# A New Iterative Algorithm for General Variational Inequality Problem with Applications

Authors: Aysha Khan; M. Akram; M. Dilshad; J. Shafi
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This study aims at investigation of a generalized variational inequality problem. We initiate a new iterative algorithm and examine its convergence analysis. Using this newly proposed iterative method, we estimate the common solution of generalized variational inequality problem and fixed points of a nonexpansive mapping. A numerical example is illustrated to verify our existence result. Further, we demonstrate that the considered iterative algorithm converges with faster rate than normalS-iterative scheme. Furthermore, we apply our proposed iterative algorithm to estimate the solution of a convex minimization problem and a split feasibility problem.


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## Body

## 1. Introduction

All through this study, we presume thatℋ is a real Hilbert space equipped with norm ⋅ induced by inner product ⋅,⋅. Let C be a nonempty closed convex subset of ℋ and f,g:C⟶ℋ be nonlinear mappings. The generalized nonlinear variational inequality is to locate a point x∗∈ℋ such that(1)fx∗,gx∗−gy∗≥0,∀y∗∈C,gx∗,gy∗∈ℋ,which was introduced by Noor [18]. We denote the set of solutions of (1) by SolC,f,g.Ifg=I, then generalized nonlinear variational inequality (1) reduces to the classical variational inequality studied by Stampacchia [23], which is to allocate a point x∗∈ℋ, such that(2)fx∗,y∗−x∗≥0,∀y∗∈ℋ.IfC∗=x∗∈ℋ:x∗,y∗≥0,∀y∗∈C is a dual cone of a convex cone C, then generalized nonlinear variational inequality (1) coincides to generalized nonlinear complementarity problem which is to locate a point x∗∈ℋ such that(3)fx∗,gx∗=0,gx∗∈C,fx∗∈C∗.It is worthy to adduce that variational inequalities which are unconventional and remarkable augmentation of variational principles provide well organized unified framework for figuring out a wide range of nonlinear problems arising in optimization, economics, physics, engineering science, operations research, and control theory, for example, [2, 8, 15, 20, 21, 24, 26, 33] and references cited therein.Next, we recall the following definitions of a nonlinear mappingf:C⊂ℋ⟶ℋ.Definition 1.
The mappingf:C⟶ℋ is said to be(i)
a-inverse strongly monotone or cocoercive if there exists a constant a>0, such that(4)fx−fy,x−y≥afx−fy2,∀x,y∈C,(ii)
L-Lipschitz continuous if there exists a constant L>0, such that(5)fx−fy≤Lx−y,∀x,y∈C.ForL=1, f is nonexpansive, and if 0<L<1, then f is a contraction. Note that a-inverse strongly monotone mapping is 1/a-Lipschitz continuous.It is customary to mention that variational inequalities, variational inclusions, and related optimization problems can be posed as fixed-point problems. This unusual formulation plays a dominant role in studying variational inequalities and nonlinear problems by employing fixed-point iterative methods.Lemma 1.
LetpC:ℋ⟶C be a projection mapping of ℋ onto C. For a given q∈ℋ,p∈C satisfies the inequality(6)p−q,r−p≥0,∀r∈Cif and only ifp=pCq.Note that the projection mappingpC is nonexpansive [16]. For more details on projection mapping pC, we refer to [12]. By utilizing Lemma 1, the generalized nonlinear variational inequality (1) can be designed as a fixed-point problem as follows:Lemma 2 (see [17]).
LetpC:ℋ⟶C be a projection mapping. For any ρ>0, x∗∈ℋ,gx∗∈C solves the generalized nonlinear variational inequality (1) if and only if(7)gx∗=pCgx∗−ρfx∗.Relation (7) can be rescripted as(8)x∗=x∗−gx∗+pCg−ρfx∗.LetT be a nonexpansive mapping and FT denotes the set of fixed points of T. If x∗∈FT∩Solℋ,f,g, then(9)x∗=Tx∗=x∗−gx∗+pCg−ρfx∗=Tx∗−gx∗+pCgx∗−ρfx∗,ρ>0.It is significant to achieve better rate of convergence if two or more iterative algorithms converge to the same point for a given problem. We recall the following concepts which are versatile tools to find finer convergence rate for different iterative methods.Definition 2.
