Iterative convergence to Cesàro means for continuous pseudocontractive mappings

Let ff be a fixed strongly pseudocontractive mapping and TT be a continuous pseudocontractive mapping with F(T)≠0̸F(T)≠0̸. The sequence {zm}{zm} is iteratively defined as follows: zm=tmf(zm)+(1−tm)1m+1∑j=0mTjzm,m≥0, where {tm}⊂(0,1){tm}⊂(0,1) satisfies the condition limm→∞tm=0limm→∞tm=0. We prove that {zm}{zm} converges strongly to the unique solution pp to some variational inequality in F(T)F(T). Our results develop and complement the corresponding ones by Matsushita–Daishi Kuroiwa [J. Math. Anal. Appl. 294 (2004) 206–214] and Shioji–Takahashi [Arch. Math., 72 (1999) 354–359] and Moore–Nnoli [J. Math. Anal. Appl. 260 (2001) 269–278] and Su–Li [Appl. Math. Comput. 181 (2006) 332–341] and Song–Chen [Appl. Math. Comput. 186 (2007) 1120–1128] and Wangkeeree [Appl. Math. Comput. 201 (2008) 239–249].