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Iterative approaches have been established to be fundamental for the creation of fractals. This paper introduces an approach to visualize Julia and Mandelbrot sets for a complex function of the form Q (z) = z p + log c t for all z ∈ ℂ , where p ∈ N ∖ { 1 } , t ∈ [ 1 , ∞) , c ∈ ℂ ∖ { 0 } , using a four-step iteration scheme extended with s -convexity. The study introduces an escape criteria for generating Julia and Mandelbrot sets using a four-step iterative method. It investigates how changes in the iteration parameters influence the shape and color of the resulting Julia and Mandelbrot sets. This approach can generate a wide range of captivating fractals and analyze them through numerical experiments. [ABSTRACT FROM AUTHOR]

This paper aims to suggest a time-fractional S P E P I P A I P S P H P R P model of the COVID-19 pandemic disease in the sense of the Atangana–Baleanu–Caputo operator. The proposed model consists of six compartments: susceptible, exposed, infected (asymptomatic and symptomatic), hospitalized and recovered population. We prove the existence and uniqueness of solutions to the proposed model via fixed point theory. Furthermore, a stability analysis in the context of Ulam–Hyers and the generalized Ulam–Hyers criterion is also discussed. For the approximate solution of the suggested model, we use a well-known and efficient numerical technique, namely the Toufik–Atangana numerical scheme, which validates the importance of arbitrary order derivative ϑ and our obtained theoretical results. Finally, a concise analysis of the simulation is proposed to explain the spread of the infection in society. [ABSTRACT FROM AUTHOR]

This research paper introduces a novel approach to visualize Julia and Mandelbrot sets by employing iterative techniques, which play a crucial role in creating fractals. The primary focus is on complex functions of the form F (z) = z p − q z 2 + r z + sin c w for all z ∈ ℂ , where p ∈ N ∖ { 1 } , q ∈ ℂ , r , c ∈ ℂ ∖ { 0 } and w ∈ [ 1 , ∞). The Mann and Picard–Mann iteration schemes with s -convexity are utilized throughout the study. Innovative escape criteria are developed to generate Julia and Mandelbrot sets using these iterative methods. These criteria serve as guidelines for determining when the iterative process should terminate, leading to the creation of captivating fractal patterns. The research investigates the impact of parameter variations within the iteration schemes on the resulting fractal's shape, size, and color. By manipulating these parameters, a wide range of captivating fractal patterns can be generated and visualized, encompassing various aesthetic possibilities. Additionally, we discuss the numerical examples related to Julia and Mandelbrot sets generated through the proposed iteration. We also delve into discussions concerning execution time and the average number of iterations. [ABSTRACT FROM AUTHOR]

This research presents an advancement in fixed point theory by extending the renowned Banach contractive condition and introducing the new convex contractive condition. Central to this work is exploring and developing new convex contraction mappings characterized by rational expressions. Our investigation not only proves fixed point results for these contractions but also highlights their uniqueness compared to well-established contractions like Banach, Kannan, Chatterjea, Jaggi, and other known contraction mappings. A significant aspect of this research is the presentation of new rational-type convex contraction mappings, demonstrating their independence from traditional contractions. Through illustrative examples, we showcase the comprehensive generalization and inherent advantages of our novel results. Furthermore, the practical applications of our findings are discussed, particularly in the context of solving matrix and functional integral equations. This is achieved by applying our theoretical results to establish the existence and uniqueness of solutions for these equations, thereby underscoring our research’s practical significance and potential impact in the broader field of mathematical studies.

AbstractIn this paper, we present a concept of semigroups of enriched Kannan mappings, which is based on enriching contractive type mappings that have been effectively used in many recent studies. We investigate some weak and strong convergence theorems for the Mann iterative process to approximate common fixed points of enriched Kannan semigroups in uniformly convex Banach spaces. In the end, by using MATLAB, we present a numerical example to support our results.

In this article, we prove the generalized Hyers-Ulam-Rassias stability for the following composite functional equation: f(f(x)−f(y))=f(x+y)+f(x−y)−f(x)−f(y),f(f\left(x)-f(y))=f\left(x+y)+f\left(x-y)-f\left(x)-f(y),where ffmaps from a (β,p)\left(\beta ,p)-Banach space into itself, by using the fixed point method and the direct method. Also, the generalized Hyers-Ulam-Rassias stability for the composite ss-functional inequality is discussed via our results.

One of the main problems in K-means clustering is setting of initial centroids which can cause misclustering of patterns which affects clustering accuracy. Recently, a density and distance-based technique for determining initial centroids has claimed a faster convergence of clusters. Motivated from this key idea, the authors study the impact of initial centroids on clustering accuracy for unsupervised feature selection. Three metrics are used to rank the features of a data set. The centroids of the clusters in the data sets, to be applied in K-means clustering, are initialized randomly as well as by density and distance-based approaches. Extensive experiments are performed on 15 datasets. The main significance of the paper is that the K-means clustering yields higher accuracies in majority of these datasets using proposed density and distance-based approach. As an impact of the paper, with fewer features, a good clustering accuracy can be achieved which can be useful in data mining of data sets with thousands of features.

