Abstract
We at first raise the so called split feasibility fixed point problem which covers the problems of split feasibility, convex feasibility, and equilibrium as special cases and then give two types of algorithms for finding solutions of this problem and establish the corresponding strong convergence theorems for the sequences generated by our algorithms. As a consequence, we apply them to study the split feasibility problem, the zero point problem of maximal monotone operators, and the equilibrium problem and to show that the unique minimum norm solutions of these problems can be obtained through our algorithms. Since the variational inequalities, convex differentiable optimization, and Nash equilibria in noncooperative games can be formulated as equilibrium problems, each type of our algorithms can be considered as a generalized methodology for solving the aforementioned problems.
Highlights
Throughout this paper, H denotes a real Hilbert space with inner product ⟨⋅, ⋅⟩ and the norm ‖ ⋅ ‖, I the identity mapping on H, N the set of all natural numbers, and R the set of all real numbers
Convex differentiable optimization, and Nash equilibria in noncooperative games can be formulated as equilibrium problems, each type of our algorithms can be considered as a generalized methodology for solving the aforementioned problems
The split feasibility problem (SFP) was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and medical image reconstruction
Summary
Throughout this paper, H denotes a real Hilbert space with inner product ⟨⋅, ⋅⟩ and the norm ‖ ⋅ ‖, I the identity mapping on H, N the set of all natural numbers, and R the set of all real numbers. Putting A = I, the previous SFFP is reduced to the common zero point problem of two maximal monotone operators: find x∗ ∈ H1 so that x∗ ∈ M−10 ∩ N−10. (iii) When H1 = H2 = H, and the bounded linear operator A is the identity mapping, SFP(1) is reduced to the convex feasibility problem (CFP): find x∗ ∈ H so that x∗ ∈ C ∩ Q,. Based on the concept of using contractions to approximate nonexpansive mappings, another type of algorithms for SFFP(5) is introduced, and the corresponding strong convergence theorem for the sequence generated by such algorithm is given too. The proposed algorithm becomes a scheme to approach the minimum norm solution of zero point problem of maximal monotone operators and the equilibrium problem. It is worth noting that as Blum and Oettli [7] showed that the variational inequalities, convex differentiable optimization, and Nash equilibria in noncooperative games can be formulated as equilibrium problems, the proposed algorithm can be considered as a generalized methodology for solving all aforementioned problems
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