Abstract
Abstract In this paper, we introduce a shrinking projection method of an inertial type with self-adaptive step size for finding a common element of the set of solutions of a split generalized equilibrium problem and the set of common fixed points of a countable family of nonexpansive multivalued mappings in real Hilbert spaces. The self-adaptive step size incorporated helps to overcome the difficulty of having to compute the operator norm, while the inertial term accelerates the rate of convergence of the proposed algorithm. Under standard and mild conditions, we prove a strong convergence theorem for the problems under consideration and obtain some consequent results. Finally, we apply our result to solve split mixed variational inequality and split minimization problems, and we present numerical examples to illustrate the efficiency of our algorithm in comparison with other existing algorithms. Our results complement and generalize several other results in this direction in the current literature.
Highlights
Let H be a real Hilbert space with inner product ⟨⋅,⋅⟩ and induced norm ∥⋅∥
Motivated by the above results and the ongoing research interest in this direction, in this paper, we present a new inertial shrinking projection algorithm, which does not require any prior knowledge of the operator norm for finding a common element of the set of solutions of split generalized equilibrium problem (SGEP) and the set of common fixed points of a countable family of nonexpansive multivalued mappings in Hilbert spaces
We apply our results to solve split mixed variational inequality problem (SMVIP) and split minimization problem (SMP), and we provide numerical examples to illustrate the efficiency of the proposed algorithm in comparison with the existing results in the current literature
Summary
Let H be a real Hilbert space with inner product ⟨⋅,⋅⟩ and induced norm ∥⋅∥. Let C be a nonempty closed convex subset of H and φ : C × C → , F : C × C → be two bifunctions. Takahashi et al [24] introduced an iterative scheme known as the shrinking projection method for finding a fixed point of a nonexpansive single-valued mapping in Hilbert spaces. Phuengrattana and Lerkchaiyaphum [26] introduced the following shrinking projection method for solving SGEP and FPP of a countable family of nonexpansive multivalued mappings: for x1 ∈ C and C1 = C, ( ( ) ) un = Tr(nF1,φ1) I − γA∗ I − Tr(nF2,φ2) A xn, zn αn(0) xn αn(1) yn(1). Motivated by the above results and the ongoing research interest in this direction, in this paper, we present a new inertial shrinking projection algorithm, which does not require any prior knowledge of the operator norm for finding a common element of the set of solutions of SGEP and the set of common fixed points of a countable family of nonexpansive multivalued mappings in Hilbert spaces.
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