\to H$ is $L$-Lipschitzian with constant $L\geq0$ and $T,V:C\to C$ are nonexpansive mappings with ${\rm Fix}(T)\neq\emptyset$. Let $0<\mu<2\eta/\kappa^2$ and $0\leq\gamma L<\tau$, 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 $\langle(I-V)z^*,z-z^*\rangle\geq0,~\forall z\in{\rm Fix}(T)$; VI (b): finding $x^*\in S$ such that $\langle(\mu F-\gamma f)x^*,x-x^*\rangle\geq0,~\forall z\in S$. 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.

Self‐Adaptive and Relaxed Self‐Adaptive Projection Methods for Solving the Multiple‐Set Split Feasibility Problem

Given nonempty closed convex subsets Ci⊆Rm, i = 1, 2, …, t and nonempty closed convex subsets Qj⊆Rn, j = 1, 2, …, r, in the n‐ and m‐dimensional Euclidean spaces, respectively. The multiple‐set split feasibility problem (MSSFP) proposed by Censor is to find a vector such that , where A is a given M × N real matrix. It serves as a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator’s range. MSSFP has a variety of specific applications in real world, such as medical care, image reconstruction, and signal processing. In this paper, for the MSSFP, we first propose a new self‐adaptive projection method by adopting Armijo‐like searches, which dose not require estimating the Lipschitz constant and calculating the largest eigenvalue of the matrix ATA; besides, it makes a sufficient decrease of the objective function at each iteration. Then we introduce a relaxed self‐adaptive projection method by using projections onto half‐spaces instead of those onto convex sets. Obviously, the latter are easy to implement. Global convergence for both methods is proved under a suitable condition.

The Equivalence of Convergence Results of Modified Mann and Ishikawa Iterations with Errors without Bounded Range Assumption

Let E be an arbitrary uniformly smooth real Banach space, let D be a nonempty closed convex subset of E, and let T : D → D be a uniformly generalized Lipschitz generalized asymptotically Φ‐strongly pseudocontractive mapping with q ∈ F(T) ≠ ∅. Let {an}, {bn}, {cn}, {dn} be four real sequences in [0,1] and satisfy the conditions: (i) an + cn ≤ 1, bn + dn ≤ 1; (ii) an, bn, dn → 0 as n → ∞ and cn = o(an); (iii) . For some x0, z0 ∈ D, let {un}, {vn}, {wn} be any bounded sequences in D, and let {xn}, {zn} be the modified Ishikawa and Mann iterative sequences with errors, respectively. Then the convergence of {xn} is equivalent to that of {zn}.

Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces

Let C be a nonempty closed convex subset of a real uniformly smooth Banach space X, an infinite family of nonexpansive mappings with the nonempty set of common fixed points , and f : C → C a contraction. We introduce an explicit iterative algorithm xn+1 = αnf(xn)+(1 − αn)Lnxn, where and wk &gt; 0 with . Under certain appropriate conditions on {αn}, we prove that {xn} converges strongly to a common fixed point x* of , which solves the following variational inequality: , where J is the (normalized) duality mapping of X. This algorithm is brief and needs less computational work, since it does not involve W‐mapping.

On Unification of the Strong Convergence Theorems for a Finite Family of Total Asymptotically Nonexpansive Mappings in Banach Spaces

We unify all known iterative methods by introducing a new explicit iterative scheme for approximation of common fixed points of finite families of total asymptotically I‐nonexpansive mappings. Note that such a scheme contains a particular case of the method introduced by (C. E. Chidume and E. U. Ofoedu, 2009). We construct examples of total asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Note that no such kind of examples were known in the literature. We prove the strong convergence theorems for such iterative process to a common fixed point of the finite family of total asymptotically I‐nonexpansive and total asymptotically nonexpansive mappings, defined on a nonempty closed‐convex subset of uniformly convex Banach spaces. Moreover, our results extend and unify all known results.

