Let C be a nonempty, closed and convex subset of a reflexive real Banach space E with dual space E*. Throughout this paper, we shall denote by |·| and 〈·,·〉 the norm and the duality pairing between the elements of E and E*, respectively. The Equilibrium Problem (EP) is formulated as:

(1.1)Find x*∈C​​such that​​g(x*,y)≥0,​​∀​y∈C,

where g:C×C→R is a bifunction satisfying g(x,x)=0 for all x∈C. The EP provides a unified framework for the study of various problems arising in pure and applied sciences such as complimentarity problems, fixed point problems, optimization problems, variational inequality problems and so on [4, 15, 31]. For instance, if g(x,y)=〈Fx,y-x〉 for all x,y∈C where F:C→E* is a mapping, then the EP becomes the Variational Inequality Problem (VIP) (see [16, 35]) which consists of finding a point x*∈C such that

(1.2)〈F(x*),x*-y〉≥0,​​∀​y∈C.

In the study of EP, research is split into existence results and the development of iterative algorithms for approximating solutions. For more on existence results, we refer the readers to the works of [4, 15] and the references therein. Iterative schemes for approximating solutions of equilibrium problems have been studied in both finite and infinite dimensional spaces (see [3, 12, 31]). In recent years, there have been several results about approximating the solution of equilibrium problems involving pseudomonotone and strongly pseudomonotone bifunctions (see [10, 12, 14, 19, 22, 32]). We note that in most of these results, there is the requirement that the bifunction g satisfies a certain Lipschitz-type condition on the set C. A bifunction g:C×C→R is said to satisfy Lipschitz-type condition, if there exist constants c1,c2>0 such that for all x,y,z∈C

(1.3)g(x,y)+g(y,z)≥g(x,z)-c1|x-y|2-c2|y-z|2.

The Lipschitz condition (1.3) does not hold in general, and when it does, it is not always easy to find the Lipschitz constants c1 and c2. This may affect the efficiency of the method (see [13, 20, 23]). In addition to this, previous methods require solving two strongly convex programming problems; this is not efficient when the feasible set or the bifunction have complex structures. To reduce some of these drawbacks, Vinh and Gibali [17] recently introduced gradient projection type algorithms for solving the EP with a pseudomonotone bifunction in real Hilbert spaces as follows:

Algorithm 1.1. Inertial Gradient projection method (IGPM) for EP

Initialization: Take θn∈[0,1) and a positive sequence {βn}n=0∞ satisfying

(1.4)∑n=0∞βn=+∞, ∑n=0∞βn2<+∞.

Select initial points x0,​x1∈C and set n=1.

Iterative step: Given xn-1 and xn (n≥1), choose αn such that 0≤θn≤θ¯n

where

θ¯n=minθ,βn2‖xn −xn−1 ‖,ifxn≠xn−1θ,          otherwise.

Compute

wn=xn+θn(xn−xn−1).

Take g(xn)∈∂(xn,·)(xn)(n≥1). Calculate

ηn=max{1,‖g(xn)‖},λn=βnηn

and

xn+1=PC(wn−αng(xn)).

Stopping criterion If xn+1=wn=xn for some n≥1 then stop. Otherwise set n:=n+1 and return to Iterative step.

Although, Algorithm 1.1 does not require a Lipschitz-like condition on the pseudomonotone bifunction, its convergence requires condition (1.4) which significantly affects the convergence rate of the algorithm. Rehman et al [32] proposed an explicit algorithm which does not requires condition (1.4) but rather the Lipschitz-like condition (1.3) in real Hilbert space as follows:

Algorithm 1.2. Inertial explicit subgradient extragradient method (IESEM)

Initialization: Choose x-1,​y-1,​x0,​y0∈H, α1, μ∈(0,h(θn)) and let αn∈[0,5-2) be a nondecreasing sequence.

Iterative step: Given xn-1, {yn-1}, {xn}, {yn} and αn(n≥0).

