The expansion of the universe has been accepted by scientists for more than a century. However, since the 1990s, observations have suggested that the universe is expanding at an accelerating rate [5,6]. Although the source of this acceleration is still unknown, cosmologists call it dark energy; a type of mysterious energy which exerts a negative pressure on the universe. In this paper, applying different approach from those of classical mathematical physics based mainly on functional analysis, we propose an answer to solve this mystery. In fact, we first establish a theorem for multivalued mappings; and as a result we apply this theorem to show the existence of a remnant field of high energy in a continuously expanding system of energy. The results may also be applied to the early universe to show the existence of a possible source for dark energy. Our answer is also confirmed partially by a recent paper published in Nature Astronomy [7].
In this paper, we investigate the common fixed point property for commutative nonexpansive mappings on τ-compact convex sets in normed and Banach spaces, where τ is a Hausdorff topological vector space topology that is weaker than the norm topology. As a consequence of our main results, we obtain that the set of common fixed points of any commutative family of nonexpansive self-mappings of a nonempty clm-compact (resp. weak* compact) convex subset C of L 1 (µ) with a σ-finite µ (resp. the James space J 0 ) is a nonempty nonexpansive retract of C.
Abstract. The Markov-Kakutani fixed point theorem has been considered as one of the most remarkable theorems due to considerable diversity in its applications in the history of functional analysis. Different approaches have been investigated to prove this theorem; however, the condition of compactness of the underlying set is essentially used. In this paper, we develop a new method, based on Zermelo's well-ordering theorem, to weaken the compactness condition.
This paper presents a framework of iterative methods for finding specific common fixed points of a nonexpansive self-mappings semigroup in a Banach space. We prove, with appropriate conditions, the strong convergence to the solution of some variational inequalities.
We establish the first common fixed point theorem for commutative set-valued convex mappings. This may help to generalize common fixed point theorems in single-valued setting to those in set-valued. We also prove the existence of a fixed point in a continuously expanding sets under a none convex upper semicontinuous set-vaued mapping; as a result we answer positively to a question of Lau and Yao.
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