In this paper, we introduce a new type of metric spaces called controlled G-metric spaces, denoted, G_ζ , which are a generalisation of G-metric spaces. This is an extension of the work by Aghajani et al. in the article “Common fixed point of generalised weak contractive mappings in partially ordered G_b- metric spaces. Filomat 2014, 28, 1087–1101”. We do this by employing a control function ζ (x,y,z) to the right-hand side of the G_b- triangle inequality so that the new triangle inequality becomes G_ζ (x,y,z) ≤ζ (x,a,a)G_ζ (x,a,a) + ζ (a,y,z)G_ζ (a,y,z) for all a, x,y,z ∈ X . Examples of controlled G- metric spaces which are not G_b- metric spaces in the sense of Aghajani et al. are given to show that our extension of G- metric spaces is different. We give some convergence properties on this new space. In order to further illustrate the usefulness of this new structure, a generalised Banach contraction principle is introduced and new fixed point results are developed and proved. Some examples are presented to support main result proved therein. These results improve, unify and generalize already well-known results on G- metric spaces and G_b- metric spaces.

In metric type spaces, we established some new fixed point results for multivalued contractive mappings defined on a generalized distance. Consequently, our findings not only consolidate various existing metric fixed point results but also extend and generalize such results in the existing literature. We present an example to reinforce the results proved herein.

The higher dimensional Fokas equation is the integrable expansion of the Davey–Stewartson and Kadomtsev–Petviashvili equations. In wave theory, the Fokas model plays a crucial role in explaining the physical phenomena of waves both inside and outside of water. The (4+1)-dimensional fractional-order Fokas equation is the subject of this article. Two effective approaches are employed to obtain the solutions for the considered equation: the generalized auxiliary equation technique and the G′/(bG′+G+a) technique. Several novel soliton solutions are obtained, including periodic solitary waves, bright solitons, and dark solitons. Various parametric values are employed to produce these new soliton waves at certain fractional order levels α. Furthermore, the bilinear version of the equation helps to develop its two-wave, three-wave, and multi-wave, as well as lump and rogue wave solutions. The properties of the solutions to the underlying problem are most effectively analyzed through the use of graphical representations. These outcomes and techniques can be used to study various fractional-order problems that emerge in wave theory, such as those in physics, hydraulics, optical technology, quantum mechanics, and plasma particles.

This work rigorously proves the existence of attractors of a finite collection of generalized multivalued mappings which are generalized mapping that are defined in the setup of partial metric spaces. We hereby put forward Generalized Multivalued Iterated Function Systems (GMIFS) and we obtain corresponding results under different types of assumptions known as generalized contractive circumstances. We construct a few examples that can be used to illustrate the results obtained in this manuscript. Additionally, this work extends several outcomes documented in prior studies within the fields of Iterated Function Systems (IFS).

In this paper, fixed point results with respect to generalized rational contractive mappings in semi-metric spaces endowed with a directed graph are proved. Some examples are provided to illustrate the results. The obtained results extend, improve and generalize many results in the existing literature.

The aim of this paper is to study the sufficient conditions for the existence of fixed points of Perov type T-contractive mappings in the setup of complete cone b-metric space associated with generalized c-distance. Some examples are presented to support our main results and concepts defined herein. The results proved in the paper extend and generalize various well known results in the existing literature.

In this paper, we first prove the existence and uniqueness of the solution to a variable-order Caputo–Fabrizio fractional stochastic differential equation driven by a multiplicative white noise, which describes random phenomena with non-local effects and non-singular kernels. The Euler–Maruyama scheme is extended to develop the Euler–Maruyama method, and the strong convergence of the proposed method is demonstrated. The main difference between our work and the existing literature is the fact that our assumptions on the nonlinear external forces are those of one-sided Lipschitz conditions on both the drift and the nonlinear intensity of the noise as well as the proofs of the higher integrability of the solution and the approximating sequence. Finally, to validate the numerical approach, current results from the numerical implementation are presented to test the efficiency of the scheme used in order to substantiate the theoretical analysis.

In this study, we explore a mathematical model of human immunodeficiency virus type 1 (HIV-1) infection in CD4+ T-cells, considering fractional-order dynamics in the Caputo sense. Fractional models are of great significance due to their capacity to predict disease outbreaks while incorporating memory and non-local characteristics. To establish the existence and uniqueness of solutions for the fractional model, a fixed-point theorem and an iterative technique are employed. Furthermore, it is confirmed that the solutions of the model are both positive and bounded. The basic reproduction number, denoted as R0, is calculated using the next-generation matrix approach, and the equilibrium points of the model, including disease-free and endemic equilibrium points, are presented. A sensitivity analysis of R0 is performed by varying different parameters. To analyze the local stability of equilibrium points, the Routh-Hurwitz technique is employed. The global stability of equilibrium points is demonstrated using the Lyapunov function and LaSalle's principle, as well as through UlamHyers (UH) stability and the generalized UH stability conditions. The proposed numerical scheme Adams-Bashforth predictor-corrector is validated by comparing it with the fourth-order Runge-Kutta (RK4) technique. We examine the influence of the fractional order (ξ) by conducting numerical simulations for different values of ξ using the Adams-Bashforth predictor-corrector approach. Furthermore, graphical presentations illustrate how different crucial model parameters impact disease dynamics. The results obtained from this study demonstrate that the adopted strategies significantly improve prediction accuracy.