(see [3]). Let pn and qn be two real sequences converging to p and q, respectively. Suppose that limn⟶∞pn−p/qn−q=l exists. Then,(i)
pn converges faster than qn if l=0(ii)
pn and qn converges with identical rates if 0<l<∞Definition 3 (See [3]).
Letpn and qn be two real sequences converging to the same fixed point t. If un and vn are two sequences of positive real numbers converging to 0 such that pn−t≤un and qn−t≤vn for all n∈ℕ. Then, pn converges to t faster than qn if un converges faster than vn.Lemma 3 (see [4]).
Letφn and ψn be nonnegative sequences of real numbers satisfying(10)φn+1≤σφn+ψn,∀n∈ℕ,where σ∈0,1 and limn⟶∞ψn=0. Then limn⟶∞φn=0.Lemma 4 (see [31]).
Letϕn,φn, and ψn be nonnegative sequences of real numbers satisfying(11)ϕn+1≤1−φnϕn+ψn,∀n∈ℕ,where φn∈0,1,∑n=1∞φn=∞, and ψn=oφn. Then, limn⟶∞ϕn=0.Mann, Ishikawa, and Halpern iterative methods are fundamental tools for solving fixed-point problems of nonexpansive mappings. In recent past, a number of fixed point iterative methods have been constructed and implemented to solve various classes of nonlinear mappings [2, 9, 10, 19, 22, 25, 28–30, 34]. Agarwal and others [1] introduced the S-iteration method which converges faster than some well-known iterative algorithms such as Mann, Ishikawa, and Picard for contraction as well as nonexpansive mappings. Due to the super convergence rate, it attracted number of researchers to study fixed-point problems, minimization problems, variational inclusions, variational inequalities, and alternate points problems in different settings. In [18], Noor utilized formulation (9) to propose following iterative algorithm:(12)u0∈C,un+1=1−anun+anTun−gun+pCgun−ρfun,where an is a sequence in 0,1. The author proved strong convergence of the proposed iterative algorithm. Furthermore, it is customary that the normal S-iterative algorithm converges faster than the Mann and Picard iterative algorithm. Owing to its uncomplicated nature and faster convergence rate, Gursoy and others [14] investigated the following normal S-iterative algorithm to examine (1) as follows:(13)p0∈C,pn+1=Tqn−gqn+pCgqn−ρfqn,qn=1−ξnpn+ξnTpn−gpn+pCgpn−ρfpn,ξn∈0,1.Recently, Ullah and Arshad [27] introduced a more efficient iterative algorithm called the M-iterative method for Suzuki’s generalized nonexpansive mappings as follows:(14)u1∈C,wn=1−anun+anTun,vn=Twn,un+1=Tvn,where an is a sequence in 0,1. They analyzed convergence and showed that their iterative procedure converges faster than the Picard S [13] and S-iteration process [1]. In recent work, Garodia and Uddin [11] developed a new iterative algorithm for Suzuki’s generalized nonexpansive mappings as follows:(15)u1∈C,wn=Tun,vn=T1−anwn+anTwn,un+1=Tvn,where an is a sequence in 0,1. The authors approximated fixed-points and inspected the convergence. Also, they proved that the posed iterative method converges with faster rate than that of the M-iterative method.Stimulated by the work discussed in above-mentioned references, in this study, we investigate algorithm (15) to estimate the common solution of fixed points of a nonexpansive mapping T and the generalized nonlinear variational inequality (1) as follows:(16)u1∈C,wn=Tun−gun+pCgun−ρfun,rn=1−anwn+anTwn−gwn+pCgwn−ρfwn,vn=Trn−grn+pCgrn−ρfrn,un+1=Tvn−gvn+pCgvn−ρfvn,where an is a sequence in 0,1 satisfying certain assumptions. We analyze strong convergence of our proposed iterative algorithm (16) under some mild assumptions. We also pose a modified form of our iterative algorithm (16) to investigate convex optimization and split feasibility problems. Theoretical findings are validated by an illustrative numerical example. Our existence and convergence results can be seen as generalizations and prevalent of some known results.


