Abstract
In a real Hilbert space, we aim to investigate two modified Mann subgradient-like methods to find a solution to pseudo-monotone variational inequalities, which is also a common fixed point of a finite family of nonexpansive mappings and an asymptotically nonexpansive mapping. We obtain strong convergence results for the sequences constructed by these proposed rules. We give some examples to illustrate our analysis.
Highlights
Let the h·, ·i and k · k represent the inner product and induced norm in a real Hilbert space H, respectively
We denote by PC the nearest point projection from H onto C, where
Given T : C → H a nonlinear mapping, we denote by Fix( T ) the fixed point set of T, i.e., Fix( T ) = { x ∈ C : x = Tx }
Summary
Let the h·, ·i and k · k represent the inner product and induced norm in a real Hilbert space H, respectively. In study [22], Reich et al suggested the modified projection-type method for solving the VIP with the pseudo-monotone and uniformly continuous mapping A, given a sequence {αn } ⊂ (0, 1) and a contraction f : C → C with constant $ ∈ [0, 1). Ceng, Yao and Shehu [21] proposed a Mann-type method of (2) to solve pseudo-monotone variational inequalities and the common fixed point problem of many finitely nonexpansive self-mappings { Ti }iN=1 on C and an asymptotically nonexpansive self-mapping T0 := T on C. In a real Hilbert space H, let the VIP and CFPP represent the pseudo-monotone variational inequality problem with uniformly continuous mapping A, the common fixed point problem of a finite family of nonexpansive mappings { Ti }iN=1 , and an asymptotically nonexpansive mapping T0 := T, respectively. The strong convergence criteria for the sequence { xn } in this paper are more convenient and more beneficial in comparison with those of studies [9,21]
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