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On Split Monotone Variational Inclusion Problem with Multiple Output Sets with Fixed Point Constraints

  • Victor Amarachi Uzor , Timilehin Opeyemi Alakoya und Oluwatosin Temitope Mewomo ORCID logo EMAIL logo
Veröffentlicht/Copyright: 1. März 2023
Computational Methods in Applied Mathematics
Aus der Zeitschrift Computational Methods in Applied Mathematics Band 23 Heft 3

Abstract

In this paper, we introduce and study the concept of split monotone variational inclusion problem with multiple output sets (SMVIPMOS). We propose a new iterative scheme, which employs the viscosity approximation technique for approximating the solution of the SMVIPMOS with fixed point constraints of a nonexpansive mapping in real Hilbert spaces. The proposed method utilises the inertial technique for accelerating the speed of convergence and a self-adaptive step size so that its implementation does not require prior knowledge of the operator norm. Under mild conditions, we obtain a strong convergence result for the proposed algorithm and obtain a consequent result, which complements several existing results in the literature. Moreover, we apply our result to study the notions of split variational inequality problem with multiple output sets with fixed point constraints and split convex minimisation problem with multiple output sets with fixed point constraints in Hilbert spaces. Finally, we present some numerical experiments to demonstrate the implementability of our proposed method.

Keywords: Forward-Backward Splitting Method; Split Monotone Variational Inclusion Problem; Fixed Point Problem; Nonexpansive Mappings; Adaptive Step Size; Inertial Technique
MSC 2010: 47H09; 47H10; 47J25; 90C2

Funding source: National Research Foundation

Award Identifier / Grant number: 119903

Funding statement: The research of the second author is wholly supported by the University of KwaZulu-Natal, Durban, South Africa Postdoctoral Fellowship. He is grateful for the funding and financial support. The third author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF.

A Appendix

Algorithm A.1

Algorithm A.1 ([45, Algorithm 3.1])

For any x 0 ∈ H , let H 0 = H , A 0 = I H , B 0 = B , and let { x n } be the sequence generated by

u n = ∑ i = 0 N β n , i ⁢ [ x n − τ n , i ⁢ A i * ⁢ ( I H i − J λ n , i B i ) ⁢ A i ⁢ x n ] , x n + 1 = α n ⁢ f ⁢ ( x n ) + ( 1 − α n ) ⁢ u n , n ≥ 0 ,

where { α n } ⊂ ( 0 , 1 ) , and { β n , i } , { λ i , n } , i = 0 , 1 , … , N , are sequences of positive real numbers such that

{ β n , i } ⊂ [ c , d ] ⊂ ( 0 , 1 ) and ∑ i = 0 N β n , i = 1 for each ⁢ n ≥ 0 ,

and

τ n , i = ψ n , i ⁢ ∥ ( I H i − J λ n , i B i ) ⁢ A i ⁢ x n ∥ 2 ∥ A i * ⁢ ( I H i − J λ n , i B i ) ⁢ A i ⁢ x n ∥ 2 + ϕ n , i ,

where { ψ n , i } ⊂ [ e , f ] ⊂ ( 0 , 2 ) and { ϕ n , i } is a sequence of positive real numbers for each i = 0 , 1 , … , N , and f : H → H is a strict contraction mapping into itself with the coefficient ρ ∈ [ 0 , 1 ) .

Algorithm A.2

Algorithm A.2 ([45, Algorithm 3.4])

For any x 0 ∈ H , let H 0 = H , A 0 = I H , B 0 = B , and let { x n } be the sequence generated by

u n , i = J λ n , i B i ⁢ ( A i ⁢ x n ) , i = 0 , 1 , … , N , choose ⁢ i n ⁢ such that ⁢ ∥ A i n ⁢ x n − u n , i n ∥ = max i = 0 , 1 , … , N ⁡ { ∥ A i ⁢ x n − u n , i ∥ } , u n = x n − τ n , i n ⁢ A i n * ⁢ ( A i n ⁢ x n − u n , i n ) , x n + 1 = α n ⁢ f ⁢ ( x n ) + ( 1 − α n ) ⁢ u n , n ≥ 0 ,

where { α n } ⊂ ( 0 , 1 ) , and { β n , i } , { λ n , i } , i = 0 , 1 , … , N , are sequences of positive real numbers such that

