Our results extend and improve many recent results in thisdirection.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper, we study the problem of finding a common solution to split generalized mixed equilibriumproblem and fixed point problem for quasi-φ-nonexpansive mappings in 2-uniformly convex and uniformly smoothBanach space E1 and a smooth, strictly convex, and reflexive Banach space $E_2$ .
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Key Findings & Data:Finally,numerical experiments are presented to demonstrate the practicability, efficiency, and performance of our algorithm incomparison with other existing algorithms in the literature.
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Conclusion & Impact:Our results extend and improve many recent results in thisdirection.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters
Abstract
In this paper, we study the problem of finding a common solution to split generalized mixed equilibriumproblem and fixed point problem for quasi-φ-nonexpansive mappings in 2-uniformly convex and uniformly smoothBanach space E1 and a smooth, strictly convex, and reflexive Banach space $E_2$ . An iterative algorithm with Armijolinesearch rule for solving the problem is presented and its strong convergence theorem is established. The convergenceresult is obtained without using the hybrid method which is mostly used when strong convergence is desired. Finally,numerical experiments are presented to demonstrate the practicability, efficiency, and performance of our algorithm incomparison with other existing algorithms in the literature. Our results extend and improve many recent results in thisdirection.
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Cite This Article
OYEWOLE, O. K., MEWOMO, O. T., JOLAOSO, L. O., KHAN, S. H. (2022). An extragradient algorithm for split generalized equilibrium problem and the set of fixed points of quasi-φ-nonexpansive mappings in Banach spaces. Turkish Journal of Mathematics, Cilt: 44 - Sayı: 4, 1146–1170. https://ajindex.com/articles/607490
Agarwal RP, O’Regan D, Saha DR. Fixed Point Theory for Lipschitizian-Type Mappings with Applications. New York, NY, USA: Springer, 2009
Alber YI. Metric and generalized projection operators in Banach spaces: properties and applications In: Kartsatos AG (editor). Theory and Applications of Nonlinear Operators and Accretive and Monotone Type. Lecture Notes in Pure and Applied Mathematics, Dekker, New York; 1996; 178: 15-50
Ansari QH. Rehan A. Split feasibility and fixed point problems. In: Ansari QH (editor), Nonlinear Analysis: Approximation Theory: Optimization and Application, Springer, New York 2014: 281-322
Antipin AS. The convergence of proximal methods to fixed points of external mappings and estimates of their rates of convergence. Computational Mathematics and Mathematical Physics 1995; 35: 539-551
Blum E, Oetlli W. From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994; 63: 123-145
Bryne C. Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Problems 2002; 18: 441-453
Butnariu D, Reich S, Zaslavski AJ. Asymptotic behavior of relatively nonexpansive operators in Banach spaces, Journal of Applied Analysis, 2001; 7: 151-174
Butnariu D, Reich, S, Zaslavki AJ. Weak convergence of orbits of nonlinear operators in reflexive Banach spaces, Numerical Functional Analysis and Optimization, 2003; 24: 489-508
Ceng LC, Yao JC. A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. Journal of Computational and Applied Mathematics 2008; 214: 186-201
Censor Y, Reich S. Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 1996; 37: 323-339
Chidume CE, Nnakwe MO. Iterative algorithms for split variational Inequalities and generalized split feasibility problems with applications. Journal of Nonlinear Varaitional Inequality 2019; 3: 127-140
Cioranescu I. Geometry of Banach spaces, Duality Mappings and Nonlinear. Dordrecht, the Netherlands: Kluwer, 1990
Cohen G. Auxiliary problem principle and decomposition of optimization problems. Journal of Optimization Theory and Applications 1980; 32:277-305
Cohen G. Auxiliary principle extended to variational inequalities. Journal of Optimization Theory and Applications 1988; 59: 325-333
Dinh BV, Son DX, Jiao L, Kim DS. Linesearch algorithms for split equilibrium pr