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The principal results in this article deal with the existence of fixed points of a new class of generalized F-contraction. In our approach, by visualizing some non-trivial examples we will obtain better geometrical interpretation. Our main results substantially improve the theory of F-contraction mappings and the related fixed point theorems. In section-4, application to graph theory is entrusted and proved results are endorsed by an example through graph. The presented new techniques give the possibility to justify the existence problems of the solutions for some class of integral equations. For the future aspects of our study, an open problem is suggested regarding discretized population balance model, whose solution may be derived from the established techniques.

In this article, we develop a new notion that combines fixed-point theory and graph theory: graphical bipolar b-metric spaces. We demonstrate fixed-point solutions in the framework of graphical bipolar b-metric spaces, employing covariant and contravariant mapping contractions, which is a new addition to this end. This article also features illustrative examples drawn from various contexts to further demonstrate our findings. This is a significant study since it melds ideas from graph theory with those from generalized bipolar metric spaces, and considers that the symmetry of the edges of the underlying graphs connected with the enunciated metric spaces is essential in the graphical metric spaces.

Based on the impressive feature of convex orbital β-Lipschitz mappings, we are interested in presenting two ways to expand this idea to more generalizations. First, convex orbital β-Lipschitz mappings are extended to quadratic weak orbital β-Lipschitz mappings. Moreover, the new type of decreasing mappings in inner product spaces is presented, and fixed point results for quadratic weak orbital β-Lipschitz mappings in Hilbert spaces are proved with the help of the proposed decreasingness. In a second way, some open question on convex orbital β-Lipschitz mappings is answered in Banach spaces.

We discuss the relation of graphic contraction and perimetric contractions: mappings contracting perimeters of triangles, generalized Kannan-type mappings, generalized Chatterjea-type mappings, and generalized ĆRR-type mappings. Generalized Kannan-type and ĆRR-type mappings are graphic contractions, while mappings contracting perimeters of triangles and generalized Chatterjea-type mappings mappings are graphic contractions under properly introduced remetrization that preserves completeness.

In the present article, first, we introduce the notation of graphical rectangular b-metric spaces and scrutinize some basic and topological properties of the underlying spaces. Some examples endowed with appropriate graphs are propounded to validate the established results, thereby giving a better insight into the corresponding concepts and investigations. Our results extend and improve several results in the literature. Second, by means of the obtained results, an application to the vibrations of a vertical heavy hanging cable is entrusted to manifest the viability of the obtained results. Finally, some open problems are also stated for the future scope of the study.

In the commenced work, we establish some novel results concerning graph contractions in a more generalized setting. Furthermore, we deliver some examples to elaborate and explain the usability of the attained results. By virtue of nontrivial examples, we show our results improve, extend, generalize, and unify several noteworthy results in the existing state-of-art. We adopt computer simulation validating our results. To arouse further interest in the subject and to show its efficacy, we devote this work to recent applications which emphasize primarily the applications for the existence of the solution of various models related to engineering problems viz. fourth-order two-point boundary value problems describing deformations of an elastic beam, ascending motion of a rocket, and a class of integral equations. This approach is entirely new and will open up some new directions in the underlying graph structure.

Helmholtz’s problem helps us to completely understand how sound behaves in a cylinder that is closed from one of its ends and opened at another. This paper aims to employ some novel convergence results to the Helmholtz problem with mixed boundary conditions and demonstrate the existence and uniqueness of the solution by applying graph-controlled contractions. For this purpose, we enunciate graphically Reich type and graphically Ćirić type contractions in the realm of graphical-controlled metric type spaces. In our study, we showcase the existence and uniqueness of fixed point results by employing these graphical contractions. This is demonstrated through extensive examples that a graphicalcontrolled metric-type space is distinct from traditional controlled metric-type spaces. We also exhibit an example of a graphically Reich contraction that is not a classical Reich contraction. Similarly, a decent example of graphical Ćirić contraction is presented, which is distinct from the classical Ćirić contraction. Concrete illustrative examples solidify our theoretical framework.

This study focuses on establishing the existence of fixed points in graphical ôv(s) spaces using generalized contractions based on Reich and Edelstein-type conditions. The proposed framework extends classical contraction principles, allowing broader applications. Examples are provided to illustrate the results and enhance their clarity. The study further explores the solution of a fourth-order differential equation modeling the deformation of a cantilever beam under a uniformly distributed load, offering insights into its mechanical behavior.

Our aim is to prove some new fixed point theorems for a hybrid pair of multivalued$ \\alpha _{\\ast } $ -dominated mappings involving a generalized$ Q $ -contraction in a complete modular-like metric space. Further results involving graphic contractions for a pair of multi-graph dominated mappings have been considered. Applying our obtained results, we resolve a system of nonlinear integral equations.

In this paper we consider ( s − q ) -graphic contraction mapping in b-metric like spaces. By using our new approach for the proof that a Picard sequence is Cauchy in the context of b-metric-like space, our results generalize, improve and complement several approaches in the existing literature. Moreover, some examples are presented here to illustrate the usability of the obtained theoretical results.