In this paper, we study the generalized Hyers–Ulam stability of Euler–Lagrange-type cubic functional equation of the form 2mf(x+my)+2f(mx-y)=(m3+m)[f(x+y)+f(x-y)]+2(m4-1)f(y)for all x,y?X, where mis a fixed scalar such that m?0,1, and fis a map from a quasi-normed space Xto a quasi-Banach space Yover the same field with Xby applying the alternative fixed point theorem.

In this article, we prove the generalized Hyers-Ulam stability for the following additive-quartic functional equation:f(x+3y)+f(x−3y)+f(x+2y)+f(x−2y)+22f(x)+24f(y)=13[f(x+y)+f(x−y)]+12f(2y),f(x+3y)+f(x-3y)+f(x+2y)+f(x-2y)+22f(x)+24f(y)=13{[}f(x+y)+f(x-y)]+12f(2y),where fmaps from an additive group to a complete non-Archimedean normed space.

In recent times, fixed point theory has undergone a great development, especially through the use of new auxiliary functions and also by weakening the axioms that characterize a metric space. In this work, in the setting of b-metric spaces, the notions of a modified R-function and an Rs-contraction are introduced and some new fixed point results for such contractions involving modified R-functions are established. Also, we give some example for supporting our results. Finally, we prove some application by using the main results.

Let (X, ⊥) be an orthogonality module in the sense of Rätz over a unital Banach algebra Aand Ybe a real Banach module over A. In this paper, we apply the alternative fixed point theorem for proving the Hyers-Ulam stability of the orthogonally generalized k-quadratic functional equation of the form

We present some useful techniques to revise and extend the main result on Hyers–Ulam stability of homomorphisms in p-Banach algebras [9] to quasi-Banach algebras. The techniques may be used to revise and extend other results on the stability of homomorphisms in p-Banach algebras in the literature to quasi-Banach algebras.

In this work, we present the viscosity method for the implicit double midpoint rule for finding the fixed points of nonexpansive mappings in real Hilbert spaces. The strong convergence theorem of this method is proved under some specific assumption imposed on the control parameters. Also, we present some numerical results which are presented to illustrate the proposed method and convergence results. Moreover, we provide the applications to the variational inequality problems, constrained convex minimization problems, nonlinear evolution equations, and Fredholm integral equations.

The purpose of this paper is to improve the notion of a $$\mathcal {Z}$$ Z -contraction mapping with respect to a b-simulation function and also prove fixed point results on b-metric spaces endowed with only a transitive relation. Our results can reduce to several important results in the literature. As applications, we extend and generalize Ran and Reurings’s results to prove nonlinear matrix equations by giving new notions concerning b-simulation functions. Finally, we give some numerical examples to support some results of our applications.

In this work, we improve the Ri’s fixed point theorem in Ri (Indag Math 27:85–93, 2016) by creating the $$\varphi $$ φ -fixed point version of such result in metric spaces. These results generalize and extend several well-known comparable results in the literature. Several applications and some examples of our theorems are also given.

In this paper, we introduce a new family of mappings from $$\mathbb {R}_{+}^{4}$$ R+4 to $$\mathbb {R}_{+}$$ R+ and it depends on fixed function $$\uppsi :\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$$ ψ:R+→R+ , where $$\mathbb {R}_{+}=[0,\infty )$$ R+=[0,∞) . We call it a $$\Phi _\uppsi $$ Φψ family. Then we define a new contractive type condition for multi-valued mappings involving a function $$\upphi \in \Phi _\uppsi $$ ϕ∈Φψ . We also establish some fixed point theorems using our new contractive condition. Our results generalize many existing fixed point theorems.

Let $$\mathbb {K}$$ K , $$\mathbb {F}$$ F be two fields of real or complex numbers and Xbe a normed space over field $$\mathbb {F}$$ F . In this paper, we consider the following generalized logarithmic functional equation: $$\begin{aligned} f(x^a y^b)= Af(x) + Bf(y), \end{aligned}$$ f(xayb)=Af(x)+Bf(y), where fmaps from $$\mathbb {K}$$ K into X, $$a,b\in \mathbb {Z}{\setminus }\{0\}$$ a,b∈Z\{0} and $$A, B\in \mathbb {F}$$ A,B∈F . By using the fixed point method, the generalized Hyers–Ulam stability results for such functional equation are proved.