CONVERGENCE OF ITERATIVE SCHEMES FOR NONEXPANSIVE MAPPINGS

Let K be a nonempty closed convex subset of a real uniformly convex Banach space X and suppose T : K → K is a nonexpansive mapping with the nonempty fixed point set Fix(T). Let [Formula: see text], [Formula: see text] and [Formula: see text] be sequences in [0, 1] such that [Formula: see text][Formula: see text][Formula: see text] for some constants a, b, α, β, and γ. Let x0 ∈ K be any initial point. Then it is proved that the implicit iteration [Formula: see text] defined by [Formula: see text] converges weakly to a fixed point of T. Furthermore, it is generalized that if [Formula: see text] is a finite family of nonexpansive self-mappings of K with the nonempty common fixed points set [Formula: see text] and if the parameters [Formula: see text], [Formula: see text], and [Formula: see text] satisfy the conditions (0.1), (0.2), (0.3), and [Formula: see text] for some constant c , then the modified implicit iteration [Formula: see text] defined by [Formula: see text] where Tn = Tn(modN), converges weakly to a common fixed point of the family [Formula: see text]. The results presented in this paper improve and extend the corresponding results of Z. Opial [11], S. Reich [13], H.-K. Xu and R. G. Ori [20], and Zhao et al. [21].

Strong convergence theorems by a hybrid extragradient-like approximation method for asymptotically nonexpansive mappings in the intermediate sense in Hilbert spaces

Abstract Let C be a nonempty closed convex subset of a real Hilbert space H. Let S : C → C be an asymptotically nonexpansive map in the intermediate sense with the fixed point set F(S). Let A : C → H be a Lipschitz continuous map, and VI(C, A) be the set of solutions u ∈ C of the variational inequality ⟨ A u , v - u ⟩ ≥ 0 , ∀ v ∈ C . The purpose of this study is to introduce a hybrid extragradient-like approximation method for finding a common element in F(S) and VI(C, A). We establish some strong convergence theorems for sequences produced by our iterative method. AMS subject classifications: 49J25; 47H05; 47H09.

A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem

Let C be a nonempty closed convex subset of a real Hilbert space H . Let f : C → H be a ρ -contraction. Let S : C → C be a nonexpansive mapping. Let B , B ˜ : H → H be two strongly positive bounded linear operators. Consider the triple-hierarchical constrained optimization problem of finding a point x ∗ such that x ∗ ∈ Ω , 〈 ( B ˜ − γ f ) x ∗ − ( I − B ) S x ∗ , x − x ∗ 〉 ≥ 0 , ∀ x ∈ Ω , where Ω is the set of the solutions of the following variational inequality: x ∗ ∈ E P ( F , A ) , 〈 ( B ˜ − S ) x ∗ , x − x ∗ 〉 ≥ 0 , ∀ x ∈ E P ( F , A ) , where E P ( F , A ) is the set of the solutions of the equilibrium problem of finding z ∈ C such that F ( z , y ) + 〈 A z , y − z 〉 ≥ 0 , ∀ y ∈ C . Assume Ω ≠ 0̸ . The purpose of this paper is the solving of the above triple-hierarchical constrained optimization problem. For this purpose, we first introduce an implicit double-net algorithm. Consequently, we prove that our algorithm converges hierarchically to some element in E P ( F , A ) which solves the above triple-hierarchical constrained optimization problem. As a special case, we can find the minimum norm x ∗ ∈ E P ( F , A ) which solves the monotone variational inequality 〈 ( I − S ) x ∗ , x − x ∗ 〉 ≥ 0 , ∀ x ∈ E P ( F , A ) .

Two-step projection methods for a system of variational inequality problems in Banach spaces

Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E and let ? C be a sunny nonexpansive retraction from E onto C. Let the mappings $${T, S: C \to E}$$ be ? 1-strongly accretive, μ 1-Lipschitz continuous and ? 2-strongly accretive, μ 2-Lipschitz continuous, respectively. For arbitrarily chosen initial point $${x^0 \in C}$$ , compute the sequences {x k } and {y k } such that $${\begin{array}{ll} \quad y^k = \Pi_C[x^k-\eta S(x^k)],\\ x^{k+1} = (1-\alpha^k)x^k+\alpha^k\Pi_C[y^k-\rho T(y^k)],\quad k\geq 0, \end{array}}$$ where {? k } is a sequence in [0,1] and ?, ? are two positive constants. Under some mild conditions, we prove that the sequences {x k } and {y k } converge to x* and y*, respectively, where (x*, y*) is a solution of the following system of variational inequality problems in Banach spaces: $${\left\{\begin{array}{l}\langle \rho T(y^*)+x^*-y^*,j(x-x^*)\rangle\geq 0, \quad\forall x \in C,\\\langle \eta S(x^*)+y^*-x^*,j(x-y^*)\rangle\geq 0,\quad\forall x \in C.\end{array}\right.}$$ Our results extend the main results in Verma (Appl Math Lett 18:1286---1292, 2005) from Hilbert spaces to Banach spaces. We also obtain some corollaries which include some results in the literature as special cases.

