A Tseng-Type Algorithm with Self-Adaptive Techniques for Solving the Split Problem of Fixed Points and Pseudomonotone Variational Inequalities in Hilbert Spaces

In this paper, we survey the split problem of fixed points of two pseudocontractive operators and variational inequalities of two pseudomonotone operators in Hilbert spaces. We present a Tseng-type iterative algorithm for solving the split problem by using self-adaptive techniques. Under certain assumptions, we show that the proposed algorithm converges weakly to a solution of the split problem. An application is included.


Introduction
In this paper, we survey the variational inequality (in short, VI(C, φ)) of seeking an element p † ∈ C such that where C is a nonempty closed convex set in a real Hilbert space H, ·, · means the inner product of H, and φ : H → H is a nonlinear operator. Denote by Sol(C, φ) the solution set of variational inequality (1).
In order to abate the restriction (2), Korpelevich's extragradient algorithm ( [26]) was proposed in 1976 where proj C denotes the orthogonal projection from H onto C and the step-size τ is in (0, 1 κ ). Extragradient algorithm (4) guarantees the convergence of the sequence {x k } provided φ is monotone. Extragradient algorithm and its variant have been investigated extensively, see [27][28][29][30][31]. However, we have to compute (i) twice proj C at two different points and (ii) two values φ(x k ) and φ(y k ). Two important modifications of extragradient algorithm have been made. One is proposed in [32] by Censor, Gibali and Reich and another is the following remarkable algorithm proposed in [33] by Tseng where γ ∈ (0, 1 κ ). On the other hand, if φ is not Lipschitz continuous or its Lipschitz constant κ is difficult to estimate, then algorithms (4) and (5) are invalid. To avoid this obstacle, Iusem [34] used a self-adaptive technique without prior knowledge of Lipschitz constant κ of φ for solving (1). Some related works on self-adaptive methods for solving (1), please refer to [35][36][37][38].
Let H 1 and H 2 be two real Hilbert spaces. Let C and Q be two nonempty closed and convex subsets of H 1 and H 2 , respectively. Let S : C → C, T : Q → Q, f : H 1 → H 1 and g : H 2 → H 2 be four nonlinear operators. We consider the classical split problem which is to find a point x * ∈ C such that where Fix(S) := {u † |u † = Su † } and Fix(T) := {v † |v † = Tv † } are the fixed point sets of S and T, respectively. The solution set of (6) is denoted by Γ, i.e., Let f and g be the null operators in C and Q, respectively. Then, the split problem (6) becomes to the split fixed point problem studied in [39,40] which is to find an element point x * ∈ C such that x * ∈ Fix(S) and Ax * ∈ Fix(T).
Let S and T be the identity operators in C and Q, respectively. Then, the split problem (6) becomes to the split variational inequality problem studied in [41] which is to find an element x * ∈ C such that x * ∈ Sol(C, f ) and Ax * ∈ Sol(Q, g).
The solution set of (8) is denoted by Γ 1 , i.e., The split problems (6)-(8) have a common prototype that is the split feasibility ( [42]) problem of finding a point x * such that x * ∈ C and Ax * ∈ Q.
The split problems have emerged their powerful applications in image recovery and signal processing, control theory, biomedical engineering and geophysics. Some iterative algorithms for solving the split problems have been studied and extended by many scholars, see [43][44][45][46][47].
Motivated by the work in this direction, in this paper, we further survey the split problem (6) in which S and T are two pseudocontractive operators and f and g are two pseudomonotone operators. We present a Tseng-type iterative algorithm for solving the split problem (6) by using self-adaptive techniques. Under certain conditions, we show that the proposed algorithm converges weakly to a solution of the split problem (6).

Preliminaries
Let H be a real Hilbert space equipped with inner product ·, · and the induced norm defined by x → x := (x, x). For any x, x † ∈ H and constant η ∈ R, we have The symbol " denotes the weak convergence and the symbol " → denotes the strong convergence. Use ω w (u k ) to denote the set of all weak cluster points of the sequence • Weakly sequentially continuous, if H u k ũ implies that φ(u k ) φ(ũ). Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that an operator S : C → C is said to be pseudocontractive if For given u † ∈ H, there exists a unique point in C, denoted by proj C [u † ] such that It is known that proj C is firmly nonexpansive, that is, proj C satisfies It is obvious that proj C is nonexpansive, i.e., proj Moreover, proj C satisfies the following inequality ( [48]) Lemma 1 ([49]). Let C be a nonempty, convex and closed subset of a Hilbert space H. Assume that the operator S : C → C is pseudocontractive and κ-Lipschitz continuous. Then, for allũ ∈ C and u † ∈ Fix(S), we have where α is a constant in (0, ).

