Let H be a real Hilbert space with the inner product 〈⋅,⋅〉 and induced norm ‖⋅‖. A "Zero-Point Problem" (ZPP) for monotone operators is defined as follows: find x*∈H such that

(1.1)0∈Tx*,

where T:H→H is a monotone operator. Several authors have focused on the convergence of iterative methods in order to locate a zero-point for monotone operators in Hilbert spaces. Martinet [9] constructed the "Proximal Point Algorithm" (PPA) to solve problem (1.1). The PPA is depicted as:

(1.2)xn+1=(I+λnT)−1xn,   ∀n≥1,

where I is the identity mapping, and {λn} is a sequence of positive real numbers. After Martinet [9], several algorithms were developed to solve (ZPP). The reader can see [10] - [14] and for more information, refer to the citations provided in this article.

The "Monotone Inclusion Problem" (MIP) is to obtain x∗∈H such that

(1.3)0∈(D+M)x∗.

where D:H→H is a single-valued mapping and M:H⇉H is a multi-valued mapping. The MIP (1.3) can be written as the ZPP (1.1) by setting T:=D+M.

According to Lions and Mercier [6], the most common way to solve the problem (1.3) is to use the forward-backwrad splitting method, defined as follows:

(1.4)xn+1=(I+λM)−1(I−λD)xn,   ∀n≥1,

where x1∈H is arbitrarily chosen and λ>0. In algorithm (1.4), the operator D is called a forward operator and M is called a backward operator. For more details about the forward-backward splitting method used to solve the MIP (1.3), the reader is directed to see [5, 7, 14].

Polyak [11] introduced the inertial extrapolation method to speed up the rate of convergence of the iteration. This is sometimes called the heavy ball method. Many scholars have exploited this notion to combine algorithms with inertial terms to speed up the rate of convergence.

In 2015, Lorenz and Pock [7] studied the MIP (1.3) and proposed the inertial forward-backward algorithm for monotone operators, which combines the heavy ball method and forward-backward method. The algorithm is defined as:

(1.5)wn=xn+θn(xn−xn−1)xn+1=(I+λM)−1(I−λD)wn,   ∀n≥1,

where x0,x1∈H are arbitrarily chosen, and D:H→H and M:H⇉H are single and multi-valued mappings, respectively. The sequence {xn} generated by algorithm (1.5) converges weakly to a solution of MIP (1.3) with some suitable assumptions.

There are many ways to solve the MIP other than using an algorithm combined with the heavy ball idea. Tseng [15] introduced the modified forward-backward splitting method, a powerful iterative method for solving the monotone inclusion problem (1.3). In short, it is known as Tseng's splitting algorithm. Let C be a closed and convex subset of a real Hilbert space H. Tseng's splitting algorithm is defined as:

(1.6)yn=(I+λnM)−1(I−λnD)xnxn+1=PC(yn−λn(Dyn−Dxn)),   ∀n≥1,

where x1∈H is arbitrarily chosen, λn is chosen to be the largest λ∈{δ,δl,δl2,..} satisfying λ‖Dyn−Dxn‖≤μ‖xn−yn‖ where δ>0,l∈(0,1),μ∈(0,1) and PC is the projection onto a closed convex subset C of H.

Kitkaun and Kumam [5] combined the forward -backward splitting method with the viscosity approximation method for solving MIP (1.3). It is called the inertial viscosity forward-backward splitting algorithm, which is defined as:

(1.7)wn=xn+θn(xn−xn−1)xn+1=αnδf(xn)+(1−αn)(I−λnM)−1(I−λnD)wn,   ∀n≥1,

where x0,x1∈H are arbitrarily chosen, f:H→ℝ is a differentiable function such that its gradient δf is a contraction with constant ρ∈(0,1) and D:H→H and M:H⇉H are an inverse strongly monotone and a maximal monotone operator, respectively. The sequence generated by the algorirthm (1.7) converges strongly to a solution of MIP (1.3) under suitable conditions.

Now we define a new type of monotone inclusion problem known as "Cayley's Inclusion Problem" (CIP). Find x*∈H such that

(1.8)0∈(ℭλM+M)x*

where M:H⇉H is a multi-valued maximal monotone mapping, ℭλM:=2JλM−I is the single valued mapping and known as "Cayley operator" and JλM:=(I+λM)−1 is the "resolvent operator" associated with maximal monotone mapping M with λ>0. The solution set of CIP is denoted as:

Ω:={x∈H:0∈ℭλM(x)+M(x)}.

