Nonempty Convex Subset Research Articles

Relative numerical ranges

Relying on the ideas of Stampfli [14] and Magajna [12] we introduce, for operators S and T on a separable complex Hilbert space, a new notion called the numerical range of S relative to T at r∈σ(|T|). Some properties of these numerical ranges are proved. In particular, it is shown that the relative numerical ranges are non-empty convex subsets of the closure of the ordinary numerical range of S. We show that the position of zero with respect to the relative numerical range of S relative to T at ‖T‖ gives an information about the distance between the involved operators. This result has many interesting corollaries. For instance, one can characterize those complex numbers which are in the closure of the numerical range of S but are not in the spectrum of S.

Hyperspaces of convex bodies of constant width

For every n≥1, let cc(Rn) denote the hyperspace of all non-empty compact convex subsets of the Euclidean space Rn endowed with the Hausdorff metric topology. For every non-empty convex subset D of [0,∞) we denote by cwD(Rn) the subspace of cc(Rn) consisting of all compact convex sets of constant width d∈D and by crwD(Rn) the subspace of the product cc(Rn)×cc(Rn) consisting of all pairs of compact convex sets of constant relative width d∈D. In this paper we prove that cwD(Rn) and crwD(Rn) are homeomorphic to D×Rn×Q, whenever D≠{0} and n≥2, where Q denotes the Hilbert cube. In particular, the hyperspace cw(Rn) of all compact convex bodies of constant width as well as the hyperspace crw(Rn) of all pairs of compact convex sets of constant relative positive width are homeomorphic to Rn+1×Q.

Local maximum points of explicitly quasiconvex functions

This work concerns generalized convex real-valued functions defined on a nonempty convex subset of a real topological linear space. Its aim is twofold: first, to show that any local maximum point of an explicitly quasiconvex function is a global minimum point whenever it belongs to the intrinsic core of the function’s domain and second, to characterize strictly convex normed spaces by applying this property for a particular class of convex functions.

Fixed points by certain iterative schemes with applications

The main aim of this paper is to present the concept of general Mann and general Ishikawa type double-sequences iterations with errors to approximate fixed points. We prove that the general Mann type double-sequence iteration process with errors converges strongly to a coincidence point of two continuous pseudo-contractive mappings, each of which maps a bounded closed convex nonempty subset of a real Hilbert space into itself. Moreover, we discuss equivalence from the -stabilities point of view under certain restrictions between the general Mann type double-sequence iteration process with errors and the general Ishikawa iterations with errors. An application is also given to support our idea using compatible-type mappings. MSC:47H10, 54H25.

Convergence theorems for fixed points of uniformly continuous Φ-pseudo-contractive-type operator

Let E be a real normed linear space, let K be a nonempty convex subset of E, and let T:K→E be a uniformly continuous Φ-pseudo-contractive-type mapping. It is proved that both the Mann-type and Ishikawa-type iteration schemes converge strongly to the unique fixed point of T, without requiring that the sequences associated with the schemes be bounded. Our theorems are significant improvements on the results of Chang et al. (Iterative methods for nonlinear operators in Banach spaces, 2002), those of Gu (Northeast Math. J. 17(3):340-346, 2001), and those of a host of other authors.MSC:47H06, 47H09, 47J05, 47J25.

Implicit iteration scheme for two phi-hemicontractive operators in arbitrary Banach spaces

The purpose of this paper is to characterize conditions for the convergence of the implicit Ishikawa iterative scheme with errors in the sense of Agarwal et al. (J. Math. Anal. Appl. 272:435-447, 2002) to a common fixed point of two ϕ-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space.MSC:47H09, 47J25.

Iteration scheme for common fixed points of hemicontractive and nonexpansive operators in Banach spaces

Abstract The purpose of this paper is to characterize the conditions for the convergence of the iterative scheme in the sense of Agarwal et al. (J. Nonlinear Convex. Anal. 8(1): 61-79, 2007), associated with nonexpansive and ϕ-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space.

Implicit iteration scheme for phi-hemicontractive operators in arbitrary Banach spaces

The purpose of this paper is to characterize the conditions for the convergence of the implicit Mann iterative scheme with error term to the unique fixed point of φ-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space.

On the convergence of Mann iteration scheme for the finite family of \phi-hemicontractive mappings

The purpose of this paper is to characterize conditions for the convergence of the Mann type iterative scheme to the common fixed point of a family of φ-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space.

