Strong Convergence of a New Three Step Iterative Scheme in Banach Spaces

By Renu Chugh, Vivek Kumar, Sanjay Kumar. 2012

135 citations recorded by the source corpus.

In this paper, we suggest a new type of three step iterative scheme called the CR iterative scheme and study the strong convergence of this iterative scheme for a certain class of quasi-contractive operators in Banach spaces. We show that for the aforementioned class of operators, the CR iterative scheme is equivalent to and faster than Picard, Mann, Ishikawa, Agarwal et al., Noor and SP iterative schemes. Moreover, we also present various numerical examples using computer programming in C++ for the CR iterative scheme to compare it with the other above mentioned iterative schemes. Our results show that as far as the rate of convergence is concerned 1) for increasing functions the CR iterative scheme is best, while for decreasing functions the SP iterative scheme is best; 2) CR iterative scheme is best for a certain class of quasi-contractive operators.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/13eab3707df560ef9b8ff4c6fa12df09bcbaaf2e

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In Computational Theory and Mathematics

1. Faik Gursoy
2. Muhammad Arshad
3. Vatan Karakaya
4. Kifayat Ullah

In Geometry and Topology

1. Muhammad Arshad
2. Kifayat Ullah
3. Faik Gursoy
4. Vatan Karakaya

In Numerical Analysis

1. Muhammad Arshad
2. Kifayat Ullah
3. Faik Gursoy
4. Vatan Karakaya

Citations in the corpus

• Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process (2018, methodology): “In [4], the authors claimed that CR iteration process is equivalent to and faster than Picard, Mann, Ishikawa, Agarwal, Noor and SP iterative processes for quasi-contractive operators in Banach spaces.”
• SOME FIXED POINT RESULTS FOR A NEW THREE STEPS ITERATION PROCESS IN BANACH SPACES (2017, background): “7) converges faster than all S [2], CR [5], Ishikawa [9], S* [10], Mann [12], Noor [13], SP [15], Picard [16], Picard-Mann [17], Picard-S (1.”
• A PICARD-S HYBRID TYPE ITERATION METHOD FOR SOLVING A DIFFERENTIAL EQUATION WITH RETARDED ARGUMENT (2014, methodology): “[10] shows that CR iterative method (1.”