Let S:H→H be a nonexpansive mapping. A fixed point of S is a point x∈H such that

(1.9)S(x)=x.

The set of all fixed point of S is denoted by Fix(S)={x∈H:S(x)=x}.

The objective of this article is to propose an algorithm consisting of Tseng's technique with inertial extrapolation and applying the viscosity iterative method in order to obtain the common solution of CIP and Fix(S) in the framework of real Hilbert space. Moreover, we show that the sequences generated by the algorithm are strongly convergent to the point in the solution set Σ:=Ω∩ Fix(S). In particular, we give numerical illustrations of the algorithm.

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. The weak convergence of {xn}n=1∞ to x is denoted by xn⇀x as n→∞, while the strong convergence of {xn}n=1∞ to x is written as xn→x as n→∞. Now assume that x, y, z ∈H the following relations are valid for inner product spaces,

(2.1)‖x+y‖2≤‖x‖2+2〈y,x+y〉,
(2.2)‖αx+(1−α)y‖2=α‖x‖2+(1−α)‖y‖2−α(1−α)‖x−y‖2,
(2.3)‖αx+βy+γz‖2=α‖x‖2+β‖y‖2+γ‖z‖2                      −αβ‖x−y‖2−αγ‖x−z‖2−βγ‖y−z‖2

for any α,β,γ∈[0,1] such that α+β+γ=1.

The following definitions are required to obtain the desired results.

Definition 2.1. Let D:H→H be a single-valued mapping, then it is said to be

(i) nonexpansive if for all x,y∈H,

‖D(x)−D(y)‖≤‖x−y‖,

(ii) firmly nonexpansive if for all x,y∈H,

‖D(x)−D(y)‖2≤〈Dx−Dy,x−y〉,

(iii) L-Lipschitz continuous if for all x,y∈H, there exists L>0 such that

‖D(x)−D(y)‖≤L‖x−y‖,

(iv) α-strongly monotone if for all x,y∈H, there exists a constant α>0 such that

〈D(x)−D(y),x−y〉≥α‖x−y‖2,

(v) µ- inverse strongly monotone if for all x,y∈H, there exists a constant μ>0 such that

〈D(x)−D(y),x−y〉≥μ‖D(x)−D(y)‖2.

Definition 2.2. Let M:H⇉H be a multi-valued mapping, then it is said to be

(i) monotone if for all x,y∈H,u∈M(x),v∈M(y) such that

〈x−y,u−v〉≥0,

(ii) strongly monotone if for all x,y∈H,u∈M(x),v∈M(y), there exist θ>0 such that

〈x−y,u−v〉≥θ‖x−y‖2,

(iii) maximal monotone if M is monotone and (I+λM)(H)=H for all λ>0, where I is the identity mapping on H.

Definition 2.3 Let M:H⇉H be a multi-valued mapping, then the resolvent operator associated with M is defined as:

JλM(x)=[I+λM]−1(x),∀x∈H.

Here λ>0 and I is the identity mapping.

Remark 2.3. The resolvent operator JλM has the following properties:

(i) it is single-valued and nonexpansive, i.e.,

‖JλM(x)−JλM(y)‖≤‖x−y‖,∀x,y∈H,

(ii) it is 1-inverse strongly monotone, i.e,

‖JλM(x)−JλM(y)‖2≤〈x−y,JλM(x)−JλM(y)〉,∀x,y∈H.

Definition 2.4. Let M:H⇉H be a multi-valued mapping and JλM be the resolvent operator associated with M, then the Cayley operator ℭλM is defined as:

(2.4)ℭλM(x)=[2JλM(x)−I],∀x∈H.

Remark 2.5. Using Remark 2.3.(i), it can be easily seen that the Cayley operator ℭλM is 3-Lipschitz continuous. Now, for the sake of convenience we shall denote ℭλM by ℭ throughout the paper.

Lemma 2.6. Let M:H⇉H be a maximal monotone mapping and B:H→H be a Lipschitz continuous mapping. Then a mapping B+M:H⇉H is a maximal monotone mapping.

