Algorithm for split variational inequality, split equilibrium problem and split common fixed point problem

1.   Introduction

Split equilibrium problem (SEP), introduced by He ^[20] in 2012, which is defined as: find the solution of equilibrium problem (EP) whose image is also a solution of another EP under a given bounded linear operator. Let $\mathbb{H}_{1}$ and $\mathbb{H}_{2}$ be two real Hilbert spaces and $C$ and $Q$ be closed convex subsets of $\mathbb{H}_{1}$ and $\mathbb{H}_{2}$, respectively and $B : \mathbb{H}_{1} \to \mathbb{H}_{2}$ be a linear and bounded operator. Let $f$ and $F$ be two functions from $C \times C$ and $Q \times Q$ to $\mathbb R$, respectively, then SEP is to get a point $c \in C$ such that

and

Equation (1.1) is named as classical Equilibrium Problem given by Blum ^[2] and its solution set is denoted by $EP(f)$.

SEP provides us a way to split the solution between two different subsets such that the solution of one problem and its image implies the solution of another problem under the imposed bounded linear operator. As the special case of SEP, split variational inequality problem (SVIP) was introduced by Censor et al. ^[17], in 2012. The SVIP is stated as: consider $\mathrm {g}$ and $\mathrm {g}^{\prime}$ be two operator for $\mathbb{H}_{1}$ and $\mathbb{H}_{2}$, two Hilbert Spaces, respectively, $B : \mathbb{H}_{1}\to \mathbb{H}_{2}$ be a linear and bounded operator, $C$ and $Q$ be the same as defined above, then

and

and the equation (1.2) separately gives classical variational inequality (VI).

Split common fixed point problem (SCFPP), introduced by Censor and Segal ^[18], in 2009. The SCFPP problem is to find a point $c \in Fix(\mathrm{S})$ gives $B c \in Fix(\mathrm {T})$, where $B : \mathbb{H}_{1} \to \mathbb{H}_{2}$ is linear and bounded operator with $\mathrm{S}: \mathbb{H}_{1} \to \mathbb{H}_{1}$ and $\mathrm {T}: \mathbb{H}_{2} \to \mathbb{H}_{2}$ are the general operators and $Fix()$ denoted the solution set of fixed point of the considered mapping. SCFPP is the generalization of Split Feasibility Problem, given by Censor and Elfving ^[19], in 1994 and this problem formulate a point $c \in C$ with $B c \in Q$ where $C$ and $Q$ are convex subsets of $\mathbb{H}_{1}$ and $\mathbb{H}_{2}$, respectively.

Korpelevich ^[3], in 1976, introduced the extragradient method for solving Eq (1.2) when $\mathrm {g}$ is monotone and k-Lipschitz continuous in the finite dimensional Euclidean space. In 2003, Takahashi and Toyoda ^[13] introduced the following method to calculate the common solution of VIP and fixed point problem (FPP)

where $S:C \to C$ is nonexpansive mapping and $f:C \to \mathbb{H}_{1}$ is a $\nu$-inverse strongly monotone mapping.

After that, in 2006 Nadezhkina and Takahashi ^[14] suggested the modified extragradient method to prove the weak convergence of the defined iteration to the common solution of VIP and FPP:

where $S:C \to C$ is nonexpansive mapping and $f:C \to \mathbb{H}_{1}$ is a monotone and $k$-Lipschitz continuous mapping. Later on, in 2011, Kangtunyakarn ^[1] proved the convergence theorem for calculating the common point of the three sets of solutions of equilibrium problem, variational inequality and the fixed point problems by practicing with a newly developed mapping achieved by infinite family of real numbers and of nonexpansive mappings.

In 2017, Tian et al. ^[11] proposed an algorithm for finding an element to solve the class of SVIP by combining extragradient method with CQ algorithm. In 2018, Lohawech et al. ^[12] proved the existence of common solution of the SVIP and FPP by the introduced iterative method which was inspired from Nadezhkina and Takahashi's ^[14] modified extragradient method and Xu's ^[7] algorithm and proved its weak convergence.

Inspired from work of Tian et al. ^[11] and Lohewech et al. ^[12], we extend results in the direction of findings the common solution of SVIP and SEP with SCFPP.

2.   Preliminaries

Lemma 2.1. ^[9] Given $a \in \mathbb{H}$ and $z \in C$, then the following statements are equivalent:

(i) $z = P_{ C }a$;

(ii) $\langle a-z, z-b\rangle \geq 0\ \forall \ b \in C$;

(iii) $\left\langle a-P_{ C } a, b-P_{ C } a\right\rangle \leq 0 \ \forall \ a\in \mathbb{H}, b \in C;$

(iv) $\lVert a-b\rVert^2 \geq \lVert a-z\rVert^2+\lVert b-z\rVert^2\ \forall \ b\in C$.

