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This manuscript is devoted to constructing a novel iterative scheme and reckoning of fixed points for generalized contraction mappings in hyperbolic spaces. Also, we establish Δ\Delta and strong convergence results by the considered iteration under the class of mappings satisfying condition (E). Moreover, some qualitative results of the suggested iteration, like weak w2{w}^{2}-stability and data dependence results, are discussed. Furthermore, to test the efficiency and effectiveness of the proposed iteration, practical experiments are given. To support the theoretical results, illustrative examples are presented. Finally, our results improve and generalize several classical results in the literature of fixed point iterations.

Using the method of asymptotic integration of the equations of the theory of elasticity, the axisym- metric problem of the theory of elasticity for a radially inhomogeneous transversally isotropic cylinder of small thickness is studied. Suppose that the elastic moduli are arbitrary continuous functions of the radius of the cylinder. It is assumed that the side part of the cylinder is fixed, and stresses are set at the ends of the cylinder, leaving the cylinder in equilibrium. Solutions have been determined having the nature of a boundary layer and are localized at the ends of the cylinder. The first terms of its asymptotic expansion coincide with the Saint-Venant edge effect in the theory of plates. The nature of the stress-strain state has been studied. It is shown that some boundary layer solutions may not be extinguished by propagating far from the ends. [ABSTRACT FROM AUTHOR]

This study concerns a numerical analysis of the meshing temperature for a multicomponent nylon gear by incorporating typical viscoelastic responses of polymers. Firstly, the generalized Maxwell model parameters were calibrated by the dynamic mechanical analysis to capture the viscoelasticity of polyamide 66. Next, the finite element model predicting the meshing temperature was established, including the heat generation analysis model, which calculates the generation of the frictional and hysteretic heat, and the heat propagation analysis model, which simulates the heat transmission. Different material models were used for polyamide 66 to investigate the influences of temperature dependence, strain rate dependence, and hysteretic effect. The effect of the iteration sequence on the meshing temperature calculation was also analyzed, i.e., how the heat generation should be updated as the loading cycles increase. A reasonable iteration scheme was determined by balancing the computational precision and consumption. It is noteworthy that the evolution of the ambient temperature around the gear and the worm within the housing was also considered. Finally, the accuracy of such meshing temperature analysis was validated on a self-developed gear test rig, in terms of the peak temperature evolution on the gear side surface. The experimental results and the simulation with the ambient temperature correction match well and the maximum error is 5 °C. Graphical Abstract This is a visual representation of the abstract. [ABSTRACT FROM AUTHOR]

We present several classes of nonlinear difference equations solvable in closed form, which can be obtained from some known iteration processes, and for some of them we give some generalizations by presenting methods for constructing them. We also conduct several analyses and give many comments related to the difference equations and iteration processes.

In this paper, we establish a new iterative process for approximation of fixed points for contraction mappings in closed, convex metric space. We conclude that our iterative method is more accurate and has very fast results from previous remarkable iteration methods like Picard-S, Thakur new, Vatan Two-step and K-iterative process for contraction. Stability of our iteration method and data dependent results for contraction mappings are exact, correspondingly on testing our iterative method is advanced. Finally, we prove enquiring results for some weak and strong convergence theorems of a sequence which is generated from a new iterative method, Suzuki generalized non-expansive mappings with condition (C) in uniform convexity of metric space. Our results are addition, enlargement over and above generalization for some well-known conclusions with literature for theory of fixed point.

In this paper, we introduce a new iterative process for approximation fixed points. We show the stability of our proposed iteration process. We prove some weak and strong convergence theorems for generalized α -nonexpansive mappings in the framework of uniformly convex Banach spaces. We also provide an example of a generalized α -nonexpansive mapping to show numerically that our iteration is faster than various prominent iteration processes. [ABSTRACT FROM AUTHOR]

The aim of this article is to design a new iteration process for solving certain fixed-point problems. In particular, we prove weak and strong convergence theorems for generalized nonexpansive mappings in the framework of uniformly convex Banach spaces. In addition, we discuss the stability of the solution under mild conditions. Further, we provide some numerical examples to indicate that the proposed method works properly.

The motive of this article is to study a modified iteration scheme for monotone nonexpansive mappings in the class of uniformly convex Banach space and establish some convergence results. We obtain weak and strong convergence results. In addition, we present a nontrivial numerical example to show the convergence of our iteration scheme. To demonstrate the utility of our scheme, we discuss the solution of nonlinear integral equations as an application, which is again supported by a nontrivial example.