Step 1 Construct a half space     Cn={w∈H:〈wn−αnvn−yn,w−yn〉≤0},    where vn∈∂f(yn−1,yn) and compute     xn+1=argmin{αnf(yn,y)+12‖wn−y‖2:y∈Cn},    where wn=xn+θn(xn−xn−1).Step 2  Set     αn+1=minαn,μ(‖yn−1−yn‖2+‖xn+1−yn‖2)2[f(yn−1 ,xn+1 )−f(yn−1 ,yn )−f(yn ,xn+1 )]+    and compute     xn+1=argmin{αn+1f(yn,y)+12‖wn−y‖2:y∈C},

where wn+1=xn+1+θn+1(xn+1-xn).

Stopping criterion If xn+1=wn and yn=yn-1 for some n≥0 then stop. Otherwise set n:=n+1 and return to Step 1.

The authors proved that the sequences {xn} and {yn} generated by Algorithm 1.2 converges weakly to a solution of the EP.

Let us also mention that the inertial extrapolation term in Algorithms 1.1 and 1.2 is used as a means of speeding up the convergence properties of the algorithms. This method was first introduced by Polyak [30] and has been adopted by many other authors, see for instance [11, 19, 22, 27, 32].

Motivated by the above results, in this paper, we are concerned with finding an iterative method which does not involve the condition (1.4) and a Lipschitz-like condition for solving pseudomonotone EP in reflexive Banach space. We introduce a new inertial self-adaptive Bregman projection method which does not require the bifunction satisfying the Lipschitz-like condition and its convergence is proved without using condition (1.4). We also use the Bregman distance techniques which generalizes the Euclidean distance popularly used by other authors.

The rest of the paper is organized as follows. In Section 2, we collect some basic definitions and preliminary results required in our main results. In Section 3, we introduce our algorithm and prove a strong convergence result for the sequence generated by the algorithm. In Section 4, we give an application of our result to variational inequality problems. We provide some numerical reports and also compare the performance of our method with other methods in the literature in Section 5. We give some concluding remarks in Section 6.

In this section, we give some definitions and preliminary results which will be used in our convergence analysis. Let C be a nonempty, closed and convex subset of a real Banach space E with the norm |·| and dual space E*. We denote the weak and strong convergence of a sequence {xn}⊂E to a point x∈E by xn⇀x and xn→x, respectively.

A function f:E→(-∞,+∞] is said to be

• (i)

  proper, if dom(f)={x∈E:f(x)<∞}≠∅;

• (ii)

  f is strongly convex with strongly convexity constant ρ>0, i.e

  f(x)≥f(y)−〈∇f(y),x−y〉+ρ2‖x−y‖2,∀x,y∈E 

• (iii)

  Gâteaux differentiable at x∈E, if there exists an element in E denoted by f′(x) or ∇f(x) such that

  limt→0f(x+ty)−f(x)t=〈y,f′(x)〉,∀y∈E,

  where f′(x) or ∇f(x) is called the Gâteaux differential or gradient of f at x. We note that ∇f(∇f*(x*))=x* for all x*∈E* 

• (iv)

  Fréchet differentiable at x, if the limit in (iii) above exists uniformly on the unit sphere of E.

Lemma 2.1. ([33]) Let f:E→R be uniformly Fréchet differentiable and bounded on bounded subsets of E, then ∇f is uniformly continuous on bounded subsets of E from strong topology of E to the strong topology of E*.

The subdifferential set f at a point x denoted by ∂f is defined by

∂f(x):={x*∈E*:f(x)-f(y)≤〈y-x,x*〉,​​y​∈E}.

Every element x*∈∂f(x) is called a subgradient of f at x. If f is continuously differentiable, then ∂f(x)={∇f(x)}, which is the gradient of f at x. The Fénchel conjugate of f is the convex functional f*:E*→R∪{+∞} defined f*(x*)=sup{〈x*,x〉-f(x):x∈E}. Let E be a reflexive Banach space, the function f is said to be Legendre if and if only it satisfies the following two conditions (see [1]):

• (L1)

  int dom(f)≠∅ and ∂f is single-valued on its domain ;

• (L2)

  int dom(f)≠∅ and ∂f* is single-valued on its domain.