Lemma 2.2. ^[2] Let the function $f : C \times C \to \mathbb R$ satisfy the following conditions:

(i) $f (u, u) = 0 \; \forall\; u \in C;$

(ii) $f$ is monotone, i.e. $f (u, v)+ f (v, u) \leq 0\ \forall\; u, v \in C;$

(iii) for each $u, v, w \in C, \lim_{t\to 0} f (tw + (1-t)u, v) \leq f (u, v);$

(iv) for each $u \in C, f (u, .)$ is convex and lower semicontinuous.$

Then $EP(f)\neq \phi.$

Lemma 2.3. ^[2] Let $r > 0, u \in \mathbb{H}$, and $f$ satisfy the conditions (i)-(iv) in Lemma (2.2), then there exists $w \in C$ such that $f (w, v) + \frac{1}{r}\langle v-w, w-u\rangle \geq 0 \forall\; v \in C$.

Lemma 2.4. ^[2] Let $r > 0, u \in \mathbb{H}$, and $f$ satisfy the conditions (i)-(iv) in Lemma (2.2). Define a mapping $\mathrm {T}_r:\mathbb{H} \to C$ as $\mathrm {T}_r(u) = \{ w \in C : f (w, v)+ \frac{1}{r}\langle v-w, w-u\rangle \geq 0 \forall\; v \in C \}$. Then the following hold:

(i) $\mathrm {T}_r$ is single-valued;

(ii) $\mathrm {T}_r$ is firmly nonexpansive, i.e., $\lVert \mathrm {T}_r u- \mathrm {T}_rv \rVert \leq \langle \mathrm {T}_ru- \mathrm {T}_rv, u-v \rangle$ for all $u, v \in \mathbb{H}$;

(iii) $EP(f) = F(\mathrm {T}_r)$, where $F(\mathrm {T}_r)$ denotes the sets of fixed point of $\mathrm {T}_r$;

(iv) $EP(f)$ is closed and convex.

Definition 2.1. ^[15] Let $A : \mathbb{H} \to \mathbb{H}$ be a set valued mapping with the effective domain $D(A) = \{x \in \mathbb{H}: A x\neq \phi\}$.

The set valued mapping $A$ is said to be monotone if, for each $x, y \in D(A), u \in A x$ and $v \in A y$, we have

As, graph of $A$ is defined by $G(A) = \{(x, y): y \in A x\}$ and the mapping $A$ is maximal if its graph $G(A)$ is not appropriately contained in the graph of any other mapping which is of same type as $A$. The accompanying nature of the maximal monotone mappings is advantageous and supportive to utilize:

A monotone mapping $A$ is maximal if and only if, for $(x, u) \in \mathbb{H} \times \mathbb{H},$

For maximal monotone set-valued mapping $A$ on $\mathbb{H}$ and $r > 0$, the operator

is called the resolvent of $A$.

Consider $f : C \to \mathbb{H}$ be a monotone and $k$-Lipschitz continuous mapping. In ^[5], normal cone to $C$ is specified by

is maximal monotone and resolvent of $N_{ C }$ is $P_{ C }$.

Lemma 2.5. ^[14] Let $\mathbb{H}_{1}$ and $\mathbb{H}_{2}$ be real Hilbert spaces. Let $A : \mathbb{H}_{1} \to \mathbb{H}_{1}$ be a maximal monotone mapping and $J_r$ be the resolvent of $A$ for $r < 0$. Suppose that $\mathrm {T}: \mathbb{H}_{2} \to \mathbb{H}_{2}$ is a nonexpansive mapping and $B : \mathbb{H}_{1} \to \mathbb{H}_{1}$ is a bounded linear operator. Assume that $A ^{-1}0 \cap B ^{-1}Fix(\mathrm {T})\neq \phi.$ Let $r, \alpha > 0$ and $z \in \mathbb{H}_{1}$. Then the following statements are equivalent:

(i) $z = J_r(I-\alpha B ^*(I- \mathrm {T}) B)z;$

(ii) $0 \in B ^*(I-T) B z+ A z;$

(iii) $z \in A ^{-1}0\cap B ^{-1}Fix(\mathrm {T}).$

Lemma 2.6. ^[16] Let $\{\beta_n\}$ be a real sequence satisfying $0 < a\leq \beta_n \leq b < 1$ for all $n \geq 0$, and let $\{\mu_n\}$ and $\{\nu_n\}$ be two sequences in $\mathbb{H}$ such that, for some $\eta \geq 0$,

Lemma 2.7. ^[6] Let $\{a_n\}$ be a sequence in $\mathbb{H}$ satisfying the properties:

(i) $\lim_{n \to\infty} \lVert a_n -a\rVert$ exists for each $a \in C$;

(ii) $\omega_w(a_n)\subset C$, where $\omega_w(a_n)$ represents the set of all weak cluster points of $\{a_n\}$.

