In this manuscript, fixed point results without continuity via triangular notion on fuzzy bipolar metric spaces are established. The paper includes tangible examples which display the motivation for such investigations as those presented here. We solve an integral equation in this setting. The present work is a generalization of some published works.

Using Atangana-Baleanu ($ AB $) fractional integral operators, we first establish some fractional Hermite-Hadamard-Mercer inclusions for interval-valued functions in this study. In this, some fresh developments of the Hermite-Hadamard inequality for fractional integral operators are presented. A few instances are also given to support our conclusions. The resulting results could provide new insight into a variety of integral inequalities for fuzzy interval-valued functions, fractional interval-valued functions, and the optimization issues they raise. Finally, matrices-related applications are also shown.

The aims of this paper are (a) to introduce the concept of the 0-complete m-metric spaces, (b) to obtain the results for $m^w$-Caristi mapping using Kirk’s approach, (c) to investigate the problem of non-cooperative equilibrium (abbreviated as NCE) in two- and three-person games in the structure of game theory and find the solution by employing coupled and tripled fixed-point results within the framework of 0-complete m-metric spaces (m-metric spaces, respectively), and (d) to establish some coupled fixed-point results which extend the scope of metric fixed point theory. We provide some examples to support the concepts and results presented in this paper. As an application of our results in this paper, we obtain the existence of a solution for a nonlinear integral equation.

This paper performed an investigation into the s-embedding of the Lie superalgebra $\overrightarrow{(}{S}^{1\mid 1})$, a representation of smooth vector fields on a (1,1)-dimensional super-circle. Our primary objective was to establish a precise definition of the s-embedding, effectively dissecting the Lie superalgebra into the superalgebra of super-pseudodifferential operators ( $S\psi D\odot$) residing on the super-circle $S^{1|1}$. We also introduce and rigorously define the central charge within the framework of $\overrightarrow{(}{S}^{1\mid 1})$, leveraging the canonical central extension of $S\psi D\odot$. Moreover, we expanded the scope of our inquiry to encompass the domain of fuzzy Lie algebras, seeking to elucidate potential connections and parallels between these ostensibly distinct mathematical constructs. Our exploration spanned various facets, including non-commutative structures, representation theory, central extensions, and central charges, as we aimed to bridge the gap between Lie superalgebras and fuzzy Lie algebras. To summarize, this paper is a pioneering work with two pivotal contributions. Initially, a meticulous definition of the s-embedding of the Lie superalgebra $\overrightarrow{(}{S}^{1|1})$ is provided, emphasizing the representationof smooth vector fields on the (1,1)-dimensional super-circle, thereby enriching a fundamental comprehension of the topic. Moreover, an investigation of the realm of fuzzy Lie algebras was undertaken, probing associations with conventional Lie superalgebras. Capitalizing on these discoveries, we expound upon the nexus between central extensions and provide a novel deformed representation of the central charge.

We establish the existence and uniqueness of solutions to a class of second-order nonlinear boundary value problems involving impulses, delay, and possible singularities. The approach leverages the recent notion of paired-Chatterjea-type contractions. Under a smallness condition ensuring the associated integral operator is a Banach contraction with constant $\mu <{\displaystyle \frac{1}{3}}$, we show that it is also a Chatterjea, and hence, a paired-Chatterjea contraction. By the fixed point theorem of Chand, this guarantees at most two fixed points; a supplementary uniqueness argument then ensures a unique solution in the Banach space $P{C}^1\left([a,b]\right)$.

In this present work, we prove a fixed point result for generalized contraction mappings in the setting of a double control metric space (DCMS) by using α-orbital admissibility. The uniqueness of the fixed point is established by adding further hypotheses. The presented result is supported by a concrete example. Moreover, we ensure the existence of a solution of a Fredholm type integral equation via a fixed point technique. Some consequences are also presented to make effective the obtained results.