A wide range of new research articles in artificial intelligence, logic programming, and other applied sciences are based on fixed-point theorems. The aim of this article is to present an approximation method for finding the fixed point of generalized Suzuki nonexpansive mappings on hyperbolic spaces. Strong and Δ -convergence theorems are proved using the Noor iterative process for generalized Suzuki nonexpansive mappings (GSNM) on uniform convex hyperbolic spaces. Due to the richness of uniform convex hyperbolic spaces, the results of this paper can be used as an extension and generalization of many famous results in Banach spaces together with C A T (0) spaces. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new iterative process for approximation of fixed points for contraction mappings in closed, convex metric space. We conclude that our iterative method is more accurate and has very fast results from previous remarkable iteration methods like Picard-S, Thakur new, Vatan Two-step and K-iterative process for contraction. Stability of our iteration method and data dependent results for contraction mappings are exact, correspondingly on testing our iterative method is advanced. Finally, we prove enquiring results for some weak and strong convergence theorems of a sequence which is generated from a new iterative method, Suzuki generalized non-expansive mappings with condition (C) in uniform convexity of metric space. Our results are addition, enlargement over and above generalization for some well-known conclusions with literature for theory of fixed point. [ABSTRACT FROM AUTHOR]

In this paper, we study multi-valued generalized ffnonexpansive mappings in uniformly convex Banach spaces. We introduce a new multi-valued iterative process and prove some weak and strong convergence results in uniformly convex Banach space. We also study the stability of this iteration process. Further, we provide a numerical example of the multi-valued generalized α-nonexpansive mapping. Finally, the convergence of this iteration process to the fixed point for multi-valued generalized α-nonexpansive mapping is discussed on this numerical example. [ABSTRACT FROM AUTHOR]

We present several classes of nonlinear difference equations solvable in closed form, which can be obtained from some known iteration processes, and for some of them we give some generalizations by presenting methods for constructing them. We also conduct several analyses and give many comments related to the difference equations and iteration processes. [ABSTRACT FROM AUTHOR]

The aim of this article is to design a new iteration process for solving certain fixed-point problems. In particular, we prove weak and strong convergence theorems for generalized nonexpansive mappings in the framework of uniformly convex Banach spaces. In addition, we discuss the stability of the solution under mild conditions. Further, we provide some numerical examples to indicate that the proposed method works properly. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iteration scheme, named as the S**-iteration scheme, for approximation of fixed points of Suzuki's generalized non-expansive mappings. We also put forward some weak and strong convergence theorems for Suzuki's generalized nonexpansive mappings in the setting of uniformly convex Banach spaces. We prove the stability of our instigated scheme and give numerical examples to show that this scheme is faster than Agarwal, Abbas, Thakur and Ullah iteration schemes. Our results comprehend, improve and consolidate many results in the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new iteration process which is faster than the leading S [J. Nonlinear Convex Anal. 8, no. 1 (2007), 61-79], Thakur et al. [App. Math. Comp. 275 (2016), 147-155] and M [Filomat 32, no. 1 (2018), 187-196] iterations for numerical reckoning fixed points. Using new iteration process, some fixed point convergence results for generalized α-nonexpansive mappings in the setting of uniformly convex Banach spaces are proved. At the end of paper, we offer a numerical example to compare the rate of convergence of the proposed iteration process with the leading iteration processes.

The motive of this paper is to propose a new iterative algorithm for evaluating a solution of constrained minimization problem, variational inequality and split feasibility problem and common fixed point of two generalized α -nonexpansive mappings. We obtain few convergence results in the setting of uniformly convex Banach space. We also present some numerical examples for supporting our main results and to demonstrate the convergence behaviour of the obtained process. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish weak and strong convergence theorems of new three-step iterations for Iasymptotically nonexpansive mappings in Banach space.Also we introduce and study convergence theorems of the three-step iterative sequence for three I-asymptotically nonexpansive mappings in an uniformly convex Banach space. The results obtained in this paper extend and improve the recent ones announced by Chen and Guo [1], S. Temir [14], Yao and Noor[16] and many others. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold: (a) to revisit the results in Iqbal et al. (Numer Algorithms 1–21, 2019. https:doi.org/10.1007/s11075-019-00854-z) and prove some convergence and stability results for the subclass of multivalued contractive operators under some mild conditions (b) to introduce a multivalued Reich-Suzuki type α -nonexpansive mappings and present some fixed point results for this class of mappings. We considered a more natural notion of stability called weak w 2 -stability instead of simple stability in Iqbal et al. (Numer Algorithms 1–21, 2019. https:doi.org/10.1007/s11075-019-00854-z). Some illustrative examples to support the results obtained herein are also presented. [ABSTRACT FROM AUTHOR]