Lemma 2.7. Let M:H⇉H be a set-valued maximal monotone mapping and λ>0. Then the following statements hold:

• 1. JλM is a single-valued and firmly nonexpansive mapping;

• 2. Fix(JλM)=M−1(0);

• 3. ‖x−JλM‖≤2‖x−JγM‖, 0<λ≤γ,∀x∈H;

• 4. (I−JλM) is firmly nonexapansive mapping;

• 5. Suppose that M−1(0)≠ϕ. Then

  ‖JλM(x)−z‖2≤‖x−z‖2−‖JλM(x)−x‖2  for all  x∈H and z∈M−1(0)

  and

  〈x−JλM,JλM−z〉≥0  for all  x∈H and z∈M−1(0).

Lemma 2.8. Let {an} and {cn} be nonnegative sequences of real numbers such that ∑ n=1∞cn<∞, and let {bn} be a sequence of real numbers such that limsupn→∞bn≤0. If there exists n0∈ℕ such that,for any n≥n0,

an+1≤(1−αn)an+αnbn+cn,

where {αn} is a sequence in (0,1) such that ∑ n=0∞αn=∞, then limn→∞an=0.

Lemma 2.9. ([4]) Let C be a closed convex subset of H and S:C→C a nonexapansive mapping with Fix(S)≠ϕ. If there exists {xn} in C satisfying xn⇀z and ‖xn−Sxn‖→0, then z={S}z.

Lemma 2.10. ([8]) Let {γn} is a sequence of real numbers. Suppose that there is subsequence {γnj}j≥0 of {γn} satisfying γnj≤γnj+1 for each j≥0. Let {ϕ(n)}n≥n* be a sequence of integers defined by

(2.5)ϕ(n):=max{k≤n:γk<γk+1}.

Then {ϕ(n)}n≥n* is a nondecreasing with limn→∞ϕ(n)=∞. Moreover, for each n≥n*, we have γϕ(n)≤γϕ(n)+1 and γn≤γϕ(n)+1.

In this section, we present an inertial Tseng type algorithm for solving our problem. For the convergence analysis of the proposed method, we consider the following assumptions in order to accomplish our goal.

Assumption 1. H is a real Hilbert space, ℭ:H→H is L-Lipschitz continuous and monotone, and M:H⇉H is a maximal monotone operator.

Assumption 2. Σ:=Ω∩F(S) is nonempty.

Assumption 3. {θn}⊂[0,θ),{βn}⊂(β*,β')⊂(0,1−αn) for some θ>0,β*>0,β'>0, and {αn}⊂(0,1) satisfies limn→∞αn=0 and ∑ n=1∞αn=∞.

Assumption 4. f:H→H is a ρ-contractive mapping.

Next the algorithm is presented.

Lemma 3.1. Assume that Assumptions 1-4 hold, then any sequence {λn} in Algorithm is nonincreasing and converges to λ such that min{λ1,μL}≤λ.

Proof. See [17, Lemma 3.1].

Algorithm 3.2. Inertial-viscosity Tseng type algorithm

Initialization: Given λ1>0 and μ∈(0,1). Select arbitrary elements x0,x1∈H and set n:=1.

Iterative Steps: Construct {xn} by using the following steps:

Step 1. Set

wn=xn+θn(xn−xn−1)

and compute,

yn=JλnM(I−λnℭ)wn.

If w_n=y_n, then stop and wn∈Σ. Otherwise

Step 2. Compute

zn=yn−λn(ℭyn−ℭwn)

and

Step 3. Compute

xn+1=αnf(xn)+(1−αn−βn)xn+βnS(zn)

and update,

λn+1=min{μ‖wn−yn‖‖ℭwn−ℭyn‖,λn}   if  ℭwn≠ℭyn;λn                                   otherwise.

Replace n by n+1 and then repeat Step 1.

Lemma 3.3. Let q∈Σ. As given in the algorithm together with all four Assumptions, the following inequalities are true.

(3.1)‖zn−q‖2≤‖wn−q‖2−(1−μ2λn2λn+12)‖wn−yn‖2

and

(3.2)‖zn−yn‖≤μλnλn+1‖wn−yn‖.

Proof. In the same manner as [3, Lemma 6], we obtain that inequalities (3.1) and (3.2) hold.

Lemma 3.4. Suppose that limn→∞‖wn−yn‖=0.If there exists a weakly convergent subsequences {wnj} of {wn}, then under Assumptions 1-4, we have that the limit of {wnj} belongs to Σ.

Proof The proof is similar to the proof of [3, Lemma7].

With the above results we are now ready for the main convergence theorem.

Theorem 3.5. Suppose that limn→∞θnαn‖xn−xn−1‖=0, then under Assumptions 1-4, we have xn→q as n→∞, where q=PΣ∘f(q).

Proof. For the sake of simplicity, we divide the proof into four claims.

Claim 1. {xn} is bounded sequence.

We observe from (3.1) that limn→∞(1−μ2λn2λn+12)=1−μ2>0. Let q′∈Σ and indeed, thanks to Lemma (3.2), we get

(3.3)‖zn−q′‖≤‖wn−q′‖.

Also, from the definition of {yn} and nonexansiveness of JλM, we have

(3.4)‖yn−q′‖≤‖JλnM(I−λnℭ)wn−q′‖≤‖wn−q′‖

By the sequence {θnαn‖xn−xn−1‖} converges to 0, we have that there exists a constant M₁ such that, for all n∈ℕ,

θnαn‖xn−xn−1‖≤M1.

From the definition of w_n and (3.3), we obtain

(3.5)‖zn−q′‖≤‖wn−q′‖=‖xn+θn(xn−xn−1)‖≤‖xn−q′‖+θn‖xn−xn−1‖≤‖xn−q′‖+θnαn‖xn−xn−1‖αn≤‖xn−q′‖+αnM1.

By Assumption 4, nonexpansiveness of S and using (3.10), the following relation is obtained:

‖xn+1−q'‖=‖αnf(xn)+(1−αn−βn)xn+βnS(zn)−q'‖≤‖αn(f(xn)−q')+(1−αn−βn)(xn−q')+βn(Szn−q')‖≤αn‖f(xn)−q'‖+(1−αn−βn)‖xn−q'‖+βn‖Szn−q'‖≤αn‖f(xn)−q'‖+(1−αn−βn)‖xn−q′‖+βn‖zn−q'‖≤αn‖f(xn)−f(q')‖+αn‖f(q')−q'‖+(1−αn)‖xn−q'‖+αnβnM1≤[1−αn(1−ρ)]‖xn−q′‖+αn(‖f(q')−q'‖+M1)=[1−αn(1−ρ)]‖xn−q'‖+αn(1−ρ)‖f(q')−q'‖+M11−ρ≤max{‖xn-x‖,‖f(q')-q'‖+M11-ρ}≤max{‖xn-1-x‖,‖f(q')-q'‖+M11-ρ}⋮≤max{‖x0-x‖,‖f(q')-q'‖+M11-ρ}.

This leads to a conclusion that ‖xn+1−q'‖≤max{‖x0-x‖,‖f(q')-q'‖+M11-ρ}. Consequently, the sequence {xn} is bounded. In addition, {f(xn)} is also bounded. Since Σ is closed and convex set, PΣ∘f is a ρ- contractive mapping. Now, we can uniquely find q∈Σ with q=PΣ∘f(q) duq' to the Banach fixed point theorem. We also get, that for any q′∈Σ,

〈f(q)−q,q′−q〉≤0.

Claim 2. There is M0>0 such that

βn(1−αn−βn)‖xn−Szn‖2≤γn−γn+1+αn(‖f(xn)−q‖2+M0).

Applying (3.5), we have

(3.6)‖zn−q‖2≤(‖xn−q‖+αnM1)2=γn+αn(2M1‖xn−q‖+αnM12)≤γn+αnM0

for some M0>0. It follows from the assumption on f, (2.3) and (3.6) that

γn+1=‖αn(f(xn)−q)+(1−αn−βn)(xn−q)+βn(Szn−q)‖2=αn‖f(xn)−q‖2+(1−αn−βn)γn+βn‖Szn−q‖2−αn(1−αn−βn)‖f(xn)−xn‖2−βn(1−αn−βn)‖xn−Szn‖2−αnβn‖f(xn)−Szn‖2≤αn‖f(xn)−q‖2+(1−αn−βn)γn+βn‖zn−q‖2−βn(1−αn−βn)‖xn−Szn‖2≤αn‖f(xn)−q‖2+(1−αn)γn+αnβnM0−βn(1−αn−βn)‖xn−Szn‖2≤γn+αn(‖f(xn)−q‖2+M0)−βn(1−αn−βn)‖xn−Szn‖2.

