Consider two problems in a Hilbert space: A constrained convex optimization (CCO) and an equilibrium problem (EP) associated with a bifunction satisfying certain properties. It is known that CCO can be solved by the gradient-projection algorithm (GPA). It is also known that EP is equivalent to a fixed point problem. Therefore, both problems can be solved by fixed point algorithms.

Let H be a real Hilbert space and C a nonempty closed convex subset of H. A self-mapping T of C is said to be nonexpansive if $\Vert Tx-Ty\Vert \le \Vert x-y\Vert$ for all x,y∈ C. A self-mapping f of C is said to be a κ-contraction if $\Vert f(x)-f(y)\Vert \le \kappa \Vert x-y\Vert$ for all x,y ∈ C, where $\kappa \in [0,1)$ is a constant. We use Fix(T) to denote the set of fixed points of T; i.e., $Fix(T)=\{x\in C:Tx=x\}$.

Recall that the equilibrium problem (EP) associated to a bifunction $\varphi :C\times C\to ℝ$ is to find a point u∈ C with the property

(1.1)$$\varphi (u,v)\ge 0,v\in C.$$

The set of solutions of EP (1.1) is denoted by EP(ϕ).

If ϕ is of the form $\varphi (u,v)=⟨Au,v-u⟩$ for all u,v∈ C, where A: C→ H is a mapping, then $u\in EP(\varphi )$ if and only if u ∈ C is a solution to the variational inequality (VI):

$$⟨Au,v-u⟩\ge 0,v\in C.$$

Consequently, EP (1.1) includes, as special cases, numerous problems from various areas such as in physics, optimization and economics, and has been received a lot of attention; see, e.g., [7, 8, 5, 9, 12, 13, 14, 16, 17, 18, 19, 22, 28] and the references therein.

In 2010, Tian [20] considered the following iterative method:

(1.2)$$x_{n+1}={\alpha }_n\gamma f(x_n)+(I-\mu {\lambda }_{n}F)T{x}_n,n\ge 0,$$

where F is Lipschitz and strongly monotone and $\gamma ,\mu >0$ are some constants. He proved that if the parameters $\left\{{\alpha }_n\right\}$ and $\left\{{\lambda }_n\right\}$ satisfy certain appropriate conditions, the sequence $\left\{x_n\right\}$ generated by (1.2) converges strongly to the unique solution x^*∈ Fix(T) of the variational inequality

$$⟨(\gamma f-\mu F)x^{*},x-x^{*}⟩\le 0,\text{for all}x\in Fix(T).$$

Consider the constrained convex minimization problem:

(1.3)$$\text{Minimize }\{g(x):x\in C\},$$

where $g:C\to ℝ$ is a real-valued convex function. We denote by U the set of solutions of (1.3). It is well known that the gradient-projection algorithm (GPA) can be used to solve (1.3). If g is continuously differentiable, then GPA generates a sequence $\left\{x_n\right\}$ via the recursive formula:

(1.4)$$x_{n+1}=P_C(x_n-{\lambda }_n\nabla g(x_n)),n\ge 0,$$

where the initial guess x₀∈ C is chosen arbitrarily, and the parameters $\left\{{\lambda }_n\right\}$ are positive real numbers satisfying certain conditions. The convergence of GPA (1.4) depends on the behavior of the gradient $\nabla g$. As a matter of fact, it is known that if $\nabla g$ is α-strongly monotone and L-Lipschitzian with constants $\alpha >0$ and $L\ge 0$, then the operator

(1.5)$$W_{\lambda }:=P_C(I-\lambda \nabla g)$$

is a contraction for $0<\lambda <2\alpha /L^2$. Consequently, the sequence $\left\{x_n\right\}$ generated by PGA (1.4) converges in norm to the unique minimizer of (1.3) provided $({\lambda }_n)$ is contained in a compact subset of $(0,2\alpha /L^2)$, in particular, ${\lambda }_n\equiv \lambda \in (0,2\alpha /L^2)$.

However, if the gradient $\nabla g$ is not strongly monotone, the operator $W_{\lambda }$ is no longer contractive (in general). As a result, PGA (1.4) fails to converge strongly in an infinite-dimensional Hilbert space (see a counterexample in [23]). Nevertheless, if $\nabla g$ is Lipschitzian, PGA (1.4) can still converge in the weak topology.

In 2012, Tian and Liu [21] studied a composite iterative scheme by the viscosity approximation method [15, 26] for finding a common solution of an equilibrium problem and a constrained convex minimization problem:

(1.6)$$\left\{\begin{array}{ll}\varphi (u_n,y)+\frac{1}{r_n}⟨y-u_n,u_n-x_n⟩\ge 0,\hfill & y\in C,\hfill \\ [3.3mm]x_{n+1}={\alpha }_{n}f(x_n)+(1-{\alpha }_n)T_n{u}_n,\hfill & n\ge 1,\hfill \end{array}\right.$$

where $\varphi :C\times C\to ℝ$ is a bifunction, $\nabla g$ is L-Lipschitzian with L≥0 such that $U\cap EP(\varphi )\ne \varnothing$, f is a contraction, x₁∈ C, $\left\{{\alpha }_n\right\}\subset (0,1)$, $\left\{r_n\right\}\subset (0,\infty )$, $u_n=Q_{r_n}{x}_n$, T_n is determined by the relation: $P_C(I-{\lambda }_n\nabla g)=s_{n}I+(1-s_n)T_n$, $s_n=\frac{2-{\lambda }_{n}L}{4}$ and $\left\{{\lambda }_n\right\}\subset (0,\frac{2}{L})$. They proved that the sequence $\left\{x_n\right\}$, generated by (1.6), converges strongly to a point in $U\cap EP(\varphi )$ under certain conditions.

