VISCOSITY APPROXIMATION METHOD FOR MODIFIED SPLIT GENERALIZED EQUILIBRIUM AND FIXED POINT PROBLEMS

. We introduce a viscosity iterative algorithm for approximating a common solution of a modiﬁed split generalized equilibrium problem and a ﬁxed point problem for a quasi-pseudocontractive mapping which also solves some variational inequality problems in real Hilbert spaces. The proposed iterative algorithm is constructed in such a way that it does not require the prior knowledge of the operator norm. Furthermore, we prove a strong conver- gence theorem for approximating the common solution of the aforementioned problems. Finally, we give a numerical example of our main theorem. Our result complements and extends some related works in the literature.


Introduction
Let C and Q be nonempty closed convex subsets of real Hilbert spaces H 1 and H 2 respectively. The Split Feasibility Problem (SFP), first introduced in [7] by Censor and Elfving, requires finding a point in a nonempty closed convex subset in one space such that its image under a bounded linear operator is in another nonempty closed convex subset in the image space. That is, find x * ∈ C such that where A : H 1 → H 2 is a bounded linear operator. The SFP arises in many fields in the real world, such as signal processing, image reconstruction, and intensitymodulated radiation therapy problems. For example, see [8,9,23] and the references therein. Many well-known iterative algorithms have been established for the SFP; for instance, Byrne [5] proposed the CQ algorithm to study the SFP; Qu and Xiu [20] considered a modified CQ algorithm to study the SFP; and Xu [26]  introduced a regularized algorithm for studying the SFP and proved a strong convergence result.
The introduction of the SFP to fixed point theory has also yielded some optimization problems such as the split equilibrium problem, the split variational inequality problem, the split inclusion poblem, among others.
Let A : C → H be a mapping. The Variational Inequality Problem (VIP) is to find u ∈ C such that Au, v − u ≥ 0, (1.1) for all v ∈ C. The solution set of (1.1) is denoted by VIP (C, A). The VIP has emerged as a fascinating branch of mathematical and engineering sciences, with a wide range of applications in industry, finance, economics, ecology, and pure and applied sciences; see, for instance, [11,17,25]. Another optimization problem which includes the VIP is the Equilibrium Problem (EP), first introduced and studied by Blum and Oettli [4]; see also [24]. Many problems in physics, optimization, and economics can be reduced to finding the solution of EP, which is defined as follows: find x ∈ C such that F (x, y) ≥ 0, (1.2) for all y ∈ C, where F : C × C → R is a bifunction. We denote by EP (1.2) the solution set of (1.2).
Let F : C × C → R be a bifunction and f : H → H a mapping. The Generalized Equilibrium Problem (GEP) is to find x ∈ C such that F (x, y) + f (x), y − x ≥ 0, (1.3) for all y ∈ C. We denote by EP(F, f ) the solution set of (1.3). In 2013, Kazmi and Rizvi [18] introduced and studied the following Split Equilibrium Problem (SEP), which is to find x * ∈ C such that (1.4) and such that y * = Ax * ∈ Q solves F 2 (y * , y) ≥ 0, ∀ y ∈ Q, (1.5) where F 1 : C × C → R and F 2 : Q × Q → R are nonlinear bifunctions. The SEP (1.4)-(1.5) reduces to EP (1.2), if H 1 ≡ H 2 , F 1 ≡ F 2 , A ≡ I, and C = Q.
The Split Variational Inequality Problem (SVIP) was introduced and studied by Censor et al. [10], who defined the problem as follows: find x * ∈ C such that (1.6) and such that