Let f be a strictly convex and Gâteaux differentiable function. The function Df : dom (f)×int dom (f)→[0,∞) defined by

Df(x,y)=f(x)-f(y)-〈x-y,∇f(y)〉,

is called the Bregman distance with respect to the function f. It is worthy of mentioning that the bifunction Df is not a metric in the usual sense because it does not satisfy the symmetry and triangle inequality properties. However, it posses the following important property called the three points identity:

(2.1)Df(x,y)+Df(y,z)-Df(x,z)=〈∇f(z)-∇f(y),x-y〉,

for x∈ dom(f) and y,z∈ int dom(f). Also, from the strong convexity of f and the definition of the Bregman distance, we have that

(2.2)Df(x,y)≥ρ2|x-y|2.

The Bregman distance function has been widely used by many authors in the literature (see [1, 7, 8] and the references therein).

Remark 2.2. Practical important examples of Bregman distance functions can be found in [2]. For example, if f(x)=12|·|, then Df(x,y)=12|x-y|2 which is the Euclidean distance. Also, if f(x)=-∑j=1mxjlog(xj) which is the Shannon's entropy for the non-negative orthant R++m:={x∈Rm:xj>0}, we obtain the Kullback-Leibler cross entropy defined by

(2.3)Df(x,y)=∑ j=1mxjlog xj yj −1+∑ j=1myj.

Definition 2.3. ([5]) Let C be a nonempty, closed and convex subset of a reflexive real Banach space E. A Bregman projection of x∈ int dom(f) onto C⊂ int dom(f) is the unique vector ΠCx∈C which satisfies

Df(ΠCx,x)=inf{Df(y,x):y∈C}.

Lemma 2.4. ([9]) Let C be a nonempty, closed and convex subset of E and x∈E. Let f:E→R be a Gâteaux differentiable and totally convex function. Then

• (i)

  q=ΠCx if and only if 〈∇f(x)-∇f(q),y-q〉≤0, for all y∈C 

• (ii)

  Df(y,ΠCx)+Df(ΠC(x),x)≤Df(y,x), for all y∈C.

Definition 2.5. ([6, 9]) The bifunction vf : int dom(f)×[0,+∞) defined by

vf(x,t):=inf{Df(y,x):y∈dom(f),|y-x|=t}

is called the modulus of total convexity at x. The function f is called totally convex at x∈ int dom(f) if vf(x,t) is positive for any t>0. The modulus of total convexity of f on C is the bifunction vf:intdom(f)×[0,+∞), defined by

vf(C,t):=inf{vf(x,t):x∈C∩intdom(f)}.

he function f is called totally convex on bounded subsets if vf(C,t)>0 for any nonempty and bounded subset C and any t>0. Also, f is said to be coercive, if lim‖x‖→+∞f(x)‖x‖=+∞.

Proposition 2.6. ([6]) If x∈ int dom(f), then the following are equivalent:

• (i)

  the function f is totally convex at x,

• (ii)

  f is sequentially consistent, i.e., for any sequence {yn}⊂ dom(f),

limn→∞Df(yn,x)=0⇒limn→∞‖yn−x‖=0.

We also recall (see [6]) that the function f is called sequentially consistent, if for any two sequences {xn} and {yn} in

E such that the first one is bounded,

(2.4)limn→∞Df(xn,yn)=0⇒limn→∞‖xn−yn‖=0.

Proposition 2.7. ([6]) If dom(f) contains at least two points, then the function f is totally convex on bounded sets if and only if the function f is sequentially consistent.

Proposition 2.8. ([34]) Let f:E→R be a Gâteaux differentiable and totally convex function. If x¯∈E and the sequence {Df(xn,x¯)} is bounded, then the sequence {xn} is also bounded.

