Convergence theorems for fixed points of uniformly continuous generalized Φ-hemi-contractive mappings

Abstract Let E be a real normed linear space, K be a nonempty subset of E and T : K → E be a uniformly continuous generalized Φ-hemi-contractive mapping, i.e., F ( T ) : = { x ∈ K : T x = x } ≠ Φ , and there exist x ∗ ∈ F ( T ) and a strictly increasing function Φ : [ 0 , ∞ ) → [ 0 , ∞ ) , Φ ( 0 ) = 0 such that for all x ∈ K , there exists j ( x − x ∗ ) ∈ J ( x − x ∗ ) such that 〈 T x − x ∗ , j ( x − x ∗ ) 〉 ⩽ ‖ x − x ∗ ‖ 2 − Φ ( ‖ x − x ∗ ‖ ) . (a) If y ∗ ∈ K is a fixed point of T, then y ∗ = x ∗ and so T has at most one fixed point in K. (b) Suppose there exists x 0 ∈ K , such that the sequence { x n } defined by x n + 1 = a n x n + b n T x n + c n u n , ∀ n ⩾ 0 , is contained in K, where { a n } , { b n } and { c n } are real sequences satisfying the following conditions: (i) a n + b n + c n = 1 ; (ii) ∑ n = 0 ∞ ( b n + c n ) = ∞ ; (iii) ∑ n = 0 ∞ ( b n + c n ) 2 ∞ ; (iv) ∑ n = 0 ∞ c n ∞ ; and { u n } is a bounded sequence in E. Then { x n } converges strongly to x ∗ . In particular, if y ∗ is a fixed point of T in K, then { x n } converges strongly to y ∗ . A related result deals with the iterative approximation of the zeros of uniformly continuous generalized Φ-quasi-accretive mappings.