An accelerated hybrid projection method with a self‐adaptive step‐size sequence for solving split common fixed point problems

Abstract

This paper attempts to solve the split common fixed point problem for demicontractive mappings. We give an accelerated hybrid projection algorithm that combines the hybrid projection method and the inertial technique. The strong convergence theorem of this algorithm is obtained under mild conditions by a self-adaptive step-size sequence, which does not need prior knowledge of the operator norm. Some numerical experiments in infinite-dimensional Hilbert spaces are provided to illustrate the reliability and robustness of the algorithm and also to compare it with existing ones. KEYWORDS demicontractive mapping, hybrid projection method, inertial technique, self-adaptive step-size sequence, split common fixed point problem MSC CLASSIFICATION 47H09; 47H10; 47J25; 65K15 INTRODUCTIONLet  be a Hilbert space with the norm || • || and the inner product ⟨• , •⟩ and F(K) be the fixed point set of a mapping K. Recall that the mapping K ∶  →  is said to be a strictly pseudocontractive if there exists a constant ∈ [0, 1) such thatBoth the demicontractive mapping and the strictly pseudocontractive mapping were studied by many authors. [1][2][3][4] Besides, Suparatulatorn et al. 5 and Yao et al. 6 considered the other case, which is the demicontractive mapping with coefficient ∈ [0, 1). Obviously, this case is contained in the demicontractive mapping with coefficient ∈ (− ∞, 1) that we are going to study. Meanwhile, the strictly pseudocontractive mapping with F(K) ≠ ∅ is the demicontractive mapping. The opposite is contradictory. Further, according to the properties of the norm and the inner product in Hilbert spaces, we know that the fixed point set of a demicontractive mapping is closed and convex.

Inertial algorithms with adaptive stepsizes for split variational inclusion problems and their applications to signal recovery problem

“…Here, S F 1 and S F 2 are defined as (2) and 𝜆 n is defined as (11). If the solution set 𝛷 of SVIP* is nonempty, the sequence {z n } converges in norm to z * ∈ 𝛷 and z * = P 𝛷 (0), that is, the minimum-norm element of 𝛷.…”

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Accelerated inertial subgradient extragradient algorithms with non-monotonic step sizes for equilibrium problems and fixed point problems

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