A mathematical proof for the existence of a possible source for dark energy

Abstract

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].

The first common fixed point theorem for commutative set-valued mappings

“…Next, we generalize Theorem 2.2 in [7] for an arbitrary upper semicontinuous set-valued mapping in one dimesional Euclidean spaces.…”

“…Then, (∆, ⊆), where ⊆ is inclusion, is a partially ordered set. Also, by Lemma 2.1 in [7], every descending chain in ∆ has a lower bound in ∆. Therefore, by Zorn's lemma, ∆ has a minimal element, say U 0 .…”

“…) by commutativity of T 1 and T 2 and definition of composition for set-valued mappings. Since T 1 (x) is a nonempty convex compact subset of X, by Theorem 2.2 in [7], T 1 has a fixed point in T 2 (x). That is , there exists y ∈ T 2 (x) such that y ∈ T 1 (y).…”

“…In 1941, Kakutani [9] showed that if C is a nonempty convex compact subset of an n-dimentional Euclidean space R n and T from C into 2 C is an upper semicontinuous mapping such that T (x) is a nonempty convex closed subset of C for all x ∈ C; then, T possesses a fixed point in C. In 1951, Glicksberg [5] and in 1952, Fan [4], independently, generalized Kakutani's fixed point theorem [5] from Euclidean spaces to locally convex vector spaces. In [7], we showed that for a continuously expanding compact and convex subset of a locally convex vector space, under an upper semicontinuous set-valued convex mapping, there exists at least one point that remains fixed under the expansion. In this work we generalize this result to an arbitrary upper semicontinuous set-valued mapping in one dimensional Euclidean space R.…”