Lemma 2.9. ([29]) If f:E→(-∞,+∞] is a proper lower semicontinuous function, then f*:E*→(-∞,+∞] is a proper weak​* lower semicontinuous and convex function. Thus for all y∈E, we have

Dfy,∇f*∑i=1Nλi∇f(xi)≤∑i=1NλiDf(y,xi)

where {xi}iN⊂E and {λi}iN⊂(0,1) with ∑i=1Nλi=1.

A Banach space is said to satisfy the Opial property [28], if for any sequence {xn}⊂E such that xn⇀x for some x∈E, we have

limn→∞‖xn−x‖<limn→∞‖xn−y‖

for all y∈E with y≠x. We note that all Hilbert spaces, all finite dimensional Banach spaces and the Banach space ℓp (1≤p<∞) satisfy the Opial property. It is also worthy of mentioning that not every Banach space satisfy the Opial property (see [18]). In order to extend this property to cover all Banach spaces, Huang et al. [25] established the following lemma.

Lemma 2.10. ([25]) Let E be a Banach space and f:E→(-∞,+∞] be a proper strictly convex function that is Gâteaux differentiable and {xn} is a sequence in E such that xn⇀x for some x∈E. Then

limn→∞Df(x,xn)<limn→∞Df(y,xn)

for all y∈domf with y≠x.

Lemma 2.11. ([17]) Let {an} and {bn} be two nonnegative real sequences such that

an+1≤an−bn.

Then, {an} is bounded and ∑n=1∞bn<∞.

Definition 2.12. Let C be a nonempty, closed and convex subset of a Banach space E and g:C×C→R be a bifunction, g is said to be:

• (i)

  strongly γ-monotone on C, if there exists γ>0 such that

  g(x,y)+g(y,x)≤-γ|x-y|2,​​∀​x,y∈C 

• (ii)

  monotone on C, if

  g(x,y)+g(y,x)≤0,​​∀​x,y∈C 

• (iii)

  strongly γ-pseudomonotone on C, if there exists γ>0 such that for any x,y∈C

  g(x,y)≥0⇒g(y,x)≤-γ|x-y|2 

• (iv)

  pseudomonotone on C, if

  g(x,y)≥0⇒g(y,x)≤0,∀x,y∈C.

It is easy to see that (i)⇒(ii)⇒(iv) and (i)⇒(iii)⇒(iv) but the converse implications are not always true, see, for instance [17, 20, 23].

In this section, we give a concise and precise statement of our algorithm and discuss some of its elementary properties convergence analysis. Let C be a nonempty, closed and convex subset of a reflexive Banach space E and f:E→R∪{+∞} be a bounded Legendre function which is uniformly Fréchet differentiable, strongly coercive, strongly convex and totally convex on bounded subsets of E. Let g:C×C→R be a bifunction satisfying the following conditions:

• (A1)

  g(x,·) is convex and lower semicontinuous for every x∈E 

• (A2)

  g is pseudomonotone on C;

• (A3)

  EP(g,C)≠∅ 

• (A4)

  If {xn}n=0∞⊂E is bounded, then the sequence {h(xn)∈∂g(xn,·)(xn)}n=0∞ is bounded.

• (A5)

  For L>0, we assume h(x)∈∂g(x,·)(x) is L-Lipschitz continuous. However, the knowledge of L is not required in execution and practice of our method.

Remark 3.1. We note that condition (A4) is a standard assumption and holds when g(x,·) is bounded on bounded sets (see, for instance [9, Proposition 1.1.11]).

Next, we present our algorithm as follows:

Algorithm 3.2. Modified Bregman Popov Extragradient Method for EP Initialization: Choose x0,x1,y0 and y1∈C and α1>0, μ∈(0,ρ(2-1)), θ∈(0,1).

