# Weak and Strong Convergence of an Iterative Algorithm for Lipschitz Pseudo-Contractive Maps in Hilbert Spaces

ABSTRACT Let H be a real Hilbert space and K a nonempty, closed convex subset of H.Let T : K → K be Lipschitz pseudo-contractive map with a nonempty fixed points set. We introduce a modified Ishikawa iterative algorithm for Lipschitz pseudo-contractive maps and prove that our new iterative algorithm converges strongly to a fixed point of T in real Hilbert space.

Contents

Certification ii

Dedication iii

Acknowledgement iv

Abstract viii

1 Introduction 1

1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Demiclosedness Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Nonlinear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Iterative Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 The Picard iteration Method . . . . . . . . . . . . . . . . . . . . . . 12

1.4.2 Krasnoselskii Iteration Method [6] . . . . . . . . . . . . . . . . . . . 13

1.4.3 The Mann Iteration Process [21] . . . . . . . . . . . . . . . . . . . . 14

1.4.4 The Ishikawa Iteration Process [19] . . . . . . . . . . . . . . . . . . 18

1.4.5 Mann Iteration Process with Errors in the Sense of Liu . . . . . . . 19

1.4.6 Ishikawa Iteration Process with Errors in the Sense of Liu . . . . . 19

vi

1.4.7 The Agarwal-O’Regan-Sahu Iteration Process . . . . . . . . . . . . 20

1.5 ORGANIZATION OF THESIS . . . . . . . . . . . . . . . . . . . . . . . . 21

2 Preliminaries 22

2.1 Definitions and Technical Results About Convergent Sequences of Real

Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.1 Definition (Strong Convergence) . . . . . . . . . . . . . . . . . . . . 22

2.1.2 Definition (Weak Convergence) . . . . . . . . . . . . . . . . . . . . 22

2.2 Projections onto Convex Set . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3 Some Definitions and Results Used in the Main Work . . . . . . . . . . . . 36

2.4 Corollary (Demiclosedness Principle) . . . . . . . . . . . . . . . . . . . . . 38

3 Weak and Strong Convergence of an Iterative Algorithm for Lipschitz

Pseudo-Contractive Maps in Hilbert Spaces 44

3.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

References 52

Subscribe to access this work and thousands more
Overall Rating

## 0

5 Star
(0)
4 Star
(0)
3 Star
(0)
2 Star
(0)
1 Star
(0)
APA

VELUHAN, E (2022). Weak and Strong Convergence of an Iterative Algorithm for Lipschitz Pseudo-Contractive Maps in Hilbert Spaces. Afribary. Retrieved from https://afribary.com/works/weak-and-strong-convergence-of-an-iterative-algorithm-for-lipschitz-pseudo-contractive-maps-in-hilbert-spaces

MLA 8th

VELUHAN, EZEKIEL "Weak and Strong Convergence of an Iterative Algorithm for Lipschitz Pseudo-Contractive Maps in Hilbert Spaces" Afribary. Afribary, 18 Oct. 2022, https://afribary.com/works/weak-and-strong-convergence-of-an-iterative-algorithm-for-lipschitz-pseudo-contractive-maps-in-hilbert-spaces. Accessed 09 Aug. 2024.

MLA7

VELUHAN, EZEKIEL . "Weak and Strong Convergence of an Iterative Algorithm for Lipschitz Pseudo-Contractive Maps in Hilbert Spaces". Afribary, Afribary, 18 Oct. 2022. Web. 09 Aug. 2024. < https://afribary.com/works/weak-and-strong-convergence-of-an-iterative-algorithm-for-lipschitz-pseudo-contractive-maps-in-hilbert-spaces >.

Chicago

VELUHAN, EZEKIEL . "Weak and Strong Convergence of an Iterative Algorithm for Lipschitz Pseudo-Contractive Maps in Hilbert Spaces" Afribary (2022). Accessed August 09, 2024. https://afribary.com/works/weak-and-strong-convergence-of-an-iterative-algorithm-for-lipschitz-pseudo-contractive-maps-in-hilbert-spaces

##### Document Details
Field: Mathematics Type: Thesis 65 PAGES (15657 WORDS) (pdf)