Abstract
Abstract In this paper we introduce a multi-step implicit iterative scheme with regularization for finding a common solution of the minimization problem (MP) for a convex and continuously Fréchet differentiable functional and the common fixed point problem of an infinite family of nonexpansive mappings in the setting of Hilbert spaces. The multi-step implicit iterative method with regularization is based on three well-known methods: the extragradient method, approximate proximal method and gradient projection algorithm with regularization. We derive a weak convergence theorem for the sequences generated by the proposed scheme. On the other hand, we also establish a strong convergence result via an implicit hybrid method with regularization for solving these two problems. This implicit hybrid method with regularization is based on the CQ method, extragradient method and gradient projection algorithm with regularization. MSC:49J30, 47H09, 47J20.
Highlights
Let H be a real Hilbert space with the inner product ·, · and the norm ·, let C be a nonempty closed convex subset of H and let PC be the metric projection of H onto C
We aim to find a common solution of the minimization problem (MP) for a convex and continuously Fréchet differentiable functional and the common fixed point problem of an infinite family of nonexpansive mappings in the setting of Hilbert spaces
Motivated and inspired by the research going on in this area, we propose two iterative schemes for this purpose
Summary
Let H be a real Hilbert space with the inner product ·, · and the norm · , let C be a nonempty closed convex subset of H and let PC be the metric projection of H onto C. If the mapping A is pseudomonotone Lipschitz-continuous, T is not necessarily a maximal monotone operator. To overcome this difficulty, Ceng et al [ ] suggested another iterative method. Ceng et al [ ] suggested another iterative method We aim to find a common solution of the minimization problem (MP) for a convex and continuously Fréchet differentiable functional and the common fixed point problem of an infinite family of nonexpansive mappings in the setting of Hilbert spaces.
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