Iterative step: Having xn-1,xn,yn-1,yn and αn. Calculate xn+1,yn+1 and αn+1 for each n≥1 as follows

(3.1)wn=∇f*(∇f(xn)+θ(∇f(xn−1)−∇f(xn)),xn+1=ΠTn∇f*((∇f(wn)−αnh(yn))),αn+1=minαn,μ‖yn −yn−1 ‖‖h(yn )−h(yn−1 )‖, if‖h(yn)−h(yn−1)‖>0,αn,              otherwise,yn+1=ΠC∇f*((∇f(xn+1)−αn+1h(yn))),

where

Tn={y∈E:〈∇f(wn)-αnh(yn-1)-∇f(yn),y-yn〉≤0}

and h(x)∈∂g(x,·)(x) for each x∈C.

Stopping criterion If xn+1=wn and yn+1=yn=yn-1 for some n≥1 then stop. Otherwise set n:=n+1 and return to Iterative step.

Remark 3.3. The sequence {αn} given in (3.1) is nonincreasing and

limn→∞αn=α≥minα1,μL.

Proof: It follows from the definition of {αn} that αn+1≤αn. Thus, {αn} is non-increasing. Now, since |h(yn)-h(yn-1)|≤L|yn-yn-1| for L>0, we get that

μ‖yn−yn−1‖‖h(yn)−h(yn−1)‖≥μL, if‖h(yn)−h(yn−1)‖>0.

This together with (3.1) implies

αn≥minα1,μL.

Thus, the sequence {αn} is lower bounded. The conclusion follows.

The following inequality is satisfied for {xn} and {yn}.

Lemma 3.4. Let {xn} and {yn} be defined by Algorithm 3.2, then for every x*∈ EP(g,C) the following inequality holds

Df(x*,xn+1)≤Df(x*,xn)−1−2μαn ραn+1 Df(xn+1,yn)
(3.2)−1−(1+2)μαnραn+1Df(yn,wn)+θ(Df(x*,xn−1)−Df(x*,xn)).

Proof: Since x*∈ EP(g,C) and xn+1=ΠTn∇f*(∇f(wn)-αnh(yn)), we have by Lemma 2.4 (i), that

〈∇f(wn)−αnh(yn)−∇f(xn+1),x*−xn+1〉≤0,

this implies

(3.3)〈∇f(wn)-∇f(xn+1),x*-xn+1〉≤αn〈h(yn),x*-xn+1〉.

Using the three points identity (2.1) and subdifferential of g in the second variable in (3.3), we obtain

Df(x*,xn+1)=Df(x*,wn)−Df(xn+1,wn)+〈∇f(wn)−∇f(xn+1),x*−xn+1〉≤Df(x*,wn)−Df(xn+1,wn)+αn〈h(yn),x*−xn+1〉
(3.4)=Df(x*,wn)−Df(xn+1,wn)+αn〈h(yn),yn−xn+1〉+αn〈h(yn),x*−yn〉≤Df(x*,wn)−Df(xn+1,wn)+αn〈h(yn),yn−xn+1〉+αng(yn,x*)=Df(x*,wn)−Df(xn+1,wn)+αn〈h(yn−1),yn−xn+1〉
(3.5)+αn〈h(yn)−h(yn−1),yn−xn+1〉+αng(yn,x*).

Note that

αn〈h(yn−1),yn−xn+1〉=〈∇f(wn)−αnh(yn−1)−∇f(yn),xn+1−yn〉+〈∇f(yn)−∇f(wn),xn+1−yn〉.

Since xn+1∈Tn, 〈∇f(wn)-αnh(yn-1)-∇f(yn),xn+1-yn〉≤0 implies that

(3.6)αn〈h(yn-1),yn-xn+1〉≤〈∇f(yn)-∇f(wn),xn+1-yn〉.

Using the three points identity (2.1), we have

〈∇f(yn)-∇f(wn),xn+1-yn〉=Df(xn+1,wn)-Df(xn+1,yn)-Df(yn,wn).

Substituting this into (3.6), we get

(3.7)αn〈h(yn-1),yn-xn+1〉≤Df(xn+1,wn)-Df(xn+1,yn)-Df(yn,wn).

