Hybrid Iterative Scheme Research Articles

A novel fixed point iteration process applied in solving the Caputo type fractional differential equations in Banach spaces

We introduce the modified Picard–Ishikawa hybrid iterative scheme and establish some strong convergence results for the class of asymptotically generalized [Formula: see text]-pseudocontractive mappings in the intermediate sense in Banach spaces and approximate the fixed point of this class of mappings via the newly introduced iteration scheme. We construct some numerical examples to support our results. Furthermore, we apply the Picard–Ishikawa hybrid iteration scheme in solving the nonlinear Caputo type fractional differential equations. Our results generalize, extend and unify several existing results in literature.

Convergence Analysis of Picard Thakur Hybrid Iterative Scheme for α-Nonexpansive Mappings in Uniformly Convex Banach Spaces

In this study, we investigate the convergence behavior of fixed points for generalized α-nonexpansive mappings using the Picard-Thakur hybrid iterative scheme. We obtain weak and strong convergence results for generalized α-nonexpansive mappings in a uniformly convex Banach space. Numerically, we demonstrate that the Picard-Thakur hybrid iterative scheme converges more rapidly than other well-known schemes. Additionally, we present findings on data dependence and provide a numerical example to illustrate the concept. The obtained results are expanded and generalized to be consistent with relevant findings in the existing literature.

Fast hybrid iterative schemes for solving variational inclusion problems

Tseng's forward‐backward‐forward splitting method for finding zeros of the sum of Lipschitz continuous monotone and maximal monotone operators is known to converge weakly in infinite dimensional Hilbert spaces. The inertial and viscosity approximation techniques are the techniques widely used to accelerate iterative algorithms and obtain strong convergence, respectively. In this paper, we propose two fast, strongly convergent modifications of Tseng's method and present some consequences and applications of our results. Moreover, we illustrate the performance and computability of our algorithms with relevant numerical examples. The two hybrid techniques incorporate the inertial and viscosity techniques. Unlike the conventional inertial‐viscosity hybrid techniques in the literature, our new algorithms compute both the inertial extrapolation and viscosity approximation simultaneously at the first step of each iteration.

A fixed point iterative scheme based on Green's function for numerical solutions of singular BVPs

<abstract><p>We suggest a novel iterative scheme for solutions of singular boundary value problems (SBVPs) that is obtained by embedding Green's function into the Picard-Mann Hybrid (PMH) iterative scheme. This new scheme we call PMH-Green's iterative scheme and prove its convergence towards a sought solution of certain SBVPs. We impose possible mild conditions on the operator or on the parameters involved in our scheme to obtain our main outcome. After this, we prove that this new iterative scheme is weak $ w^{2} $-stable. Eventually, using two different numerical examples of SBVPs, we show that our new approach suggests highly accurate numerical solutions as compared the corresponding Picard-Green's and Mann-Green's iterative schemes.</p></abstract>

Computational comparative analysis of fixed point approximations of generalized $ \alpha $-nonexpansive mappings in hyperbolic spaces

<abstract><p>In this article, we use the Picard-Thakur hybrid iterative scheme to approximate the fixed points of generalized $ \alpha $-nonexpansive mappings. For generalized $ \alpha $-nonexpansive mappings in hyperbolic spaces, we show several weak and strong convergence results. It is proved numerically and graphically that the Picard-Thakur hybrid iterative scheme converges more faster than other well-known hybrid iterative methods for generalized $ \alpha $-nonexpansive mappings. We also present an application to Fredholm integral equation.</p></abstract>

Corrigendum to “Strong Convergence of a New Hybrid Iterative Scheme for Nonexpensive Mappings and Applications”

Corrigendum to “Strong Convergence of a New Hybrid Iterative Scheme for Nonexpensive Mappings and Applications”

Strong Convergence of a New Hybrid Iterative Scheme for Nonexpensive Mappings and Applications

In the article, we have proposed a new type of hybrid iterative scheme which is a hybrid of Picard and Thakur et al. repetitive schemes. This new hybrid iterative scheme converges faster than all leading schemes like Picard-S∗ hybrid, Picard-S, Picard-Ishikawa hybrid, Picard-Mann hybrid, Thakur et al. and Abbas and Nazir, S-iterative, Ishikawa and Mann iterative schemes for contraction mapping. By using the Picard-Thakur hybrid iterative scheme, we can find the solution of delay differential equations and also prove some convergence results for nonexpansive mapping in a uniformly convex Banach space.