Then $\{a_n\}$ converges weakly to a point in $C$.

3.   Main results

Throughout, we consider $C$ and $Q$ both are nonempty closed convex subsets of real Hilbert spaces $\mathbb{H}_{1}$ and $\mathbb{H}_{2}$, respectively. Suppose that $B : \mathbb{H}_{1} \to \mathbb{H}_{2}$ is a non-zero bounded linear operator, $f : C \times C \to \mathbb R$ and $F : Q \times Q \to \mathbb R$ be two functions satisfiy the conditions (i) to (iv) of Lemma (2.2), $\mathrm {g}: C \to \mathbb{H}_{1}$ is a monotone and $k$-Lipschitz continuous mapping and $\mathrm {g}^{\prime}: \mathbb{H}_{2} \to \mathbb{H}_{2}$ is a $\delta$- inverse strongly monotone mapping. Suppose $\mathrm {T} : \mathbb{H}_{2} \to \mathbb{H}_{2}$ and $\mathrm{S}: C \to C$ are nonexpansive mappings. Let $\{\mu_n\}, \{\alpha_n\} \subset (0, 1), \{\gamma_n\}\subset [a, b]$ for some $a, b \in \left(0, \frac{1}{\lVert B \rVert^2}\right)$ and $\{\lambda_n\} \subset [c, d]$ where $c, d \in (0, \frac{1}{k})$.

Initially, we define an algorithm for solving VIP, SCFPP and SEP, the purpose of which is to discover an element $a^*$ in such a way that

Theorem 3.1. Fix $\daleth = \{u \in VI(C, \mathrm {g})\cap Fix(\mathrm{S})\cap EP(f): B u\in Fix(\mathrm {T})\cap EP(F)\}$ and consider that $\daleth \neq \phi$. Let the sequences $\{u_n\}, \{v_n\}, \{a_n\}, \{b_n\}$ and $\{c_n\}$ be defined by $a_1 = a \in C$ and

for all $n \in \mathbb N$. Then, the sequence $\{a_n\}$ weakly converges to an element $u \in \daleth$, where $u = \lim_{n \to \infty}P_{ \daleth}a_n$.

Proof. Let $a \in \daleth$ and consider $\{ \mathrm {T}_{r_n}\}$ be a sequence of mapping as stated in Lemma (2.4), gives $a = P_{ C }(a-\lambda_n B a) = \mathrm {T}_{r_n}^{ f }a$, also $a = \mathrm {T}_{r_n}^{ F } B a$ as $a\in EP(f)$ and $B a \in EP(F)$.

From Theorem 3.1 of ^[11] that $P_{ C }(I-\gamma_n B ^*(I- \mathrm {T}) B)$ is $\frac{1+\gamma_n\lVert B \rVert^2}{2}$ averaged. It is clear to see from Lemma 2.2 of ^[10] that $\mu_n I +(1-\mu_n)P_{ C }(I-\gamma_n B ^* (I- \mathrm {T}) B)$ is $\mu_n+(1-\mu_n) \frac{1+\gamma_n\lVert B \rVert^2}{2}$ averaged. So, $c_n$ can be written as

where $\beta_n = \mu_n+(1-\mu_n) \frac{1+\gamma_n\lVert B \rVert^2}{2}$ and $V_n$ is a nonexpansive mapping for each $n \in \mathbb N$.

Let $a \in \daleth$ and from Lemma (2.4), we obtain

and hence,

Also,

From (3.5)

implies

Now, set $z_n = P_{ C }(c_n-\lambda_n \mathrm {g}(b_n))$ for all $n \geq 0$. It pursue from Lemma (2.1) that

Using the monotonicity of $\mathrm {g}$ and $a$ is solution of $VIP(\mathrm {g}, C)$, we have

From Eqs (3.8) and (3.9), we obtain

Using condition (iii) of Lemma (2.1) again, this yields

and so,

for each $n \in \mathbb N$, we obtain that

that is,

implies

Now, by Eqs (3.6) and (3.10), we have

Hence, there exists a constant $s \geq 0$ such that

implies $\{a_n\}$ is bounded. This gives us that, $\{b_n\}, \{c_n\}, \{u_n\}$ and $\{v_n\}$ are all bounded.