Lemma 2 ([50]
). Let C be a nonempty closed convex subset of a real Hilbert space H. Let f : C → H be a continuous and pseudomonotone operator. Then p † ∈ Sol(C, f ) iff p † solves the following variational inequality f (u), u − p † ≥ 0, for all u ∈ C.

Lemma 3 ([51]
). Let C be a nonempty, convex and closed subset of a Hilbert space H. Let the operator S : C → C be continuous pseudocontractive. Then, S is demiclosed, i.e., u k ũ and S(u k ) → u † as k → +∞ imply that S(ũ) = u † .

Lemma 4 ([52]
). Let Γ be a nonempty closed convex subset of a real Hilbert space H. Let {x k } ⊂ H be a sequence. If the following assumptions are satisfied Then the sequence {x k } converges weakly to some point in Γ.

Main Results
In this section, we present our main results. Let H 1 and H 2 be two real Hilbert spaces. Let C ⊂ H 1 and Q ⊂ H 2 be two nonempty closed convex sets. Let S : C → C, T : Q → Q, f : H 1 → H 1 and g : H 2 → H 2 be four nonlinear operators. Let A : C → Q be a bounded linear operator with its adjoint A * .
Next, we introduce an iterative algorithm for solving the split problem (6). In order to demonstrate the convergence analysis of Algorithm 1, we add some conditions on the operators and the parameters.
Step 1. Assume that the present iterate x k and the step-sizes γ k and τ k are given. Compute (12) Step 2. Compute the next iterate x k+1 by the following form Step 3. Increase k by 1 and go back to Step 1. Meanwhile, update and Suppose that (c1): S and T are two pseudocontractive operators with Lipschitz constants L 1 and L 2 , respectively; (c2): the operator f is pseudomonotone on H 1 , weakly sequentially continuous and κ 1 -Lipschitz continuous on C; (c3): the operator g is pseudomonotone on H 2 , weakly sequentially continuous and κ 2 -Lipschitz continuous on Q.

