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(ii) If (H_{1}=H_{2}) and L is the identity operator, then the problem (1.9) is reduced to the problem of finding an element of the set ((A+B)^{-1}0cap F(T)).

In the last decade, many author studied the problem for finding an element of the set of fixed points of a nonlinear mapping; see, for instance, [12 14].

In particular, compared with Theorems and in [16], our results (i.e., Theorems 3.2 and 4.2 in this paper) extend the problem of finding an element of to the one of finding an element of.

Our Theorems 3.1 and 4.2 are the extension and improvements of Theorems 1.1 and 1.2 in the following way: (i the problem of finding an element of includes the one of finding an element of as a special case; (ii the algorithms in this paper are very different from those in [17] because of considering the complexity involving the problem of finding an element of.

The problem of finding an element of Γ is more general than the problem of finding a solution of the SFP in [15], Theorem 3.6 and the problem of finding an element of (Gamma_{0}cap operatorname{Fix}(S)) with (S Crightarrow C) being a nonexpansive mapping in [23], Theorem 3.2.

Since for a nonexpansive mapping, the mapping is -inverse strongly monotone, the problem of finding an element of, where denotes the set of fixed points of the nonexpansive mapping, is equivalent to that of finding an element of, where denotes the set of solutions of the mapping, and contained in the class of problem (1.9).

Also, let us notice that the problem of finding an element of F ( T ) is equivalent to the problem of finding an element of x ∈ S ( I − T ).

(i) The combination of the problem of finding an element of Fix ( S ) ∩ VI ( C, A ) in [[9], Theorem 3.1] and the one of finding an element of ⋂ i = 1 N Fix ( S i ) ∩ VI ( C, A ) in [[11], Theorem 3.1] is extended to develop the one of finding an element of ⋂ i = 1 ∞ Fix ( S i ) ∩ Γ in our Theorem 3.1.

The problem of finding an element of (Gamma_{0}cap operatorname{Fix}(T)) with (T Crightarrow C) being a pseudo-contractive mapping is more general than the problem of finding a solution of the SFP in [15], Theorem 3.6 and the problem of finding an element of (Gamma_{0}cap operatorname{Fix}(S)) with (S Crightarrow C) being a nonexpansive mapping in [23], Theorem 3.2.

Meantime, the algorithms in this paper are very different from those in [16] (because of considering the complexity involving the problem of finding an element of ).

(iii) The problem of finding an element of Fix ( S ) ∩ Ξ ∩ Γ in Theorems 4.1 and 4.2 is more general than the corresponding problems in [[23], Theorem 5.7] and [[34], Theorem 3.1], respectively.