Solving a General Split Equality Problem Without Prior Knowledge of Operator Norms in Banach Spaces
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Results in Mathematics
Solving a General Split Equality Problem Without Prior Knowledge of Operator Norms in Banach Spaces Gholamreza Zamani Eskandani and Masoumeh Raeisi Abstract. In this paper, using Bregman distance, we introduce an iterative algorithm for approximating a common solution of Split Equality Fixed Point Problem and Split Equality Equilibrium Problem in p-uniformly convex and uniformly smooth Banach spaces that are more general than Hilbert spaces. The advantage of the algorithm is that it is done without the prior knowledge of Bregman Lipschitz coefficients and operator norms. The strong convergence of the algorithm is established under mild assumptions. As special cases, we shall utilize our results to study the Split Equality Null point Problems and Split Equality Variational Inequality Problems. A numerical example is given to demonstrate the convergence of the algorithm. Our results complement and extend some related results in the literature. Mathematics Subject Classification. 47H05, 47H09, 47H10. Keywords. Split equality fixed point problem, pseudomonotone bifunction, bregman projection, duality mapping, variational inequality.
1. Introduction Let A : H1 → H3 and B : H2 → H3 be two bounded linear operators, T : H1 → H1 and S : H2 → H2 be two nonlinear operators, where Hi (1 ≤ i ≤ 3) is a Hilbert space. The Split Equality Fixed Point Problem (SEFPP) studied by Moudafi in [35] is to find: x∗ ∈ F (T ), y ∗ ∈ F (S) such that Ax∗ = By ∗ .
(1)
which allows asymmetric and partial relations between the variables x and y. The interest is to cover many situations, for instance, in decomposition methods for PDE’s, applications in game theory and in intensity-modulated 0123456789().: V,-vol
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G. Z. Eskandani and M. Raeisi
Results Math
radiation therapy (IMRT). In decision sciences, this allows to consider agents who interplay only via some components of their decision variables (see [3]). In IMRT, this amounts to envisaging a weak coupling between the vector of doses absorbed in all voxels and that of the radiation intensity ([16]). If H2 = H3 and B = I, then the Split Equality Fixed Point Problem (1) reduces to the following Split Common Fixed Point Problem (SCFPP) introduced by Censor and Segal in [15]: Find x∗ ∈ F (T ) such that Ax∗ ∈ F (S).
(2)
SCFPP is in itself at the core of the modelling of many inverse problems in various areas of mathematics and physical sciences and has been used to model significant real-world inverse problems in sensor networks, in radiation therapy treatment planning, in resolution enhancement, in wavelet-based denoising, in antenna design, in computerized tomography, in materials science, in watermarking, in data compression, in magnetic resonance imaging, in holography, in colour imaging, in optics and neural networks and in graph matching (see [17]). Due to its extraordinary utility and broad applicability in many areas of applied mathematics, algorithms for solving the SCFPP receive great attention (see [9,10,22,24,32,36,39,45,46] and references ther
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