ON CONVERGENCE THEOREMS OF ITERATIVE SCHEMES FOR PSEUDOCONTRACTIVE MAPPINGS

Abstract. In this paper, we establish the strong convergence for theMann iterative scheme associated with uniformly continuous pseudocon-tractive mappings in real Banach spaces. Moreover, our technique ofproofs is of independent interest. 1. Introduction and preliminariesLet Ebe a real Banach space and Kbe a nonempty convex subset of E.Let Jdenote the normalized duality mapping from Eto 2 E de ned byJ(x) = ff 2E : hx;fi= jjxjj 2 and jjfjj= jjxjjg;where E denotes the dual space of Eand h;idenotes the generalized dualitypairing. We shall denote the single-valued duality map by j.Let T: D(T) ˆE!Ebe a mapping with domain D(T) in E:De nition 1.1. Tis said to be Lipschitzian if there exists L>1 such that forall x;y2D(T)kTx TykLkx yk:De nition 1.2. T is said to be nonexpansive if for all x, y 2D(T), thefollowing inequality holds:kTx Tykkx ykfor all x;y2D(T):De nition 1.3. T is said to be pseudocontractive if there exists j(x y) 2J(x y) such thathTx Ty;j(x y)ikx yk 2 for all x;y2D(T); n1:

Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings

Suppose that E is a real normed linear space, C is a nonempty convex subset of E, T : C → C is a Lipschitzian mapping, and x* ∈ C is a fixed point of T. For given x0 ∈ C, suppose that the sequence {xn} ⊂ C is the Mann iterative sequence defined by xn+1 = (1 − αn)xn + αnTxn, n ≥ 0, where {αn} is a sequence in [0, 1], , . We prove that the sequence {xn} strongly converges to x* if and only if there exists a strictly increasing function Φ : [0, ∞)→[0, ∞) with Φ(0) = 0 such that .

A Multiplayer Pursuit Differential Game on a Closed Convex Set with Integral Constraints

We study a simple motion pursuit differential game of many pursuers and many evaders on a nonempty convex subset of ℝn. In process of the game, all players must not leave the given set. Control functions of players are subjected to integral constraints. Pursuit is said to be completed if the position of each evader yj, j ∈ {1, 2, …k}, coincides with the position of a pursuer xi, i ∈ {1, …, m}, at some time tj, that is, xi(tj) = yj(tj). We show that if the total resource of the pursuers is greater than that of the evaders, then pursuit can be completed. Moreover, we construct strategies for the pursuers. According to these strategies, we define a finite number of time intervals [θi−1, θi] and on each interval only one of the pursuers pursues an evader, and other pursuers do not move. We derive inequalities for the resources of these pursuer and evader and, moreover, show that the total resource of the pursuers remains greater than that of the evaders.

Properties of the solutions of those equations for which the Krasnoselskii iteration converges

Let (X, +, R, →) be a vectorial L-space, Y ⊂ X a nonempty convex subset of X and f : Y → Y be an operator with Ff := {x ∈ Y | f(x) = x} 6= ∅. Let 0 < λ < 1 and let fλ be the Krasnoselskii operator corresponding to f, i.e., fλ(x) := (1 − λ)x + λf(x), x ∈ Y. We suppose that fλ is a weakly Picard operator (see I. A. Rus, Picard operators and applications, Sc. Math. Japonicae, 58 (2003), No. 1, 191-219). The aim of this paper is to study some properties of the fixed points of the operator f: Gronwall lemmas and comparison lemmas (when (X, +, R, →, ≤) is an ordered L-space) and data dependence (when X is a Banach space). Some applications are also given.

Strengthening of strong and approximate convexity∗

Let X be a real linear space and \(D\subseteq X\) be a nonempty convex subset. Given an error function E:[0,1]×(D−D)→ℝ∪{+∞} and an element \(t\in\left]0,1\right[\), a function f:D→ℝ is called (E,t)-convex if $$f(tx+(1-t)y)\le tf(x)+(1-t)f(y)+E(t,x-y)$$ for all x,y∈D. The main result of this paper states that, for all a,b∈(ℕ∪{0})+{0,t,1−t} such that {a,b,a+b}∩ℕ≠∅, every (E,t)-convex function is also \(\big(F,\frac{a}{a+b}\big)\)-convex, where $$F(s,u):=\frac{{(a+b)}^2s(1-s)}{t(1-t)}E\left(t,\frac{u}{a+b}\right),\qquad (u\in (D-D), \, s\in\left]0,1\right[).$$ As a consequence, under further assumptions on E, the strong and approximate convexity properties of (E,t)-convex functions can be strengthened.

