A New Iterative Method for the Set of Solutions of Equilibrium Problems and of Operator Equations with Inverse-Strongly Monotone Mappings

and Applied Analysis 3 Set A i = I − T i . Obviously, A i are λ i -inverse-strongly monotone; that is, ⟨A i (x) − A i (y) , x − y⟩ ≥ λi 󵄩󵄩󵄩󵄩A i (x) − A i (y) 󵄩󵄩󵄩󵄩 2 , ∀x, y ∈ D (A i ) , λ i = 1 − k i


Introduction
Let  be a real Hilbert space with the inner product ⟨⋅, ⋅⟩ and the norm ‖ ⋅ ‖, respectively.Let  be a nonempty closed convex subset of , and let  be a bifunction from  ×  into (−∞, +∞).The equilibrium problem for  is to find  * ∈  such that  ( * , V) ≥ 0, ∀V ∈ . ( The set of solutions of ( 1) is denoted by EP().Equilibrium problem (1) includes the numerous problems in physics, optimization, economics, transportation, and engineering, as special cases.
Assume that the bifunction  satisfies the following standard properties.
Let {  },  = 1, . . ., , be a finite family of   -strictly pseudocontractive mappings from  into  with the set of fixed points (  ); that is,  (  ) = { ∈  :    = } . ( Assume that The problem of finding an element is studied intensively in .
Recall that a mapping  in  is said to be a -strictly pseudocontractive mapping in the terminology of Browder and Petryshyn [28] if there exists a constant 0 ≤  < 1 such that      − 2

Abstract and Applied Analysis
We know that the class of -strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings.
In the case that  ≡ 0 and  = 1, (4) is a problem of finding a fixed point for a -strictly pseudocontractive mapping in  and is given by Marino and Xu [17].
Theorem 1 (see [17]).Let  be a nonempty closed convex subset of a real Hilbert space .Let  :  →  be a -strictly pseudocontractive mapping for some 0 ≤  < 1, and assume that Let {  } be the sequence generated by the following algorithm: Assume that the control sequence {  } is chosen so that   < 1 for all .Then {  } converges strongly to  ()  0 , the projection of  0 onto ().
For the case that  ≡ 0 and  > 1, (4) is a problem of finding a common fixed point for a finite family of   -strictly pseudocontractive mappings   in  and is studied in [27].
Let  0 ∈  and {  }, {  }, and {  } three sequences in [0, 1] satisfying   +   +   = 1 for all  ≥ 1, and let {  } be a sequence in .Then the sequence {  } generated by is called the implicit iteration process with mean errors for a finite family of strictly pseudocontractive mappings {  }  =1 .The scheme (9) can be expressed in the compact form as where   =   mod  .
If  is an arbitrary bifunction satisfying Assumption A and  = 1, then ( 4) is a problem of finding a common element of the fixed point set for a -strictly pseudocontractive mapping in  and of the solution set of equilibrium problem for  (see [26]).
Theorem 3 (see [26]).Let  be a nonempty closed convex subset of a real Hilbert space .Let  be a bifunction from  ×  to (−∞, +∞) satisfying Assumption A, and let  be a nonexpansive mapping of  into  such that Let  be a contraction of  into itself and let {  } and {  } be sequences generated by  1 ∈  and for all  ∈ N, where Then, {  } and {  } converge strongly to  ∈ () ∩ (), where Set   =  −   .Obviously,   are   -inverse-strongly monotone; that is, From now on, let {  }  =1 be a finite family of   -inversestrongly monotone mappings in  with  ⊂ ⋂  =1 (  ) and   > 0,  = 1, . . ., .On the other hand, if there exists  0 ∈ {1, 2, . . ., } such that And hence,   0 has only one solution and, consequently, the stated problem does not have sense.So, without loss of generality, assume that 0 <   ≤ 1,  = 1, . . ., . Set where Assume that EP() ∩  ̸ = 0. Our problem is to find an element Since the mapping  =  −  is (1/2)-inverse-strongly monotone for each nonexpansive mapping , the problem of finding an element  * ∈ , which is not only a solution of a variational inequality involving an inverse-strongly monotone mapping but also a fixed point of a nonexpansive mapping, is a particular case of (18).
Theorem 5 (see [34]).Let  be a nonempty closed convex subset of a real Hilbert space .Let  be a -inverse-strongly monotone mapping of  into , and let  be a nonexpansive nonself-mapping of  into  such that Suppose that  1 =  ∈  and {  } is given by for every  = 1, 2, . .., where {  } is a sequence in [0, 1) and then {  } converges strongly to  ()∩(,) .
Some similar results are also considered in [36,37].
We construct a regularization solution   of the following single equilibrium problem: find   ∈  such that where and {  } is the positive sequence of regularization parameters that converges to 0, as  → +∞.
The first one is the following theorem.
Theorem 7 (see [38]).For each   > 0, problem (28) has a unique solution   such that where  is a positive constant.
Next, we introduce the second result.Let {c  } and {  } be some sequences of positive numbers, and let  0 and  1 be two arbitrary elements in .Then, the sequence {  } of iterations is defined by the following equilibrium problem: find  +1 ∈  such that c ( ( +1 , V) Theorem 8 (see [38]).Assume that the parameters c ,   , and   are chosen such that Then, the sequence {  } defined by (31) converges strongly to the element  * , as  → +∞.
In this paper, we consider the new another iteration method: for an arbitrary element  0 in , the sequence {  } of iterations is defined by finding   ∈  such that  (  , ) + ⟨  −   ,  −   ⟩ ≥ 0, ∀ ∈ , where   is the metric projection of  onto  and {  } and {  } are sequences of positive numbers.
The strong convergence of the sequence {  } defined by ( 32) is proved under some suitable conditions on {  } and {  } in the next section.

Main Results
We formulate the following lemmas for the proof of our main theorems.
Lemma 12 (see [38]).Let  be any inverse-strongly monotone mapping from  into  with the solution set   := { ∈  : () = 0}, and let  0 be a closed convex subset of  such that Then, the solution set of the following variational inequality is coincided with   ∩  0 .
From Lemma 9, we can consider the firmly nonexpansive mapping  0 defined by From Lemma 10, we know that  0 is nonexpansive.Consequently,  0 :=  −  0 is (1/2)-inverse-strongly monotone.Let Then,  0 = EP() and problem (18) are equivalent to finding Now, we construct a regularization solution   for (40) by solving the following variational inequality problem: find   ∈  such that where the positive regularization parameter   → 0, as  → +∞.Now we are in a position to introduce and prove the main results.
Theorem 13.Let  be a nonempty closed convex subset of a real Hilbert space .Let  be a bifunction from  ×  to (−∞, +∞) satisfying Assumption A and let {  }  =1 be a finite family of   -inverse-strongly monotone mappings in  with  ⊂ ⋂  =1 (  ) and   > 0,  = 1, . . ., , such that where () denotes the set of solutions for (1) and where  is some positive constant.
Proof.From Lemma 12, we know that  0 is the set of solutions for the following variational inequality problem: find  * ∈  such that If we define the new bifunction  0 (, V) by then problem (41) is the same as (28) with a new (, V), and the proof for the theorem is a complete repetition of the proof for Theorem 2.1 in [38].Set where () denotes the set of solutions for (1) and Suppose that   ,   satisfy the following conditions: Then, the sequence {  } defined by (32) converges strongly to  * ∈ () ∩ ; that is, Proof.Let   be the solution of (41).Then, We note that, for  > 0,  > 0,  > 0, the inequality ( + )