scholarly journals A ROBUST ITERATIVE APPROACH FOR SOLVING NONLINEAR VOLTERRA DELAY INTEGRO–DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 7 (2) ◽  
pp. 59
Author(s):  
Austine Efut Ofem ◽  
Unwana Effiong Udofia ◽  
Donatus Ikechi Igbokwe
Keyword(s):  
Differential Equations ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Stability Result ◽  
Iterative Approach ◽  
Weak And Strong Convergence ◽  
Mild Conditions ◽  
Convergence Results ◽  
The Stability

This paper presents a new iterative algorithm for approximating the fixed points of multivalued generalized \(\alpha\)–nonexpansive mappings. We study the stability result of our new iterative algorithm for a larger concept of stability known as weak \(w^2\)–stability. Weak and strong convergence results of the proposed iterative algorithm are also established. Furthermore, we show numerically that our new iterative algorithm outperforms several known iterative algorithms for multivalued generalized \(\alpha\)–nonexpansive mappings. Again, as an application, we use our proposed iterative algorithm to find the solution of nonlinear Volterra delay integro-differential equations. Finally, we provide an illustrative example to validate the mild conditions used in the result of the application part of this study. Our results improve, generalize and unify several results in the existing literature.

Some convergence results of M iterative process in Banach spaces

2019 ◽  
pp. 2150017 ◽  
Author(s):  
Kifayat Ullah ◽  
Faiza Ayaz ◽  
Junaid Ahmad
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Process ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Weak And Strong Convergence ◽  
Convergence Results

In this paper, we prove some weak and strong convergence results for generalized [Formula: see text]-nonexpansive mappings using [Formula: see text] iteration process in the framework of Banach spaces. This generalizes former results proved by Ullah and Arshad [Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process, Filomat 32(1) (2018) 187–196].

An Iterative algorithm for $\eta$-(anti)-Hermitian least-squares solutions of quaternion matrix equations

2015 ◽  
Vol 30 ◽  
Author(s):  
Fatemeh Beik ◽  
Salman Ahmadi-Asl
Keyword(s):  
Least Squares ◽  
Iterative Algorithm ◽  
Numerical Experiments ◽  
Iterative Algorithms ◽  
Iterative Approach ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Hermitian Matrices ◽  
Convergence Properties ◽  
Least Squares Problems

Recently, some research has been devoted to finding the explicit forms of the η-Hermitian and η-anti-Hermitian solutions of several kinds of quaternion matrix equations and their associated least-squares problems in the literature. Although exploiting iterative algorithms is superior than utilizing the explicit forms in application, hitherto, an iterative approach has not been offered for finding η-(anti)-Hermitian solutions of quaternion matrix equations. The current paper deals with applying an efficient iterative manner for determining η-Hermitian and η-anti-Hermitian least-squares solutions corresponding to the quaternion matrix equation AXB + CY D = E. More precisely, first, this paper establishes some properties of the η-Hermitian and η-anti-Hermitian matrices. These properties allow for the demonstration of how the well-known conjugate gradient least- squares (CGLS) method can be developed for solving the mentioned problem over the η-Hermitian and η-anti-Hermitian matrices. In addition, the convergence properties of the proposed algorithm are discussed with details. In the circumstance that the coefficient matrices are ill-conditioned, it is suggested to use a preconditioner for accelerating the convergence behavior of the algorithm. Numerical experiments are reported to reveal the validity of the elaborated results and feasibility of the proposed iterative algorithm and its preconditioned version.

scholarly journals On the stability and boundedness of solutions of nonlinear third order differential equations with delay

Filomat ◽  
2010 ◽  
Vol 24 (3) ◽  
pp. 1-10 ◽  
Author(s):  
Cemil Tunç
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Relevant Literature ◽  
Stability Result ◽  
Boundedness Of Solutions ◽  
Third Order ◽  
Functional Differential ◽  
The Stability ◽  
Equations With Delay