In this paper, motivated by the above results, we will propose an iterative fixed point algorithm to find a point which is a common solution to an equilibrium problem associated with a bifunction and a constrained convex minimization problem. We will prove strong convergence of the algorithm under certain conditions and identify the limit of the sequence generated by the algorithm as the unique solution of some variational inequality.

Let H be a real Hilbert space with inner product $⟨\cdot ,\cdot ⟩$ and norm $\Vert \cdot \Vert$, respectively. Weak and strong convergence are denoted by the symbols $\rightharpoonup$ and $\to$, respectively. In a real Hilbert space H, we have the identity

$$\Vert \lambda x+(1-\lambda )y{\Vert }^2=\lambda \Vert x{\Vert }^2+(1-\lambda )\Vert y{\Vert }^2-\lambda (1-\lambda )\Vert x-y{\Vert }^2$$

for all x,y ∈ H and $\lambda \in ℝ$. Let C be a nonempty closed convex subset of H. Then the (nearest point or metric) projection on C is defined by

$$P_{C}x:=\arg {\min }_{y\in C}\Vert x-y{\Vert }^2,x\in H.$$

Note that P_C is nonexpansive and characterized by the inequality (for z ∈ C)

$$z=P_{C}x\iff ⟨x-z,z-y⟩\ge 0,y\in C.$$

The following inequality is convenient in applications. Recall that for all x,y in a Hilbert space H we have

$$\Vert x+y{\Vert }^2\le \Vert x{\Vert }^2+2⟨y,x+y⟩.$$

Next, given a Hilbert space H and a function $\phi :H\to ℝ$, we recal that φ is weakly lower semicontinuous l.s.c. considering H with its weak topology, that is, $\phi (\widehat{u})\le \lim \inf \phi (u_n)$ whenever u_n converges weakly to $\widehat{u}$.

Lemma 2.1. ([10]) Let H be a real Hilbert space, C a closed convex subset of H and T:C → C a nonexpansive mapping with $Fix(T)\ne \varnothing$. If $\left\{x_n\right\}$ is a sequence in C such that $x_n\rightharpoonup x$ and $(I-T)x_n\to y$, then (I-T)x=y. [This is known as the demiclosedness principle of nonexpansive mappings.]

Let $\varphi :C\times C\to ℝ$ be a bifunction. Throughout the paper we always assume that ϕ satisfies the following (standard) conditions [2]:

(A₁) ϕ(x,x)=0 for all x∈ C;

(A₂) ϕ is monotone, i.e., $\varphi (x,y)+\varphi (y,x)\le 0$, for all x,y∈ C;

(A₃) for each $x,y,z\in C,$ $\underset{t\downarrow 0}{\lim }\varphi (tz+(1-t)x,y)\le \varphi (x,y);$

(A₄) for each x∈ C, the mapping $y\mapsto \varphi (x,y)$ is convex, weakly lower semicontinuous (l.s.c.).

Under these conditions, it is easy to see that the solution set EP(ϕ) of the equilibrium problem (1.1) is closed and convex.

Following Combettes and Hirstoaga [6], we can define, for each fixed $r>0$, a mapping $Q_r:H\to C$ by the equation

(2.1)$$Q_{r}x=z$$

for x∈ H, where z∈ C is the unique solution of the inequality:

(2.2)$$\varphi (z,y)+\frac{1}{r}⟨y-z,z-x⟩\ge 0,y\in C.$$

Lemma 2.2. ([6]) The mapping Q_r possesses the properties:

(i) Q_r is firmly nonexpansive, namely,

$$\Vert Q_{r}x-Q_{r}y{\Vert }^2\le ⟨Q_{r}x-Q_{r}y,x-y⟩,x,y\in H;$$

(ii) $Fix(Q_r)=EP(\varphi )$.

Property (ii) shows an equivalence of EP (1.1) to the fixed point problem of $Q_{r}x=x$ and thus EP (1.1) can be solved by fixed point methods.

Definition 2.3. ([21]) A mapping $T:H\to H$ is said to be an averaged mapping (av, for short) if $T=(1-\alpha )I+\alpha S$, where $\alpha \in (0,1)$, I is the identity map and S:H→ H is nonexpansive. In this case, we say that T is α-averaged (α-av, for short).

Note that firmly nonexpansive mappings (e.g., projections) are $\frac{1}{2}$-av.

Proposition 2.4. ([4, 23])} The composite of finitely many averaged mappings is averaged. That is, if each of the mappings ${\left\{T_i\right\}}_{i=1}^N$ is averaged, then so is the composite $T_1\cdots T_N$. In particular, if T₁ is α₁-av, and T₂ is α₂-av, where ${\alpha }_1,{\alpha }_2\in (0,1)$, then the composite T₁T₂ is α-av, where $\alpha ={\alpha }_1+{\alpha }_2-{\alpha }_1{\alpha }_2$.

Definition 2.5. A nonlinear operator G with domain D(G) and range R(G) both in H is said to be:

(i) β-strongly monotone if there exists $\beta >0$ such that

$$⟨x-y,Gx-Gy⟩\ge \beta \Vert x-y{\Vert }^2,x,y\in D(G),$$

(ii) ν-inverse strongly monotone (for short, ν-ism) if there exists $\nu >0$ such that

$$⟨x-y,Gx-Gy⟩\ge \nu \Vert Gx-Gy{\Vert }^2,x,y\in D(G).$$

It can be easily seen that any projection P_C is a 1-ism.

Inverse strongly monotone (also referred to as co-coercive) operators have extensively been used in practical problems from various areas, for instance, traffic assignment problems; see, for example, [3, 11] and references therein.

Proposition 2.6. ([4, 23]) Let T be an operator of H into itself.

• (i) T is nonexpansive if and only if the complement I-T is $\frac{1}{2}$-ism.

• (ii) If T is ν-ism, then for $\mu >0$, $\mu T$ is $\frac{\nu }{\mu }$-ism.