Approximating fixed points for nonself mappings in CAT(0) spaces

AbstractSuppose K is a nonempty closed convex subset of a complete CAT(0) space X with the nearest point projection P from X onto K. Let T : K → X be a nonself mapping, satisfying Condition (E) with F(T): = {x ∈ K : Tx = x} ≠ ∅. Suppose {x n } is generated iteratively by x1 ∈ K, x n +1 = P ((1 - α n )x n ⊕ α n TP [(1 - β n )x n ⊕ β n Tx n ]),n ≥ 1, where {α n } and {β n } are real sequences in [ε, 1 - ε] for some ε ∈ (0, 1). Then, {x n } Δ-converges to some point x⋆ in F(T). This extends a result of Laowang and Panyanak [Fixed Point Theory Appl. 367274, 11 (2010)] for nonself mappings satisfying Condition (E).

A common fixed point theorem for a commuting family of nonexpansive mappings one of which is multivalued

AbstractBruck [Pac. J. Math. 53, 59-71 1974 Theorem 1] proved that for a nonempty closed convex subset E of a Banach space X, if E is weakly compact or bounded and separable and suppose that E has both (FPP) and (CFPP), then for any commuting family S of nonexpansive self-mappings of E, the set F(S) of common fixed points of S is a nonempty nonexpansive retract of E. In this paper, we extend the above result when one of its elements in S is multivalued. The result extends previously known results (on common fixed points of a pair of single valued and multivalued commuting mappings) to infinite number of mappings and to a wider class of spaces.2000 Mathematics Subject Classification: 47H09; 47H10

Approximations for nonlinear mappings by the hybrid method in Hilbert spaces

The hybrid method in mathematical programming was introduced by Haugazeau (1968) [1] and he proved a strong convergence theorem for finding a common element of finite nonempty closed convex subsets of a real Hilbert space. Later, Bauschke and Combettes (2001) [2] proposed some condition for a family of mappings (the so-called coherent condition) and established interesting results by the hybrid method. The authors (Nakajo et al., 2009) [10] extended Bauschke and Combettes’s results. In this paper, we introduce a condition weaker than the coherent condition and prove strong convergence theorems which generalize the results of Nakajo et al. (2009) [10]. And we get strong convergence theorems for a family of asymptotically κ -strict pseudo-contractions, a family of Lipschitz and pseudo-contractive mappings and a one-parameter uniformly Lipschitz semigroup of pseudo-contractive mappings.

Regularized algorithms for hierarchical fixed-point problems

Let C be a nonempty closed convex subset of a real Hilbert space H . Let S : C → C be a non-expansive mapping and { T i } i = 1 ∞ : C → C be an infinite family of non-expansive mappings. The purpose of this paper is to find the minimum norm solution of the following general hierarchical fixed point problem Find x ̃ ∈ ⋂ n = 1 ∞ F i x ( T n ) such that 〈 x ̃ − S x ̃ , x ̃ − x 〉 ≤ 0 , ∀ x ∈ ⋂ n = 1 ∞ F i x ( T n ) . We introduce an explicit regularized algorithm with strong convergence for finding the minimum norm solution of the above hierarchical fixed point problem.

Strong convergence theorems for system of equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense

Let be N uniformly continuous asymptotically λ i -strict pseudocontractions in the intermediate sense defined on a nonempty closed convex subset C of a real Hilbert space H. Consider the problem of finding a common element of the fixed point set of these mappings and the solution set of a system of equilibrium problems by using hybrid method. In this paper, we propose new iterative schemes for solving this problem and prove these schemes converge strongly. MSC: 47H05; 47H09; 47H10.