## 2. Convergence Results

Theorem 1.
Letf,g:C⟶ℋ be a1, a2-inverse strongly monotone mappings, respectively, and T:ℋ⟶C be a nonexpansive mapping such that FT∩SolC,f,g≠ϕ. Suppose that the assumption(17)ρ−a1<a11−Υ,holds, where Υ=2a2−1/a2. Then, the iterative sequence un defined by (16) converges strongly to u∗∈FT∩SolC,f,g with the following error estimates:(18)un+1−u∗≤ζ3n+1u0−u∗∏k=0n1−ak1−ζ,∀n∈ℕ,where(19)ζ=2a2−1a2+a1−ρa1.Proof.
Note that(20)u∗=Tu∗−gu∗+pCgu∗−ρfu∗.
Sincef being a1-inverse strongly monotone is 1/a1-Lipschitz continuous mapping, T and pC are the nonexpansive mappings. Then, from (16 and 20), we obtain(21)wn−u∗=Tun−gun+pCgun−ρfun−Tu∗−gu∗+pCgu∗−ρfu∗≤2un−u∗−gun−gu∗+un−u∗−ρfun−fu∗.
Sincef is a1-inverse strongly monotone mapping, then we have(22)un−u∗−ρfun−fu∗2=un−u∗2+ρ2fun−fu∗2−2ρun−u∗,fun−fu∗≤un−u∗2+ρ2a12un−u∗2−2ρa1fun−fu∗2≤a1−ρa12un−u∗2.
Also,g is a2-inverse strongly monotone mapping; then we have(23)un−u∗−gun−gu∗2=un−u∗2+gun−gu∗2−2un−u∗,gun−gu∗≤un−u∗2+1a22un−u∗2−2a2gun−gu∗2≤a2−1a22un−u∗2.
Thus, from (21) to (23), we have(24)wn−u∗≤2a2−1a2+a1−ρa1un−u∗=ζun−u∗,where ζ is defined by (19), and from (17), we have ζ<1. Again, following the same steps (21)–(24) and from (16), we obtain(25)vn−u∗≤ζrn−u∗.
Next, we estimate(26)rn−u∗=1−anwn+anTwn−gwn+pCgwn−ρfwn−u∗≤1−anwn−u∗+anζwn−u∗=1−an1−ζwn−u∗,which amounts to say(27)vn−u∗≤ζ1−an1−ζwn−u∗,un+1−u∗=Tvn−gvn+pCgvn−ρfvn−Tu∗−gu∗+pCgu∗−ρfu∗≤ζvn−u∗≤ζ31−an1−ζun−u∗.
Since,1−an1−ζ<1. Therefore, we get un+1−u∗≤ζ3un−u∗,∀n∈ℕ. By repeating the process in this fashion, we obtain(28)un+1−u∗≤ζ3n+1u0−u∗,∀n∈ℕ,which gives that limn⟶∞un−u∗=0.Now, we exemplify the existence of solution.Example 1.
Letℋ=ℝ,C=1,2 be equipped with norm u=u and inner product u,v=u.v. Let f,g,T:1,2⟶ℝ be defined by(29)fu=u2,gu=u34+34,Tu=u2+u316+78.
Then, for allu,v∈C, observe that(30)fu−fv,u−v=u−v2u+v≥2u−v2,gu−gv,u−v=14u−v2u2+uv+v2≥34u−v2,Tu−Tv=116u−vu2+uv+v2+u+v≤u−v.
Then,f and g are 2 and 3/4-inverse strongly monotone mapping, respectively, and T is nonexpansive mapping. One can easily verify that u∗=1∈C is the unique fixed point of T. Also,(31)fu∗,gv−gu∗=v3−14≥0,for allv∈C.
Thus, we haveu∗=1∈FT∩SolC,f,g.Theorem 2.
Letℋ be a real Hilbert space and C be a nonempty closed convex subset of ℋ. Let f,g,T, and ζ be same as defined in Theorem 1. Let pn and un be the sequences defined by (13) and (16), respectively. Suppose that (17) holds and FT∩C,f,g≠ϕ. Then, the following statements hold:(i)
If1+ζ3/ξn is bounded and ∑n=0∞an=∞, then the sequence un−pn converges strongly to 0 with following error estimates:(32)un+1−pn+1≤1−ξn1−ζun−pn+1+ζ3un−u∗,∀n∈ℕ.pn converges strongly to u∗∈FT∩SolC,f,g.(ii)
Ifpn converges strongly to u∗∈FT∩SolC,f,g, then pn−un converges strongly to 0 with following error estimates:(33)pn+1−un+1≤ζ3pn−un+1+ζ3pn−u∗,∀n∈ℕ.Proof.
(i)
