Strong convergence algorithm for solving an equilibrium problem and a fixed point problem using the Bergman distance in Banach spaces

Subject Areas : Analyze

Mostafa Ghadampour ^{1 *} , ٍEbrahim Soori ²

1 - Department of Mathematics, Payame Noor University, Tehran, Iran
2 - Department of Mathematics, Lorestan University, Lorestan, Khoramabad, Iran.

Received: 2022-04-19 Accepted : 2022-11-08 Published : 2025-04-08

Keywords: مسئله نقطه ثابت, نقطه ثابت مجانبی, نگاشت غیر انبساطی برگمن, نابرابری تغییراتی, فرشه دیفرانسیل پذیر,

Abstract :

In this paper, using the Bergman distance, we introduce a new projection-type algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points. Then, the strong convergence of the sequence generated by the algorithm under suitable conditions is proved. In fact, we prove that the sequence generated by the algorithm converges to the projection of an element in the intersection of the fixed points set and the solutions set of an equilibrium problem. For this purpose, we introduce a Bergman Lipschitz-type condition for a pseudomonotone bifunction. Then, we see an application for a variational inequality problem and we apply our result for finding a common element of the solution set of a variational inequality problem and the set of fixed points of a nonexpansive mapping. Finally, using MATLAB software, we provide a numerical example to illustrate the convergence performance of the main algorithm.

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