Minimizing Irregular Convex Functions: Ulam Stability for Approximate Minima

The main concern of this article is to study Ulam stability of the set of e-approximate minima of a proper lower semicontinuous convex function bounded below on a real normed space X, when the objective function is subjected to small perturbations (in the sense of Attouch & Wets). More precisely, we characterize the class all proper lower semicontinuous convex functions bounded below such that the set-valued application which assigns to each function the set of its e-approximate minima is Hausdorff upper semi-continuous for the Attouch–Wets topology when the set \(\mathcal{C}(X)\) of all the closed and nonempty convex subsets of X is equipped with the Hausdorff topology. We prove that a proper lower semicontinuous convex function bounded below has Ulam-stable e-approximate minima if and only if the boundary of any of its sublevel sets is bounded.

Iterative algorithm for multi-valued pseudocontractive mappings in Banach spaces

Let D be nonempty open convex subset of a real Banach space E. Let T : D ¯ → K C ( E ) be a continuous pseudocontractive mapping satisfying the weakly inward condition and let u ∈ D ¯ be fixed. Then for each t ∈ ( 0 , 1 ) there exists y t ∈ D ¯ satisfying y t ∈ t T y t + ( 1 − t ) u . If, in addition, E is reflexive and has a uniformly Gâteaux differentiable norm, and is such that every closed convex bounded subset of D ¯ has fixed point property for nonexpansive self-mappings, then T has a fixed point if and only if { y t } remains bounded as t → 1 ; in this case, { y t } converges strongly to a fixed point of T as t → 1 − . Moreover, an explicit iteration process which converges strongly to a fixed point of T is constructed in the case that T is also Lipschitzian.

A Characterization of Convex Fuzzy Mappings

As any real-valued functions can be regarded as a fuzzy mapping, the research on fuzzy mappings extends the research on real-valued functions. In this paper, a characterization of convex fuzzy mappings is obtained. Supposeis a fuzzy mapping, whereis a non-empty convex subset ofandis the set of all fuzzy numbers. With respect to the fuzzy-max order, is convex if and only if it is both quasi-convex and intermediate-point convex.

On generalized implicit vector equilibrium problems in Banach spaces

Let X and Y be real Banach spaces, K be a nonempty convex subset of X , and C : K → 2 Y be a multifunction such that for each u ∈ K , C ( u ) is a proper, closed and convex cone with int C ( u ) ≠ 0̸ , where int C ( u ) denotes the interior of C ( u ) . Given the mappings T : K → 2 L ( X , Y ) , A : L ( X , Y ) → L ( X , Y ) , f 1 : L ( X , Y ) × K × K → Y , f 2 : K × K → Y , and g : K → K , we introduce and consider the generalized implicit vector equilibrium problem: Find u ∗ ∈ K such that for any v ∈ K , there is s ∗ ∈ T u ∗ satisfying f 1 ( A s ∗ , v , g ( u ∗ ) ) + f 2 ( v , g ( u ∗ ) ) ∉ − int C ( u ∗ ) . By using the KKM technique and the well-known Nadler’s result, we prove some existence theorems of solutions for this class of generalized implicit vector equilibrium problems. Our theorems extend and improve the corresponding results of several authors.

A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings

LetXbe a real uniformly convex Banach space andCa closed convex nonempty subset ofX. Let{Ti}i=1rbe a finite family of nonexpansive self-mappings ofC. For a givenx1∈C, let{xn}and{xn(i)},i=1,2,…,r, be sequences definedxn(0)=xn,xn(1)=an1(1)T1xn(0)+(1-an1(1))xn(0),xn(2)=an2(2)T2xn(1)+an1(2)T1xn+(1-an2(2)-an1(2))xn,…, xn+1=xn(r)=anr(r)Trxn(r-1)+an(r-1)(r)Tr-1xn(r-2)+⋯+an1(r)T1xn+(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))xn,n≥1, whereani(j)∈[0,1]for allj∈{1,2,…,r},n∈ℕandi=1,2,…,j. In this paper, weak and strong convergence theorems of the sequence{xn}to a common fixed point of a finite family of nonexpansive mappingsTi (i=1,2,…,r)are established under some certain control conditions.

A New Iteration Process for Approximation of Common Fixed Points for Finite Families of Total Asymptotically Nonexpansive Mappings

LetEbe a real Banach space, andKa closed convex nonempty subset ofE. LetT1,T2,…,Tm:K→Kbemtotal asymptotically nonexpansive mappings. A simple iterative sequence{xn}n≥1is constructed inEand necessary and sufficient conditions for this sequence to converge to a common fixed point of{Ti}i=1mare given. Furthermore, in the case thatEis a uniformly convex real Banach space, strong convergence of the sequence{xn}n=1∞to a common fixed point of the family{Ti}i=1mis proved. Our recursion formula is much simpler and much more applicable than those recently announced by several authors for the same problem.