It follows from Theorem1 that limn⟶∞un−u∗=0. Next, we prove that limn⟶∞pn−u∗=0. Following (13) and (16) and the steps as in (21)–(24), we obtain(34)un+1−pn+1=Tvn−gvn+pCgvn−ρfvn−Tqn−gqn+pCgqn−ρfqn≤ζvn−qn,whereζ is same as in (19). Again, utilizing (13), (16), and (34), we have(35)un+1−pn+1≤ζvn−1−ξnpn−ξnTpn−gpn+pCgpn−ρfpn≤ζvn−u∗+1−ξnζpn−u∗+ξnζTpn−gpn+pCgpn−ρfpn−Tu∗−gu∗+pCgu∗−ρfu∗≤ζvn−u∗+1−ξnζpn−u∗+ξnζ2pn−u∗=ζvn−u∗+1−ξn1−ζpn−u∗≤ζζ21−an1−ζun−u∗+1−ξn1−ζpn−u∗=ζζ21−an1−ζun−u∗+1−ξn1−ζun−u∗+1−ξn1−ζun−pn≤ζ1−ξn1−ζun−pn+1+ζ3max1−an1−ζ,1−ξn1−ζun−u∗≤1−ξn1−ζun−pn+1+ζ3un−u∗.Letϕn=un−pn,φn=ξn1−ζ,ψn=1+ζ3un−u∗, and δn=un−u∗,∀n∈ℕ. It follows from assumption of the theorem that 1+ζ3/ξn is bounded; therefore, 1+ζ3/ξn1−ζ is also bounded. Then, there exists a constant M>0, such that 1+ζ3/ξn1−ζ<M,∀n∈ℕ. Since limn⟶∞δn=0 and 1+ζ3/ξn1−ζ is bounded, therefore, 1+ζ3/ξn1−ζδn⟶0 as n⟶∞, i.e., limn⟶∞ψn/φn=0, which amounts to say that ψn=oφn. Thus, all the assumptions of Lemma 4 are fulfilled. Hence, limn⟶∞un−pn=0 and pn−u∗≤un−pn+un−u∗. Thus, we have limn⟶∞pn−u∗=0.(ii)
Next, we estimate thatpn−un⟶0. Since pn converges to u∗∈FT∩SolC,f,g, then following the same arguments as in (34) and (35), we obtain(36)pn+1−un+1≤ζ1−ξn1−ζpn−u∗+ζζ21−an1−ζun−u∗≤ζ31−an1−ζpn−un+ζ31−an1−ζpn−u∗+ζ1−ξn1−ζpn−u∗≤ζ31−an1−ζpn−un+ζ3pn−u∗+pn−u∗≤ζ3pn−un+1+ζ3pn−u∗.
Letϕn′=pn−un,ψn′=1+ζ3pn−u∗,∀n∈ℕ. By the assumption pn converges to u∗ and utilizing the fact that 1+ζ3 is bounded, we obtain that ψn′⟶0 as n⟶∞. Thus, all the assumptions of Lemma 3 are fulfilled. Hence, limn⟶∞pn−un=0. Also, we know that un−u∗≤pn−un+pn−u∗,∀n∈ℕ. Thus, limn⟶∞un−u∗=0. Hence, pn−un⟶0 as n⟶∞.Theorem 3.
Letℋ be a real Hilbert space and C be a closed convex subset of ℋ. Suppose f,g,T, and ζ are identical as in Theorem 1. Let pn and un be sequences defined by (13) and (16), respectively. Suppose that assumption (17) holds and FT∩SolC,f,g≠ϕ. If p0=u0, then un converges faster than pn to u∗, such that u∗∈FT∩SolC,f,g.Proof.
It follows from (27) that(37)un+1−u∗≤ζ31−an1−ζun−u∗.
Sincean is a sequence in 0,1, we can choose a constant a∈ℝ, such that 0<a≤an<1,∀n∈ℕ. Then,(38)un+1−u∗≤ζ31−a1−ζun−u∗.
By repeating the process, we obtain(39)un+1−u∗≤ζ3n+11−a1−ζn+1u0−u∗,∀n∈ℕ.
Also, it follows from (13) that(40)pn+1−u∗=Tqn−gqn+pCgqn−ρfqn−Tu∗−gu∗+pCgu∗−ρfu∗≤qn−u∗−gqn−gu∗+gqn−gu∗−ρfqn−fu∗≤2qn−u∗−gqn−gu∗+qn−u∗−ρfqn−fu∗.
By following the arguments as discussed from (21)to(24), we have(41)pn+1−u∗≤ζqn−u∗.
Also,(42)qn−u∗=1−ξnpn+ξnTpn−gpn+pCgpn−ρfpn−u∗≤1−ξnpn−u∗+2ξnpn−u∗−gpn−gu∗+ξnpn−u∗−ρfpn−fu∗≤1−ξnpn−u∗+ξnζpn−u∗=1−ξn1−ζpn−u∗.
By combining (41) and (42), we get(43)pn+1−u∗≤ζ1−ξn1−ζpn−u∗.
Sinceξn is a sequence in 0,1, we can choose a constant ξ∈ℝ, such that 0<ξ≤ξn<1,∀n∈ℕ. Then,(44)pn+1−u∗≤ζ1−ξ1−ζpn−u∗.
Thus, by repeating the process, we obtain(45)pn+1−u∗≤ζn+11−ξ1−ζn+1p0−u∗,∀n∈ℕ,
Setan=ζ3n+11−a1−ζn+1u0−u∗ and bn=ζn+11−ξ1−ζn+1p0−u∗; then,(46)An=anbn=ζ3n+11−a1−ζn+1u0−u∗ζn+11−ξ1−ζn+1p0−u∗⟶0asn⟶∞.
Hence,un converges faster that pn.