From Eqs (3.7) and (3.11), we deduce

By using Eq (3.12), we find

From Eq (3.3), we calculate

Using Eqs (3.5), (3.6) in (3.11), we get

Again, using Eqs (3.4), (3.5), (3.6) in (3.11), we obtain

By triangle inequality, we find

Also,

Equations (3.6), (3.11) and (3.17), implies

and so

Thus,

Also, by definition of $\{b_n\}$, we have

implies, $b_n-z_n\to 0$ as $n \to \infty.$ Again using triangle inequality, we have

and

gives, when $n \to \infty$

From the definition of $\{c_n\}$, we implies

Thus, Equation (3.13) gives

Let $z \in \omega_w(u_n)$. Then there exists a subsequence $\{u_{n_i}\}$ of $\{u_n\}$ which weakly convergent to $z$. We acquire that $\{ B ^*(I- \mathrm {T}) B u_n\}$ is bounded, reason being $B ^*(I- \mathrm {T}) B$ is $\frac{1}{2\lVert B \rVert^2}$ inverse strongly monotone. From the firm nonexpansive nature of $P_{ C }$, we have

Without loss of generality, we assume that $\gamma_{n_i} \to \hat\gamma \in \left(0, \frac{1}{\lVert B \rVert^2}\right)$ and so,

Consider

In particular

From Eqs (3.18), (3.20) and (3.21) in (3.23), we obtain

Now, using Eq (3.16), (3.24) in (3.22), we find

By the demiclosedness principle ^[8], (3.24) and (3.25), respectively, implies

From Corollary 2.9 ^[11] and Lemma (2.4), we obtain

Now we claim that $z \in VI(C, \mathrm {g})$. From Eqs (3.13), (3.14), (3.17), (3.18) and (3.19), we obtain $b_{n_i} \to u$, $c_{n_i} \to u, z_{n_i} \to u, u_{n_i} \to u$ and $v_{n_i} \to u$. Interpret the set-valued mapping $A : \mathbb{H} \to \mathbb{H}$ by

In 2006, Takahashi ^[14] suggested that $A$ is maximal monotone and for this $0\in A v$ iff $v \in VI(C, \mathrm {g})$. For $(v, w)\in D(A)$ we have $w \in A v = \mathrm {g}(v)+N_{ C }v$ and implies $w- \mathrm {g}(v) \in N_{ C }v$. Therefore, for any $x \in C$, we get

As $v\in C$. The explanation of $b_n$ and Lemma (2.1) implies that

Continuing

By Equation (3.25) with $\{b_{n_i}\}$, we obtain

Thus,

By considering $i \to \infty$, we have

By maximal monotonicity of $A$, we obtain $0 \in A z$ and then $z \in VI(C, \mathrm {g})$.

Now, we will exhibit that $z \in Fix(\mathrm{S})$.

From Eqs (3.6), (3.10) and the nonexpansive nature of $\mathrm{S}$, we get

By taking limit superior

and

Further

Thus, Lemma (2.6), implies

Again from the fact of

By Eqs (3.19) and (3.27), we find

This infers that

Thus, we have $z \in Fix(\mathrm{S}).$

Now, we prove $z \in EP(f)$. As $u_n = \mathrm {T}_{r}^{ f }a_n$ and

From monotonicity of f, we have

and hence

Since $\frac{u_{n_i}-a_{n_i}}{r} \to 0$ as $u_{n_i}\to 0$ weakly lower semicontinuity of $f (a, y)$ on second variable $y$, we have

For $k$ with $0\leq k \leq 1$ and $z \in C$, let $z_t = kz+(1-t)u$

Consequently, $w_w(a_n) \subset \daleth$. From the Lemma (2.7), the sequence $\{a_n\}$ converges weakly to an element $u \in \daleth$ and Lemma 3.2 ^[13] satisfies $u = \lim_{n \to \infty} P_{ \daleth} a_n.$

The successive result gives us the suitable conditions to obtain the presence of a common solution of the split variational inequality problems, fixed point problems and split equilibrium problems, that is, to discover an element $a^*$ in such a way that

Theorem 3.2. Set $\daleth = \{u \in VI(C, \mathrm {g})\cap Fix(\mathrm{S})\cap EP(f): B u\in VI(Q, \mathrm {g}^{\prime})\cap EP(F)\}$ and consider that $\daleth \neq \phi$. Let the sequences $\{u_n\}, \{v_n\}, \{a_n\}, \{b_n\}$ and $\{c_n\}$ be defined by $a_1 = a \in C$ and{

for all $n \in \mathbb N$. Then, the sequence $\{a_n\}$ weakly converges to an element $u \in \daleth$, where $u = \lim_{n \to \infty}P_{ \daleth}a_n$.