Retinal vessel segmentation is part of the morphological extraction of retinal blood vessels that plays an essential role in medical image processing. Manual segmentation is possible to do, but it is time-consuming and requires special operators. Moreover, the possibility of variability between operators is vast. This study aims to answer the shortcomings of the manual segmentation process by automatically segmenting retinal blood vessels. The main contribution of this study is the use of a simple method to iteratively segment retinal blood vessels. All processes in the segmentation are simulated using Matlab. The algorithm was evaluated by comparing the results of the automatic segmentation with 20 manually segmented images from the STARE dataset. The result show specificity 98.13%, accuracy 93.60%, sensitivity 56.42%, precision 80.48%, and the dice coefficient 64.06%. In conclusion, the automatic retinal blood vessel image segmentation process worked well.

A wide range of new research articles in artificial intelligence, logic programming, and other applied sciences are based on fixed-point theorems. The aim of this article is to present an approximation method for finding the fixed point of generalized Suzuki nonexpansive mappings on hyperbolic spaces. Strong and Δ-convergence theorems are proved using the Noor iterative process for generalized Suzuki nonexpansive mappings (GSNM) on uniform convex hyperbolic spaces. Due to the richness of uniform convex hyperbolic spaces, the results of this paper can be used as an extension and generalization of many famous results in Banach spaces together with CAT(0) spaces.

In this paper, we propose a new iteration process which is faster than the leading; S [J. Nonlinear Convex Anal. 8, no. 1 (2007), 61{79], Thakur et al. [App. Math. Comp. 275 (2016), 147{155] and M [Filomat 32, no. 1 (2018), 187{196] iterations for numerical reckoning fixed points. Using this new iteration process, some fixed point convergence results for generalized ff-nonexpansive mappings in the setting of uniformly convex Banach spaces are proved. At the end of paper, we offer a numerical example to compare the rate of convergence of the proposed iteration process with the leading iteration processes. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new iterative algorithm and show that the newly introduced iterative algorithm converges faster than a number of existing iterative algorithms. We present a numerical example followed by graphs to validate our claim. We prove strong and weak convergence results for approximating fixed points of Suzuki generalized nonexpansive mappings. Again we reconfirm our results by example and table. Further, we utilize our proposed algorithm to solve delay differential equation.

For a family (A x) x ∈ (0 , 1) of integral quasi-arithmetic means satisfying certain measurability-type assumptions we search for an integral mean K such that K ( (A x (P)) x ∈ (0 , 1) ) = K (P) for every compactly supported probability Borel measure P . Also some results concerning the uniqueness of invariant means will be given. [ABSTRACT FROM AUTHOR]

In this paper, we study the convergence of the newly defined SP*-iteration to fixed point for the generalized a-nonexpansive mappings in CAT(0) spaces. Our results improve and extend some recently results in the literature of fixed point theory in CAT(0) spaces. [ABSTRACT FROM AUTHOR]

In this paper, we use the AP iteration method in the context of CAT(0) space to approximate invariant points for contraction maps. We also prove that our iteration process for contraction map is faster than the leading Picard-S iteration process to demonstrate the proposed algorithm's convergence behaviour. To support the analytic proofs, numerical examples are given. Furthermore, we demonstarte that the AP iteration method is T-stable. [ABSTRACT FROM AUTHOR]

In this paper, we study Picard-S iteration scheme for monotone nonexpansive mappings in the setting of CAT(0) spaces. We prove Δ and strong convergence results. Further, we provide a non trivial numerical example to illustrate the convergence of our iteration scheme and its stability with respect to the different parameters and initial values. [ABSTRACT FROM AUTHOR]

We consider a wider class of nonexpansive type mappings and present some fixed point results for this class of mappingss in hyperbolic spaces. Indeed, first we obtain some existence results for this class of mappings. Next, we present some convergence results for an iteration algorithm for the same class of mappings. Some illustrative non-trivial examples have also been discussed.

