Approximation of zeros of bounded maximal monotone mappings, solutions of Hammerstein integral equations and convex minimization problems

Let E be a real normed space with dual space E∗$E^{*}$ and let A:E→2E∗$A:E\rightarrow2^{E^{*}}$ be any map. Let J:E→2E∗$J:E\rightarrow2^{E^{*}}$ be the normalized duality map on E. A new class of mappings, J-pseudocontractive maps, is introduced and the notion of J-fixed points is used to prove that T:=(J−A)$T:=(J-A)$ is J-pseudocontractive if and only if A is monotone. In the case that E is a uniformly convex and uniformly smooth real Banach space with dual E∗$E^{*}$, T:E→2E∗$T: E\rightarrow2^{E^{*}}$ is a bounded J-pseudocontractive map with a nonempty J-fixed point set, and J−T:E→2E∗$J-T :E\rightarrow2^{E^{*}}$ is maximal monotone, a sequence is constructed which converges strongly to a J-fixed point of T. As an immediate consequence of this result, an analog of a recent important result of Chidume for bounded m-accretive maps is obtained in the case that A:E→2E∗$A:E\rightarrow2^{E^{*}}$ is bounded maximal monotone, a result which complements the proximal point algorithm of Martinet and Rockafellar. Furthermore, this analog is applied to approximate solutions of Hammerstein integral equations and is also applied to convex optimization problems. Finally, the techniques of the proofs are of independent interest.

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