Proof. From the $\delta$-inverse strongly monotonicity of $\mathrm {g}^{\prime}$, it is $\frac{1}{\delta}$- Lipschitz continuous and so, $\theta \in(0, 2\delta)$, we find that $I-\theta \mathrm {g}^{\prime}$ is nonexpansive. Also, $P_{ Q }$ is firmly nonexpansive, implies $P_{ Q }(I-\theta \mathrm {g}^{\prime})$ is nonexpansive. With replacement of $\mathrm {T} = P_{ Q }(I-\theta \mathrm {g}^{\prime})$ in Theorem (3.1), we get that $\{a_n\}$ is weakly convergent to an element $u \in VI(C, \mathrm {g})\cap Fix(\mathrm{S})\cap EP(f)$ and $B u\in Fix(P_{ Q }(I-\theta \mathrm {g}^{\prime})\cap \mathrm {T}_{r_n}^{ F })$. We pursue from $B u = P_{ Q }(I-\theta \mathrm {g}^{\prime}) B u$ and $B u \in EP(F)$ and Lemma (2.1) that $B u \in VI(Q, \mathrm {g}^{\prime}) \cap EP(F)$. This completes the proof.

The following results are the direct consequences of Theorem (3.1).

Theorem 3.3. Let $A : \mathbb{H}_{2} \to \mathbb{H}_{2}$ be a maximal monotone mapping with $D(A)\neq \phi$. Consider $\daleth = \{u\in VI(C, \mathrm {g})\cap Fix(\mathrm{S})\cap EP(f): B u\in A ^{-1}0\cap EP(F)\}$ and assume $\daleth \neq \phi$. Let the sequences $\{u_n\}, \{v_n\}, \{a_n\}, \{b_n\}$ and $\{c_n\}$ be defined by $a_1 = a \in C$ and{

for all $n \in \mathbb N$, where $J-r$ is resolvent of $A$ for $r > 0$. Then, the sequence $\{a_n\}$ weakly converges to an element $u\in \daleth$, where $u = \lim_{n \to\infty}P_{ \daleth}a_n.$

Proof. From the firmly nonexpansive nature of $J_r$ and $Fix(J_r) = A ^{-1}0$, the proof remains the same as of Theorem (3.1) by considering $J_r = \mathrm {T}$.

Theorem 3.4. Let $A : \mathbb{H}_{2} \to \mathbb{H}_{2}$ be a maximal monotone mapping with $D(A)\neq \phi$ and $\mathrm G : \mathbb{H}_{2} \to \mathbb{H}_{2}$ be a $\delta$-inverse strongly monotone mapping. Set $\daleth = \{u\in VI(C, \mathrm {g})\cap Fix(\mathrm{S})\cap EP(f): B u\in (A + \mathrm G)^{-1}0 \cap EP(F)\}$ and assume $\daleth \neq \phi$. Let the sequences $\{u_n\}, \{v_n\}, \{a_n\}, \{b_n\}$ and $\{c_n\}$ be defined by $a_1 = a \in C$ and{

}for all $n \in \mathbb N$, where $J-r$ is resolvent of $A$ for $r > 0$. Then, the sequence $\{a_n\}$ weakly converges to an element $u\in \daleth$, with $u = \lim_{n \to\infty}P_{ \daleth}a_n.$

Proof. From the $\delta-$ strongly inverse monotone nature of $\mathrm G$ implies that $I-r \mathrm G$ is nonexpansive. Also, from the nonexpansive nature of $J_r$, we get that $J_r(I-r \mathrm G)$ is also nonexpansive. As $u \in (A + \mathrm G)^{-1}0$ if and only if $u = J_r(I-r \mathrm G)u$. Thus, the proof remains the same as of Theorem (3.1) by considering $J_r(I-r \mathrm G) = \mathrm {T}$.

4.   Conclusions

We obtained the weak convergence of the defined algorithm for solving variational inequality, split common fixed point and split equilibrium problems, by extending the results of Tian et al. ^[11] and Lohewech et al. ^[12].

Acknowledgments

Special thanks to CSIR PUSA Delhi to provide scholarship for the research under the file no-09/382(0187)/2017-EMR-I.

Conflict of interest

There is no conflict of interest.