We introduce multivalued generalized α-nonexpansive mappings and present a fixed point result. The multivalued version of the iteration process (Piri et al., Numerical Algorithms, 1–20, 2018) is proposed and some weak and strong convergence results in uniformly convex Banach space are established. Further, we also study the stability of the modified iteration process. Finally, we compare the rate of convergence of suggested multivalued version of iteration process with several known iteration processes through a numerical example. [ABSTRACT FROM AUTHOR]

An initial boundary value problem is considered for the dynamic beam system wtt=cd−a+b∫01wx2dxwxx−cdψx,ψtt=cψxx−c2d(ψ−wx). Its solution is found by means of an algorithm, the constituent parts of which are the finite element method, the implicit symmetric difference scheme used to approximate the solution with respect to the spatial and time variables, and also a Picard type iteration process for solving the system of nonlinear equations obtained by discretization. Errors of three parts of the algorithm are estimated and, as a result, its total error estimate is obtained. A numerical example is solved. [ABSTRACT FROM AUTHOR]

Diabetic retinopathy is one of the most frequent complications of diabetes, which leads to severe consequences, including rapid and irreversible vision loss. The laser coagulation procedure to treat diabetic retinopathy consists in applying a series of microburns to the fundus to deal with macular edema. The existing hardware/software packages are primarily based on the use of a predetermined pattern for coagulate arrangement. However, due to the complex form of edema and vascular system, this approach leads to an uneven arrangement. To solve the problem, we propose a new approach based on the application of sphere packing algorithms (circle packing in two-dimensional images) in the specified area of interest. Since one of the main requirements for the laser coagulation procedure is that it should have the minimum duration, a problem of the computational complexity of the developed algorithms arises. This requirement is completely ignored by the existing approaches, therefore the development of new high-performance coagulate arrangement algorithms is highly relevant. In the paper, we propose seven new algorithms for coagulate arrangement and provide a detailed analysis of key characteristics of the algorithms. The characteristics considered have made it possible to extract information relating to the algorithm properties. Regularity is determined by the median, asymmetry and the kurtosis; determinism is determined by the variance and the mean.

In this paper, we present a new modified iteration process in the setting of uniformly convex Banach space. The newly obtained iteration process can be used to approximate a common fixed point of three nonexpansive mappings. We have obtained strong and weak convergence results for three nonexpansive mappings. Additionally, we have provided an example to support the theoretical proof. In the process, several relevant results are improved and generalized.

In this paper, we are interested in solving minimization problem and common fixed point problem of finite family consisting asymptotically nonexpansive mappings in Hadamard spaces. For finding common solutions of their problems, we introduce a new modified proximal point algorithm involving convex combination technique. Under suitable conditions, Δ-convergence and strong convergence theorems are proved. Furthermore, we provide some numerical examples which show the efficiency and implementation of the proposed algorithm as well as show computational overview by comparing it with previously known algorithms. [ABSTRACT FROM AUTHOR]

The numerical approach for computer simulation of femtosecond laser pulse interaction with a semiconductor is considered under the formation of 3D contrast time‐dependent spatiotemporal structures. The problem is governed by the set of nonlinear partial differential equations describing a semiconductor characteristic evolution and a laser pulse propagation. One of the equations is a Poisson equation concerning electric field potential with Neumann boundary conditions that requires fulfillment of the well‐known condition for Neumann problem solvability. The Poisson equation right part depends on free‐charged particle concentrations that are governed by nonlinear equations. Therefore, the charge conservation law plays a key role for a finite‐difference scheme construction as well as for solvability of the Neumann difference problem. In this connection, the iteration methods for the Poisson equation solution become preferable than using direct methods like the fast Fourier transform. We demonstrate the following: if the finite‐difference scheme does not possess the conservatism property, then the problem solvability could be broken, and the numerical solution does not correspond to the differential problem solution. It should be stressed that for providing the computation in a long‐time interval, it is crucial to use a numerical method that possessing asymptotic stability property. In this regard, we develop an effective numerical approach—the three‐stage iteration process. It has the same economic computing expenses as a widely used split‐step method, but, in contrast to the split‐step method, our method possesses conservatism and asymptotic stability properties. Computer simulation results are presented. [ABSTRACT FROM AUTHOR]