Hybrid Iterative Scheme for Variational Inequality Problem Involving Pseudo-monotone Operator with Application in Signal Recovery

Hybrid Iterative Scheme for Variational Inequality Problem Involving Pseudo-monotone Operator with Application in Signal Recovery

Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces

Abstract We get the strong and Δ \Delta -convergence of the Picard-Krasnoselskii hybrid iteration scheme to a fixed point of a self-map endowed with the condition ( B γ , μ ) \left({B}_{\gamma ,\mu }) . We use the nonlinear context of CAT(0) spaces for establishing these results. We present a new example of a self-map endowed with ( B γ , μ ) \left({B}_{\gamma ,\mu }) condition and prove that its Picard-Krasnoselskii hybrid iterative process is more effective than the Picard and Krasnoselskii hybrid iterative processes. This improves and extends some recently announced results of the current literature.

New Convergence Theorems for Maximal Monotone Operators in Banach Spaces

The purpose of this paper is to introduce a new hybrid iterative scheme for resolvents of maximal monotone operators in Banach spaces by using the notion of generalized fprojection. Next, we apply this result to the convex minimization and variational inequality problems in Banach spaces. The results presented in this paper improve and extend important recent results in the literature.

FIXED POINT THEOREMS IN COMPLEX VALUED CONVEX METRIC SPACES

Our purpose in this paper is to introduce the concept of complex valued convex metric spaces and introduce an analogue of the Picard-Ishikawa hybrid iterative scheme, recently proposed by Okeke [24] in this new setting. We approximate (common) fixed points of certain contractive conditions through these two new concepts and obtain several corollaries. We prove that the Picard-Ishikawa hybrid iterative scheme [24] converges faster than all of Mann, Ishikawa and Noor [23] iterative schemes in complex valued convex metric spaces. Also, we give some numerical examples to validate our results.

Modified matrix-free methods for solving system of nonlinear equations

Some hybrid derivative-free methods for nonlinear system of equations via acceleration and correction parameters are presented. These methods are based on applying the three term hybrid iterative scheme on an enhance matrix free method. The step length was determined using an inexact line search procedure. Moreover, the proposed methods proved to be globally convergent under mild conditions. The comprehensive numerical results presented in this work, show the efficiency of the proposed methods by comparing them with some existing methods in the previous literature.

Hybrid iterative scheme for solving split equilibrium and hierarchical fixed point problems

We suggest and analyze a hybrid projected subgradient–proximal iterative scheme to approximate a common solution of a split equilibrium problem for pseudomonotone and monotone bifunctions and a hierarchical fixed point problem for nonexpansive and quasi-nonexpansive mappings. We prove that sequences generated by the proposed scheme converge weakly to a common solution of these problems. Further, we discuss some consequences of the main result and a numerical example for the proposed scheme.

Polarimetric SAR Speckle Reduction by Hybrid Iterative Filtering

Speckle filtering in synthetic aperture radar polarimetry (PolSAR) is essential for the extraction of significant information. In this study, the authors introduced a hybrid iterative filtering scheme. The proposed iterative filter is initialized by a polarimetric filter ensuring a high speckle reduction level. Then, to enhance the spatial details, the iterative filter is applied for few iterations. The key parameter b k which traduced the variability of the pixels has been studied. An expression that takes into account the following important numerical considerations is proposed. 1) The variability of the pixel is measured by coefficient variation (CV) rather than the variance. 2) The statistics are computed using non-local neighborhood instead of local square one. 3) A new normalizing function (i. e. hyperbolic tangent) is implemented. 4) The dynamic of range of b k is increased by multiplying the CVs of the filtered and the originals images. Comparison with various state of the art PolSAR filters demonstrated the effectiveness of the proposed iterative method. Simulated, one-look and multilook real PolSAR data were used for validation.

