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In this article, we present an inertial subgradient extragradient-type method that uses a non-monotone step size rule to find a numerical solution to equilibrium problems in real Hilbert spaces. The presented iterative scheme is based on an extragradient subgradient method and an inertial-type scheme. In fact, the proposed iterative scheme is effective in terms of performance, and the key advantage derives directly from the use of the variable step size rule, which is revised by each iteration on the basis of the Lipschitz-type constants as well as certain prior iterations. We obtain a weak convergence theorem for a new method by using mild conditions on a bifunction. Applications of the main results are given to solve various nonlinear problems. Several numerical findings are given in order to illustrate the numerical behaviour of the proposed method and to compare it to others. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in solving minimization problem and common fixed point problem of finite family consisting asymptotically nonexpansive mappings in Hadamard spaces. For finding common solutions of their problems, we introduce a new modified proximal point algorithm involving convex combination technique. Under suitable conditions, Δ-convergence and strong convergence theorems are proved. Furthermore, we provide some numerical examples which show the efficiency and implementation of the proposed algorithm as well as show computational overview by comparing it with previously known algorithms. [ABSTRACT FROM AUTHOR]