Research

19 pages, 335 KiB

Open AccessArticle

Convergence Analysis for Yosida Variational Inclusion Problem with Its Corresponding Yosida Resolvent Equation Problem through Inertial Extrapolation Scheme

by Arvind Kumar Rajpoot, Mohd Ishtyak, Rais Ahmad, Yuanheng Wang and Jen-Chih Yao

Mathematics 2023, 11(3), 763; https://0-doi-org.brum.beds.ac.uk/10.3390/math11030763 - 02 Feb 2023

Cited by 2 | Viewed by 1091

Abstract

In this paper, we study a Yosida variational inclusion problem with its corresponding Yosida resolvent equation problem. We mention some schemes to solve both the problems, but we focus our study on discussing convergence criteria for the Yosida variational inclusion problem in real [...] Read more.

In this paper, we study a Yosida variational inclusion problem with its corresponding Yosida resolvent equation problem. We mention some schemes to solve both the problems, but we focus our study on discussing convergence criteria for the Yosida variational inclusion problem in real Banach space and for the Yosida resolvent equation problem in q-uniformly smooth Banach space. For faster convergence, we apply an inertial extrapolation scheme for both the problems. An example is provided. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications)

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12 pages, 274 KiB

Open AccessArticle

Algorithms for Approximating Solutions of Split Variational Inclusion and Fixed-Point Problems

by Li-Jun Zhu and Yonghong Yao

Mathematics 2023, 11(3), 641; https://0-doi-org.brum.beds.ac.uk/10.3390/math11030641 - 27 Jan 2023

Cited by 13 | Viewed by 1023

Abstract

In this paper, the split fixed point and variational inclusion problem is considered. With the help of fixed point technique, Tseng-type splitting method and self-adaptive rule, an iterative algorithm is proposed for solving this split problem in which the involved operators S and [...] Read more.

In this paper, the split fixed point and variational inclusion problem is considered. With the help of fixed point technique, Tseng-type splitting method and self-adaptive rule, an iterative algorithm is proposed for solving this split problem in which the involved operators S and T are demicontractive operators and g is plain monotone. Strong convergence theorem is proved under some mild conditions. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications)

14 pages, 307 KiB

Open AccessArticle

Fixed Points for (ξ,ω)-Weakly Cyclic Type Generalized Contraction Condition in Metric Spaces with an Application

by Penumarthy Parvateesam Murthy, Pusplata Sahu, Amr Elsonbaty, Khizar Hyatt Khan, Rajagopalan Ramaswamy and Stojan Radenović

Mathematics 2023, 11(1), 166; https://0-doi-org.brum.beds.ac.uk/10.3390/math11010166 - 28 Dec 2022

Viewed by 782

Abstract

In the present work, we have introduced a new type of

(ξ, ω)

-weakly cyclic generalized contraction in the setting of metric spaces and established some fixed-point results. Fixed-point results are useful in establishing the existence of unique solution to [...] Read more.

In the present work, we have introduced a new type of

(ξ, ω)

-weakly cyclic generalized contraction in the setting of metric spaces and established some fixed-point results. Fixed-point results are useful in establishing the existence of unique solution to differential equations. We have supplemented the derived results with suitable non-trivial examples with an application to the Boundary Value Problem, generalizing some known results. The analytical result has been verified with numerical simulation. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications)

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18 pages, 322 KiB

Open AccessArticle

Stationary Condition for Borwein Proper Efficient Solutions of Nonsmooth Multiobjective Problems with Vanishing Constraints

by Hui Huang and Haole Zhu

Mathematics 2022, 10(23), 4569; https://0-doi-org.brum.beds.ac.uk/10.3390/math10234569 - 02 Dec 2022

Viewed by 720

Abstract

This paper discusses optimality conditions for Borwein proper efficient solutions of nonsmooth multiobjective optimization problems with vanishing constraints. A new notion in terms of contingent cone and upper directional derivative is introduced, and a necessary condition for the Borwein proper efficient solution of [...] Read more.

This paper discusses optimality conditions for Borwein proper efficient solutions of nonsmooth multiobjective optimization problems with vanishing constraints. A new notion in terms of contingent cone and upper directional derivative is introduced, and a necessary condition for the Borwein proper efficient solution of the considered problem is derived. The concept of

ε

proper Abadie data qualification is also introduced, and a necessary condition which is called a strictly strong stationary condition for Borwein proper efficient solutions is obtained. In view of the strictly strong stationary condition, convexity of the objective functions, and quasi-convexity of constrained functions, sufficient conditions for the Borwein proper efficient solutions are presented. Some examples are given to illustrate the reasonability of the obtained results. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications)

17 pages, 421 KiB

Open AccessArticle

Solving System of Mixed Variational Inclusions Involving Generalized Cayley Operator and Generalized Yosida Approximation Operator with Error Terms in q-Uniformly Smooth Space

by Rais Ahmad, Mohd Ishtyak, Arvind Kumar Rajpoot and Yuanheng Wang

Mathematics 2022, 10(21), 4131; https://0-doi-org.brum.beds.ac.uk/10.3390/math10214131 - 05 Nov 2022

Viewed by 922

Abstract

In this paper, we solve a system of mixed variational inclusions involving a generalized Cayley operator and the generalized Yosida approximation operator. An iterative algorithm is suggested to discuss the convergence analysis. We have shown that our system admits a unique solution by [...] Read more.

