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1
Content available remote The Knaster-Tarski theorem versus monotonne nonexpansive mappings
EN
Let X be a partially ordered set with the property that each family of order intervals of the form [a, b], [a,→) with the finite intersection property has a nonempty intersection. We show that every directed subset of X has a supremum. Then we apply the above result to prove that if X is a topological space with a partial order ⪯ for which the order intervals are compact, F is a nonempty commutative family of monotone maps from X into X and there exists c ∈ X such that c ⪯ Tc for every T ∈ F, then the set of common fixed points of F is nonempty and has a maximal element. The result, specialized to the case of Banach spaces, gives a general fixed point theorem for monotone mappings that drops many assumptions from several recent results in this area. An application to the theory of integral equations of Urysohn’s type is also given.
EN
In this paper we are concerned with the asymptotic behavior of random (unrestricted) infinite products of nonexpansive selfmappings of closed and convex subsets of a complete hyperbolic space. In contrast with our previous work in this direction, we no longer assume that these subsets are bounded. We first establish two theorems regarding the stability of the random weak ergodic property and then prove a related generic result. These results also extend our recent investigations regarding nonrandom infinite products.
EN
In this paper we apply the de la Vallee Poussin sum to a combinatorial Chebyshev sum by Ziad S. Ali in [1]. One outcome of this consideration is the main lemma proving the following combinatorial identity: with Re(z) standing for the real part of z we have (wzór). Our main lemma will indicate in its proof that the hypergeometric factors 2F1(1, 1/2 + n; 1 + n; 4); and 2F1(1, 1/2 + 2n; 1 + 2n; 4) are complex, each having a real and imaginary part. As we apply the de la Vallee Poussin sum to the combinatorial Chebyshev sum generated in the Key lemma by Ziad S. Ali in [1], we see in the proof of the main lemma the extreme importance of the use of the main properties of the gamma function. This represents a second important consideration. A third new outcome are two interesting identities of the hypergeometric type with their new Meijer G function analogues. A fourth outcome is that by the use of the Cauchy integral formula for the derivatives we are able to give a dierent meaning to the sum: (wzór). A fifth outcome is that by the use of the Gauss-Kummer formula we are able to make better sense of the expressions (wzór) by making use of the series denition of the hypergeometric function. As we continue we notice a new close relation of the Key lemma, and the de la Vallee Poussin means. With this close relation we were able to talk about P the de la Vallee Poussin summability of the two innite series (wzór). Furthermore the application of the de la Vallee Poussin sum to the Key lemma has created two new expansions representing the following functions: (wzór).
EN
A very interesting approach in the theory of fixed point is some general structures was recently given by Jachymski by using the context of metric spaces endowed with a graph. The purpose of this article is to present some new fixed point results for G-nonexpansive mappings defined on an ultrametric space and non-Archimedean normed space which are endowed with a graph. In particular, we investigate the relationship between weak connectivity graph and the existence of fixed point for these mappings.
EN
Let C be a convex compact subset of a uniformly convex Banach space. Let {Tt}t≥0 be a strongly-continuous nonexpansive semigroup on C. Consider the iterative process defined by the sequence of equations xk+1=ckTtk+1(xk+1)+(1−ck)xk. We prove that, under certain conditions on {ck} and {tk}, the sequence {xk}∞n=1 converges strongly to a common fixed point of the semigroup {Tt}t≥0. There are known results on convergence of such iterative processes for nonexpansive semigroups in Hilbert spaces and Banach spaces with the Opial property, and also weak convergence results in Banach spaces that are simultaneously uniformly convex and uniformly smooth. In this paper, we do not assume the Opial property or uniform smoothness of the norm.
6
Content available remote On nonlinear differential equations in generalized Musielak-Orlicz spaces
EN
We consider ordinary differential equations u′(t)+(I−T)u(t)=0, where an unknown function takes its values in a given modular function space being a generalization of Musielak-Orlicz spaces, and T is nonlinear mapping which is nonexpansive in the modular sense. We demonstrate that under certain natural assumptions the Cauchy problem related to this equation can be solved. We also show a process for the construction of such a solution. This result is then linked to the recent results of the fixed point theory in modular function spaces.
EN
This paper is concerned with weak uniformly normal structure and the structure of the set of fixed points of Lipschitzian mappings. It is shown that in a Banach space X with weak uniformly normal structure, every asymptotically regular Lipschitzian semigroup of self-mappings defined on a weakly compact convex subset of X satisfies the (ω)-fixed point property. We show that if X has a uniformly Gâteaux differentiable norm, then the set of fixed points of every asymptotically nonexpansive mapping is nonempty and sunny nonexpansive retract of C. Our results improve several known fixed point theorems for the class of Lipschitzian mappings in a general Banach space.
EN
Let C be a ρ-bounded, ρ-closed, convex subset of a modular function space L_ρ. We investigate the existence of common fixed points for semigroups of nonlinear mappings Tt : C→ C, i.e. a family such that To(x) = x, Ts+t = Ts(Tt(x)), where each T_t is either ρ-contraction or -nonexpansive. We also briefly discuss existence of such semigroups and touch upon applications to differential equations.
9
Content available remote Strict pseud-contraction strong convergence theorems for strict pseud-contractions
EN
In this paper, we prove two strong convergence theorems for strict pseudocontractions in Hilbert spaces by hybrid methods. Our results extend and improve the recent ones announced by Nakajo and Takahashi [K. Nakajo, W. Takahashi,. Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003), 372-379], Marino and Xu [G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), 336-346], Martinez-Yanes and Xu [C. Martinez-Yanes, H.K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anai. 64 (2006), 2400-2411] and some others.
10
Content available remote Common fixed point results with applications in convex metric space
EN
Sufficient conditions for the existence of a common fixed point for uniformly Cq— commuting mappings satisfying a generalized contractive conditions in the framework of a convex metric space are obtained. As an application, related results on best approximation are derived. Our results generalize various known results in the literature.
11
EN
Implicit and explicit processes for eonstructing the unique sunny nonexpansive retraction onto the common fixed point set of either a finite or infmite family of nonexpansive mappings in a Banach space are proposed and corresponding convergence theorems are established.
EN
The purpose of this paper is to study the weak and strong convergence of an new implicit iteration process to a common fixed point for a finite family of asymptotically nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve important known results in [1], [2], [4]-[9], [11]-[15] and others.
13
Content available remote A basic fixed point theorem
EN
The paper contains a fixed point theorem for stable mappings in metric discus spaces (Theorem 10). A consequence is Theorem 11 which is a far-reaching extension of the fundamental result of Browder, Göhde and Kirk for non-expansive mappings.
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