In this paper, we study the generalized F-iterated function system in G-metric space. Several results of common attractors of generalized iterated function systems obtained by using generalized F-Hutchinson operators are also established. We prove that the triplet of F-Hutchinson operators defined for a finite number of general contractive mappings on a complete G-metric space is itself a generalized F-contraction mapping on a space of compact sets. We also present several examples in 2-D and 3-D for our results.

The aim of this paper is to study the sufficient conditions for the existence of attractor of a generalized cyclic iterated function system composed of a complete metric space and a finite collection of generalized cyclic F-contraction mappings. Some examples are presented to support our main results and concepts defined herein. The results proved in the paper extend and generalize various well known results in the existing literature.

There has been a worldwide epidemic of heroin that has affected people, families, societies, and cultures across the world. Now, the heroin epidemic has transitioned from heroin abuse to the overuse of synthetic narcotics, which are widely accessible and inexpensively produced. In this work, a novel mathematical approach is applied to investigate the dynamics of the heroin epidemic model and its harmful effect on society with different population data. A heroin model has been constructed with the importance of a non-singular kernel in the sense of a generalized Mittag-Leffler kernel. The well-posedness of the proposed model is proven via fixed-point theory. To examine the heroin model, two equilibrium states have been determined. These equilibrium states are proven to be locally and globally asymptotically stable. To analyze the behavior of heroin, a basic reproduction number and sensitivity analysis are used to determine the impact of different parameters mathematically as well as through simulations. To find the approximate solution, we implement the Toufik–Atangana numerical method at different fractional order values. The sensitivity of the heroin model is carried out, and 3-D graphs show the significance of the parameter involved in the model. Finally, the numerical outcomes are presented with different values of fractional parameters.

We investigate the existence of attractors for generalized cyclic iterated function systems generated in dislocated metric spaces. This is achieved by employing a finite family of generalized cyclic F-contraction mappings. We provide non-trivial examples that meet the conditions of generalized cyclic .F- Hutchinson operators, demonstrating that the sequence of compact iterates converge to the attractors. The results presented in this paper extend and generalize various well-known results related to fixed points, iterated function systems, and attractor-based problems in the existing literature.

The aim of this paper is to introduce to a pair of fuzzy graphic rational F-contraction multivalued mappings and to study the necessary condition for the existence of common fixed points of fuzzy multivalued mappings in the setup of generalized parametric metric space endowed with a directed graph. A non-trivial example is presented to support the results presented herein. Our results improve and extend some recent results in the existing literature.

We investigate the existence of fixed point problems on a partial metric space. The results obtained are for set contractions in the domain of sets and the pattern for the partial metric space is constructed on a directed graph. Essentially, our main strategy is to employ generalized $ \phi $-contractions in order to prove our results, where the fixed points are investigated with a graph structure. Moreover, we state and prove the well-posedness of fixed point based problems of the generalized $ \phi $-contractive operator in the framework of a partial metric space. We illustrate the main results in this manuscript by providing several examples.

Using the setting of ordered metric spaces, we obtain common end point of two multivalued mappings satisfying a generalized $(\psi,\varphi)$-weak contractive condition. Under comparative condition on the set of end points of multivalued mappings, our results assure the uniqueness of the end point. These results generalize and improve several recent results on single-valued as well as multivalued mappings.

Several generalised contractive type conditions are established for existence, uniqueness and well-posedness of the fixed point results, limit shadowing property, and also forNazir, Talat Silvestrov, Sergei the property of coincidence of sets of periodic points and fixed points for cyclic contractive maps on multiplicative metric spaces.

Using the setting of $G$-metric spaces, common fixed point theorems for four maps satisfying the weakly commuting conditions are obtained for various generalized contractive conditions. Several examples are also presented to show the validity of main results.

We generate a fractal using a finite collection of generalized cyclic contraction mappings, belonging to a particular category of mappings defined on a partial metric space. As a consequence, different results are attained for iterated function system that satisfy a different set of generalized cyclic contraction conditions. To substantiate the proven results, an example together with some applications are presented. With these results, we extend, unify and generalize some common results in contemporary literature.