In this paper, we will give extended versions of two standard fixed point principles: one for Hardy-Rogers type operators and the other one forĆirić type operators in complete metric space. Our results generalize similar theorems given in [9].

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In this paper, we propose a variable metric version of Tseng's algorithm (the forward-backward-forward algorithm: FBF) combined with extrapolation from the past that includes error terms for finding a zero of the sum of a maximally monotone operator and a monotone Lipschitzian operator in Hilbert spaces. This can be seen as the optimistic gradient descent ascent (OGDA) algorithm endowed with variable metrics and error terms. Primal-dual algorithms are also proposed for monotone inclusion problems involving compositions with linear operators. The primal-dual problem occurring in image deblurring demonstrates an application of our theoretical results.