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9 pages, 272 KiB

Open AccessArticle

Fixed-Points of Interpolative Ćirić-Reich–Rus-Type Contractions in b-Metric Spaces

by Pradip Debnath and Manuel de La Sen

Symmetry 2020, 12(1), 12; https://doi.org/10.3390/sym12010012 - 19 Dec 2019

Cited by 25 | Viewed by 2848

The concept of symmetry is inherent in the study of metric spaces due to the presence of the symmetric property of the metric. Significant results, such as with the Borsuk–Ulam theorem, make use of fixed-point arguments in their proofs to deal with certain symmetry principles. As such, the study of fixed-point results in metric spaces is highly correlated with the symmetry concept. In the current paper, we first define a new and modified Ćirić-Reich–Rus-type contraction in a b-metric space by incorporating the constant s in its definition and establish the corresponding fixed-point result. Next, we adopt an interpolative approach to establish some more fixed-point theorems. Existence of fixed points for

ω

-interpolative Ćirić-Reich–Rus-type contractions are investigated in this context. We also illustrate the validity of our results with some examples. Full article

16 pages, 302 KiB

Open AccessArticle

PPF-Dependent Fixed Point Results for Multi-Valued ϕ-F-Contractions in Banach Spaces and Applications

by Mohammed M. M. Jaradat, Babak Mohammadi, Vahid Parvaneh, Hassen Aydi and Zead Mustafa

Symmetry 2019, 11(11), 1375; https://doi.org/10.3390/sym11111375 - 6 Nov 2019

Cited by 2 | Viewed by 1957

Abstract

The solutions for many real life problems is obtained by interpreting the given problem mathematically in the form of

f (x) = x

. One of such examples is that of the famous Borsuk–Ulam theorem, in which using some fixed point [...] Read more.

The solutions for many real life problems is obtained by interpreting the given problem mathematically in the form of

f (x) = x

. One of such examples is that of the famous Borsuk–Ulam theorem, in which using some fixed point argument, it can be guaranteed that at any given time we can find two diametrically opposite places in a planet with same temperature. Thus, the correlation of symmetry is inherent in the study of fixed point theory. In this paper, we initiate

ϕ - F

-contractions and study the existence of PPF-dependent fixed points (fixed points for mappings having variant domains and ranges) for these related mappings in the Razumikhin class. Our theorems extend and improve the results of Hammad and De La Sen [Mathematics, 2019, 7, 52]. As applications of our PPF dependent fixed point results, we study the existence of solutions for delay differential equations (DDEs) which have numerous applications in population dynamics, bioscience problems and control engineering. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

12 pages, 780 KiB

Open AccessArticle

Common Fixed Point Results of Set Valued Maps for A_φ-Contraction and Generalized ϕ-Type Weak Contraction

by Murchana Neog, Mohammed M. M. Jaradat and Pradip Debnath

Symmetry 2019, 11(7), 894; https://doi.org/10.3390/sym11070894 - 8 Jul 2019

Viewed by 2439

Abstract

The solutions for many real life problems may be obtained by interpreting the given problem mathematically in the form

f (x) = x

. One such example is that of the famous Borsuk–Ulam theorem in which, using some fixed point argument, [...] Read more.

The solutions for many real life problems may be obtained by interpreting the given problem mathematically in the form

f (x) = x

. One such example is that of the famous Borsuk–Ulam theorem in which, using some fixed point argument, it can be guaranteed that at any given time we can find two diametrically opposite places in a planet with same temperature. Thus, the correlation of symmetry is inherent in the study of fixed point theory. In this article, some new results concerning coincidence and a common fixed point for an

A_{φ}

-contraction and a generalized

ϕ

-type weak contraction are established. We prove our results for set valued maps without using continuity of the corresponding maps and completeness of the relevant space. Our results generalize and extend several existing results. Some new examples are given to demonstrate the generality and non-triviality of our results. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

14 pages, 1260 KiB

Open AccessReview

A Symmetric Approach Elucidates Multisensory Information Integration

by Arturo Tozzi and James F. Peters

Information 2017, 8(1), 4; https://doi.org/10.3390/info8010004 - 27 Dec 2016

Cited by 5 | Viewed by 9535

Abstract

Recent advances in neuronal multisensory integration suggest that the five senses do not exist in isolation of each other. Perception, cognition and action are integrated at very early levels of central processing, in a densely-coupled system equipped with multisensory interactions occurring at all temporal and spatial stages. In such a novel framework, a concept from the far-flung branch of topology, namely the Borsuk-Ulam theorem, comes into play. The theorem states that when two opposite points on a sphere are projected onto a circumference, they give rise to a single point containing their matching description. Here we show that the theorem applies also to multisensory integration: two environmental stimuli from different sensory modalities display similar features when mapped into cortical neurons. Topological tools not only shed new light on questions concerning the functional architecture of mind and the nature of mental states, but also provide an empirically assessable methodology. We argue that the Borsuk-Ulam theorem is a general principle underlying nervous multisensory integration, resulting in a framework that has the potential to be operationalized. Full article

(This article belongs to the Special Issue Symmetry and Information)

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14 pages, 1572 KiB

Open AccessReview

Symmetries, Information and Monster Groups before and after the Big Bang

by Arturo Tozzi and James F. Peters

Information 2016, 7(4), 73; https://doi.org/10.3390/info7040073 - 21 Dec 2016

Cited by 3 | Viewed by 8133

Abstract

The Monster group, the biggest of the sporadic groups, is equipped with the highest known number of dimensions and symmetries. Taking into account variants of the Borsuk–Ulam theorem and a novel topological approach cast in a physical fashion that has the potential to be operationalized, the universe can be conceived as a lower-dimensional manifold encompassed in the Monster group. Our universe might arise from spontaneous dimension decrease and symmetry breaking that occur inside the very structure of the Monster Module. We elucidate how the energetic loss caused by projection from higher to lower dimensions and by the Monster group’s non-abelian features is correlated with the present-day asymmetry in the thermodynamic arrow. By linking the Monster Module to its theoretical physical counterparts, it is then possible to calculate its enthalpy and Lie group trajectories. Our approach also reveals how a symmetry break might lead to a universe based on multi-dimensional string theories and CFT/AdS (anti-de Sitter/conformal field theory) correspondence. Full article

(This article belongs to the Special Issue Symmetry and Information)

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