scholarly journals Generalized Yosida approximation operator with an application to a system of Yosida inclusions

2018 ◽  
Vol 2018 (1) ◽  
pp. 1-20 ◽  
Keyword(s):  
Approximation Operator ◽  
Yosida Approximation

scholarly journals Solving Yosida inclusion problem in Hadamard manifold

2019 ◽  
Vol 9 (2) ◽  
pp. 357-366 ◽  
Author(s):  
Mohammad Dilshad
Keyword(s):  
Approximate Solution ◽  
Vector Field ◽  
Variational Inequalities ◽  
Hadamard Manifold ◽  
Inclusion Problem ◽  
Approximation Operator ◽  
Hadamard Manifolds ◽  
Yosida Approximation ◽  
Type Algorithm ◽  
Monotone Vector Field

Abstract We consider a Yosida inclusion problem in the setting of Hadamard manifolds. We study Korpelevich-type algorithm for computing the approximate solution of Yosida inclusion problem. The resolvent and Yosida approximation operator of a monotone vector field and their properties are used to prove that the sequence generated by the proposed algorithm converges to the solution of Yosida inclusion problem. An application to our problem and algorithm is presented to solve variational inequalities in Hadamard manifolds.

Composite relaxed resolvent operator and Yosida approximation operator for solving a system of Yosida inclusions

2018 ◽  
Vol 24 (2) ◽  
pp. 185-195
Author(s):  
Rais Ahmad ◽  
Vishnu Narayan Mishra ◽  
Mohd. Ishtyak ◽  
Mijanur Rahaman
Keyword(s):  
Resolvent Operator ◽  
Approximation Operator ◽  
Yosida Approximation

Abstract In this paper, we first study a composite relaxed resolvent operator and prove some of its properties. After that, we introduce a Yosida approximation operator based on the composite relaxed resolvent operator and demonstrate some properties of the Yosida approximation operator. Finally, we obtain the solution of a system of Yosida inclusions by applying these concepts. We construct a conjoin example in support of many concepts derived in this paper. Our concepts and results are new in the literature and can be used for further research.

scholarly journals Cayley Inclusion Problem Involving XOR-Operation

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 302 ◽  
Author(s):  
Imran Ali ◽  
Rais Ahmad ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Resolvent Operator ◽  
Convergence Result ◽  
Inclusion Problem ◽  
Approximation Operator ◽  
Yosida Approximation ◽  
Xor Operation

In this paper, we study an absolutely new problem, namely, the Cayley inclusion problem which involves the Cayley operator and a multi-valued mapping with XOR-operation. We have shown that the Cayley operator is a single-valued comparison and it is Lipschitz-type-continuous. A fixed point formulation of the Cayley inclusion problem is shown by using the concept of a resolvent operator as well as the Yosida approximation operator. Finally, an existence and convergence result is proved. An example is constructed for some of the concepts used in this work.

scholarly journals Yosida Complementarity Problem with Yosida Variational Inequality Problem and Yosida Proximal Operator Equation Involving XOR-Operation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Rais Ahmad ◽  
Arvind Kumar Rajpoot ◽  
Imran Ali ◽  
Ching-Feng Wen
Keyword(s):  
Variational Inequality ◽  
Operator Equation ◽  
Complementarity Problem ◽  
Variational Inequality Problem ◽  
Approximation Operator ◽  
Operator Technique ◽  
Yosida Approximation ◽  
Proximal Operator ◽  
Inequality Problem ◽  
Xor Operation

Due to the importance of Yosida approximation operator, we generalized the variational inequality problem and its equivalent problems by using Yosida approximation operator. The aim of this work is to introduce and study a Yosida complementarity problem, a Yosida variational inequality problem, and a Yosida proximal operator equation involving XOR-operation. We prove an existence result together with convergence analysis for Yosida proximal operator equation involving XOR-operation. For this purpose, we establish an algorithm based on fixed point formulation. Our approach is based on a proximal operator technique involving a subdifferential operator. As an application of our main result, we provide a numerical example using the MATLAB program R2018a. Comparing different iterations, a computational table is assembled and some graphs are plotted to show the convergence of iterative sequences for different initial values.

Graph convergence and generalized Yosida approximation operator with an application

2017 ◽  
Vol 11 (2) ◽  
pp. 155-163 ◽  
Author(s):  
Rais Ahmad ◽  
Mohd. Ishtyak ◽  
Mijanur Rahaman ◽  
Iqbal Ahmad
Keyword(s):  
Approximation Operator ◽  
Yosida Approximation ◽  
Graph Convergence

scholarly journals Parametric Matroid of Rough Set

2015 ◽  
Vol 23 (06) ◽  
pp. 893-908 ◽  
Author(s):  
Yanfang Liu ◽  
Hong Zhao ◽  
William Zhu
Keyword(s):  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Attribute Reduction ◽  
Independent Set ◽  
Approximation Operator ◽  
Approximation Number ◽  
Lower Approximation ◽  
The One

Rough set is mainly concerned with the approximations of objects through an equivalence relation on a universe. Matroid is a generalization of linear algebra and graph theory. Recently, a matroidal structure of rough sets is established and applied to the problem of attribute reduction which is an important application of rough set theory. In this paper, we propose a new matroidal structure of rough sets and call it a parametric matroid. On the one hand, for an equivalence relation on a universe, a parametric set family, with any subset of the universe as its parameter, is defined through the lower approximation operator. This parametric set family is proved to satisfy the independent set axiom of matroids, therefore a matroid is generated, and we call it a parametric matroid of the rough set. Through the lower approximation operator, three equivalent representations of the parametric set family are obtained. Moreover, the parametric matroid of the rough set is proved to be the direct sum of a partition-circuit matroid and a free matroid. On the other hand, partition-circuit matroids are well studied through the lower approximation number, and then we use it to investigate the parametric matroid of the rough set. Several characteristics of the parametric matroid of the rough set, such as independent sets, bases, circuits, the rank function and the closure operator, are expressed by the lower approximation number.

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