In 2009, Ceng et al. [17] introduced an implicit-type algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudo-contraction in a real Hilbert space.
there exists x 0 in X such that ( x 0, T x 0 ) ∈ E ( G ), T is an implicit relation type G-contraction.
A simple and natural way to unify and prove in a simple manner several metrical fixed point theorems is to consider an implicit contraction type condition instead of the usual explicit contractive conditions.
T is α-admissible, there exists x 0 in X such that α ( x 0, x 0 ) ≥ 1 and α ( x 0, T x 0 ) ≥ 1, T is an implicit relation type modified α-contraction.
there exists x 0 in X such that ( x 0, T x 0 ) ∈ E ( G ), T is an implicit relation type G-contraction, if { x n } is a sequence in X such that ( x n, x n + 1 ) ∈ E ( G ) and x n → x as n → + ∞, then ( x n, x ) ∈ E ( G ) for all n ∈ N. Then T has a fixed point.
Assume that T : X → X is a continuous self-mapping satisfying the following conditions: (i) T is α-admissible, (ii) there exists x 0 in X such that α ( x 0, x 0 ) ≥ 1 and α ( x 0, T x 0 ) ≥ 1, (iii) T is an implicit relation type modified α-contraction. .
Let { x n } be an implicit Mann type iteration defined by (1.6), where α n ∈ ( 0, 1 ) and lim n → ∞ α n = 0, then the sequence { x n } converges strongly to a fixed point of T. Proof Firstly, let ∀ p ∈ F ( T ), ∀ n ≥ 1, we show that lim n → ∞ ∥ x n − p ∥ exists and { x n } is bounded.
T is α-admissible, there exists x 0 in X such that α ( x 0, x 0 ) ≥ 1 and α ( x 0, T x 0 ) ≥ 1, T is an implicit relation type modified α-contraction, if { x n } is a sequence in X such that α ( x n, x n + 1 ) ≥ 1 and x n → x as n → + ∞, then α ( x, x ) ≥ 1 and α ( x n, x ) ≥ 1 for all n ∈ N. Then T has a fixed point.
