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Recently Berinde and Păcurar [Approximating fixed points of enriched contractions in Banach spaces. J. Fixed Point Theory Appl. 22 (2020), no. 2., 1--10], first introduced the idea of enriched contraction mappings and proved the existence of a fixed point of an enriched contraction mapping using the well-known fact that any fixed point of the averagedmapping Tλ, where λ ∈ (0, 1], is also a fixed point of the initialmapping T. In this work, we introduce the idea of weak enriched contraction mappings, and a new generalization of an averaged mapping called double averaged mapping. The first attempt is to prove the existence and uniqueness of the fixed point of a double averaged mapping associated with a weak enriched contraction mapping. Based on this result on Banach spaces, we give some sufficient conditions for the equality of all fixed points of a double averaged mapping and the set of all fixed points of a weak enriched contraction mapping. Moreover, our results show that an appropriate Kirk's iterative algorithm can be used to approximate a fixed point of a weak enriched contraction mapping. An illustrative example for showing the efficiency of our results is given. [ABSTRACT FROM AUTHOR]

Chemical graph theory (CGT) is a field of mathematical science that applies classical graph theory to chemical structures and processes. Chemical graphs are the principal data format used in cheminformatics to illustrate chemical interactions. Several researchers have addressed boundary-value problems using star graphs. Star graphs were used since their method requires a central point linked to other vertices but not to itself. Our objective is to expand the mechanism by introducing the idea of an isobutane graph that has the chemical formula C 4 H 10 and CAS number 75-28-5. By using the appropriate fixed point theory findings, this paper investigates the existence of solutions to fractional boundary value problems of Caputo type on such graphs. Additionally, two examples are provided to strengthen our important conclusions. [ABSTRACT FROM AUTHOR]

The analysis of iterative differential equations is often related to the various applications of calculus, which support all mathematical sciences. These equations are critical when it comes to interpreting the infection models. Furthermore, the inclusion of self-mapping raises the complexity of determining the existence of solutions for the iterative differential equations. This paper considers a particular type of second-order iterative differential equations and uses the Banach fixed point theorem to find the existence and uniqueness of the proposed differential equation's solution. We discuss the Hyers-Ulam and Hyers-Ulam-Rassias type stability of a solution to the proposed iterative boundary-value problem and present three illustrative examples to support our main results. [ABSTRACT FROM AUTHOR]

Chemical graph theory is a branch of mathematics that combines graph theory and chemistry and discusses the effect of research on "serious" or "pure" mathematics. A range of new graph ideas can be identified in the current development of mathematical chemistry and chemical graph theory. These advances include chemical kinetics as well as biomacromolecules. On star graphs, a few researchers have studied fractional differential equations. They used star graphs because their approach requires a common point that has edges with other nodes, whereas there are no edges between them. Our aim is to broaden the approach by using the idea of an ethane graph, which is an organic chemical compound with the chemical formula C 2 H 6 and having more than one junction nodes. In this work, we analyze the existence of solutions on such graphs for the fractional boundary value problem in the sense of Caputo fractional derivative. Inspired by a graph describing ethane's chemical compound, we assume a graph with labeled vertices of 0 or 1 and describe fractional differential equations on each edge of this graph. We use the notable fixed point theorems to find the existence and uniqueness of a solution to the proposed fractional differential equation. An example is also presented to support our main results. [ABSTRACT FROM AUTHOR]