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MODIFIED SPLIT GENERALIZED EQUILIBRIUM PROBLEM 391 where f 1 : C → H 1 and f 2 : Q → H 2 are nonlinear mappings. The SVIP has already been used in practice as a model in intensity-modulated radiation therapy (IMRT) treatment planning and the modelling of many inverse problems arising from phase retrieval and other real-world problems such as data compression or sensor networks in computerized tomography; see for example [14]. Very recently, Cheawchan and Kangtunyakarn [13] introduced the Modified Split Generalized Equilibrium Problem (MSGEP), which is to find x * ∈ C such that (1.8) and such that For solving EP, we assume that the bifunction F : C × C → R satisfies the following conditions: (A3) for each x, y, z ∈ C, lim sup t→0 F (tz + (1 − t)x, y) ≤ F (x, y); (A4) for each x ∈ C, y → F (x, y) is convex and lower semi-continuous.
Let r > 0 and x ∈ H. Then, there exists z ∈ C such that Let C be a nonempty closed convex subset of a real Hilbert space H. For every point x ∈ H, there exists a unique nearest point in C, denoted by P C x, such that P C is called the metric projection of H onto C. It is well known that P C is a nonexpansive mapping of H onto C and satisfies Moreover, P C (x) is characterized by the following properties: For all x, y ∈ H, it is well known that every nonexpansive operator T : H → H satisfies the inequality and therefore, we have that for all x ∈ H and y ∈ F (T ).
We now give some definitions that will be needed later.
(iv) firmly quasi-nonexpansive, if F (T ) = ∅ and (v) strictly pseudo-contractive if there exists k ∈ (0, 1] such that (vi) demicontractive, if F (T ) = ∅ and there exists k ∈ [0, 1) such that From the definitions stated above, we notice that the class of demicontractive mappings includes many nonlinear mappings, such as quasi-nonexpansive and strictly pseudo-contractive with nonempty fixed points sets, as special cases.
In 2015, Chang et al. [12] introduced a new type of nonlinear mapping called quasi-pseudo-contractive mapping, as follows: It is obvious that this class of mappings contains the class of demicontractive mappings, see [12].
Definition 1.6. Let H be a real Hilbert space and C be a nonempty closed convex subset of H. A mapping T : C → C is said to be demiclosed at 0 if for any bounded sequence {x n } ⊂ C such that {x n } converges weakly to x and lim n→∞ x n − T x n = 0, we have that T x = x.
The viscosity iterative algorithm is one of the algorithms that have been used extensively by authors to approximate solutions of fixed point problems and optimization problems. The algorithm is constructed in such a way that it also solves some variational inequality problem (see [6,21,28] and the references therein). In 2017, Deepho et al. [16] considered the viscosity iterative algorithm to approximate a common element of the set of solutions of a split variational inclusion problem of a finite family of k-strictly pseudo-contractive nonself mappings. They proved a strong convergence result under suitable conditions, which also solves some variational inequality problem. The following iteration process was used to approximate the aforementioned problems: where α n , β n ∈ (0, 1), λ > 0, g is a contraction mapping with coefficient ρ ∈ (0, 1), is a finite family of k i -strictly pseudo-contraction mappings, and J Bi λ (i = 1, 2) is the resolvent of the maximal monotone mappings. In 2018, Abass et al. [1] introduced an iterative algorithm that does not require the prior knowledge of the operator norm to approximate the common solution of SEP and fixed point problem for an infinite family of quasi-nonexpansive multivalued mappings. Using their iterative algorithm, they prove a strong convergence result.
Very recently, Cheawchan and Kangtunyakarn [13] introduced a new iterative algorithm for finding a common element of the set of solutions of variational inequality problems and the set of solutions of MSGEP without assuming the demiclosedness condition. They proved the following theorem. Theorem 1.7 ([13]). Let C and Q be nonempty closed convex subsets of real Hilbert spaces H 1 and H 2 respectively. Let A : Suppose the following conditions hold: Motivated by the works of Abass et al. [1], Deepho et al. [16], Cheawchan and Kangtunyakarn [13], we propose a viscosity iterative algorithm that does not require any knowledge of the spectral radii to approximate a common solution of MSGEP and fixed point problem for a quasi-pseudo-contractive mapping, which is also a solution of some variational inequality problem. We prove a strong convergence of the iterative scheme to a solution of the aforementioned problems in the framework of real Hilbert spaces. Furthermore, we give a numerical example of our main result.

