Abstract
In this article, we study the extended split equality problem and extended split equality fixed point problem, which are extensions of the convex feasibility problem. For solving the extended split equality problem, we present two self-adaptive stepsize algorithms with internal perturbation projection and obtain the weak and the strong convergence of the algorithms, respectively. Furthermore, based on the operators being quasinonexpansive, we offer an iterative algorithm to solve the extended split equality fixed point problem. We introduce a way of selecting the stepsize which does not need any prior information about operator norms in the three algorithms. We apply our iterative algorithms to some convex and nonlinear problems. Finally, several numerical results are shown to confirm the feasibility and efficiency of the proposed algorithms.
Highlights
Let H1, H2, and H3 be three real Hilbert spaces and C ⊂ H1 and Q ⊂ H2 be two nonempty, closed, and convex sets
The split feasibility problem (SFP) was first introduced by Censor and Elfving [1], which was used in modeling various inverse problems arising from phase retrievals and medical image reconstruction and further studied by many researchers
We introduce two simultaneous iterative algorithms with internal perturbation projection to solve extended split equality problem (ESEP) (5) and define the solution set of ESEP (5) as Ω1: Algorithm 9
Summary
Let H1, H2, and H3 be three real Hilbert spaces and C ⊂ H1 and Q ⊂ H2 be two nonempty, closed, and convex sets. The extended split equality problem is to find x1 ∈ C1, x2 ∈ C2, ⋯, xn ∈ Cn such that A1x1 ð5Þ = A2x2 = ⋯ = Anxn, where Ai : Hi ⟶ H are linear operators They presented the following simultaneous iterative algorithm:. To solve problem (7), Che and Li [23] proposed the following iterative algorithm: They established the weak convergence of scheme (8) under the conditions that the operators S and T are quasinonexpansive mappings. Che et al [19] proposed the following extended split equality fixed point problem (ESEFPP), which is to find x1 ∈ FixðG1Þ, x2 ∈ FixðG2Þ, ⋯, xn ∈ FixðGnÞ such that A1x1 = A2x2 = ⋯ = Anxn, ð9Þ and presented the following simultaneous iterative algorithm: Journal of Function Spaces. Several numerical results are shown to confirm the effectiveness of our algorithms
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