• (iii) T is averaged if and only if the complement I-T is ν-ism for some $\nu >\frac{1}{2}$.

Indeed, for $\alpha \in (0,1)$, T is α-av if and only if I-T is $\frac{1}{2\alpha }$-ism.

Lemma 2.7. ([1, 25]) Assume $\left\{a_n\right\}$ is a sequence of nonnegative real numbers such that

$$a_{n+1}\le (1-{\gamma }_n)a_n+{\gamma }_n{\nu }_n+{\mu }_n,$$

where $\left\{{\gamma }_n\right\}$ is a sequence in [0,1], $\left\{{\mu }_n\right\}$ is a sequence of nonnegative real numbers and $\left\{{\nu }_n\right\}$ is a sequence in $ℝ$ such that

$${\displaystyle \sum _{n=1}^{\infty }{\gamma }_n}=\infty ,\underset{n\to \infty }{\lim \sup }{\nu }_n\le 0,{\displaystyle \sum _{n=1}^{\infty }{\mu }_n}<\infty .$$

Then $\underset{n\to \infty }{\lim }{a}_n=0$.

In this section we introduce an iterative fixed point algorithm for finding a point in U ∩ EP(ϕ), that is, a common solution to the equilibrium and optimization problems (1.1) and (1.3).

In the rest of this paper, we always assume that C is a nonempty closed convex subset of a Hilbert space H and $g:C\to ℝ$ is a real-valued continuously differentiable, convex function such that the gradient ∇ g is L-Lipschitz for some L≥ 0. Recall that U denotes the set of solutions of the minimization problem (1.3). Then we have that a point x^*∈ U if and only if $x^{*}\in Fix(W_{\lambda })$, where W_λ is defined in (1.5); that is,

$$x^{*}=P_C(I-\lambda \nabla g)x^{*}$$

for each $\lambda >0$. Further helpful properties of the mapping $P_C(I-\lambda \nabla g)$ are outlined in [23] and summarized below.

Lemma 3.1. Suppose a continuously differentiable, convex function g has L-Lipschitz continuous gradient $\nabla g$

and let $\lambda \in (0,2/L)$ be given.

• (i) The mapping $P_C(I-\lambda \nabla g)$ is α-av, where $\alpha =\frac{2+\lambda L}{4}\in (0,1)$; namely, one can write $P_C(I-\lambda \nabla g)=sI+(1-s)T$, where $s=\frac{2-\lambda L}{4}$ and $T=\frac{1}{2+\lambda L}[4P_C(I-\lambda \nabla g)-(2-\lambda L)I]$ is nonexpansive.

• (ii) The function $h(\lambda )x:=P_C(I-\lambda \nabla g)x$ is uniformly Lipschitz continuous in $\lambda \in [0,\infty )$ over bounded x ∈ C.

Proof. (i) has been proved in [23]. To prove (ii) take $\lambda ,{\lambda }^{\prime }\in [0,\infty )$ and let $r>0$ be given. Setting $M_r:=\sup \{\Vert \nabla g(x)\Vert :\Vert x\Vert \le r,x\in C\}$ and noting that P_C is nonexpansive, we get

$$\begin{array}{l}\Vert h(\lambda )-h({\lambda }^{\prime })\Vert =\Vert P_C(I-\lambda \nabla g)x-P_C(I-{\lambda }^{\prime }\nabla g)x\Vert \\ \le |\lambda -{\lambda }^{\prime }|\Vert \nabla g(x)\Vert \le M_r|\lambda -{\lambda }^{\prime }|.\end{array}$$

Lemma 3.2. ([27, Lemma 3.1]) Suppose $F:C\to H$ is α-Lipschitz and η-strongly monotone and let $0<\mu <\frac{2\eta }{{\alpha }^2}$. Then, for $\nu \in (0,1)$, the mapping $U^{\nu }$ defined by

(3.1)$$U^{\nu }x:=x-\mu \nu Fx,x\in C$$

is a $(1-\xi \nu )$-contraction from C into H, with $\xi :=1-\sqrt{1-\mu (2\eta -\mu {\alpha }^2)}\in (0,1]$.

The following is the main result of this paper.

Theorem 3.3. Assume the bifunction $\varphi :C\times C\to ℝ$ satisfies the standard conditions (A₁)-(A₄), and the objective function $g:C\to ℝ$ is continuously differentiable and convex such that the gradient $\nabla g$ is L-Lipschitz with L≥0. Assume $U\cap EP(\varphi )\ne \varnothing$. Let Q_r be defined by (2.1) for $r>0$. Let f be a κ-contraction of C with $\kappa \in [0,1)$ and F: C→ H an α-Lipschitz and η-strongly monotone operator on C with $\alpha >0$ and $\eta >0$. Assume $0<\mu <\frac{2\eta }{{\alpha }^2}$ and $0<\gamma <\frac{\xi }{\kappa }$ with $\xi =1-\sqrt{1-\mu (2\eta -\mu {\alpha }^2)}$. Suppose $\left\{{\alpha }_n\right\}$, $\left\{{\beta }_n\right\}$ and $\left\{r_n\right\}$ are real sequences satisfying the following conditions:

(B₁) $\left\{{\alpha }_n\right\}\subset [0,1]$, $\underset{n\to \infty }{\lim }{\alpha }_n=0$ and ${\displaystyle \sum _{n=1}^{\infty }{\alpha }_n}=\infty ;$

(B₂) $\left\{{\beta }_n\right\}\subset [0,1]$, $\underset{n\to \infty }{\lim }{\beta }_n=0$ and ${\displaystyle \sum _{n=1}^{\infty }|}{\beta }_{n+1}-{\beta }_n|<\infty ;$

(B₃) $\left\{r_n\right\}\subset (0,\infty )$, $\underset{n\to \infty }{\lim \inf }{r}_n>0$ and ${\displaystyle \sum _{n=1}^{\infty }|}{r}_{n+1}-r_n|<\infty ;$

(B₄) ${\displaystyle \sum _{n=1}^{\infty }|}{\alpha }_{n+1}-{\alpha }_n|<\infty$ or $\underset{n\to \infty }{\lim }\frac{{\alpha }_n}{{\alpha }_{n+1}}=1$.

Let {x_n} be a sequence generated by the iteration process:

(3.2)$$\left\{\begin{array}{ll}{u}_n=Q_{r_n}{x}_n,\hfill & \hfill \\ y_n=P_C({\alpha }_n\gamma f(x_n)+(I-{\alpha }_n\mu F)T_n{u}_n),\hfill & \hfill \\ x_{n+1}=(1-{\beta }_n)y_n+{\beta }_n{T}_n{y}_n,\hfill & n\ge 1,\hfill \end{array}\right.$$

where the initial point x₁ ∈ C is selected arbitrarily, $\left\{{\lambda }_n\right\}\subset (0,\frac{2}{L})$, and T_n is determined by the relation $P_C(I-{\lambda }_n\nabla g)=s_{n}I+(1-s_n)T_n$ with $s_n=\frac{2-{\lambda }_{n}L}{4}$ (cf. Lemma 3.1 (i)). Namely,

(3.3)$$T_n=\frac{1}{1-s_n}(P_C(I-{\lambda }_n\nabla g)-s_{n}I).$$

Suppose $\left\{{\lambda }_n\right\}$ satisfies the condition

(B₅) $0<\underset{\_}{\lambda }\le {\lambda }_n<\frac{2}{L}$ for all n, and ${\displaystyle \sum _{n=1}^{\infty }|}{\lambda }_{n+1}-{\lambda }_n|<\infty$.

Then, the sequences {x_n} and {u_n} defined by (3.2) converge strongly to a point $q\in U\cap EP(\varphi )$, where $q=P_{U\cap EP(\varphi )}(I-\mu F+\gamma f)(q)$ is the unique solution to the following variational inequality:

$$⟨(\mu F-\gamma f)q,q-x⟩\le 0,x\in U\cap EP(\varphi ).$$

To prove Theorem 3.3 we first establish some lemmas.

Lemma 3.4. Let T_n be defined by (3.3). Let $r>0$ and set

$$M(r):=\sup \{4\Vert \nabla g(y)\Vert +L\Vert P_{C}y-y\Vert :\Vert y\Vert \le r,y\in C\}<\infty .$$

Then, for $y\in C$ such that $\Vert y\Vert \le r$, we have

$$\Vert T_{n+1}y-T_{n}y\Vert \le M(r)|{\lambda }_{n+1}-{\lambda }_n|.$$

Proof. By definition of T_n, we have for $y\in C$

$$\begin{array}{l}{T}_{n+1}y-T_{n}y=\frac{1}{1-s_{n+1}}{P}_C(y-{\lambda }_{n+1}\nabla g(y))-\frac{s_{n+1}}{1-s_{n+1}}y\\ -\frac{1}{1-s_n}{P}_C(y-{\lambda }_n\nabla g(y))+\frac{s_n}{1-s_n}y\\ =\frac{1}{1-s_{n+1}}[P_C(y-{\lambda }_{n+1}\nabla g(y))-P_C(y-{\lambda }_n\nabla g(y))]\\ +\left(\frac{1}{1-s_{n+1}}-\frac{1}{1-s_n}\right)P_C(y-{\lambda }_n\nabla g(y))+\left(\frac{s_n}{1-s_n}-\frac{s_{n+1}}{1-s_{n+1}}\right)y\\ =\frac{1}{1-s_{n+1}}[P_C(y-{\lambda }_{n+1}\nabla g(y))-P_C(y-{\lambda }_n\nabla g(y))]\\ +\frac{s_{n+1}-s_n}{(1-s_n)(1-s_{n+1})}(P_C(y-{\lambda }_n\nabla g(y))-y).\end{array}$$

Since P_C is nonexpansive and $s_n=(2-{\lambda }_{n}L)/4$ (thus $1-s_n=\frac{2+{\lambda }_n}{4}\ge \frac{1}{2}$),

it turns out that (noting ${\lambda }_{n}L<2$)

$$\begin{array}{l}\Vert T_{n+1}y-T_{n}y\Vert \le 2\Vert \nabla g(y)\Vert |{\lambda }_{n+1}-{\lambda }_n|\\ +L|{\lambda }_{n+1}-{\lambda }_n|(\Vert P_C(y-{\lambda }_n\nabla g(y))-P_{C}y\Vert +\Vert P_{C}y-y\Vert )\\ \le 2\Vert \nabla g(y)\Vert |{\lambda }_{n+1}-{\lambda }_n|\\ +L|{\lambda }_{n+1}-{\lambda }_n|({\lambda }_n\Vert \nabla g(y)\Vert +\Vert P_{C}y-y\Vert )\\ \le (4\Vert \nabla g(y)\Vert +L\Vert P_{C}y-y\Vert )|{\lambda }_{n+1}-{\lambda }_n|.\end{array}$$

This finishes the proof.

Lemma 3.5. For $x,x^{\prime }\in C$ and $r,r^{\prime }>0$, we have

(3.4)$$\Vert Q_{r}x-Q_{r^{\prime }}x\Vert \le \left|1-\frac{r}{r^{\prime }}\right|\Vert Q_{r^{\prime }}x-x\Vert$$

and

(3.5)$$\Vert Q_{r}x-Q_{r^{\prime }}{x}^{\prime }\Vert \le \Vert x-x^{\prime }\Vert +\left|1-\frac{r}{r^{\prime }}\right|\Vert Q_{r^{\prime }}x-x\Vert .$$

Proof. Set $u=Q_{r}x$ and $u^{\prime }=Q_{r^{\prime }}x$. By definition, we have

$$\varphi (u,y)+\frac{1}{r}⟨y-u,u-x⟩\ge 0,\text{for all}y\in C.$$

In particular,

$$\varphi (u,u^{\prime })+\frac{1}{r}⟨u^{\prime }-u,u-x⟩\ge 0.$$

Similarly, we have

$$\varphi (u^{\prime },u)+\frac{1}{r^{\prime }}⟨u-u^{\prime },u^{\prime }-x⟩\ge 0.$$

Adding up the last two relations and using the monotonicity $\varphi (u,u^{\prime })+\varphi (u^{\prime },u)\le 0$, we obtain

$$⟨u^{\prime }-u,\frac{1}{r}(u-x)-\frac{1}{r^{\prime }}(u^{\prime }-x)⟩\ge 0.$$

This can be rewritten as

$$\begin{array}{l}\frac{1}{r}\Vert u^{\prime }-u{\Vert }^2\le \left(\frac{1}{r}-\frac{1}{r^{\prime }}\right)⟨u^{\prime }-u,u^{\prime }-x⟩\\ \le \left|\frac{1}{r}-\frac{1}{r^{\prime }}\right|\Vert u^{\prime }-u\Vert \Vert u^{\prime }-x\Vert \end{array}$$

and (3.4) follows immediately.

Next using the nonexpansivity of $Q_{r^{\prime }}$, we get

$$\begin{array}{l}\Vert Q_{r}x-Q_{r^{\prime }}{x}^{\prime }\Vert \le \Vert Q_{r^{\prime }}{x}^{\prime }-Q_{r^{\prime }}x\Vert +\Vert Q_{r}x-Q_{r^{\prime }}x\Vert \\ \le \Vert x-x^{\prime }\Vert +\left|1-\frac{r}{r^{\prime }}\right|\Vert Q_{r^{\prime }}x-x\Vert .\end{array}$$

The proof of the lemma is complete.

Proof. Theorem 3.3First we make a convention: For the sake of convenience, we will use $M>0$ to stand for an appropriate constant for several estimates from various places throughout the proof which is divided into five steps.

Step 1. The sequences {x_n} and {u_n} are bounded. To see this, we take a point $p\in U\cap EP(\varphi )$ to derive from (3.2) that (noting Q_rp=p and T_np=p for all $r>0$ and $n\ge 1$, and P_C is nonexpansive)

$$\begin{array}{l}\Vert x_{n+1}-p\Vert \le (1-{\beta }_n\Vert y_n-p\Vert +{\beta }_n\Vert T_n{y}_n-p\Vert \le \Vert y_n-p\Vert \\ \le \Vert {\alpha }_n\gamma f(x_n)+(I-{\alpha }_n\mu F)T_n{u}_n-p\Vert \\ =\Vert {\alpha }_n\gamma [f(x_n)-f(p)]+{\alpha }_n[\gamma f(p)-\mu F(p)]\\ +(I-{\alpha }_n\mu F)T_n{u}_n-(I-{\alpha }_n\mu F)(p)\Vert .\end{array}$$

Now since f is κ-contraction, $(I-{\alpha }_n\mu F)$ is $(1-{\alpha }_n\xi )$-contraction, T_n is nonexpansive, and $\Vert u_n-p\Vert =\Vert Q_{r_n}{x}_n-p\Vert \le \Vert x_n-p\Vert$, it follows from the last relation that

$$\begin{array}{l}\Vert x_{n+1}-p\Vert \le (1-{\alpha }_n(\xi -\gamma \kappa ))\Vert x_n-p\Vert +{\alpha }_n\Vert \gamma f(p)-\mu F(p)\Vert \\ \le \max \{\Vert x_n-p\Vert ,\frac{\Vert \gamma f(p)-\mu F(p)\Vert }{\xi -\gamma \kappa }\}.\end{array}$$

As a result, we get, by induction,

$$\Vert x_n-p\Vert \le \max \{\Vert x_1-p\Vert ,\frac{\Vert \gamma f(p)-\mu F(p)\Vert }{\xi -\gamma \kappa }\}$$

for all $n\ge 1$. Hence, $\left\{x_n\right\}$ is bounded, which implies that $\left\{u_n\right\}$, $\left\{y_n\right\}$, $\{f(x_n)\}$, $\{\mu F(T_n{u}_n)\}$ and $\left\{T_n{y}_n\right\}$ are all bounded.

Step 2. Asymptotic regularity of $\left\{x_n\right\}$: $\underset{n\to \infty }{\lim }\Vert x_{n+1}-x_n\Vert =0$. To see this we use the definition of the algorithm (3.2) to derive that, after some manipulations,

(3.6)$$\begin{array}{l}{x}_{n+2}-x_{n+1}=(1-{\beta }_{n+1})y_{n+1}+{\beta }_{n+1}{T}_{n+1}{y}_{n+1}-(1-{\beta }_n)y_n-{\beta }_n{T}_n{y}_n\\ =(1-{\beta }_{n+1})(y_{n+1}-y_n)+{\beta }_{n+1}(T_{n+1}{y}_{n+1}-T_{n+1}{y}_n)\\ +({\beta }_n-{\beta }_{n+1})(y_n-T_{n+1}{y}_n)+{\beta }_n(T_{n+1}{y}_n-T_n{y}_n).\end{array}$$

Since T_n+1 is nonexpansive and (y_n) is bounded, and by Lemma 3.4, we obtain from (3.6),

(3.7)$$\Vert x_{n+2}-x_{n+1}\Vert \le \Vert y_{n+1}-y_n\Vert +(|{\beta }_{n+1}-{\beta }_{n+1}|+{\beta }_n|{\lambda }_{n+1}-{\lambda }_n|)M,$$

where $M>0$ is a big enough constant.

To estimate $\Vert y_{n+1}-y_n\Vert$, we again use (3.2) and the nonexpansiveness of P_C to get

(3.8)$$\begin{array}{l}\Vert y_{n+1}-y_n\Vert \le \Vert {\alpha }_{n+1}\gamma f(x_{n+1})+(I-{\alpha }_{n+1}\mu F)T_{n+1}{u}_{n+1}\\ -{\alpha }_n\gamma f(x_n)-(I-{\alpha }_n\mu F)T_n{u}_n\Vert \\ =\Vert ({\alpha }_{n+1}-{\alpha }_n)[\gamma f(x_{n+1})-\mu F(T_{n+1}{u}_{n+1})]\\ +{\alpha }_n\gamma [f(x_{n+1})-f(x_n)]\\ +(I-{\alpha }_n\mu F)T_{n+1}{u}_{n+1}-(I-{\alpha }_n\mu F)T_n{u}_n\Vert .\end{array}$$

Observe by Lemmas 3.4 and 3.5

(3.9)$$\begin{array}{l}\Vert T_{n+1}{u}_{n+1}-T_n{u}_n\Vert \le \Vert T_{n+1}{u}_{n+1}-T_{n+1}{u}_n\Vert +\Vert T_{n+1}{u}_n-T_n{u}_n\Vert \\ \le \Vert u_{n+1}-u_n\Vert +\Vert T_{n+1}{u}_n-T_n{u}_n\Vert \\ =\Vert Q_{r_{n+1}}{x}_{n+1}-Q_{r_n}{x}_n\Vert +\Vert T_{n+1}{u}_n-T_n{u}_n\Vert \\ \le \Vert x_{n+1}-x_n\Vert +|1-\frac{r_n}{r_{n+1}}|\Vert Q_{r_{n+1}}{x}_n-x_n\Vert +M|{\lambda }_{n+1}-{\lambda }_n|\\ \le \Vert x_{n+1}-x_n\Vert +\left(|1-\frac{r_n}{r_{n+1}}|+|{\lambda }_{n+1}-{\lambda }_n|\right)M.\end{array}$$

Now since f is κ-contraction and $I-{\alpha }_n\mu F$ is $(1-\xi {\alpha }_n)$-contraction, also using (3.9), we get from (3.8)

(3.10)$$\begin{array}{l}\Vert y_{n+1}-y_n\Vert \le M|{\alpha }_{n+1}-{\alpha }_n|+\kappa \gamma {\alpha }_n\Vert x_{n+1}-x_n\Vert \\ +(1-\xi {\alpha }_n)[\Vert x_{n+1}-x_n\Vert +(|1-\frac{r_n}{r_{n+1}}|+|{\lambda }_{n+1}-{\lambda }_n|)M]\\ =[1-(\xi -\kappa \gamma ){\alpha }_n]\Vert x_{n+1}-x_n\Vert \\ +(|{\alpha }_{n+1}-{\alpha }_n|+|1-\frac{r_n}{r_{n+1}}|+|{\lambda }_{n+1}-{\lambda }_n|)M.\end{array}$$

Substituting (3.10)) into (3.7) yields

(3.11)$$\Vert x_{n+2}-x_{n+1}\Vert \le (1-{\widehat{\alpha }}_n)\Vert x_{n+1}-x_n\Vert +{\nu }_n+{\delta }_n,$$

where ${\widehat{\alpha }}_n=(\xi -\kappa \gamma ){\alpha }_n$, ${\nu }_n=M|{\alpha }_{n+1}-{\alpha }_n|$ and

$${\delta }_n=(|{\beta }_{n+1}-{\beta }_n|+|1-\frac{r_n}{r_{n+1}}|+(1+{\beta }_n)|{\lambda }_{n+1}-{\lambda }_n|)M.$$

By condition (B₁), we have ${\displaystyle \sum _n{\widehat{\alpha }}_n}=\infty$. By conditions (B₂)-(B₅), we always have ${\displaystyle \sum _n{\delta }_n}<\infty$, and moreover, ${\displaystyle \sum _n{\nu }_n}<\infty$ if ${\displaystyle \sum _n|}{\alpha }_{n+1}-{\alpha }_n|<\infty$ or $\underset{n}{\lim }(\nu /{\widehat{\alpha }}_n)=0$ if $\underset{n}{\lim }({\alpha }_n/{\alpha }_{n+1})=1$. So Lemma 2.7 is applicable to (3.11), which yields $\Vert x_{n+1}-x_n\Vert \to 0$, as claimed.

Step 3. We claim that

(3a) $\Vert x_n-y_n\Vert \to 0$,

(3b) $\Vert x_n-u_n\Vert \to 0$,

(3c) $\Vert P_C(I-{\lambda }_n\nabla g)x_n-x_n\Vert \to 0$.

Indeed, since

$$\begin{array}{l}\Vert x_n-y_n\Vert \le \Vert x_n-x_{n+1}\Vert +\Vert x_{n+1}-y_n\Vert \\ =\Vert x_n-x_{n+1}\Vert +{\beta }_n\Vert y_n-T_n{y}_n\Vert \\ =\Vert x_n-x_{n+1}\Vert +M{\beta }_n\to 0(as{\beta }_n\to 0)\end{array}$$

and (3a) follows. To show (3b), we first observe that the firm nonexpansivity of $Q_{r_n}$ implies that (noting $u_n=Q_{r_n}{x}_n$ and $Fix(Q_{r_n})=EP(\varphi )$)

$$\Vert u_n-p{\Vert }^2\le \Vert x_n-p{\Vert }^2-\Vert u_n-x_n{\Vert }^2,p\in EP(\varphi ).$$

For the sake of brevity, we shall use the $O({\alpha }_n)$ notation in our argument below. For $p\in U\cap EP(\varphi )$, we have (noting again that P_C is nonexpansive)

$$\begin{array}{l}\Vert x_{n+1}-p{\Vert }^2=\Vert (1-{\beta }_n)(y_n-p)+{\beta }_n(T_n{y}_n-p){\Vert }^2\le \Vert y_n-p{\Vert }^2\\ \le \Vert {\alpha }_n\gamma f(x_n)+(I-{\alpha }_n\mu F)T_n{u}_n-p{\Vert }^2\\ =\Vert {\alpha }_n(\gamma f(x_n)-\mu F(p))+(I-{\alpha }_n\mu F)T_n{u}_n-(I-{\alpha }_n\mu F)p{\Vert }^2\\ ={\alpha }_n^2\Vert \gamma f(x_n)-\mu F(p){\Vert }^2+\Vert (I-{\alpha }_n\mu F)T_n{u}_n-(I-{\alpha }_n\mu F)p{\Vert }^2\\ +2{\alpha }_n⟨\gamma f(x_n)-\mu F(p),(I-{\alpha }_n\mu F)T_n{u}_n-(I-{\alpha }_n\mu F)p⟩\\ \le {\alpha }_n^{2}M+{(1-\xi {\alpha }_n)}^2\Vert u_n-p{\Vert }^2+2{\alpha }_{n}M(1-\xi {\alpha }_n)\Vert u_n-p\Vert \\ \le \Vert u_n-p{\Vert }^2+O({\alpha }_n)\\ \le \Vert x_n-p{\Vert }^2-\Vert u_n-x_n{\Vert }^2+O({\alpha }_n).\end{array}$$

It turns out that

$$\begin{array}{l}\Vert x_n-u_n{\Vert }^2\le \Vert x_n-p{\Vert }^2-\Vert x_{n+1}-p{\Vert }^2+O({\alpha }_n)\\ =O(\Vert x_{n+1}-x_n\Vert )+O({\alpha }_n)\to 0.\end{array}$$

This proves (3b). To verify (3c), it suffices to verify that $‖P_C(I-{\lambda }_n▽g)u_n-u_n‖\to 0$. Since $P_C(I-{\lambda }_n▽g)=s_{n}I+(1-s_n)T_n$, it follows that

$$\begin{array}{l}‖P_C(I-{\lambda }_n▽g)u_n-u_n‖=(1-s_n)‖T_n{u}_n-u_n‖\\ \le \Vert T_n{u}_n-y_n\Vert +\Vert y_n-u_n\Vert \\ \le {\alpha }_n\Vert \gamma f(x_n)-\mu F(T_n{x}_n)\Vert +\Vert y_n-u_n\Vert \\ =O({\alpha }_n)+\Vert y_n-u_n\Vert \to 0.\end{array}$$

Step 4. We have the following asymptotic variational inequality:

(3.12)$$\underset{n\to \infty }{\lim \sup }⟨\gamma f(q)-\mu F(q),y_n-q⟩\le 0,$$

where q is the unique fixed point of the contraction $P_{U\cap EP(\varphi )}(I-\mu F+\gamma f)$; namely, $q{=}_{U\cap EP(\varphi )}(I-\mu F+\gamma f)q$. Alternatively, q is the unique solution of the variational inequality

(3.13)$$⟨\gamma f(q)-\mu F(q),y-q⟩\le 0,y\in U\cap EP(\varphi ).$$

To prove (3.12), take a subsequence $(y_{n_i})$ of (y_n) weakly convergent to a point $\widehat{y}\in C$ and such that

$$\begin{array}{l}\underset{n\to \infty }{\lim \sup }⟨\gamma f(q)-\mu F(q),y_n-q⟩=\underset{i\to \infty }{\lim }⟨\gamma f(q)-\mu F(q),y_{n_i}-q⟩\\ =⟨\gamma f(q)-\mu F(q),\widehat{y}-q⟩.\end{array}$$

By virtue of VI (3.13), it suffices to show $\widehat{y}\in U\cap EP(\varphi )$. To see $\widehat{y}\in U$, we use (3c) to get (noticing $\Vert x_n-y_n\Vert \to 0$)

(3.14)$$‖P_C(I-{\lambda }_{n_i}▽g)y_{n_i}-y_{n_i}‖\to 0.$$

We may assume ${\lambda }_{n_i}\to \widehat{\lambda }$; note that $\widehat{\lambda }\in (0,2/L]$ due to condition (B₅). It is then not hard (see Lemma 3.1(ii)) to find from (3.14) that

(3.15)$$‖P_C(I-\widehat{\lambda }▽g)y_{n_i}-y_{n_i}‖\to 0.$$

The demiclosedness principle of nonexpansive mappings then ensures that $\widehat{y}\in Fix(P_C(I-\widehat{\lambda }▽g)=U$. It remains to show $\widehat{y}\in EP(\varphi )$. Since $u_n=Q_{r_n}{x}_n$, it follows from the definition of Q_r and the monotonicity of ϕ that

$$\frac{1}{r_n}⟨y-u_n,u_n-x_n⟩\ge -\varphi (u_n,y)\ge \varphi (y,u_n),y\in C.$$

This results in $\underset{n\to \infty }{\lim \sup }\varphi (y,u_n)\le 0$ for each $y\in C$, due to the facts $\Vert u_n-x_n\Vert \to 0$ and $\underset{n}{\inf }{r}_n>0$. As $\varphi (y,\cdot )$ is l.s.c. and $u_{n_i}\rightharpoonup \widehat{y}$, it turns out that $\varphi (y,\widehat{y})\le 0$ for each $y\in C$. Now set $y_t=ty+(1-t)\widehat{y}\in C$ with $t\in (0,1)$. We then have by the standard conditions (A₁)-(A₄)

$$0=\varphi (y_t,ty+(1-t)\widehat{y})\le t\varphi (y_t,y)+(1-t)\varphi (y_t,\widehat{y})\le t\varphi (y_t,y).$$

Hence, $\varphi (y_t,y)\ge 0$ and $\varphi (y,y_t)\le 0$. Letting $t\to 0$ yields $\varphi (y,\widehat{y})\le 0$ for each $y\in C$. This asserts $\widehat{y}\in EP(\varphi )$ and the proof of Step 4 is complete.

