Computational and Applied Mathematics 2020 39:38 https://doi.org/10.1007/s40314-019-1014-2 A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem L. O. Jolaoso 1 · A. Taiwo 1 · T. O. Alakoya 1 · O. T. Mewomo 1 Received: 3 May 2019 / Revised: 16 October 2019 / Accepted: 14 November 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019 Abstract In this paper, we propose a new extragradient method consisting of the hybrid steepest descent method, a single projection method and an Armijo line searching the technique for approximating a solution of variational inequality problem and finding the fixed point of demicontractive mapping in a real Hilbert space. The essence of this algorithm is that a single projection is required in each iteration and the step size for the next iterate is determined in such a way that there is no need for a prior estimate of the Lipschitz constant of the underlying operator. We state and prove a strong convergence theorem for approximating common solutions of variational inequality and fixed points problem under some mild conditions on the control sequences. By casting the problem into an equivalent problem in a suitable product space, we are able to present a simultaneous algorithm for solving the split equality problem without prior knowledge of the operator norm. Finally, we give some numerical examples to show the efficiency of our algorithm over some other algorithms in the literature. Keywords Variational inequality · Extragradient method · Split equality problem · Hyrbid-steepest descent · Armijo line search Mathematics Subject Classification 65K15 · 47J25 · 65J15 · 90C33 Communicated by Pablo Pedregal. B O. T. Mewomo mewomoo@ukzn.ac.za L. O. Jolaoso 216074984@stu.ukzn.ac.za; lateefjolaoso89@gmail.com A. Taiwo 218086816@stu.ukzn.ac.za T. O. Alakoya 218086823@stu.ukzn.ac.za; timimaths@gmail.com 1 School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa 123