Preliminaries
We state some known and useful results which will be needed in the proof of our main theorem. In what follows, we denote strong and weak convergence by "→" and " ", respectively. Lemma 2.1. Let H be a real Hilbert space. Then for all x, y ∈ H and α ∈ (0, 1) we have Recall that a Banach space X is said to satisfy Opial's condition if for any sequence {x n } in X which converges weakly to x * ,

Lemma 2.3 ([15]
). Let C be a nonempty closed convex subset of a real Hilbert space H and F : C × C → R be a bifunction satisfying (A1)-(A4). For r > 0 and x ∈ H, define a mapping T F r : H → C as follows: for all x ∈ H. Then the following hold:

Lemma 2.5 ([12]). Let H be a real Hilbert space and T : H → H be an L-
Then the following conclusions hold: Lemma 2.6 ( [19]). Let C be a nonempty closed convex subset of a real Hilbert space H. Assume that f : C → C is a contraction with coefficient µ ∈ (0, 1) and D is a strongly positive linear bounded operator with a coefficient σ > 0. Then, for Lemma 2.7 ([26]). Assume that {a n } is a sequence of nonnegative real numbers such that a n+1 ≤ (1 − σ n )a n + σ n δ n , n > 0, where {σ n } is a sequence in (0, 1) and {δ n } is a real sequence satisfying Then lim n→∞ a n = 0.

Main result
Similarly, we obtain Adding the last two inequalities and using condition (A2) we obtain That is, Thus, from Lemma 2.1 we obtain Since r1 r2 ≤ 1, we get which implies that α n ∈ (0, 1), and the step size γ n is chosen in such a way that for some > 0,
Thus from (3.9) we have that Hence, from condition (i) of Theorem 3.3, we obtain Let w n = x n + γ n A * (T F2 sn (I − s n f 2 ) − I)Ax n . Applying inequality (3.6), we have Using the property of inverse strongly monotone operator and (3.12), we have (3.13) From Theorem 3.3, we have that (3.14) Substituting (3.13) into (3.14), we obtain Hence, Therefore, from condition (i) of Lemma 3.2, we obtain By the firm nonexpansivity of T F1 rn , we have That is, (3.16) From (3.8), (3.12) and (3.16), we obtain which yields Since Again, (3.20) From (3.1), we have On substituting (3.20) into (3.21), we obtain which yields Again, from (3.19) and (3.24) we obtain Since {x n } is bounded, there exists a weakly convergent subsequence {x nj } of {x n } such that x nj x * . Since every Hilbert space has the Opial property, we have x n x * . On the other hand, from (3.19) we have that u n x * . Using (3.23) and the demiclosedness property of K, we have that Kx * = x * . Hence x * ∈ F (T ). Next, we show that x * ∈ Ω. Assume that x * / ∈ EP (F 1 , f 1 ); since EP (F 1 , f 1 ) = F (T F1 r (I − rf 1 )), we obtain x * = T F1 r (I − rf 1 )x * . Using Opial's condition and (3.17), Lemma 2.3, and Lemma 3.1 we obtain This is a contradiction, therefore x * ∈ EP (F 1 , f 1 ).
We now show that lim sup j→∞ (D − τ g)x, x − x n ≤ 0, where x = P Γ (I + τ g − D)x. Indeed, the subsequence {x nj } of {x n } converges weakly to x * . We obtain, by the property of metric projection P Γ , Also, we show the uniqueness of a solution of the variational inequality (3.26) Suppose that x ∈ Γ and x * ∈ Γ are both solutions of (3.26); then Adding the last two inequalities we have Since D − τ g is strongly monotone by Lemma 2.6, we have that x = x * . Hence the uniqueness is proved. Lastly, we prove that x n → x * as n → ∞. From (3.1) and (3.7) we obtain Then, it follows that By conditions (i) and (ii) of Lemma 3.2, we obtain lim n→∞ x n − x * = 0 using Lemma 2.7.
These three cases are displayed in Figure 1 on the next page: