Split nonconvex variational inequality problem

Abstract

In this paper, we propose a split nonconvex variational inequality problem which is a natural extension of split convex variational inequality problem in two different Hilbert spaces. Relying on the prox-regularity notion, we introduce and establish the convergence of an iterative method for the new split nonconvex variational inequality problem. Further, we also establish the convergence of an iterative method for the split convex variational inequality problem. The results presented in this paper are new and different form the previously known results for nonconvex (convex) variational inequality problems. These results also generalize, unify, and improve the previously known results of this area.

2010 MSC

Primary 47J53, 65K10; Secondary 49M37, 90C25

Introduction

Recently, Censor et al.[1] introduced and studied the following split convex variational inequality problem (SCVIP): Let H₁ and H₂ be real Hilbert spaces with inner product 〈·,·〉 and norm ∥ · ∥. Let C and Q be nonempty, closed, and convex subsets of H₁ and H₂, respectively. Let f:H₁ → H₁ and g : H₂ → H₂ be nonlinear mappings and A : H₁ → H₂ be a bounded linear operator with its adjoint operator A^∗. Then, the SCVIP is to find x^∗ ∈ C such that

$$〈f\left(x^{\ast }\right),x-x^{\ast }〉\ge 0,\forall x\in C$$

(1a)

and such that

$$y^{\ast }=A{x}^{\ast }\in Q\text{solves}〈g\left(y^{\ast }\right),y-y^{\ast }〉\ge 0,\forall y\in \mathrm{Q.}$$

(1b)

SCVIP amount to saying: find a solution of variational inequality problem (VIP) (1a), the image of which under a given bounded linear operator is a solution of VIP (1b). It is worth mentioning that SCVIP is quite general and permits split minimization between two spaces, so the image of a minimizer of a given function, under a bounded linear operator, is a minimizer of another function. The special cases of SCVIP are split zero problem and split feasibility problem which has already been studied and used in practice as a model in intensity-modulated radiation therapy treatment planning. This formulation is also at the core of the modeling of many inverse problems arising for phase retrieval and other real-world problems; for instance, in sensor networks in computerized tomography and data compression; see[2–5].

In this paper, we intend to generalize SCVIP (1a,b) to take into account of nonconvexity of the subsets C and Q. This new nonconvex problem is called split nonconvex variational inequality problem (SNVIP). To overcome the difficulties that arise from the nonconvexity of C and Q, we will consider the class of uniform prox-regular sets, which is sufficiently large to include the class of convex sets, p-convex sets, C^1,1 submanifolds, and many other nonconvex sets; see[6]. Using the properties of projection operator over uniformly prox-regular sets, we establish the convergence of an iterative method for SNVIP. Further, we also establish the convergence of an iterative method for the split convex variational inequality problem. The results presented in this paper are new and different form the previously known results for nonconvex (convex) variational inequality problems. These results also generalize, unify, and improve the previously known results of this area.

To begin with, let us recall the following concepts which are of common use in the context of nonsmooth analysis; see[6–9].

Throughout the rest of the paper unless otherwise stated, let C and Q be nonempty closed subsets of H₁ and H₂, respectively, not necessarily convex.

Definition 1.

The proximal normal cone of C at x ∈ H₁ is given by

$$N_C^P\left(x\right):=\{\xi \in H_1:x\in P_C(x+\mathrm{\alpha \xi }\left)\right\},$$

where α > 0 is a constant and P _C  is projection operator of H₁ onto C, that is,

$$P_C\left(x\right)=\{x^{\ast }\in C:d_C(x)=\parallel x-x^{\ast }\parallel \},$$

where d _C  (x) is the usual distance function to the subset C, that is,

$$d_C\left(x\right)=\underset{\widehat{x}\in C}{inf}\parallel \widehat{x}-x\parallel .$$

The proximal normal cone$N_C^P\left(x\right)$ has the following characterization.

Lemma 1.

Let C be a nonempty closed subset of H₁. Then,$\xi \in N_C^P\left(x\right)$ if and only if there exists a constant α :=α(ξ,x) > 0 such that

$$〈\xi ,\widehat{x}-x〉\le \alpha \parallel \widehat{x}-x{\parallel }^2,\forall \widehat{x}\in \mathrm{C.}$$

Definition 2.

The Clarke normal cone, denoted by$N_C^{\mathit{\text{cl}}}\left(x\right)$, is defined as

$$N_C^{\mathit{\text{cl}}}\left(x\right)=\overline{\text{co}}\left[N_C^P\right(x\left)\right],$$

where$\overline{\text{co}}A$ means the closure of the convex hull of A.

Poliquin and Rockafellar[7] and Clarke et al.[8] have introduced and studied a class of nonconvex sets, which are called uniformly prox-regular sets. This class of uniformly prox-regular sets has played an important role in many nonconvex applications such as optimization, dynamic systems, and differential inclusions. In particular, we have

Definition 3.

For a given r ∈ (0,∞], a subset C _r  of H₁ is said to be normalized uniformly prox-regular (or uniformly r-prox-regular) if and only if every nonzero proximal normal to C _r  can be realized by any r-ball, that is, ∀ x ∈ C _r  and$0\ne \xi \in N_{C_r}^P\left(x\right)$, one has

$$〈\frac{\xi }{\parallel \xi \parallel },\widehat{x}-x〉\le \frac{1}{2r}\parallel \widehat{x}-x{\parallel }^2,\forall \widehat{x}\in C_r.$$

It is known that if C _r  is a uniformly r-prox-regular set, the proximal normal cone$N_{C_r}^P\left(x\right)$ is closed as a set-valued mapping. Thus, we have$N_{C_r}^{\mathit{\text{cl}}}\left(x\right)=N_{C_r}^P\left(x\right)$. We make the convention$\frac{1}{r}=0$ for r = +∞. If r = +∞, then uniformly r-prox-regularity of C _r  reduces to its convexity; see[6].

Now, let us state the following proposition which summarizes some important consequences of the uniformly prox-regularities:

Proposition 1.

Let r > 0 and let C _r  be a nonempty, closed, and uniformly r-prox-regular subset of H₁. Set U _r  = {x ∈ H₁ : d(x,C _r ) < r}.

• (1) For all x ∈ U _r , P _{C r}  (x) ≠ ∅;

• (2) For all $r^{\prime }\in (0,r),P_{C_r}$ is Lipschitz continuous with constant $\frac{r}{r-r^{\prime }}$ on $U_{r^{\prime }}=\{x\in H_1:d(x,C_r)<r^{\prime }\}$.

Split nonconvex variational inequality problem

Throughout the paper unless otherwise stated, we assume that for given r,s ∈ (0,+∞), C _r  and Q _s  are uniformly prox-regular subsets of H₁ and H₂, respectively. The SNVIP is formulated as follows:

find x^∗ ∈ C _r  such that

$$〈f\left(x^{\ast }\right),x-x^{\ast }〉+\left(\frac{\parallel f\left(x^{\ast }\right)\parallel }{2r}\right)\parallel x-x^{\ast }{\parallel }^2\ge 0,\forall x\in C_r$$

(2a)

and such that

$$\begin{array}{ll}{y}^{\ast }=& A{x}^{\ast }\in Q_s\text{solves}〈g\left(y^{\ast }\right),y-y^{\ast }〉\\ +\left(\frac{\parallel g\left(y^{\ast }\right)\parallel }{2s}\right)\parallel y-y^{\ast }{\parallel }^2\ge 0,\forall y\in Q_s.\end{array}$$

(2b)

By making use of Definition 3 and Lemma 1, SNVIP 2a,b) can be reformulated as follows:

find (x^∗,y^∗) ∈ C _r  × Q _s  with y^∗ = A x^∗ such that

$$\begin{array}{ll}\mathbf{\text{0}}& \in \mathrm{\rho f}\left(x^{\ast }\right)+N_{C_r}^P\left(x^{\ast }\right)\\ \mathbf{\text{0}}& \in \mathrm{\lambda g}\left(y^{\ast }\right)+N_{Q_s}^P\left(y^{\ast }\right),\end{array}$$

where ρ and λ are parameters with positive values and 0 denotes the zero vectors of H₁ and H₂, which in turn, since${\text{Proj}}_{C_r}={(I+N_{C_r}^P)}^{-1}$ and${\text{Proj}}_{Q_s}={(I+N_{Q_s}^P)}^{-1}$, is equivalent to find (x^∗,y^∗) ∈ C _r  × Q _s  with y^∗ = A x^∗ such that

$$\begin{array}{ll}{x}^{\ast }& ={\text{Proj}}_{C_r}(x^{\ast }-\mathrm{\rho f}(x^{\ast }\left)\right)\\ y^{\ast }& ={\text{Proj}}_{Q_s}(y^{\ast }-\mathrm{\lambda g}(y^{\ast }\left)\right),\end{array}$$

where$0<\rho <\frac{r}{1+\parallel f\left(x^{\ast }\right)\parallel }$,$0<\lambda <\frac{s}{1+\parallel g\left(y^{\ast }\right)\parallel }$, and${\text{Proj}}_{C_r}$ and${\text{Proj}}_{Q_s}$ are, respectively, projection onto C _r  and Q _s .

If r,s = +∞, then C _r  = C and Q _s  = Q, the closed convex subsets of H₁ and H₂, respectively, and hence, SNVIP (2a,b) reduces to SCVIP (1a,b) which is equivalent to find (x^∗,y^∗) ∈ C × Q with y^∗ = A x^∗ such that

$$\begin{array}{ll}{x}^{\ast }& ={\text{Proj}}_C(x^{\ast }-\mathrm{\rho f}(x^{\ast }\left)\right)\\ y^{\ast }& ={\text{Proj}}_Q(y^{\ast }-\mathrm{\lambda g}(y^{\ast }\left)\right),\end{array}$$

where Proj _C  and Proj _Q  are, respectively, projection onto C and Q.

If H₁ = H₂, f = g, A = I, identity operator, ρ = λ, r = s, and C _r  = Q _s , then SNVIP (1a,b) reduces to the nonconvex variational inequality problems (NVIP):

find x^∗ ∈ C _r  such that

$$〈f\left(x^{\ast }\right),x-x^{\ast }〉+\left(\frac{\parallel f\left(x^{\ast }\right)\parallel }{2r}\right)\parallel x-x^{\ast }{\parallel }^2\ge 0,\forall x\in C_r.$$

A number of authors developed and studied iterative methods for various classes of NVIPs; see for instance[10–12] and the references therein.

Definition 4.

A nonlinear mapping f : H₁ → H₁ is said to be

• (1) monotone, if

  $$〈f\left(x\right)-f\left(\widehat{x}\right),x-\widehat{x}〉\ge 0,\forall x,\widehat{x}\in H_1;$$

• (2) α-strongly monotone, if there exists a constant α > 0 such that

  $$〈f\left(x\right)-f\left(\widehat{x}\right),x-\widehat{x}〉\ge \alpha \parallel x-\widehat{x}{\parallel }^2,\forall x,\widehat{x}\in H_1;$$

• (3) k-inverse strongly monotone, if there exists a constant k > 0 such that

  $$〈f\left(x\right)-f\left(\widehat{x}\right),x-\widehat{x}〉\ge k\parallel f\left(x\right)-f\left(\widehat{x}\right){\parallel }^2,\forall x,\widehat{x}\in H_1;$$

• (4) β-Lipschitz continuous, if there exists a constant k > 0 such that

  $$\parallel f\left(x\right)-f\left(\widehat{x}\right)\parallel \le \beta \parallel x-\widehat{x}\parallel ,\forall x,\widehat{x}\in H_1.$$

It is easy to observe that every k-inverse strongly monotone mapping f is monotone and$\frac{1}{k}$-Lipschitz continuous.

Based on above arguments, we propose the following iterative method for approximating a solution to SNVIP (2a,b).

Algorithm 1.

Given x₀ ∈ C _r , compute the iterative sequence {x _n } defined by the iterative schemes:

$$y_n={\text{Proj}}_{C_r}(x_n-\mathrm{\rho f}(x_n\left)\right)$$

(3a)

$$z_n={\text{Proj}}_{Q_s}(A{y}_n-\mathrm{\lambda g}(A{y}_n\left)\right)$$

(3b)

$$x_{n+1}={\text{Proj}}_{C_r}[y_n+\gamma A^{\ast }(z_n-A{y}_n\left)\right]$$

(3c)

for all n = 0,1,2,.....,$0<\rho <\frac{r}{1+\parallel f\left(x_n\right)\parallel }$,$0<\lambda <\frac{s}{1+\parallel g\left(A{y}_n\right)\parallel }$ and$0<\gamma <\frac{r}{1+\parallel A^{\ast }(z_n-A{y}_n)\parallel }$.

As a particular case of Algorithm 1, we have the following algorithm for approximating a solution to SCVIP (1a,b).

Algorithm 2.

Given x₀ ∈ C, compute the iterative sequence {x _n } defined by the iterative schemes:

$$y_n={\text{Proj}}_C(x_n-\mathrm{\rho f}(x_n\left)\right)$$

(4a)

$$z_n={\text{Proj}}_Q(A{y}_n-\mathrm{\lambda g}(A{y}_n\left)\right)$$

(4b)

$$x_{n+1}={\text{Proj}}_C[y_n+\gamma A^{\ast }(z_n-A{y}_n\left)\right]$$

(4c)

for all n = 0,1,2,......

Further, we propose the following iterative method for SCVIP (1a,b) which is more general than Algorithm 2.

Let {α _n } ⊆ (0,1) be a sequence such that$\sum _{n=1}^{\infty }{\alpha }_n=+\infty$, and let ρ, λ, γ are parameters with positive values.

Algorithm 3.

Given x₀ ∈ H₁, compute the iterative sequence {x _n } defined by the iterative schemes:

$$y_n={\text{Proj}}_C(x_n-\mathrm{\rho f}(x_n\left)\right)$$

(5a)

$$z_n={\text{Proj}}_Q(A{y}_n-\mathrm{\lambda g}(A{y}_n\left)\right)$$

(5b)

$$x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n[y_n+\gamma A^{\ast }(z_n-A{y}_n\left)\right]$$

(5c)

for all n = 0,1,2,......

We remark that Algorithms 2 and 3 are different from Algorithm 5.1[1].

Results

Now, we study the convergence analysis of the Algorithm 1.

Assume that r^′ ∈ (0,r), s^′ ∈ (0,s) and denote$\delta =\frac{r}{r-r^{\prime }}$ and$\eta =\frac{s}{s-s^{\prime }}$.

Theorem 1.

For given r,s ∈ (0,+∞), let C _r  and Q _s  be uniformly prox-regular subsets of H₁ and H₂, respectively. Let f : H₁ → H₁ be α-strongly monotone and β-Lipschitz continuous and let g : H₂ → H₂ be σ-strongly monotone and μ-Lipschitz continuous. Let A:H₁ → H₂ be a bounded linear operator such that A(C _r ) ⊆ Q _s  and A^∗ be its adjoint operator. Suppose x^∗ ∈ C _r  is a solution to SNVIP (2a,b), then the sequence {x _n } generated by Algorithm 1 strongly converges to x^∗ provided that ρ,λ, and γ satisfy the following conditions:

$$\frac{2\alpha }{{\beta }^2}-\Delta <\rho <min\left\{\frac{2\alpha }{{\beta }^2}+\Delta ,\frac{r^{\prime }}{1+\parallel f\left(x_n\right)\parallel },\frac{r^{\prime }}{1+\parallel f\left(x^{\ast }\right)\parallel }\right\},$$

(6a)

$$0<\lambda <min\left\{\frac{s^{\prime }}{1+\parallel g\left(A{y}_n\right)\parallel },\frac{s^{\prime }}{1+\parallel g\left(A{x}^{\ast }\right)\parallel }\right\},$$

(6b)

$$0<\gamma <min\left\{\frac{2}{\parallel A{\parallel }^2},\frac{r^{\prime }}{1+\parallel A\ast (z_n-A{y}_n)\parallel }\right\},$$

(6c)

where

$$\begin{array}{ll}\Delta :=& \frac{1}{{\beta }^2}\left(\sqrt{4{\alpha }^2-{\beta }^2(1-d^2)}\right)\text{with}\\ 2\alpha >\beta \sqrt{1-d^2};0<r^{\prime }<r,0<s^{\prime }<s;\\ d:=& {\left[{\delta }^2\right(1+2\eta {\theta }_2\left)\right]}^{-1};{\theta }_1:=\sqrt{1-2\mathrm{\rho \alpha }+{\rho }^2{\beta }^2};\\ {\theta }_2:=& \sqrt{1-2\mathrm{\lambda \sigma }+{\lambda }^2{\mu }^2}.\end{array}$$

Proof.

Since x^∗ ∈ C _r  is a solution to SNVIP (2a,b) and the parameters ρ, λ satisfy the conditions (6a,b), then we have

$$\begin{array}{ll}{x}^{\ast }=& {\text{Proj}}_{C_r}(x^{\ast }-\mathrm{\rho f}(x^{\ast }\left)\right)\\ A{x}^{\ast }=& {\text{Proj}}_{Q_s}(A{x}^{\ast }-\mathrm{\lambda g}(A{x}^{\ast }\left)\right).\end{array}$$

From Algorithm 1 (3a) and condition (6a) on ρ, we have

$$\begin{array}{ll}\parallel y_n-x^{\ast }\parallel & =\parallel {\text{Proj}}_{C_r}(x_n-\mathrm{\rho f}(x_n\left)\right)-{\text{Proj}}_{C_r}(x^{\ast }-\mathrm{\rho f}(x^{\ast }\left)\right)\parallel \\ \le \delta \parallel (x_n-x^{\ast }-\rho (f\left(x_n\right)-f\left(x^{\ast }\right))\parallel .\end{array}$$

Now, using the fact that f is α-strongly monotone and β-Lipschitz continuous, we have

$$\begin{array}{ll}\parallel (x_n-x^{\ast }& -\rho \left(f\right(x_n)-f(x^{\ast }\left)\right){\parallel }^2\\ =\parallel (x_n-x^{\ast }{\parallel }^2-2\rho 〈f(x_n)-f(x^{\ast }),x_n-x^{\ast }〉\\ +{\rho }^2\parallel f\left(x_n\right)-f\left(x^{\ast }\right){\parallel }^2\\ \le (1-2\mathrm{\rho \alpha }+{\rho }^2{\beta }^2)\parallel (x_n-x^{\ast }{\parallel }^2.\end{array}$$

As a result, we obtain

$$\parallel y_n-x^{\ast }\parallel \le \delta {\theta }_1\parallel x_n-x^{\ast }\parallel ,$$

(7)

where${\theta }_1=\sqrt{1-2\mathrm{\rho \alpha }+{\rho }^2{\beta }^2}$.

Similarly, from Algorithm 1 (3b), condition (6b) on parameter λ and using the fact that g is σ-strongly monotone and μ-Lipschitz continuous, and A(C _r ) ⊆ Q _s , we have

$$\begin{array}{ll}\parallel z_n-A{x}^{\ast }\parallel =& \parallel {\text{Proj}}_{Q_s}(A{y}_n-\mathrm{\lambda g}(A{y}_n\left)\right)\\ -{\text{Proj}}_{Q_s}(A{x}^{\ast }-\mathrm{\lambda g}(A{x}^{\ast }\left)\right)\parallel \\ \le & \eta \parallel A{y}_n-x^{\ast }-\lambda \left(g\right(A{y}_n)-g(A{x}^{\ast }\left)\right)\parallel \end{array}$$

$$\le \eta {\theta }_2\parallel A{y}_n-A{x}^{\ast }\parallel ,$$

(8)

where${\theta }_2=\sqrt{1-2\mathrm{\lambda \sigma }+{\lambda }^2{\mu }^2}$.

Next from Algorithm 1 (3c) and condition (6c) on γ, we have

$$\begin{array}{ll}\parallel x_{n+1}-x^{\ast }\parallel & =\parallel {\text{Proj}}_{C_r}[y_n+\gamma A^{\ast }(z_n-A{y}_n\left)\right]\\ -{\text{Proj}}_{C_r}[x^{\ast }+\gamma A^{\ast }(A{x}^{\ast }-A{x}^{\ast }\left)\right]\parallel \\ \le \delta [\parallel y_n-x^{\ast }-\gamma A^{\ast }(A{y}_n-A{x}^{\ast })\parallel \\ +\gamma \parallel A^{\ast }(z_n-A{x}^{\ast })\parallel ].\end{array}$$

(9)

Further, using the definition of A^∗, the fact that A^∗ is a bounded linear operator with ∥A^∗ ∥ = ∥ A∥, and condition (6c), we have

$$\begin{array}{ll}\parallel y_n-x^{\ast }& -\gamma A^{\ast }(A{y}_n-A{x}^{\ast }){\parallel }^2\\ =\parallel y_n-x^{\ast }{\parallel }^2-2\gamma 〈y_n-x^{\ast },A^{\ast }(A{y}_n-A{x}^{\ast })〉\\ +{\gamma }^2\parallel A^{\ast }(A{y}_n-A{x}^{\ast }){\parallel }^2\\ \le \parallel y_n-x^{\ast }{\parallel }^2-\gamma (2-\gamma \parallel A{\parallel }^2)\parallel A{y}_n-A{x}^{\ast }{\parallel }^2\\ \le \parallel y_n-x^{\ast }{\parallel }^2,\end{array}$$

(10)

and using Eq. (8), we have

$$\begin{array}{ll}\parallel A^{\ast }(z_n-A{x}^{\ast })\parallel & \le \parallel A\parallel \parallel z_n-A{x}^{\ast }\parallel \\ \le \eta {\theta }_2\parallel A\parallel \parallel A{y}_n-A{x}^{\ast }\parallel \\ \le \eta {\theta }_2\parallel A{\parallel }^2\parallel y_n-x^{\ast }\parallel .\end{array}$$

(11)

Combining Eqs. (10) and (11) with inequality (9), as a result, we obtain

$$\parallel x_{n+1}-x^{\ast }\parallel \le \delta [\parallel y_n-x^{\ast }\parallel +\mathrm{\gamma \eta }{\theta }_2\parallel A{\parallel }^2\parallel y_n-x^{\ast }\parallel ].$$

Using Eq. (7), we have

$$\parallel x_{n+1}-x^{\ast }\parallel \le \theta \parallel x_n-x^{\ast }\parallel ,$$

where θ = δ² θ₁(1 + γ∥A∥²η θ₂).

Thus, we obtain

$$\parallel x_{n+1}-x^{\ast }\parallel \le {\theta }^n\parallel x_0-x^{\ast }\parallel .$$

(12)

Since γ∥A∥² < 2, hence the maximum value of (1+γ∥A∥²η θ₂) is (1 + 2η θ₂). Further, θ ∈ (0,1) if and only if

$${\theta }_1<{\left[{\delta }^2\right(1+2\eta {\theta }_2\left)\right]}^{-1}=:\mathrm{d.}$$

(13)

We also observe that d ∈ (0,1) since δ, η > 1. Finally, the inequality (13) holds from condition (6a). Thus, it follows from Eq. (12) that {x _n } strongly converges to x^∗  as n → + ∞. Since A is continuous, it follows from Eqs. (7) and (8) that y _n  → x^∗, A y _n  → A x^∗ and z _n  → A x^∗ as n → + ∞. This completes the proof. □

It is worth mentioning that in the particular case where r = + ∞, s = +∞, one has δ = η = 1 and we get the convergence result for Algorithm 3 to solve SCVIP (1a,b).

Theorem 2.

Let C and Q be nonempty closed and convex subsets of H₁ and H₂, respectively. Let f : H₁ → H₁ be α-strongly monotone and β-Lipschitz continuous and let g : H₂ → H₂ be σ-strongly monotone and μ-Lipschitz continuous. Let A : H₁ → H₂ be a bounded linear operator and A^∗ be its adjoint operator. Suppose x^∗ ∈ C is a solution to SCVIP (1a,b), then the sequence {x _n } generated by Algorithm 3 strongly converges to x^∗ provided that ρ,λ, and γ satisfy the following conditions:

$$\frac{2\alpha }{{\beta }^2}-\Delta <\rho <\frac{2\alpha }{{\beta }^2}+\Delta ,$$

(14a)

$$\gamma \in \left(0,\frac{2}{\parallel A{\parallel }^2}\right),$$

(14b)

where

$$\begin{array}{ll}\Delta :=& \frac{1}{{\beta }^2}\left(\sqrt{4{\alpha }^2-{\beta }^2(1-d^2)}\right)\text{with}2\alpha >\beta \sqrt{1-d^2};\\ d:=& {\left[{\delta }^2\right(1+2{\theta }_2\left)\right]}^{-1};{\theta }_2:=\sqrt{1-2\mathrm{\lambda \sigma }+{\lambda }^2{\mu }^2};\lambda >0.\end{array}$$

Proof.

Since x^∗ ∈ C is a solution to SCVIP (1a,b), then for ρ,λ > 0, we have

$$\begin{array}{ll}{x}^{\ast }=& {\text{Proj}}_C(x^{\ast }-\mathrm{\rho f}(x^{\ast }\left)\right)\\ A{x}^{\ast }=& {\text{Proj}}_Q(A{x}^{\ast }-\mathrm{\lambda g}(A{x}^{\ast }\left)\right).\end{array}$$

Using the same arguments used in proof of Theorem 1, we obtain

$$\parallel y_n-x^{\ast }\parallel \le \delta {\theta }_1\parallel x_n-x^{\ast }\parallel ,$$

where${\theta }_1=\sqrt{1-2\mathrm{\rho \alpha }+{\rho }^2{\beta }^2}$, and

$$\parallel z_n-A{x}^{\ast }\parallel \le {\theta }_2\parallel A{y}_n-A{x}^{\ast }\parallel ,$$

where${\theta }_2=\sqrt{1-2\mathrm{\lambda \sigma }+{\lambda }^2{\mu }^2}$.

Next from Algorithm 3 (5c), we have

$$\begin{array}{ll}\parallel x_{n+1}& -x^{\ast }\parallel \le (1-{\alpha }_n)\parallel x_n-x^{\ast }\parallel \\ +{\alpha }_n[\parallel y_n-x^{\ast }-\gamma A^{\ast }(A{y}_n-A{x}^{\ast })\parallel +\gamma \parallel A^{\ast }(z_n-A{x}^{\ast })\parallel ]\\ \le [1-{\alpha }_n(1-\theta \left)\right]\parallel x_n-x^{\ast }\parallel ,\end{array}$$

where θ = θ₁(1 + γ∥A∥²θ₂).

Thus, we obtain

$$\parallel x_{n+1}-x^{\ast }\parallel \le \prod _{j=1}^n[1-{\alpha }_j(1-\theta \left)\right]\parallel x_0-x^{\ast }\parallel .$$

(15)

It follows from condition (14a) that θ ∈ (0,1). Since$\sum _{n=1}^{\infty }{\alpha }_n=+\infty$ and θ ∈ (0,1), it implies in the light of[13] that

$$\underset{n\to +\infty }{lim}\prod _{j=1}^n[1-{\alpha }_j(1-\theta \left)\right]=0.$$

The rest of the proof is the same as the proof of Theorem 1. This completes the proof. □

Remark 1.

(1) We would like to stress that SNVIP (2a,b) can be viewed as the following split nonconvex common fixed point problem:

$$\begin{array}{ll}\text{find}{x}^{\ast }\in U:=& \text{Fix}\left({\text{Proj}}_{C_r}\right(I-\mathrm{\rho f}\left)\right)\text{such that}\\ A{x}^{\ast }\in V:=& \text{Fix}\left({\text{Proj}}_{Q_s}\right(I-\mathrm{\lambda g}\left)\right),\end{array}$$

where Fix(T) denotes the set of fixed points of mapping T.

• (2) It is of further research effort to extend the iterative methods presented in this paper for solving the split variational inclusions [2] and the split equilibrium problem [14].