Approximation of fixed points of a multivalued ρ-quasi-nonexpansive mapping for newly defined hybrid iterative scheme

The purpose of this research paper is to define a hybrid iterative scheme named Picard Noor-type hybrid iterative scheme to approximate the fixed points of a multivalued ρ-quasinonexpansive mappings for newly defined iterative scheme in modular function spaces.

Geometrically nonlinear rapid surface heating of temperature-dependent FGM arches

Based on the nonlinear dynamic analysis, thermally induced vibrations of the FGM shallow arches subjected to different sudden thermal loads are studied. Temperature and position dependence of the material properties are taken into account. Based on the uncoupled thermoelasticity assumptions, The non-linear one-dimensional transient heat conduction equation is solved numerically by a hybrid iterative GDQ method and Crank-Nicolson time marching scheme. A first order shear deformation arch theory (FSDT) is also combined with the von Kármán type of geometrical non-linearity and the Donnell kinematic assumption to obtain the equations of motion employing the Hamilton principle. Discretization of the highly coupled non-linear equations of motion is done by using the GDQ method in the arch domain. The solution of the system of the ordinary differential equations is established by means of a hybrid iterative Picard-Newmark scheme. Comparison is also made with the existing results for the case of isotropic homogeneous shallow arches, where good agreement is obtained. Also, parametric studies are proposed to show the effects of temperature dependency, geometrical non-linearity, arch thickness, power law index, and the type of thermal-mechanical boundary conditions upon the arch deflection.

Locating common fixed points of nonlinear representations of semigroups

This paper is concerned with the problem of finding common fixed points for a family of Bregman relatively weak nonexpansive mappings. The motivation is due to our finding of some gaps in a paper of K. S. Kim (Nonlinear Analysis, 73 (2010), 3413-3419), where the author was developing a hybrid iterative scheme for locating common fixed points of a nonlinear representation of a left reversible semigroup. After a brief discussion about the gaps and why they are fatal, we present a new approach by using Bergman type nonexpansive mappings. A correct version of Kim’s convergence theorem is given as a consequence of our new results, which also improve and extend some recent results in the literature.

A hybrid parametrization approach for a class of nonlinear optimal control problems

In this paper, a suitable hybrid iterative scheme for solving a class of non-linear optimal control problems (NOCPs) is proposed. The technique is based upon homotopy analysis and parametrization methods. Actually an appropriate parametrization of control is applied and state variables are computed using homotopy analysis method (HAM). Then performance index is transformed by replacing new control and state variables. The results obtained from the given method are compared with the results which are obtained using the spectral homotopy analysis method (SHAM), homotopy perturbation method (HPM), optimal homotopy perturbation method (OHPM), modified variational iteration method (MVIM) and differential transformations. The existence and uniqueness of the solution are presented. The comparison and ability of the given approach is illustrated via two examples.

Simple form of a projection set in hybrid iterative schemes for non-linear mappings, application of inequalities and computational experiments

Some relaxed hybrid iterative schemes for approximating a common element of the sets of zeros of infinite maximal monotone operators and the sets of fixed points of infinite weakly relatively non-expansive mappings in a real Banach space are presented. Under mild assumptions, some strong convergence theorems are proved. Compared to recent work, two new projection sets are constructed, which avoids calculating infinite projection sets for each iterative step. Some inequalities are employed sufficiently to show the convergence of the iterative sequences. A specific example is listed to test the effectiveness of the new iterative schemes, and computational experiments are conducted. From the example, we can see that although we have infinite choices to choose the iterative sequences from an interval, different choice corresponds to different rate of convergence.

Existence results for trifunction equilibrium problems and fixed point problems

In this paper, we establish the existence and uniqueness solutions of trifunction equilibrium problems using the generalized relaxed $$\alpha $$ -monotonicity in Banach spaces. By using the generalized f-projection operator, a hybrid iteration scheme is presented to find a common element of the solutions of a system of trifunction equilibrium problems and the set of fixed points of an infinite family of quasi- $$\phi $$ -nonexpansive mappings. Moreover, the strong convergence of our new proposed iterative method under generalized relaxed $$\alpha $$ -monotonicity is considered.