Abstract
We introduce a new projection and contraction method with inertial and self-adaptive techniques for solving variational inequalities and split common fixed point problems in real Hilbert spaces. The stepsize of the algorithm is selected via a self-adaptive method and does not require prior estimate of norm of the bounded linear operator. More so, the cost operator of the variational inequalities does not necessarily needs to satisfies Lipschitz condition. We prove a strong convergence result under some mild conditions and provide an application of our result to split common null point problems. Some numerical experiments are reported to illustrate the performance of the algorithm and compare with some existing methods.
Highlights
Let H be a real Hilbert space induced with norm k · k and inner product h·, ·i
The problem has drawn the attention of many researchers who had studied its existence of solution and proposed various iterative methods such as the extragradient method [4,5,6,7,8,9], subgradient extragradient method [10,11,12,13,14], projection and contraction method [15,16], Tseng’s extragradient method [17,18] and Bregman projection method [19,20] for approximating its solution in various dimensions
In this paper, we introduce a new inertial projection and contraction method for finding a common solution of VIP and split common fixed point problem, that is, find x∈Ω
Summary
Let H be a real Hilbert space induced with norm k · k and inner product h·, ·i. Let Ω be a nonempty, closed and convex subset of H and A : Ω → H be an operator. The authors proved that the sequence generated by (3) converges weakly to a solution of the VIP He [24] introduced a Projection and Contraction Method (PCM). The authors of Reference [27] proved that the sequence { xn } generated by Algorithm (5) converges strongly to a solution of the VIP provided the condition limn→∞ αθnn k xn − xn−1 k = 0 is satisfied. (iii) the algorithm converges weakly to a solution of (9) Motivated by these results, in this paper, we introduce a new inertial projection and contraction method for finding a common solution of VIP and split common fixed point problem, that is, find x∈Ω such that x ∈ S ∩ F (T ). Our algorithm performs only one projection onto C and no extra projection onto any subset of H
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