The aim of this manuscript is to discuss the existence and uniqueness of common solution for the following system of Urysohn integral equations: 0.1$$\begin{aligned} z(t)=\phi _{i}(t)+\int _{a}^{b}K_{i}(t,s,z(s)) ds, \end{aligned}$$ z(t)=ϕi(t)+∫abKi(t,s,z(s))ds, where $$i=1,2,3,4$$ i=1,2,3,4 , $$a,b\in \mathbb {R}$$ a,b∈R with $$a\le b$$ a≤b , $$t\in [a,b]$$ t∈[a,b] , $$z, \phi _{i} \in C([a,b],\mathbb {R}^n)$$ z,ϕi∈C([a,b],Rn) and $$K_{i}:[a,b]\times [a,b]\times \mathbb {R}^n\rightarrow \mathbb {R}^n$$ Ki:[a,b]×[a,b]×Rn→Rn is a given mapping for each $$i=1,2,3,4$$ i=1,2,3,4 . For this intention we establish common fixed point results for two pairs of weakly compatible mappings satisfying the contractive condition of rational type in the frame work of complex valued metric spaces.

In this work, we extend the concept of an almost generalized weakly contractive mapping in rectangular b-metric spaces. The establish of fixed point of such mappings is also provided. We give an example to support our results.

The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. In this paper we investigate some fixed-point theorems in modular metric spaces.

The purpose of this work is to introduce new nonlinear mappings in setup of b-metric spaces and prove fixed point theorems for such mappings. Examples are provided in order to distinguish these results from the known ones. At the end of paper, we apply our fixed point result to prove the existence of a solution for the following nonlinear integral equation: 0.1$$\begin{aligned} x(c) = \Omega (\phi (c),c ) + K(c,c,\phi (c))+\int ^b_a K(c,r,x(r))dr, \end{aligned}$$ x(c)=Ω(ϕ(c),c)+K(c,c,ϕ(c))+∫abK(c,r,x(r))dr, where $$a,b\in \mathbb {R}$$ a,b∈R with $$a

In this paper, we introduce the notion of $$\alpha $$ α -integral type G-contraction mappings to generalize the notions of Banach G-contraction and integral G-contraction mappings. We also prove some fixed point theorems for $$\alpha $$ α -integral type G-contraction mappings. By providing some example, we show that our results are real generalization of several results in literature.

In this work, we introduce the new concept of $$\alpha $$ α -multiplicative proximal contraction mappings in the setting of multiplicative metric spaces and we also solve optimal approximate solution for such mappings. Our main results improve and extend several results in the literature. We also state some illustrative examples to claim that our results properly generalizes some results in literature.

We prove a fixed point result in orbitally $$0$$ 0 -complete partial metric spaces, using a variant of almost contractive condition that involves a rational expression. Also, a common fixed point result in $$0$$ 0 -complete partial metric spaces is obtained. Several consequences are deduced and examples are presented, showing that the given result can be used for proving the existence of (common) fixed points when some known results fail.

In this paper, we prove some fixed point theorems for weakly compatible mappings in fuzzy metric spaces employing common limit range property with implicit relations. We also furnish some illustrative examples to support our main results. As an application to our main result, we derive a fixed point theorem for four finite families of self-mappings which can be utilized to derive common fixed point theorems involving any finite number of mappings. Our results improve and extend a host of previously known results including the ones contained in the paper of Gopal et al. (2011).

In this paper, we prove a new common fixed point theorem for R-weakly commuting mappings of types $$(A_{g})$$, R-weakly commuting mappings of type $$(A_{f})$$and R-weakly commuting mappings of types (P) in modified intuitionistic fuzzy metric spaces using the common limit in the range property. An example is furnished which demonstrates the validity of the hypotheses and degree of generality of our main result.

The purpose of this paper is to establish some coupled coincidence point theorems for a pair of mappings having a mixed g-monotone property in partially ordered G-metric spaces. Also, we present a result on the existence and uniqueness of coupled common fixed points. The results presented in the paper generalize and extend several well-known results in the literature. To illustrate our results, we give some examples.

In this paper, we prove some coupled coincidence point theorems for such nonlinear contraction mappings having a mixed monotone property in partially ordered metric spaces by dropping the condition of commutative. We also prove coupled common fixed point theorem for w-compatible mappings. An example of a nonlinear contraction mapping which is not applied by Lakshmikantham and Ćirić’s theorem [1]but applied by our result is given. Further, we apply our results to the existence theorem for solution of nonlinear integral equations.

In this paper, we are interested to study El-Fassi’s question on hyperstability of the general radical quintic functional equation in quasi-β-Banach spaces in [6]. First, we give some counterexamples to show that the hyperstability of a given functional equation does not occur. Then, by using the (β,p)-normalization theorem, we prove the stability for the general radical quintic functional equation in quasi-β-Banach spaces.

The main aim of this work is to introduce the new concept of λ−Υ,χ-contraction self-mappings and prove the existence of χ-fixed points for such mappings in metric spaces. Our results generalize and improve some results in existing literature. Moreover, some fixed point results in partial metric spaces can be derived from our χ-fixed points results. Finally, the existence of solutions of nonlinear integral equations is investigated via the theoretical results in this work.

We prove some common fixed point results for two mappings satisfying generalized contractive condition in b-metric space. Note that b-metric of main results in this work are not necessarily continuous. So our results extend and improve several previous works. We also present one example that shows the applicability and usefulness of our results.