Aesthetic patterns are widely used nowadays, e.g., in jewellery design, carpet design, as textures and patterns on wallpapers, etc. Most of the work during the design stage is carried out by a designer manually. Therefore, it is highly useful to develop methods for aesthetic pattern generation. In this paper, we present methods for generating aesthetic patterns using the dynamics of a discrete dynamical system. The presented methods are based on the use of various iteration processes from fixed point theory (Mann, S, Noor, etc.) and the application of an affine combination of these iterations. Moreover, we propose new convergence tests that enrich the obtained patterns. The proposed methods generate patterns in a procedural way and can be easily implemented on the GPU. The presented examples show that using the proposed methods we are able to obtain a variety of interesting patterns. Moreover, the numerical examples show that the use of the GPU implementation with shaders allows the generation of patterns in real time and the speed-up (compared with a CPU implementation) ranges from about 1000 to 2500 times.

This paper is devoted to the algorithm of the linear Cauchy problem solution for large systems of first-order ordinary differential equations using parallel calculations. The proof of the convergence of the iteration process using the solution as expansion over orthogonal polynomials for the interval [0,1] is presented. The features of this algorithm are its simplicity, the opportunity to get a solution by parallel calculations, and also the possibility to get a solution for nonlinear problems by changing the operator using the solution from the iteration process. [ABSTRACT FROM AUTHOR]

The objective of this article is to establish a new modified iteration process for nonexpansive mappings in complete CAT(κ) spaces. We prove strong and Δ‐convergence theorems of the proposed method in such spaces under some standard conditions. Furthermore, numerical experiments of non‐trivial examples are also provided to show performance and comparison speed of convergence with many previously known methods. Our main result extended and improved the corresponding recent results announced by many researchers. [ABSTRACT FROM AUTHOR]

The object of this paper is to present a new iteration process. We will show that our process is faster than the known recent iterative schemes. We discuss stability results of our iteration and prove some results in the context of uniformly convex Banach space for Suzuki generalized nonexpansive mappings. We also present a numerical example for proving the rate of convergence of our results. Our results improves many known results of the existing literature. [ABSTRACT FROM AUTHOR]

We consider a wider class of nonexpansive type mappings and present some fixed point results for this class of mappings in hyperbolic spaces. Indeed, first we obtain some existence results for this class of mappings. Next, we present some convergence results for an iteration algorithm for the same class of mappings. Some illustrative non-trivial examples have also been discussed. [ABSTRACT FROM AUTHOR]

The appropriate iteration process model is the basis for managing and optimizing the product development process. In this article, we attempt to introduce the concept of process effectiveness and process value. The relationship between rework probability and process effectiveness is discussed. The evolution function of process effectiveness is proposed to drive the overlapped iteration process of multi-coupled activities. The evolution process with input information update is studied, and a simulation model is presented to obtain the accurate iteration process of development. It is useful to analyze the risks during development, and has good flexibility and versatility. The calculation method of process value for overlapped iteration process is given, and an optimization model for product development process is provided. The model is used to improve the development process of the stamping die of a car roof. With the model, we can get a suitable overlapping rate of multi-coupled activities to improve development performance. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some properties related to double sequences. More- over, we prove some P -convergence theorems of some Halpern-type double-sequence iterations with error terms in a CAT(0) space and present an example to support these results. Our results extend and improve some recent results announced in the current literature. [ABSTRACT FROM AUTHOR]

The first goal of this paper is to propose a composite implicit iteration process for a finite family of I-nonexpansive mappings in hyperbolic spaces. Next, some strong and Δ-convergence theorems are established using the proposed iteration process. New results are obtained as corollaries to the convergence theorems. Finally, we exhibit two finite families of the mappings under consideration. [ABSTRACT FROM AUTHOR]

In this paper we propose a new iteration process, called the K * iteration process, for approximation of fixed points. We show that our iteration process is faster than the existing well-known iteration processes using numerical examples. Stability of the K * iteration process is also discussed. Finally we prove some weak and strong convergence theorems for Suzuki generalized nonexpansive mappings in the setting of uniformly convex Banach spaces. Our results are the extension, improvement and generalization of many well-known results in the literature of iterations in fixed point theory. [ABSTRACT FROM AUTHOR]