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Algorithms for Computing of Optimization Problems with a Common Minimum-Norm Fixed Point with Applications

Year: 2016 Volume: 10 Issue: 12 688 - 696 Pages

• Authors:
• Apirak Sombat
• Teerapol Saleewong
• Poom Kumam
• Parin Chaipunya
• Wiyada Kumam
• Anantachai Padcharoen
• Yeol Je Cho
• Thana Sutthibutpong

Abstract: This research is aimed to study a two-step iteration process defined over a finite family of σ-asymptotically quasi-nonexpansive nonself-mappings. The strong convergence is guaranteed under the framework of Banach spaces with some additional structural properties including strict and uniform convexity, reflexivity, and smoothness assumptions. With similar projection technique for nonself-mapping in Hilbert spaces, we hereby use the generalized projection to construct a point within the corresponding domain. Moreover, we have to introduce the use of duality mapping and its inverse to overcome the unavailability of duality representation that is exploit by Hilbert space theorists. We then apply our results for σ-asymptotically quasi-nonexpansive nonself-mappings to solve for ideal efficiency of vector optimization problems composed of finitely many objective functions. We also showed that the obtained solution from our process is the closest to the origin. Moreover, we also give an illustrative numerical example to support our results.

• Keywords:
• σ-asymptotically quasi-nonexpansive nonselfmapping
• strong convergence
• fixed point
• uniformly convex and uniformly smooth Banach space.

Approximating Fixed Points by a Two-Step Iterative Algorithm

Year: 2014 Volume: 8 Issue: 6 955 - 957 Pages

• Authors:
• Safeer Hussain Khan

Abstract: In this paper, we introduce a two-step iterative algorithm to prove a strong convergence result for approximating common fixed points of three contractive-like operators. Our algorithm basically generalizes an existing algorithm..Our iterative algorithm also contains two famous iterative algorithms: Mann iterative algorithm and Ishikawa iterative algorithm. Thus our result generalizes the corresponding results proved for the above three iterative algorithms to a class of more general operators. At the end, we remark that nothing prevents us to extend our result to the case of the iterative algorithm with error terms.

• Keywords:
• Contractive-like operator
• iterative algorithm
• fixed point
• strong convergence.

Fixed Points of Contractive-Like Operators by a Faster Iterative Process

Year: 2013 Volume: 7 Issue: 12 1754 - 1756 Pages

• Authors:
• Safeer Hussain Khan

Abstract: In this paper, we prove a strong convergence result using a recently introduced iterative process with contractive-like operators. This improves andgeneralizes corresponding results in the literature in two ways: iterativeprocess is faster, operators are more general. At the end, we indicatethat the results can also be proved with the iterative process witherror terms.

• Keywords:
• Contractive-like operator
• iterative process
• fixed point
• strong convergence.

Convergence of a One-step Iteration Scheme for Quasi-asymptotically Nonexpansive Mappings

Year: 2012 Volume: 6 Issue: 3 305 - 307 Pages

• Authors:
• Safeer Hussain Khan

Abstract: In this paper, we use a one-step iteration scheme to approximate common fixed points of two quasi-asymptotically nonexpansive mappings. We prove weak and strong convergence theorems in a uniformly convex Banach space. Our results generalize the corresponding results of Yao and Chen [15] to a wider class of mappings while extend those of Khan, Abbas and Khan [4] to an improved one-step iteration scheme without any condition and improve upon many others in the literature.

• Keywords:
• One-step iteration scheme
• asymptotically quasi non expansive mapping
• common fixed point
• condition (a')
• weak and strong convergence.