scholarly journals RELAXED Η-PROXIMAL OPERATOR FOR SOLVING A VARIATIONAL-LIKE INCLUSION PROBLEM

2015 ◽  
Vol 20 (6) ◽  
pp. 819-835 ◽  
Author(s):  
Mijanur Rahaman ◽  
Rais Ahmad ◽  
Mohd Dilshad ◽  
Iqbal Ahmad
Keyword(s):  
Iterative Algorithm ◽  
Resolvent Operator ◽  
Inclusion Problem ◽  
Proximal Operator ◽  
Nonconvex Functional

In this paper, we introduce a new resolvent operator and we call it relaxed η-proximal operator. We demonstrate some of the properties of relaxed η- proximal operator. By applying this concept, we consider and study a variational -like inclusion problem with a nonconvex functional. Based on relaxed η-proximal operator, we define an iterative algorithm to approximate the solutions of a variational-like inclusion problem and the convergence of the iterative sequences generated by the algorithm is also discussed. Our results can be treated as refinement of many previously known results. An example is constructed in support of Theorem 1.

Generalized variational-like inclusion involving relaxed monotone operators

2017 ◽  
Vol 8 (2) ◽  
Author(s):  
Syed Shakaib Irfan ◽  
Mohammad F. Khan ◽  
Ali P. Farajzadeh ◽  
Allahkaram Shafie
Keyword(s):  
Hilbert Space ◽  
Iterative Algorithm ◽  
Resolvent Operator ◽  
Recent Literature ◽  
Monotone Operators ◽  
Inclusion Problem ◽  
Proximal Operator ◽  
New Class

Abstract In this paper, we introduce a new class of resolvent operator, the η-proximal operator, and discuss some of its properties. We consider a new generalized variational-like inclusion problem involving relaxed monotone operators in Hilbert space and construct a new iterative algorithm for proving the existence of the solutions of our problem. Our results improve and generalize many corresponding results in the recent literature.

scholarly journals Convergence Analysis of a Three-Step Iterative Algorithm for Generalized Set-Valued Mixed-Ordered Variational Inclusion Problem

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 444
Author(s):  
Praveen. Agarwal ◽  
Doaa Filali ◽  
M. Akram ◽  
M. Dilshad
Keyword(s):  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Resolvent Operator ◽  
Variational Inclusion ◽  
Inclusion Problem ◽  
Existence And Uniqueness Results ◽  
Variational Inclusion Problem ◽  
Uniqueness Results ◽  
Convergence Results ◽  
The Resolvent Operator

This manuscript aims to study a generalized, set-valued, mixed-ordered, variational inclusion problem involving H(·,·)-compression XOR-αM-non-ordinary difference mapping and relaxed cocoercive mapping in real-ordered Hilbert spaces. The resolvent operator associated with H(·,·)-compression XOR-αM-non-ordinary difference mapping is defined, and some of its characteristics are discussed. We prove existence and uniqueness results for the considered generalized, set-valued, mixed-ordered, variational inclusion problem. Further, we put forward a three-step iterative algorithm using a ⊕ operator, and analyze the convergence of the suggested iterative algorithm under some mild assumptions. Finally, we reconfirm the existence and convergence results by an illustrative numerical example.

scholarly journals The Modified Inertial Iterative Algorithm for Solving Split Variational Inclusion Problem for Multi-Valued Quasi Nonexpansive Mappings with Some Applications

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 560 ◽  
Author(s):  
Pawicha Phairatchatniyom ◽  
Poom Kumam ◽  
Yeol Je Cho ◽  
Wachirapong Jirakitpuwapat ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Fixed Point ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Fixed Point Problems ◽  
Inclusion Problem ◽  
Common Elements ◽  
Inclusion Problems ◽  
Variational Inclusion Problem ◽  
Inertial Extrapolation ◽  
Appl Math

Based on the very recent work by Shehu and Agbebaku in Comput. Appl. Math. 2017, we introduce an extension of their iterative algorithm by combining it with inertial extrapolation for solving split inclusion problems and fixed point problems. Under suitable conditions, we prove that the proposed algorithm converges strongly to common elements of the solution set of the split inclusion problems and fixed point problems.

scholarly journals Approximation solvability for a system of implicit nonlinear variational inclusions with Η-monotone operators

