A Self-Adaptive Algorithm for Split Null Point Problems and Fixed Point Problems for Demicontractive Multivalued Mapping
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A Self-Adaptive Algorithm for Split Null Point Problems and Fixed Point Problems for Demicontractive Multivalued Mappings Suthep Suantai1 · Pachara Jailoka1
Received: 5 November 2019 / Accepted: 23 September 2020 © Springer Nature B.V. 2020
Abstract In this work, we study the split null point problem and the fixed point problem in Hilbert spaces. We introduce a self-adaptive algorithm based on the viscosity approximation method without prior knowledge of the operator norm for finding a common solution of the considered problem for maximal monotone mappings and demicontractive multivalued mappings. A strong convergence result of our proposed algorithm is established under some suitable conditions. Some convergence results for the split feasibility problem and the split minimization problem are consequences of our main result. Finally, we also give numerical examples for supporting our main result. Keywords Fixed points · Split null point problems · Demicontractive mappings · Maximal monotone mappings · Resolvent operators Mathematics Subject Classification 47H10 · 47J25 · 54H25
1 Introduction Throughout this work, we assume that H, H1 and H2 are real Hilbert spaces with inner products ·, · and the induced norms · , and let I be the identity operator on a Hilbert space. Denote by N and R the set of positive integers and the set of real numbers, respectively. The split inverse problem (SIP) concerns a model of finding a point x ∗ ∈ H1 that solves IP1 such that Ax ∗ ∈ H2 solves IP2 ,
(1.1)
where IP1 and IP2 denote inverse problems formulated in H1 and H2 , respectively, and A : H1 → H2 is a bounded linear operator. In 1994, Censor and Elfving [7] introduced the
B P. Jailoka
[email protected] S. Suantai [email protected]
1
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
S. Suantai, P. Jailoka
first instance of the split inverse problem which is the split feasibility problem (SFP). After that, various split problems were introduced and studied such as the split common fixed point problem (SCFPP) [8], the split common null point problem (SCNPP) [5], the split variational inequality problem (SVIP) [11], the split equilibrium problem (SEP) [23], the split minimization problem (SMP), etc. Recently, the split inverse problem has been widely studied by many authors (see [4, 5, 7–9, 11, 15, 17, 23]) due to its model can be applied in various problems related to significant real-world applications. For example, in signal and image processing can be formulated as the equation system: y = Ax + ε,
(1.2)
where y ∈ RM is the vector of noisy observations, x ∈ RN is a recovered vector, ε is the noise, and A : RN → RM is a bounded linear observation operator. It is known that (1.2) can be modeled as the constrained least-squares problem: min
x∈RN
1 y − Ax22 subject to x1 ≤ σ, 2
(1.3)
for any nonnegative real number σ . Then, we can apply SIP model (1.1) to Problem (1.3) as follows: Find a point x ∈ RN such that x1 ≤ σ and Ax = y. In addition, Kotzer et al.
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