Step 5. Strong convergence of $\left\{x_n\right\}$: $x_n\to q$ in norm, where q satisfies (3.12). Setting $z_n:={\alpha }_n\gamma f(x_n)+(I-{\alpha }_n\mu F)T_n{u}_n$ (thus, $y_n=P_C{z}_n$), we derive that

$$\begin{array}{l}\Vert x_{n+1}-q{\Vert }^2=\Vert (1-{\beta }_n)(y_n-q)+{\beta }_n(T_n{y}_n-p){\Vert }^2\le \Vert y_n-q{\Vert }^2\\ \le \Vert {\alpha }_n\gamma f(x_n)+(I-{\alpha }_n\mu F)T_n{u}_n-q{\Vert }^2\\ =\Vert [(I-{\alpha }_n\mu F)T_n{u}_n-(I-{\alpha }_n\mu F)q]+{\alpha }_n(\gamma f(x_n)-\mu F(q)){\Vert }^2\\ \le \Vert (I-{\alpha }_n\mu F)T_n{u}_n-(I-{\alpha }_n\mu F)q{\Vert }^2\\ +2{\alpha }_n⟨\gamma f(x_n)-\mu F(q),z_n-q⟩\\ \le {(1-\xi {\alpha }_n)}^2\Vert T_n{u}_n-q{\Vert }^2+2\gamma {\alpha }_n⟨f(x_n)-f(q),x_n-q⟩\\ +2\gamma {\alpha }_n⟨f(x_n)-f(q),z_n-x_n⟩+2{\alpha }_n⟨\gamma f(q)-\mu F(q),z_n-q⟩\\ \le {(1-\xi {\alpha }_n)}^2\Vert u_n-q{\Vert }^2+2\gamma \kappa {\alpha }_n\Vert x_n-q{\Vert }^2\\ +2\gamma \kappa {\alpha }_n\Vert x_n-q\Vert \Vert z_n-x_n\Vert +2{\alpha }_n⟨\gamma f(q)-\mu F(q),z_n-q⟩\\ \le (1-2(\xi -\gamma \kappa ){\alpha }_n)\Vert x_n-q{\Vert }^2+{\xi }^2{\alpha }_n^2{M}^2\\ +2\gamma \kappa {\alpha }_{n}M\Vert z_n-x_n\Vert +2{\alpha }_n⟨\gamma f(q)-\mu F(q),z_n-q⟩.\end{array}$$

Here $M\ge \Vert x_n-q\Vert$ for all n. We can rewrite the last relation as

(3.16)$$\Vert x_{n+1}-q{\Vert }^2\le (1-{\tilde{\alpha }}_n)\Vert x_n-q{\Vert }^2+{\tilde{\alpha }}_n{\tilde{\delta }}_n,$$

where ${\tilde{\alpha }}_n=2(\xi -\gamma \kappa ){\alpha }_n$ and

$${\tilde{\delta }}_n=\frac{{\xi }^2{M}^2{\alpha }_n+2\gamma \kappa M\Vert z_n-x_n\Vert +2⟨\gamma f(q)-\mu F(q),z_n-q⟩}{2(\xi -\gamma \kappa )}.$$

Since $y_n=P_C{z}_n$ and $T_n{u}_n\in C$, we obtain

$$\begin{array}{l}\Vert y_n-z_n\Vert =\Vert P_C{z}_n-z_n\Vert \le \Vert P_C{z}_n-T_n{u}_n\Vert +\Vert T_n{u}_n-z_n\Vert \\ \le 2\Vert z_n-T_n{u}_n\Vert =2{\alpha }_n\Vert \gamma f(x_n)-\mu F(T_n{u}_n)\Vert \to 0.\end{array}$$

It turns out from (3a) of Step 3 and (3.12) that $\Vert z_n-x_n\Vert \to 0$ and

(3.17)$$\underset{n\to \infty }{\lim \sup }⟨\gamma f(q)-\mu F(q),z_n-q⟩\le 0,$$

It is now immediately clear that ${\displaystyle \sum _{n=1}^{\infty }{\tilde{\alpha }}_n}=\infty$ and $\underset{n\to \infty }{\lim \sup }{\tilde{\delta }}_n\le 0$. This enables us to apply Lemma 2.7 to the relation (3.16) to arrive at $\Vert x_n-q\Vert \to 0$, that is, $x_n\to q$ in norm.

Remark 3.6. The choices of the parameters $({\alpha }_n),({\beta }_n),(r_n)$ are easy. For instance, any decreasing null sequence (β_n) satisfies (B₂). More precisely, the choices:

$${\alpha }_n=\frac{1}{n^a},{\beta }_n=\frac{1}{n^b},r_n=r+\frac{1}{n^c},n\ge 1,$$

where $0<a\le 1$ and $b,c,r>0$ satisfy (B₁)-(B₄). Also, the choice ${\lambda }_n=\underset{\_}{\lambda }+\frac{1}{n^d}$, where $\underset{\_}{\lambda }>0$ and $d>0$ satisfies (B₅) for n big enough.

Remark 3.7. Theorem 3.3 is a generalization of [21, Theorem 3.2]. In addition, we used the condition $\underset{n\to \infty }{\lim \inf }{\lambda }_n>0$ for the stepsizes $\left\{{\lambda }_n\right\}$. However, [21, Theorem 3.2] required that ${\lambda }_n\to \frac{2}{L}$, which needs the exact value L of the Lipschitz constant of the gradient $\nabla g$ of the objective function g. In practical problems, the exact value of L would be unavailable in many circumstances; consequently, verification of the condition ${\lambda }_n\to \frac{2}{L}$ turns out to be hard.