2018 ◽  
Vol 51 (1) ◽  
pp. 241-254
Author(s):  
Jong Kyu Kim ◽  
Muhammad Iqbal Bhat
Keyword(s):  
Iterative Algorithm ◽  
Resolvent Operator ◽  
Monotone Operators ◽  
Inner Product ◽  
Operator Technique ◽  
Variational Inclusions ◽  
Inclusion Problems ◽  
Resolvent Operator Technique ◽  
The Resolvent Operator ◽  
New System

AbstractIn this paper, we introduce and study a new system of variational inclusions which is called a system of nonlinear implicit variational inclusion problems with A-monotone and H-monotone operators in semi-inner product spaces. We define the resolvent operator associated with A-monotone and H-monotone operators and prove its Lipschitz continuity. Using resolvent operator technique, we prove the existence and uniqueness of solution for this new system of variational inclusions. Moreover, we suggest an iterative algorithm for approximating the solution of this system and discuss the convergence analysis of the sequences generated by the iterative algorithm under some suitable conditions.

scholarly journals Generalized Implicit Set-Valued Variational Inclusion Problem with ⊕ Operation

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 421
Author(s):  
Rais Ahmad ◽  
Imran Ali ◽  
Saddam Husain ◽  
A. Latif ◽  
Ching-Feng Wen
Keyword(s):  
Resolvent Operator ◽  
Convergence Result ◽  
Variational Inclusion ◽  
Inclusion Problem ◽  
Variational Inclusion Problem ◽  
Special Cases ◽  
The Resolvent Operator

In this paper, we consider a resolvent operator which depends on the composition of two mappings with ⊕ operation. We prove some of the properties of the resolvent operator, that is, that it is single-valued as well as Lipschitz-type-continuous. An existence and convergence result is proven for a generalized implicit set-valued variational inclusion problem with ⊕ operation. Some special cases of a generalized implicit set-valued variational inclusion problem with ⊕ operation are discussed. An example is constructed to illustrate some of the concepts used in this paper.

scholarly journals An iterative algorithm for generalized nonlinear variational inclusions with relaxed strongly accretive mappings in Banach spaces

2004 ◽  
Vol 2004 (20) ◽  
pp. 1035-1045 ◽  
Author(s):  
A. H. Siddiqi ◽  
Rais Ahmad
Keyword(s):  
Banach Spaces ◽  
Iterative Algorithm ◽  
Resolvent Operator ◽  
Existence Of Solutions ◽  
Operator Technique ◽  
Variational Inclusions ◽  
Special Cases ◽  
Accretive Mappings ◽  
Resolvent Operator Technique ◽  
The Resolvent Operator

We use Nadler's theorem and the resolvent operator technique form-accretive mappings to suggest an iterative algorithm for solving generalized nonlinear variational inclusions with relaxed strongly accretive mappings in Banach spaces. We prove the existence of solutions for our inclusions without compactness assumption and the convergence of the iterative sequences generated by the algorithm in real Banach spaces. Some special cases are also discussed.

scholarly journals Projection Algorithms for Variational Inclusions

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Youli Yu ◽  
Pei-Xia Yang ◽  
Khalida Inayat Noor
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Iterative Algorithm ◽  
Real Hilbert Space ◽  
Projection Algorithm ◽  
Variational Inclusion ◽  
Inclusion Problem ◽  
Variational Inclusions ◽  
Projection Algorithms ◽  
Variational Inclusion Problem

We present a projection algorithm for finding a solution of a variational inclusion problem in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a solution of the variational inclusion problem which also solves some variational inequality.

scholarly journals A new iterative algorithm for solving general variational inclusion problem with application

2021 ◽  
Author(s):  
M. Akram ◽  
A.F. Aljohani ◽  
M. Dilshad ◽  
Aysha Khan
Keyword(s):  
Differential Equation ◽  
Iterative Algorithm ◽  
Delay Differential Equation ◽  
Iterative Algorithms ◽  
Convergence Result ◽  
Variational Inclusion ◽  
Inclusion Problem ◽  
Numerical Example ◽  
Variational Inclusion Problem ◽  
Delay Differential

In this paper, we pose a new iterative algorithm and show that this newly constructed algorithm converges faster than some existing iterative algorithms. We validate our claim by an illustrative example. Also, we discuss the convergence of our algorithm to approximate the solution of a general variational inclusion problem. Also, we present a numerical example to verify our existence and convergence result. Finally, we apply our proposed iterative algorithm to solve a delay differential equation as an application

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