Let E be a real normed space with dual space $E^{*}$E ∗ and let $A:E\rightarrow2^{E^{*}}$A : E → 2 E ∗ be any map. Let $J:E\rightarrow2^{E^{*}}$J : E → 2 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 ∗ , $T: E\rightarrow2^{E^{*}}$T : E → 2 E ∗ is a bounded J-pseudocontractive map with a nonempty J-fixed point set, and $J-T :E\rightarrow2^{E^{*}}$J − T : E → 2 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 $A:E\rightarrow2^{E^{*}}$A : E → 2 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.

A Krasnoselskii-type algorithm is constructed and the sequence of the algorithm is proved to be an approximate fixed point sequence for a common fixed point of a suitable finite family of multi-valued strictly pseudo-contractive mappings in a real Hilbert space. Under some mild additional compactness-type condition on the operators, the sequence is proved to converge strongly to a common fixed point of the family.

LetKbe a nonempty closed and convex subset of a complete CAT(0) space. LetTi:K→CBK,i=1,2,…,m, be a family of multivalued demicontractive mappings such thatF:=⋂i=1mF(Ti)≠∅. A Krasnoselskii-type iterative sequence is shown toΔ-converge to a common fixed point of the familyTi,i=1,2,…,m. Strong convergence theorems are also proved under some additional conditions. Our theorems complement and extend several recent important results on approximation of fixed points of certain nonlinear mappings in CAT(0)spaces. Furthermore, our method of the proof is of special interest.

LetKbe a nonempty, closed, and convex subset of a real Hilbert spaceH. Suppose thatT:K→2Kis a multivalued strictly pseudocontractive mapping such thatF(T)≠∅. A Krasnoselskii-type iteration sequence{xn}is constructed and shown to be an approximate fixed point sequence ofT; that is,limn→∞d(xn,Txn)=0holds. Convergence theorems are also proved under appropriate additional conditions.

Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex real Banach space E with dual space \(E^{*}\). In this paper, a Krasnoselskii-type subgradient extragradient iterative algorithm is constructed and used to approximate a common element of solutions of variational inequality problems and fixed points of a countable family of relatively nonexpansive maps. The theorems proved are improvement of the results of Censor et al. (J. Optim. Theory Appl. 148:318–335, 2011 ).