Therefore, Claim 2 is obtained

Claim 3. There is M>0 such that

(3.7)γn+1≤[1−αn(1−ρ)]γn+αn(1−ρ)[3M1−ρ⋅θnαn‖xn−xn−1‖]+αn(1−ρ)[2M1−ρ‖xn−Szn‖+21−ρ〈f(q)−q,xn+1−q〉]

Indeed, setting tn:=(1−βn)xn+βnSzn. From inequality (3.3), nonexpansiveness of S, and the definition of w_n, we get

(3.8)‖tn−q‖≤(1−βn)‖xn−q‖+βn‖Szn−q‖≤(1−βn)‖xn−q‖+βn‖zn−q‖≤(1−βn)‖xn−q‖+βn‖wn−q‖≤‖xn−q‖+βnθn‖xn−xn−1‖

and

(3.9)‖xn−tn‖=βn‖xn−Szn‖.

Hence, from the assumption on f, and (3.1), (3.8), and (3.9), we obtain,

γn+1=‖(1−αn)(tn−q)+αn(f(xn)−f(q))−αn(xn−tn)−αn(q−f(q))‖2  ≤‖(1−αn)(tn−q)+αn(f(xn)−f(q))‖2−2αn〈xn−tn+q−f(q),xn+1−q〉  ≤(1−αn)‖tn−q‖2+αn‖f(xn)−f(q)‖2+2αn〈tn−xn,xn+1−q〉  +2αn〈f(q)−q,xn+1−q〉  ≤(1−αn)(‖xn−q‖+βnθn‖xn−xn−1‖)2+αnρ2‖xn−q‖2  +2αn‖tn−xn‖‖xn+1−q‖+2αn〈f(q)−q,xn+1−q〉  ≤(1−αn)γn+2θn‖xn−q‖‖xn−xn−1‖+θn2‖xn−xn−1‖2+αnργn  +2αnβn‖xn−Szn‖‖xn+1−q‖++2αn〈f(q)−q,xn+1−q〉  ≤[1−αn(1−ρ)]γn+θn‖xn−xn−1‖(2‖xn−x‖+θn‖xn−xn−1‖)  +2αnβn‖xn−Szn‖‖xn+1−q‖+2αn〈f(q)−q,xn+1−q〉  ≤[1−αn(1−ρ)]γn+3Mθn‖xn−xn−1‖+2Mαnβn‖xn−Szn‖  +2αn〈f(q)−q,xn+1−q〉  ≤[1−αn(1−ρ)]γn+αn(1−ρ)[3M1−ρ⋅θnαn‖xn−xn−1‖]  +αn(1−ρ)[2M1−ρ‖xn−Szn‖+21−ρ〈f(q)−q,xn+1−q〉]

for M:=supn∈ℕ{‖xn−q‖,θ‖xn−xn−1‖}>0.

Recall that our task is to show that xn→p which is now equivalent to show that γn→0 as n→∞.

Claim 4. γn→0 as n→∞.

Consider the following two cases:

Case (i). We can find N∈ℕ satisfying γn+1≤γn for each n≥N.

Since each term γn is nonnegative, it is convergent. Due to the fact that limn→∞αn=0 and limn→∞βn∈(0,1), and by Claim 2,

(3.10)limn→∞‖xn−Szn‖=0

Indeed, we immediately get

(3.11)limn→∞‖xn−θn‖=limn→∞θnαn‖xn−xn−1‖αn=0

In addition, from the definition of z_n and by using the triangle inequlity, we obtained the following inequalities:

‖zn−wn‖=‖zn−xn+xn−wn‖≤‖zn−xn‖+‖xn−wn‖

and

‖wn−yn‖≤‖wn−zn‖+‖zn−yn‖

It follows from inequality (3.2) that

(1−μλnλn+1)‖wn−yn‖≤‖zn−xn‖+‖xn−wn‖

since limn→∞[1−(1−μλnλn+1)2]=1−μ2>0, (3.10) and (3.11),

(3.12)limn→∞‖wn−yn‖=0.