From (3.7) and (3.5), we obtain that

(3.8)Df(x*,xn+1)≤Df(x*,wn)−Df(xn+1,yn)−Df(yn,wn)+αng(yn,x*)+αn〈h(yn−1)−h(yn),xn+1−yn〉.

Observe from the definition of αn, that

(3.9)αn〈h(yn−1)−h(yn),xn+1−yn〉≤αn‖h(yn−1)−h(yn)‖‖xn+1−yn‖≤μαnαn+1‖yn−1−yn‖‖xn+1−yn‖≤μαnαn+1122‖yn−1−yn‖2+12‖xn+1−yn‖2≤μαn22αn+1(2+2)‖yn−wn‖2+2‖wn−yn−1‖2+μαn2αn+1‖xn+1−yn‖2≤(1+2)μαnραn+1Df(yn,wn)+μαnραn+1Df(wn,yn−1)+2μαnραn+1Df(xn+1,yn),

where we have used 2ab≤12a2+2b2 and (a+b)2≤2a2+(2+2)b2 in separate steps and the strong convexity of f.

By substituting (3.9) into (3.8), we obtain

(3.10)Df(x*,xn+1)≤Df(x*,wn)−1−2μαn ραn+1 Df(xn+1,yn)−1−(1+2)μαn ραn+1 Df(yn,wn)+αng(yn,x*).

Since x*∈ EP(g,C) and yn∈C, we have that g(x*,yn)≥0, it follows from the fact that g is pseudomonotone that g(yn,x*)≤0. Therefore, we obtain from (3.10), that

(3.11)Df(x*,xn+1)≤Df(x*,wn)−1−2μαn ραn+1 Df(xn+1,yn)−1−(1+2)μαn ραn+1 Df(yn,wn).

Observe from (3.1) and Lemma 2.9, that

(3.12)Df(x*,wn)=Df(x*,∇f*((1−θ)∇f(xn)+θ∇f(xn−1)))≤(1−θ)Df(x*,xn)+θDf(x*,xn−1).

It therefore follows from (3.11) and (3.12), that

(3.13)Df(x*,xn+1)≤Df(x*,xn)−1−2μαn ραn+1 Df(xn+1,yn)−1−(1+2)μαn ραn+1 Df(yn,wn)+θ(D(x*,xn−1)−Df(x*,xn)).

The prove is complete.

Now, we present the weak convergence theorem for Algorithm 3.2.

Theorem 3.5. Assume that the conditions A1-A4 are satisfied and θDf(xn,xn-1)<∞, then the sequences {xn} and {yn} given by Algorithm 3.2 converge weakly to a solution x*∈ EP(g,C).

Proof. Let x*∈ EP(g,C). First, we show that {xn} is bounded. Indeed, we have from Lemma 3.4 that (3.13) satisfies an+1≤an-bn where

an=Df(x*,xn)+μαnραn+1Df(xn,yn−1)+θDf(x*,xn−1)

and

(3.14)bn=1-(1+2)μαnραn+1Df(yn,wn)+1-(1+2)μαnραn+1+μαn+1ραn+2Df(xn+1,yn).

Then, by Lemma 2.11, {an} is bounded, which implies that {xn} is bounded, limn→∞Df(xn+1,yn)=0 and limn→∞Df(yn,wn)=0. Consequently, |yn-wn|→0 and |xn+1-yn|→0 as n→∞. Again, we see from (3.1) and Lemma 2.9, that

(3.15)Df(xn,wn)≤(1−θ)Df(xn,xn)+θDf(xn,xn−1)≤θDf(xn,xn−1).

Thus, we have that Df(xn,wn)→0 as n→∞, which implies by (2.4), that |xn-wn|→0 as n→∞. Consequently, one gets that |xn+1-xn|→0 as n→∞. Furthermore

‖yn+1−yn‖≤‖yn+1−xn+1‖+‖xn+1−xn‖+‖xn−yn‖→0,asn→∞.

Since f is uniformly Fréchet differentiable, we have that |∇f(yn+1)-∇f(xn+1)|→0 as n→∞.

From the boundedness of {xn}, there exists a subsequence {xnk} of {xn} such that xnk⇀x¯. Consequently, {wnk} and {ynk} converge both converge weakly to x¯. It follows easily that x¯∈C. From the definition of yn+1 and Lemma 2.4 (i), we have

(3.16)〈∇f(xn+1)−αnh(yn)−∇f(yn+1),y−yn+1〉≤0,∀y∈C,

that is

〈∇f(xn+1)−∇f(yn+1),y−yn+1〉−αn〈h(yn),y−yn+1〉≤0,∀y∈C

and

∇f(xn+1)−∇f(yn+1)αn,y−yn+1+〈h(yn),yn+1−yn〉≤〈h(yn),y−yn〉,∀y∈C.

Hence, we have by the definition of subdifferential, that forall y∈C,

(3.17)〈∇f(xnk+1)-∇f(ynk+1)αnk,y-ynk+1〉+〈h(ynk),ynk+1-ynk〉≤g(ynk,y).

Observe that ∇f(xnk+1)-∇f(ynk+1)αnk→0 as k→∞, since αnk≥α>0. Hence, by passing limit over (3.17), we obtain that g(x¯,y)≥0,​​∀​y∈C, thus x¯∈ EP(g,C).

Finally, we show that x¯ is unique. Assume the contrary, then there exists a subsequence xni such that xni⇀x^. Following similar arguments as above, we have that x^∈ EP(g,C). It follows from the Bregman Opial-like property of E (Lemma 2.10), that

limn→∞Df(x¯,xn)=limk→∞Df(x¯,xnk)=liminfk→∞Df(x¯,xnk)<liminfk→∞Df(x^,xnk)=limk→∞Df(x^,xnk)=limn→∞Df(x^,xn).

Thus, we arrive at a contradiction. Therefore is x¯=x^.

We are now in position to establish the strong convergence of Algorithm 3.2, however we replace condition A2 of Assumption A by A2* that the bifunction g:C×C→R∪{+∞} is strongly pseudomonotone.

Strong convergence theorem for Algorithm 3.2.

Theorem 3.6. Assume that conditions A1, A2* and A3-A4 are satisfied. The sequences {xn} and {yn} of Algorithm 3.2 converge strongly to a unique solution of EP(g,C).

Proof: Let x*∈ EP(g,C). By the definition of xn+1 and Lemma 2.4, we have

〈∇f(wn)-αnh(yn)-∇f(xn+1),x*-xn+1〉≤0,

alternatively

〈∇f(wn)-∇f(xn+1),x*-xn+1〉≤αn〈h(yn),x*-xn+1〉.

By using the three points identity 2.1, we have

Df(x*,xn+1)+Df(xn+1,wn)-Df(x*,wn)≤αn〈h(yn),x*-xn+1〉,

that is

(3.18)Df(x*,xn+1)≤Df(x*,wn)-Df(xn+1,wn)+αn〈h(yn),x*-xn+1〉.

Since x*∈ EP(g,C), we have g(x*,x)≥0 for all x∈C. Using the fact that g is strongly pseudomonotone, we obtain g(x,x*)≤-γ|x-x*|2. Taking x=yn∈C, we get g(yn,x*)≤-γ|yn-x*|2. Now, using the definition of subdifferential of g at yn, we have

(3.19)〈h(yn),x*−xn+1〉≤〈h(yn),x*−yn〉+〈h(yn),yn−xn+1〉≤g(yn,x*)+〈h(yn),yn−xn+1〉≤−γ‖yn−x*‖2+〈h(yn),yn−xn+1〉.