In this paper, we solve a system of mixed variational inclusions involving a generalized Cayley operator and the generalized Yosida approximation operator. An iterative algorithm is suggested to discuss the convergence analysis. We have shown that our system admits a unique solution by using the properties of q-uniformly smooth Banach space, and we discuss the convergence criteria for sequences generated by iterative algorithm. Two examples are constructed, and an application is provided. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications)

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13 pages, 369 KiB

Open AccessArticle

System of Generalized Variational Inclusions Involving Cayley Operators and XOR-Operation in q-Uniformly Smooth Banach Spaces

by Javid Iqbal, Arvind Kumar Rajpoot, Monirul Islam, Rais Ahmad and Yuanheng Wang

Mathematics 2022, 10(16), 2837; https://0-doi-org.brum.beds.ac.uk/10.3390/math10162837 - 09 Aug 2022

Cited by 2 | Viewed by 1014

Abstract

In this paper, we consider and study a system of generalized variational inclusions involving Cayley operators and an XOR-operation in q-uniformly smooth Banach spaces. To obtain the solution of the system of generalized variational inclusions involving Cayley operators and an XOR-operation, we [...] Read more.

In this paper, we consider and study a system of generalized variational inclusions involving Cayley operators and an XOR-operation in q-uniformly smooth Banach spaces. To obtain the solution of the system of generalized variational inclusions involving Cayley operators and an XOR-operation, we use some properties of Cayley operators as well as an XOR-operation. We also discuss the convergence criterion. In support of our main result, we provide an example. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications)

26 pages, 383 KiB

Open AccessArticle

Modified Mann-Type Subgradient Extragradient Rules for Variational Inequalities and Common Fixed Points Implicating Countably Many Nonexpansive Operators

by Yun-Ling Cui, Lu-Chuan Ceng, Fang-Fei Zhang, Cong-Shan Wang, Jian-Ye Li, Hui-Ying Hu and Long He

Mathematics 2022, 10(11), 1949; https://0-doi-org.brum.beds.ac.uk/10.3390/math10111949 - 06 Jun 2022

Viewed by 1209

Abstract

In a real Hilbert space, let the CFPP, VIP, and HFPP denote the common fixed-point problem of countable nonexpansive operators and asymptotically nonexpansive operator, variational inequality problem, and hierarchical fixed point problem, respectively. With the help of the Mann iteration method, a subgradient [...] Read more.

In a real Hilbert space, let the CFPP, VIP, and HFPP denote the common fixed-point problem of countable nonexpansive operators and asymptotically nonexpansive operator, variational inequality problem, and hierarchical fixed point problem, respectively. With the help of the Mann iteration method, a subgradient extragradient approach with a linear-search process, and a hybrid deepest-descent technique, we construct two modified Mann-type subgradient extragradient rules with a linear-search process for finding a common solution of the CFPP and VIP. Under suitable assumptions, we demonstrate the strong convergence of the suggested rules to a common solution of the CFPP and VIP, which is only a solution of a certain HFPP. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications)

29 pages, 940 KiB

Open AccessArticle

Inertial Modification Using Self-Adaptive Subgradient Extragradient Techniques for Equilibrium Programming Applied to Variational Inequalities and Fixed-Point Problems

by Habib ur Rehman, Wiyada Kumam and Kamonrat Sombut

Mathematics 2022, 10(10), 1751; https://0-doi-org.brum.beds.ac.uk/10.3390/math10101751 - 20 May 2022

Cited by 5 | Viewed by 1293

Abstract

Equilibrium problems are articulated in a variety of mathematical computing applications, including minimax and numerical programming, saddle-point problems, fixed-point problems, and variational inequalities. In this paper, we introduce improved iterative techniques for evaluating the numerical solution of an equilibrium problem in a Hilbert [...] Read more.