By defining a Lyapunov functional, we investigate the stability and boundedness of solutions to nonlinear third order differential equation with constant delay, r : x'''(t) + g(x(t), x'(t))x''(t) + f (x(t - r), x'(t - r)) + h(x(t - r)) = p(t, x(t), x'(t), x(t - r), x'(t - r), x''(t)), when p(t, x(t), x'(t), x(t - r), x'(t - r), x''(t)) = 0 and ? 0, respectively. Our results achieve a stability result which exists in the relevant literature of ordinary nonlinear third order differential equations without delay to the above functional differential equation for the stability and boundedness of solutions. An example is introduced to illustrate the importance of the results obtained.

scholarly journals Almost sure exponential stability of the θ-Euler-Maruyama method for neutral stochastic differential equations with time-dependent delay when θ ∈ [0; 1 2]

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5629-5645 ◽  
Author(s):  
Maja Obradovic ◽  
Marija Milosevic
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Stability Result ◽  
Time Dependent ◽  
Almost Sure Exponential Stability ◽  
Linear Growth Condition ◽  
Highly Nonlinear ◽  
The Stability ◽  
Drift Coefficient

This paper represents a generalization of the stability result on the Euler-Maruyama solution, which is established in the paper M. Milosevic, Almost sure exponential stability of solutions to highly nonlinear neutral stochastics differential equations with time-dependent delay and Euler-Maruyama approximation, Math. Comput. Model. 57 (2013) 887 - 899. The main aim of this paper is to reveal the sufficient conditions for the global almost sure asymptotic exponential stability of the ?-Euler-Maruyama solution (? ? [0, 1/2 ]), for a class of neutral stochastic differential equations with time-dependent delay. The existence and uniqueness of solution of the approximate equation is proved by employing the one-sided Lipschitz condition with respect to the both present state and delayed arguments of the drift coefficient of the equation. The technique used in proving the stability result required the assumption ? ?(0, 1/2], while the method is defined by employing the parameter ? with respect to the both drift coefficient and neutral term. Bearing in mind the difference between the technique which will be applied in the present paper and that used in the cited paper, the Euler-Maruyama case (? = 0) is considered separately. In both cases, the linear growth condition on the drift coefficient is applied, among other conditions. An example is provided to support the main result of the paper.

scholarly journals Approximating Fixed Points Using a Faster Iterative Method and Application to Split Feasibility Problems

Computation ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 90
Author(s):  
Kifayat Ullah ◽  
Junaid Ahmad ◽  
Muhammad Arshad ◽  
Zhenhua Ma
Keyword(s):  
Iterative Scheme ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Iterative Schemes ◽  
Split Feasibility Problems ◽  
Mild Conditions ◽  
Convergence Results ◽  
Uniformly Convex Banach Spaces ◽  
The Many ◽  
Split Feasibility

In this article, the recently introduced iterative scheme of Hassan et al. (Math. Probl. Eng. 2020) is re-analyzed with the connection of Reich–Suzuki type nonexpansive (RSTN) maps. Under mild conditions, some important weak and strong convergence results in the context of uniformly convex Banach spaces are provided. To support the main outcome of the paper, we provide a numerical example and show that this example properly exceeds the class of Suzuki type nonexpansive (STN) maps. It has been shown that the Hassan et al. iterative scheme of this example is more useful than the many other iterative schemes. We provide an application of our main results to solve split feasibility problems in the setting of RSTN maps. The presented outcome is new and compliments the corresponding results of the current literature.

