Hybrid methods with regularization for minimization problems and asymptotically strict pseudocontractive mappings in the intermediate sense

and Applied Analysis 3 (b) η-strongly monotone if there exists a constant η > 0 such that ⟨Sx − Sy, x − y⟩ ≥ η 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 2 , ∀x, y ∈ C; (12) (c) α-inverse-strongly monotone (α-ism) if there exists a constant α > 0 such that ⟨Sx − Sy, x − y⟩ ≥ α 󵄩󵄩󵄩󵄩Sx − Sy 󵄩󵄩󵄩󵄩 2 , ∀x, y ∈ C. (13) Obviously, if S is α-inverse-strongly monotone, then it is monotone and (1/α)-Lipschitz continuous. It can be easily seen that if S : C → C is nonexpansive, then I − S is monotone. It is also easy to see that a projectionmappingP C is 1-ism.The inverse stronglymonotone (also known as cocoercive) operators have been applied widely in solving practical problems in various fields. We need some facts and tools which are listed in the form of the following lemmas. Lemma 6. LetX be a real inner product space. Then, 󵄩󵄩󵄩󵄩x + y 󵄩󵄩󵄩󵄩 2 ≤ ‖x‖ 2 + 2 ⟨y, x + y⟩ , ∀x, y ∈ X. (14) Lemma 7 (see [7, Proposition 2.4]). Let {x n } be a bounded sequence in a reflexive Banach spaceX. Ifω w ({x n }) = {x}, then