Convergence of a Halpern-type iteration algorithm for a class of pseudo-contractive mappings

Let EE be a real reflexive Banach space with uniformly Gâteaux differentiable norm. Let KK be a nonempty bounded closed and convex subset of EE. Let T:K→KT:K→K be a strictly pseudo-contractive map and let L>0L>0 denote its Lipschitz constant. Assume F(T)≔{x∈K:Tx=x}≠0̸F(T)≔{x∈K:Tx=x}≠0̸ and let z∈F(T)z∈F(T). Fix δ∈(0,1)δ∈(0,1) and let δ∗δ∗ be such that δ∗≔δL∈(0,1)δ∗≔δL∈(0,1). Define Snx≔(1−δn)x+δnTx∀x∈K, where δn∈(0,1)δn∈(0,1) and limδn=0limδn=0. Let {αn}{αn} be a real sequence in (0,1)(0,1) which satisfies the following conditions: C1:limαn=0;C2:∑αn=∞. For arbitrary x0,u∈Kx0,u∈K, define a sequence {xn}∈K{xn}∈K by xn+1=αnu+(1−αn)Snxnxn+1=αnu+(1−αn)Snxn. Then, {xn}{xn} converges strongly to a fixed point of TT.