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The Problem of Split Equality Fixed-Point and its Applications

Lawan Bulama Mohammed, Adem Kilicman

Abstract

It is generally known that in order to solve the split equality fixed-point problem (SEFPP), it is necessary to compute the norm of bounded and linear operators, which is a challenging task in real life, to address this issue, we studied the SEFPP involving the class of quasi-pseudocontractive mappings in Hilbert spaces and constructed novel algorithms in this regards, and we proved the algorithms' convergence both with and without prior knowledge of the operator norm for bounded and linear mappings. Additionally, we gave applications and numerical examples of our findings. A variety of well-known discoveries revealed in the literature are generalized by the findings presented in this work.

The Problem of Split Equality Fixed-Point and its Applications

Abstract

It is generally known that in order to solve the split equality fixed-point problem (SEFPP), it is necessary to compute the norm of bounded and linear operators, which is a challenging task in real life, to address this issue, we studied the SEFPP involving the class of quasi-pseudocontractive mappings in Hilbert spaces and constructed novel algorithms in this regards, and we proved the algorithms' convergence both with and without prior knowledge of the operator norm for bounded and linear mappings. Additionally, we gave applications and numerical examples of our findings. A variety of well-known discoveries revealed in the literature are generalized by the findings presented in this work.
Paper Structure (12 sections, 12 theorems, 81 equations, 2 tables)

This paper contains 12 sections, 12 theorems, 81 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. SEFPP With Prior Knowledge on Operator Norms
  5. Claim $(x_{n},y_{n})\rightharpoonup (x^{*},y^{*})$
  6. SEFPP Without Prior Knowledge on Operator Norms
  7. APPLICATIONS
  8. Application to SFP
  9. Application to Split Variational Inequality Problem (SVIP)
  10. Application to the Split Convex Minimization Problem(SCMP)
  11. NUMERICAL EXAMPLES
  12. Conclusion

Key Result

Lemma 2.2

2 Suppose $\mathbb{T}: \mathbb{H}_{1}\to \mathbb{H}_{1}$ is Lipschitz with $L>0,$ and $\mathbb{U}:= (1-\eta)I +\eta \mathbb{T}((1-\zeta)I+\zeta \mathbb{T}),$ then

Theorems & Definitions (20)

  • Remark 1.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 10 more