Equilibrium problems are articulated in a variety of mathematical computing applications, including minimax and numerical programming, saddle-point problems, fixed-point problems, and variational inequalities. In this paper, we introduce improved iterative techniques for evaluating the numerical solution of an equilibrium problem in a Hilbert space with a pseudomonotone and a Lipschitz-type bifunction. These techniques are based on two computing steps of a proximal-like mapping with inertial terms. We investigated two simplified stepsize rules that do not require a line search, allowing the technique to be carried out more successfully without knowledge of the Lipschitz-type constant of the cost bifunction. Once control parameter constraints are put in place, the iterative sequences converge on a particular solution to the problem. We prove strong convergence theorems without knowing the Lipschitz-type bifunction constants. A sequence of numerical tests was performed, and the results confirmed the correctness and speedy convergence of the new techniques over the traditional ones. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications)

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18 pages, 353 KiB

Open AccessArticle

New Fundamental Results on the Continuous and Discrete Integro-Differential Equations

by Osman Tunç, Cemil Tunç, Jen-Chih Yao and Ching-Feng Wen

Mathematics 2022, 10(9), 1377; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091377 - 20 Apr 2022

Cited by 10 | Viewed by 1308

Abstract

This work studies certain perturbed and un-perturbed nonlinear systems of continuous and discrete integro-delay differential equations (IDDEs). Using the Lyapunov–Krasovskii functional (LKF) method and the Lyapunov–Razumikhin method (LRM), uniform asymptotic stability (UAS), uniform stability (US), integrability and boundedness of solutions as well as [...] Read more.

This work studies certain perturbed and un-perturbed nonlinear systems of continuous and discrete integro-delay differential equations (IDDEs). Using the Lyapunov–Krasovskii functional (LKF) method and the Lyapunov–Razumikhin method (LRM), uniform asymptotic stability (UAS), uniform stability (US), integrability and boundedness of solutions as well as exponential stability (ES) and instability of solutions are discussed. In this paper, five new theorems and a corollary are given and three numerical applications are provided with their simulations. With this work, we aim to make new contributions to the theory of the continuous and discrete integro-differential equations. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications)

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21 pages, 350 KiB

Open AccessArticle

On Strengthened Inertial-Type Subgradient Extragradient Rule with Adaptive Step Sizes for Variational Inequalities and Fixed Points of Asymptotically Nonexpansive Mappings

by Lu-Chuan Ceng, Ching-Feng Wen, Yeong-Cheng Liou and Jen-Chih Yao

Mathematics 2022, 10(6), 958; https://0-doi-org.brum.beds.ac.uk/10.3390/math10060958 - 17 Mar 2022

Viewed by 972

Abstract

In a real Hilbert space, let the VIP denote a pseudomonotone variational inequality problem with Lipschitz continuity operator, and let the CFPP indicate a common fixed-point problem of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping. On the basis of the Mann [...] Read more.

In a real Hilbert space, let the VIP denote a pseudomonotone variational inequality problem with Lipschitz continuity operator, and let the CFPP indicate a common fixed-point problem of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping. On the basis of the Mann iteration method, the viscosity approximation method and the hybrid steepest-descent method, we propose and analyze two strengthened inertial-type subgradient extragradient rules with adaptive step sizes for solving the VIP and CFPP. With the help of suitable restrictions, we show the strong convergence of the suggested rules to a common solution of the VIP and CFPP, which is the unique solution of a hierarchical variational inequality (HVI). Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications)

12 pages, 650 KiB

Open AccessArticle

Modified Projection Method with Inertial Technique and Hybrid Stepsize for the Split Feasibility Problem

by Suthep Suantai, Suparat Kesornprom, Watcharaporn Cholamjiak and Prasit Cholamjiak

Mathematics 2022, 10(6), 933; https://0-doi-org.brum.beds.ac.uk/10.3390/math10060933 - 15 Mar 2022

Cited by 3 | Viewed by 1512

Abstract

We designed a modified projection method with a new condition of the inertial step and the step size for the split feasibility problem in Hilbert spaces. We show that our iterate weakly converged to a solution. Lastly, we give numerical examples and comparisons [...] Read more.

We designed a modified projection method with a new condition of the inertial step and the step size for the split feasibility problem in Hilbert spaces. We show that our iterate weakly converged to a solution. Lastly, we give numerical examples and comparisons that could be applied to signal recovery to show the efficiency of our method. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications)

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14 pages, 305 KiB

Open AccessArticle

On the Existence of Super Efficient Solutions and Optimality Conditions for Set-Valued Vector Optimization Problems

by Lu-Chuan Ceng, Ching-Feng Wen and Yeong-Cheng Liou

Mathematics 2022, 10(3), 316; https://0-doi-org.brum.beds.ac.uk/10.3390/math10030316 - 20 Jan 2022

Cited by 1 | Viewed by 1048

Abstract

In this paper, by using the normal subdifferential and equilibrium-like function we first obtain some properties for K-preinvex set-valued maps. Secondly, in terms of this equilibrium-like function, we establish some sufficient conditions for the existence of super minimal points of a K [...] Read more.

In this paper, by using the normal subdifferential and equilibrium-like function we first obtain some properties for K-preinvex set-valued maps. Secondly, in terms of this equilibrium-like function, we establish some sufficient conditions for the existence of super minimal points of a K-preinvex set-valued map, that is, super efficient solutions of a set-valued vector optimization problem, and also attain necessity optimality terms for a general type of super efficiency. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications)

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