Substituting (3.18) into (3.19), we get

(3.20)Df(x*,xn+1)≤Df(x*,wn)-Df(xn+1,wn)-γαn|yn-x*|2+αn〈h(yn),yn-xn+1〉,

again by using (2.1), we get

(3.21)Df(x*,xn+1)≤Df(x*,wn)−Df(wn,yn)−Df(xn+1,yn)−〈∇f(yn)−∇f(wn),xn+1−yn〉−γαn‖yn−x*‖2+αn〈h(yn),yn−xn+1〉=Df(x*,wn)−Df(wn,yn)−Df(xn+1,yn)+〈∇f(wn)−αnh(yn)−∇f(yn),xn+1−yn〉−γαn‖yn−x*‖2.

Observe that

(3.22)〈∇f(wn)−αnh(yn)−∇f(yn),xn+1−yn〉=〈∇f(wn)−αnh(yn−1)−∇f(yn),xn+1−yn〉+αn〈h(yn−1)−h(yn),xn+1−yn〉,

and 〈∇f(wn)-αnh(yn-1)-∇f(yn),xn+1-yn〉≤0, since xn+1∈Tn. Hence

(3.23)〈∇f(wn)−αnh(yn)−∇f(yn),xn+1−yn〉≤αn〈h(yn−1)−h(yn),xn+1−yn〉≤αn‖h(yn−1)−h(yn)‖‖xn+1−yn‖≤μαnαn+1‖yn−1−yn‖‖xn+1−yn‖≤μαnαn+1{122‖yn−1−yn‖2+12‖xn+1−yn‖2}≤μαn22αn(2+2)‖yn−wn‖2+2‖wn−yn−1‖2+μαn2αn+1‖xn+1−yn‖2≤(1+2)μαnραn+1Df(yn,wn)+μαnραn+1Df(wn,yn−1)+2μαnραn+1Df(xn+1,yn).

Using (3.23) in (3.21), we get

(3.24)Df(x*,xn+1)≤Df(x*,wn)−1−2μαn αn+1 Df(xn+1,yn)−1−(1+2)μαn αn+1 Df(yn,wn)+μαnαn+1Df(wn,yn−1)−γαn‖yn−x*‖2.

by using (3.12), we obtain

Df(x*,xn+1)≤Df(x*,xn)−1−2μαn αn+1 Df(xn+1,yn)−1−(1+2)μαn αn+1 Df(yn,wn)+μαnαn+1Df(wn,yn−1)+θ(Df(x*,xn−1)−Df(x*,xn))−γαn‖yn−x*‖2.

Following similar argument as in Theorem 3.5, we obtain that {xn} is bounded. It follows also that |wn-xn|→0, |yn-xn|→0, |xn+1-yn|→0 and |xn+1-xn|→0 as n→∞. Since f is continuous on bounded sets, coercive and uniformly Fréchet differentiable, we get by Lemma 2.1, that |f(xn+1)-f(wn)|→0, |f(xn)-f(xn-1)|→0 and |∇f(xn+1)-∇f(xn)|→0 as n→∞. Therefore,

(3.25)Df(x*,wn)−Df(x*,xn+1)=f(x*)−f(wn)−〈∇f(wn),x*−wn〉−f(x*)+f(xn+1)+〈∇f(xn+1),x*−xn+1〉=f(xn+1)−f(wn)+〈∇f(xn+1),x*−xn+1〉−〈∇f(wn),x*−wn〉=f(xn+1)−f(wn)+〈∇f(xn+1)−∇f(wn),x*−wn〉
(3.26)+〈∇f(xn+1),wn−xn+1〉.

Thus, passing limits over (3.26), we get

limn→∞Df(x*,wn)−Df(x*,xn+1)=0.

Also,

Df(wn,yn-1)=f(wn)-f(yn-1)-〈∇f(yn-1),wn-yn-1〉→0​​as​n→∞.

It follows from (3.24), that

γαn|yn-x*|2≤Df(x*,wn)-Df(x*,xn+1)+μαnαn+1Df(wn,yn-1)→0.

Thus, since γαn>0, we get that

limn→∞‖yn−x*‖=0.

This implies {yn} converges strongly to x*. Consequently, {xn} converges strongly to x*∈ EP(g,C).