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 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 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

scholarly journals Modified Proximal Algorithms for Finding Solutions of the Split Variational Inclusions

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 708 ◽  
Author(s):  
Suthep Suantai ◽  
Suparat Kesornprom ◽  
Prasit Cholamjiak
Keyword(s):  
Hilbert Spaces ◽  
Numerical Experiments ◽  
Variational Inclusion ◽  
Inclusion Problem ◽  
Signal Recovery ◽  
Proximal Algorithms ◽  
Split Variational Inclusion Problem ◽  
Variational Inclusion Problem ◽  
Lipschitz Constants ◽  
Theoretical Results

We investigate the split variational inclusion problem in Hilbert spaces. We propose efficient algorithms in which, in each iteration, the stepsize is chosen self-adaptive, and proves weak and strong convergence theorems. We provide numerical experiments to validate the theoretical results for solving the split variational inclusion problem as well as the comparison to algorithms defined by Byrne et al. and Chuang, respectively. It is shown that the proposed algorithms outrun other algorithms via numerical experiments. As applications, we apply our method to compressed sensing in signal recovery. The proposed methods have as a main advantage that the computation of the Lipschitz constants for the gradient of functions is dropped in generating the sequences.

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