Note that, for each n∈ℕ

(3.13)‖xn+1−xn‖≤‖xn+1−zn‖+‖zn−xn‖≤αn‖f(xn)−xn‖+(2−βn)‖xn−zn‖.

Consequently, since limn→∞αn=0 and by (3.13), limn→∞‖xn+1−xn‖=0. Next observe that, for the reason that {xn} is bounded, there is z∈H so that xnk⇀z for some subsequence {xnk} of {xn}. Then Lemma 3.3 together with (3.12) implies that z∈Σ. As a result, by the definition of q, it is straightforward to show that

limsupn→∞〈f(q)−q,xn−q〉=limk→∞〈f(q)−q,xnk−q〉=〈f(q)−q,z−q〉≤0.

Since limn→∞‖xn+1−xn‖=0, the following result obtained:

limsupn→∞〈f(q)−q,xn+1−q〉≤limsupn→∞〈f(q)−q,xn+1−xn〉+limsupn→∞〈f(q)−q,xn−q〉≤0.

Applying Lemma 2.8. to the inequality from Claim 3, we can conclude that limn→∞γn=0.

Case (ii). We can find nj∈ℕ satisfying nj≥j and γnj<γnj+1 for all j∈ℕ.

According to Lemma 2.10., the inequality γϕ(n)≤γϕ(n)+1 is obtained, where ϕ:ℕ→ℕ is defined by (2.5). This implies, by Claim 2, that

βϕ(n)(1−αϕ(n)−βϕ(n))‖xϕ(n)−Szϕ(n)‖2≤γϕ(n)−γϕ(n+1)+αϕ(n)(‖f(xϕ(n))−q‖2+M0).

Similar to case (i), since αn→0 as n→∞, we obtain

limn→∞‖xϕ(n)−zϕ(n)‖=0.

Furthermore, an argument similar in case (i) shows that

(3.14)limsupn→∞〈f(q)−q,xϕ(n)+1−q〉≤0.

Finally, from the inquality γϕ(n)≤γϕ(n)+1 and by Claim 3, we obtain

γϕ(n)+1≤[1−αϕ(n)(1−ρ)]γϕ(n)+αϕ(n)(1−ρ)[3M1−ρ⋅θϕ(n)αϕ(n)‖xϕ(n)−xϕ(n)−1‖]+αϕ(n)(1−ρ)[2M1−ρ‖xϕ(n)−Szϕ(n)‖+21−ρ〈f(q)−q,xϕ(n)+1−q〉].

Some simple calculations yield

γϕ(n)+1≤3M1−ρ⋅θϕ(n)αϕ(n)‖xϕ(n)−xϕ(n)−1‖+2M1−ρ‖xϕ(n)−Szϕ(n)‖+21−ρ〈f(q)−q,xϕ(n)+1−q〉.

From this it follows that limsupn→∞γϕ(n)+1≤0. Thus, limn→∞γϕ(n)+1=0.In addition by Lemma (2.5),

limn→∞γn≤limn→∞γϕ(n)+1=0

Hence, xn converges strongly to q. This proves our theorem.

Example 4.1. Let H=ℝ, the set of real numbers and f:ℝ→ℝ be contraction mapping and M:ℝ⇉ℝ be a set-valued map. Let f(x)=x10 and M={x5}∀x∈ℝ, then we calculate resolvent operator JλM and Cayley operator ℭλM for λ=1 as

JλM(x)=[I+λM]−1(x)=5x6.ℭλM(x)=[2JλM(x)−I]=2x3,

let S:ℝ→ℝ be defined as S(x)=x and αn=1n, βn=12n, λn=1n+3 and θn=1(n+1)2.

All the assumptions of Theorem 3.1 are satisfied and algorithm 3.1 reduces to

wn=xn+θn(xn−xn−1)yn=JλnM(I−λnℭ)wnzn=yn−λn(ℭyn−ℭwn)xn+1=1nf(xn)−2n−32nxn+12nS(zn).

The iterative sequence {xn} generated in the above algorithm is converges strongly to q=0.

All of the codes have been developed in MATLAB R2021a for simplicity. We've tried for different initial points x0=x1=1,3.5,5.0, and found that the sequence {xn} converges to the solution of the problem in each case. Graph of convergence is depicted in the fig.1.