{ β n , i } ⊂ [ c , d ] ⊂ ( 0 , 1 ) and ∑ i = 0 N β i , n = 1 for each ⁢ n ≥ 0 ,

and

τ n , i = ψ n , i ⁢ ∥ ( I H i − J λ n , i B i ) ⁢ A i ⁢ x n ∥ 2 ∥ A i * ⁢ ( I H i − J λ n , i B i ) ⁢ A i ⁢ x n ∥ 2 + ϕ n , i ,

where { ψ n , i } ⊂ [ e , f ] ⊂ ( 0 , 2 ) and { ϕ n , i } is a sequence of positive real numbers for each i = 0 , 1 , … , N , and f : H → H is a strict contraction mapping into itself with the coefficient ρ ∈ [ 0 , 1 ) .

Acknowledgements

The authors sincerely thank the reviewers for their careful reading, constructive comments and useful suggestions.

  1. Conflict of Interest: The authors declare that they have no competing interests.

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Received: 2022-04-19
Revised: 2022-12-06
Accepted: 2022-12-09
Published Online: 2023-03-01
Published in Print: 2023-07-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. A Symmetric Interior Penalty Method for an Elliptic Distributed Optimal Control Problem with Pointwise State Constraints
  3. The Tracking of Derivative Discontinuities for Delay Fractional Equations Based on Fitted L1 Method
  4. Two-Level Error Estimation for the Integral Fractional Laplacian
  5. Fractional Laplacian – Quadrature Rules for Singular Double Integrals in 3D
  6. A Priori Analysis of a Symmetric Interior Penalty Discontinuous Galerkin Finite Element Method for a Dynamic Linear Viscoelasticity Model
  7. Positivity-Preserving Numerical Method for a Stochastic Multi-Group SIR Epidemic Model
  8. Implicit-Explicit Finite Difference Approximations of a Semilinear Heat Equation with Logarithmic Nonlinearity
  9. A Novel Study Based on Shifted Jacobi Polynomials to Find the Numerical Solutions of Nonlinear Stochastic Differential Equations Driven by Fractional Brownian Motion
  10. On Split Monotone Variational Inclusion Problem with Multiple Output Sets with Fixed Point Constraints
  11. Local Discontinuous Galerkin Method for a Third-Order Singularly Perturbed Problem of Convection-Diffusion Type
  12. Simultaneous Inversion of the Space-Dependent Source Term and the Initial Value in a Time-Fractional Diffusion Equation
  13. A Posteriori Error Estimator for Weak Galerkin Finite Element Method for Stokes Problem Using Diagonalization Techniques
https://doi.org/10.1515/cmam-2022-0199
Schlagwörter für diesen Artikel
Forward-Backward Splitting Method; Split Monotone Variational Inclusion Problem; Fixed Point Problem; Nonexpansive Mappings; Adaptive Step Size; Inertial Technique

Artikel in diesem Heft

  1. Frontmatter
  2. A Symmetric Interior Penalty Method for an Elliptic Distributed Optimal Control Problem with Pointwise State Constraints
  3. The Tracking of Derivative Discontinuities for Delay Fractional Equations Based on Fitted L1 Method
  4. Two-Level Error Estimation for the Integral Fractional Laplacian
  5. Fractional Laplacian – Quadrature Rules for Singular Double Integrals in 3D
  6. A Priori Analysis of a Symmetric Interior Penalty Discontinuous Galerkin Finite Element Method for a Dynamic Linear Viscoelasticity Model
  7. Positivity-Preserving Numerical Method for a Stochastic Multi-Group SIR Epidemic Model
  8. Implicit-Explicit Finite Difference Approximations of a Semilinear Heat Equation with Logarithmic Nonlinearity
  9. A Novel Study Based on Shifted Jacobi Polynomials to Find the Numerical Solutions of Nonlinear Stochastic Differential Equations Driven by Fractional Brownian Motion
  10. On Split Monotone Variational Inclusion Problem with Multiple Output Sets with Fixed Point Constraints
  11. Local Discontinuous Galerkin Method for a Third-Order Singularly Perturbed Problem of Convection-Diffusion Type
  12. Simultaneous Inversion of the Space-Dependent Source Term and the Initial Value in a Time-Fractional Diffusion Equation
  13. A Posteriori Error Estimator for Weak Galerkin Finite Element Method for Stokes Problem Using Diagonalization Techniques
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