Weak convergence theorems for asymptotically nonexpansive nonself-mappings

Suppose that K is a nonempty closed convex subset of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T 1 , T 2 : K → E be two asymptotically nonexpansive nonself-mappings with sequences { k n } , { l n } ⊂ [ 1 , ∞ ) such that ∑ n = 1 ∞ ( k n − 1 ) < ∞ and ∑ n = 1 ∞ ( l n − 1 ) < ∞ , respectively and F ( T 1 ) ∩ F ( T 2 ) = { x ∈ K : T 1 x = T 2 x = x } ≠ 0̸ . Suppose that { x n } is generated iteratively by { x 1 ∈ K x n + 1 = P ( ( 1 − α n ) x n + α n T 1 ( P T 1 ) n − 1 y n ) y n = P ( ( 1 − β n ) x n + β n T 2 ( P T 2 ) n − 1 x n ) , ∀ n ≥ 1 , where { α n } and { β n } are two real sequences in [ ϵ , 1 − ϵ ] for some ϵ > 0 . If E also has a Fréchet differentiable norm or its dual E ∗ has the Kadec–Klee property, then weak convergence of { x n } to some q ∈ F ( T 1 ) ∩ F ( T 2 ) are obtained.

Viscosity approximation scheme for a family multivalued mapping

Let K be a nonempty closed convex subset of a real reflexive Banach space X that has weakly sequentially continuous duality mapping J φ for some gauge φ . Let T i : K → K be a family of multivalued nonexpansive mappings with F : = ∩ i = 0 ∞ F ( T i ) ≠ 0̸ which is a sunny nonexpansive retract of K with Q a nonexpansive retraction. It is our purpose in this paper to prove the convergence of two viscosity approximation schemes to a common fixed point x ̄ = Q f ( x ̄ ) of a family of multivalued nonexpansive mappings in Banach spaces. Moreover, x ̄ is the unique solution in F to a certain variational inequality.

Strong convergence of implicit viscosity approximation methods for pseudocontractive mappings in Banach spaces

Let C be a nonempty closed convex subset of a uniformly smooth Banach space X. Let f : C → C be a fixed contraction mapping, S : C → C be a nonexpansive mapping and T : C → C be a pseudocontractive mapping. Let {α n }, {β n } and {γ n } be three real sequences in (0, 1) such that α n + β n + γ n ⩽ 1. For arbitrary x 0 ∈ C, the sequence {x n } is generated by It is proven that under some conditions, {x n } converges strongly to a fixed point of T, which solves some variational inequality.

Strong convergence theorems for a semigroup of asymptotically nonexpansive mappings

Let K be a nonempty closed convex subset of a real Banach space E . Let T : = { T ( t ) : t ≥ 0 } be a strongly continuous semigroup of asymptotically nonexpansive mappings from K into K with a sequence { L t } ⊂ [ 1 , ∞ ) . Suppose F ( T ) ≠ 0̸ . Then, for a given u ∈ K there exists a sequence { u n } ⊂ K such that u n = ( 1 − α n ) 1 t n ∫ 0 t n T ( s ) u n d s + α n u , for n ∈ N , where t n ∈ R + , { α n } ⊂ ( 0 , 1 ) and { L t } satisfy certain conditions. Suppose, in addition, that E is reflexive strictly convex with a Gâteaux differentiable norm. Then, the sequence { u n } converges strongly to a point of F ( T ) . Furthermore, an explicit sequence { x n } which converges strongly to a fixed point of T is proved.

Viscosity approximation methods for monotone mappings and a countable family of nonexpansive mappings

Abstract We use viscosity approximation methods to obtain strong convergence to common fixed points of monotone mappings and a countable family of nonexpansive mappings. Let C be a nonempty closed convex subset of a Hilbert space H and P C is a metric projection. We consider the iteration process {x n} of C defined by x 1 = x ∈ C is arbitrary and $$ x_{n + 1} = \alpha _n f(x_n ) + (1 - \alpha _n )S_n P_C (x_n + \lambda _n Ax_n ) $$ where f is a contraction on C, {S n} is a sequence of nonexpansive self-mappings of a closed convex subset C of H, and A is an inverse-strongly-monotone mapping of C into H. It is shown that {x n} converges strongly to a common element of the set of common fixed points of a countable family of nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping which solves some variational inequality. Finally, the ideas of our results are applied to find a common element of the set of equilibrium problems and the set of solutions of the variational inequality problem, a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space. The results of this paper extend and improve the results of Chen, Zhang and Fan.