STRONG CONVERGENCE OF IMPLICIT ITERATIVE ALGORITHMS FOR STRICTLY PSEUDO-CONTRACTIVE MAPPINGS

2021 ◽  
Vol 10 (8) ◽  
pp. 3023-3047
Author(s):  
M.O. Aibinu ◽  
S.C. Thakur ◽  
S. Moyo
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Contractive Mapping ◽  
Optical Illusion ◽  
Implicit Algorithm ◽  
Mild Conditions ◽  
Contractive Mappings

The class of strictly pseudo-contractive mappings is known to have more powerful applications than the class of nonexpansive mappings in solving nonlinear equations such as inverse and equilibrium problems. Motivated by the potency of the class of strictly pseudo-contractive mappings, a generalized viscosity implicit algorithm is constructed for finding their fixed points in the framework of Banach spaces. The strong convergence of the newly constructed sequence to a fixed point of a strictly pseudo-contractive mapping is obtained under some mild conditions on the parameters and the fixed point is shown to solve some variational inequality problems. An example is given to illustrate the convergence analysis of the newly constructed generalized viscosity implicit algorithm for the class of strictly pseudo-contractive mappings. The example also shows that the algorithm and the conditions which are imposed on the parameters are not just optical illusion.

scholarly journals Some Convergence and Stability Results for the Kirk Multistep and Kirk-SP Fixed Point Iterative Algorithms

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Faik Gürsoy ◽  
Vatan Karakaya ◽  
B. E. Rhoades
Keyword(s):  
Fixed Point ◽  
Iterative Algorithm ◽  
Iterative Algorithms ◽  
Iterative Processes ◽  
Convergence And Stability ◽  
The Stability ◽  
Stability Results

The purpose of this paper is to introduce a new Kirk type iterative algorithm called Kirk multistep iteration and to study its convergence. We also prove some theorems related to the stability results for the Kirk multistep and Kirk-SP iterative processes by employing certain contractive-like operators. Our results generalize and unify some other results in the literature.

scholarly journals A novel iterative approach for solving common fixed point problems in Geodesic spaces with convergence analysis

2021 ◽  
Vol 37 (2) ◽  
pp. 145-160
Author(s):  
THANATPORN BANTAOJAI ◽  
CHANCHAL GARODIA ◽  
IZHAR UDDIN ◽  
NUTTAPOL PAKKARANANG ◽  
PANU YIMMUANG
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Common Fixed Point ◽  
Rate Of Convergence ◽  
Convergence Analysis ◽  
Nonexpansive Mappings ◽  
Fixed Point Problems ◽  
Iterative Approach ◽  
Mild Conditions ◽  
New Iterative Method

In this paper, we introduce a new iterative method for nonexpansive mappings in CAT(\kappa) spaces. First, the rate of convergence of proposed method and comparison with recently existing method is proved. Second, strong and \Delta-convergence theorems of the proposed method in such spaces under some mild conditions are also proved. Finally, we provide some non-trivial examples to show efficiency and comparison with many previously existing methods.

scholarly journals A Note on the Stability Analysis of Fuzzy Nonlinear Fractional Differential Equations Involving the Caputo Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Ali El Mfadel ◽  
Said Melliani ◽  
M’hamed Elomari
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Direct Method ◽  
Stability Result ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential ◽  
The Stability

In this paper, we present and establish a new result on the stability analysis of solutions for fuzzy nonlinear fractional differential equations by extending Lyapunov’s direct method from the fuzzy ordinary case to the fuzzy fractional case. As an application, several examples are presented to illustrate the proposed stability result.

scholarly journals Stability of intuitionistic fuzzy nonlinear fractional differential equations

2021 ◽  
Vol 27 (1) ◽  
pp. 83-100
Author(s):  
Said Melliani ◽  
◽  
Ali El Mfadel ◽  
Lalla Saadia Chadli ◽  
M’hamed Elomari ◽  
...  
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Stability Result ◽  
Zero Solution ◽  
Nonlinear Fractional Differential Equations ◽  
Existence And Uniqueness Results ◽  
Intuitionistic Fuzzy ◽  
Uniqueness Results ◽  
Fractional Differential ◽  
The Stability

In this paper, we study the existence and uniqueness results of solution for the intuitionistic fuzzy nonlinear fractional differential equations involving the Caputo concepts of fractional derivative. In addition, we establish essentially the Mittag-Leffler stability result for the intuitionistic fuzzy nonlinear fractional differential equations by giving some sufficient criteria to guarantee the stability of the zero solution. Finally, some examples are presented to illustrate the proposed stability theorem.

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