Remark 1.
According to (19), the sequence {γ k } is monotonically decreasing. Moreover, by the κ 1 -Lipschitz continuity of f , we obtain that . Thus, {γ k } has a lower bound min{γ 0 , ω κ 1 }. Therefore, the limit lim k→+∞ γ k exists. Similarly, the sequence {τ k } is monotonically decreasing and has a lower bound min{τ 0 , µ κ 2 }. So, the limit lim k→+∞ τ k exists. Now, we prove our main theorem. Theorem 1. Suppose that Γ = ∅. Then the sequence {x k } generated by Algorithm 1 converges weakly to some point p ∈ Γ. (10) and (12) Using Lemma 1, we obtain Combining (21) and (22) Similarly, according to (10), Lemma 1 and (17), we have the following estimate Applying the inequality (11) to (13), we obtain Since Based on (25) and (26), we get It follows that which yields By (14), we have From (10), we obtain Substituting (27) and (29) into (28), we deduce Thanks to (19) . It follows from (30) that By Remark 1, lim k→+∞ Then, there exists σ > 0 and m 1 such In combination with (31), we get This together with (23) implies that By the property (11) of proj Q and (15), we have Since Ax * ∈ Sol(Q, g) and w k ∈ Q, g(Ax * ), w k − Ax * ≥ 0. By the pseudomonotonicity of g, we obtain Taking into account (33) and (34), we obtain which is equivalent to It follows that From (14), we receive By virtue of (10), we achieve Substituting (35) and (37) into (36), we obtain Duo to (20), we have This together with (38) implies that By Remark 1, lim k→+∞ τ k τ k+1 = 1 and hence So, there exists > 0 and m 2 such that In the light of (39), we have Owing to (24) and (40), we get Observe that Combining (41) and (42), we acquire In view of (18), we have It follows from (32) and (43) that which implies that lim k→+∞ x k − x * exists. Since So, the sequences {x k }, {u k } and {v k } are all bounded. By virtue of (44), we derive By the L 1 -Lipschitz continuity of S, we have It follows that This together with (47) implies that From (12) and (47), we conclude that x k − v k → 0. Next, we show that ω w (x k ) ⊂ Γ. Pick any p † ∈ ω w (x k ). Then, there exists a subsequence {x k i } of {x k } such that x k i p † as i → +∞. In addition, y k i p † and v k i p † as i → +∞.
First, we prove that p † ∈ Sol(C, f ). In view of (11) and It follows that Noting that from (49), we have lim i→+∞ v k i − y k i = 0. Meanwhile, {y k i } and { f (v k i )} are bounded. Then, by (52), we deduce Let { j } be a positive real numbers sequence satisfying lim j→+∞ j = 0. On account of (53), for each j , there exists the smallest positive integer n j such that Moreover, for each By the pseudomonotonicity of f , we get which implies that Then, This together with (55) implies that By Lemma 2 and (56), we conclude that p † ∈ Sol(C, f ).
On the other hand, by (51), Sx k i − x k i → 0 as i → +∞. This together with x k i p † and Lemma 3 implies that p † ∈ Fix(S). Therefore, p † ∈ Fix(S) ∩ Sol(C, f ).
Next, we show that Ap † ∈ Fix(T) ∩ Sol(Q, g). Observe that It follows that This together with (48) implies that From (14), u k i p † as i → +∞. Thanks to (17) and (48), we have q k i − t k i → 0 as i → +∞. Combining with (46), we deduce that t k i Ap † . Applying Lemma 3 to (57), we obtain that Ap † ∈ Fix(T).
Next, we show that Ap † ∈ Sol(Q, g). In view of (10) and w k i = proj Q [Au k i − τ k i g(Au k i )], we achieve It follows that (58) Noting that from (r3), we have lim i→+∞ w k i − Au k i = 0. Then, by (58), we deduce Choose a positive real numbers sequence {υ j } such that lim j→+∞ υ j = 0. In terms of (59), for each υ j , there exists the smallest positive integer m j such that Moreover, for each j > 0, g(Au k i ) = 0. Setting ψ(u k i ) = g(Au k i ) g(Au k i ) 2 , we have g(Au k i ), ψ(u k i ) = 1. From (60), we have By the pseudomonotonicity of g, we get Because of g(A(u k i j )) g(Ap † ), we have Then, This together with (61) implies that By Lemma 2 and (62), we conclude that Ap † ∈ Sol(Q, g). So, p ∈ Γ and ω w (x k ) ⊂ Γ. Finally, we show that the entire sequence {x k } converges weakly to p † . As a matter of fact, we have the following facts: Thus, by Lemma 4, we deduce that the sequence {x k } weakly converges to p † ∈ Γ. This completes the proof. Corollary 1. Suppose that Γ 1 = ∅. Then the sequence {x k } generated by Algorithm 2 converges weakly to some point p 1 ∈ Γ 1 .
Step 1. Assume that the present iterate x k and the step-sizes γ k and τ k are given. Compute Step 2. Compute the next iterate x k+1 by the following form Step 3. Increase k by 1 and go back to Step 1. Meanwhile, update else. (68) else. (69)

Application to Split Pseudoconvex Optimization Problems and Fixed Point Problems
In this section, we apply Algorithm 1 to solve split pseudoconvex optimization problems and fixed point problems.
Let R n be the Euclidean space. Let C be a closed convex set in R n . Recall that a differentiable function F : R n → R is said to be pseudoconvex on C if for every pair of distinct points x, y ∈ C, Now, we consider the following optimization problem where F(x) is pseudoconvex and twice continuously differentiable. Denote by SOP(C, F) the solution set of optimization problem (70).
The following lemma reveals the relationship between the variational inequality and the pseudoconvex optimization problem.

Lemma 5 ([53]
). Suppose that F : R n → R is differentiable and pseudoconvex on C. Then if and only if x * is a minimum of F(x) in C.
Let R n and R m be two Euclidean spaces. Let C ⊂ R n and Q ⊂ R m be two nonempty closed convex sets. Let A be a given m × n real matrix. Let S : C → C and T : Q → Q be two pseudocontractive operators with Lipschitz constants L 1 and L 2 , respectively. Let F : R n → R be a differentiable function with κ 1 -Lipschitz continuous gradient which is also pseudoconvex on C. Let G : R m → R be a differentiable function with κ 2 -Lipschitz continuous gradient which is also pseudoconvex on Q.
Step 1. Assume that the present iterate x k and the step-sizes γ k and τ k are given. Compute Step 2. Compute the next iterate x k+1 by the following form x k+1 = proj C [u k + εA * (q k − Au k )].

Concluding Remarks
In this paper, we survey iterative methods for solving the split problem of fixed points of two pseudocontractive operators and variational inequalities of two pseudomonotone operators in Hilbert spaces. By using self-adaptive techniques, we construct a Tseng-type iterative algorithm for solving this split problem. We prove that the proposed Tsengtype iterative algorithm converges weakly to a solution of the split problem under some additional conditions imposed the operators and the parameters. Finally, we apply our algorithm to solve split pseudoconvex optimization problems and fixed point problems.