We extend the concept of relaxed α-monotonicity to mixed relaxed α-β-monotonicity. The concept of mixed relaxed α-β-monotonicity is more general than many existing concepts of monotonicities. Finally, we apply this concept and well known KKM-theory to obtain the solution of generalized equilibrium problem.

In this article, a nonlinear system of generalized ordered XOR-inclusion problems in Hilbert space is introduced and studied. Initially, we define the resolvent operator related to the (α,λ)-XOR-weak-ANODD multivalued mapping. Using the fixed point technique, we demonstrate the results of existence. In order to make the suggested system more realistic, we create the S-iterative algorithm and demonstrate that the sequence generated through this technique strongly converges with the proposed system’s solution. One example is provided in support of the existence result as well.

Abstract:In this paper, we introduce and study an extended mixed equilibrium problem by using auxiliary principle technique. A generalized predictor-corrector iterative algorithm is defined for solving extended mixed equilibrium problem. The convergence of the method mentioned requires some condition ( * ), g-relatively relaxed Lipschitz continuity and relatively g-relaxed monotonicity of the mappings.

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