Equilibrium problems under relaxed α-monotonicity on Hadamard manifolds

In this paper, we study the existence of solutions of both equilibrium problems and mixed equilibrium problems on Hadamard manifolds. Under relaxed α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-pseudomonotonicity assumption on the underlying bifunction we prove that the solution set of the equilibrium problem is nonempty. We also provide the existence of solution of mixed equilibrium problems with relaxed α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-monotonicity. The results presented in this paper generalize and improve some known results given in literature, see for example (Colao et al. in J Math Anal Appl 388:61–77, 2012; Jana and Nahak in Rend Circ Mat Palermo(2) 65(1):97–109, 2016; Mahato and Nahak in OPSEARCH, 2013. https://doi.org/10.1007/s12597-013-0142-5).