On the strong convergence of the proximal point algorithm with an application to Hammerstein euations
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Journal of Fixed Point Theory and Applications
On the strong convergence of the proximal point algorithm with an application to Hammerstein euations C. E. Chidume , A. Adamu, M. S. Minjibir and U. V. Nnyaba Abstract. Let E be a real normed space. A new notion of quasi-boundedness for operators A : E → 2E is introduced and the following general important result for accretive operators is proved: an accretive operator with zero in the interior of its domain is quasi-bounded. Using this result, a new strong convergence theorem for approximating a zero of an maccretive operator is proved in a uniformly smooth real Banach space. This result complements the celebrated proximal point algorithm for approximating solutions of 0 ∈ Au in a real Hilbert space where A is a maximal monotone operator. Furthermore, as an application of our theorem, a new strong convergence theorem for approximating a solution of a Hammerstein equation is proved. Finally, several numerical experiments are presented to illustrate the strong convergence of the sequence generated by our algorithm and the results obtained are compared with those obtained using some recent important algorithms. Mathematical Subject Classification. 47H09, 47H05, 47J25, 47J05. Keywords. Fixed point, accretive, uniformly smooth.
1. Introduction Let H be a real Hilbert space and A : H → 2H be an operator (possibly nonlinear). A fundamental problem in nonlinear operator theory is that of finding an element u∈H
such that
0 ∈ Au,
(1.1)
where A is monotone, i.e., A satisfies the following inequality: x − y, η − ζ ≥ 0, ∀ η ∈ Ax, ζ ∈ Ay. For example, if A is the subdifferential, ∂f : H → 2H of a proper, lower semi-continuous and convex function f : H → (−∞, ∞], defined by ∂f (x) := u ∈ H : f (y) − f (x) ≥ u, y − x, ∀y ∈ H , then, ∂f is a monotone operator and it is easy to see that a solution of the inclusion Research supported from ACBF Research Grant Funds to AUST. 0123456789().: V,-vol
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0 ∈ Au corresponds to a minimizer of f . Furthermore, for an example where solutions of 0 ∈ Au, A monotone, represent solutions of variational inequality problems, the reader may see, for example, Rockafellar [54]; and for problems where solutions of 0 ∈ Au, A monotone, represent equilibrium state of a dynamical system, the reader may see Browder, [3]. Existence theorems have been proved for problem (1.1) (see, e.g., Browder [3], Martin [42]). Also, iterative algorithms for approximating solutions of the inclusion (1.1) have been studied extensively by numerous authors (see e.g., Bruck and Reich [8], Chidume and Chidume [14], Chidume [12,13], Browder [3], Martin [42], Chidume et al. [15] and the references contained in them). One of the classical methods for approximating solution(s) of inclusion (1.1) is the celebrated proximal point algorithm (PPA) introduced by Martinet [41] and studied extensively by Rockafellar [54] and a host of other authors (see, e.g., Bruck and Reich [8] and Reich [45]). Let E be a real normed space with dual space E ∗
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