Abstract
Inspired by the works of Alvarez and Attouch (Set-Valued Anal. 9:3–11, 2001), López et al. (Inverse Probl. 28:ID085004, 2012), Takahashi (Arch. Math. (Basel) 104(4):357–365, 2015) and Suantai et al. (Appl. Gen. Topol. 18(2):345–360, 2017), as well as Promluang and Kuman (J. Inform. Math. Sci. 9(1):27–44, 2017), we propose a new inertial algorithm for solving split common null point problem without the prior knowledge of the operator norms in Banach spaces. Under mild and standard conditions, the weak and strong convergence theorems of the proposed algorithms are obtained. Also the split minimization problem is considered as the application of our results. Finally, the performances and computational examples are presented, and a comparison with related algorithms is provided to illustrate the efficiency and applicability of our new algorithm.
Highlights
In an excellent paper [6], Byrne, Censor, Gibali and Reich introduced the following split common null point problem (SCNPP) for set-valued operators: find a point x∗ ∈ H1 such that p 0 ∈ Aix∗, i=1 (1.1)and y∗j = Tjx∗ ∈ H2 such that 0 ∈ Bj y∗j, for each j = 1, 2, . . . , r, (1.2)where H1 and H2 are two real Hilbert spaces and Ai : H1 → 2H1, Bj : H2 → 2H2 are maximal monotone operators, Tj : H1 → H2 are bounded linear operators.The split common null point problem is motivated by many related problems
The first is the split inverse problem (SIP) which is formulated in Censor, Gibali and Reich [7]
3.3 Strong convergence analysis for Algorithm 3.2 For the strong convergence theorem of Algorithm 3.2, which we present we recall the minimum-norm element of Ω, which is a solution of the following problem: argmin x : x ∈ H, x ∈ A–1(0) and y = Tx ∈ B–1(0) ⊂ E
Summary
The first is the split inverse problem (SIP) which is formulated in Censor, Gibali and Reich [7]. It concerns a model in which two vector spaces X and Y and a linear operator A : X → Y are given. The first one, denoted by IP1, Tang Journal of Inequalities and Applications (2019) 2019:17 is formulated in the space X, and the second one, denoted by IP2, is formulated in the space Y Given these data, the SIP is formulated as follows: find a point x∗ ∈ X that solves IP1 and such that the point y = Tx∗ ∈ Y solves IP2
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