## 3. Applications

### 3.1. Convex Minimization Problem

Now, we solve convex minimization problem as an application of Theorem1.LetC be a closed convex subset of a real Hilbert space ℋ, ρC:ℋ⟶C be a projection, and F:C⟶ℝ be a convex, Frechet differentiable mapping. We consider the following convex minimization problem:(47)minu∗∈CFu∗.Clearly,u∗∈C is a solution of pCI−ρ∇F if and only if(48)∇Fu∗,u−u∗≥0,∀u∈C.More precisely,u∗∈C solves problem (47) if and only if u∗ is a fixed point of the projection mapping pCI−ρ∇F, i e.,(49)u∗=ρCu∗−ρ∇Fu∗,where ∇F is the gradient of mapping F. This formulation is known as gradient projection, which plays a key role in solving problem (47). So far, several iterative methods have been employed to solve minimization problems [7, 26, 32]. By considering f:=∇F and assuming T=g=I, the identity mapping, we propose the following modified gradient projection algorithm for solving pCI−ρ∇F as follows:(50)u1∈C,wn=pCun−ρ∇Fun,rn=1−anwn+anpCwn−ρ∇Fwn,vn=pCrn−ρ∇Frn,un+1=pCvn−ρ∇Fvn,where an is a sequence in 0,1. Now, we approximate the proposed algorithm (50) to estimate the solution of (47).Theorem 4.
LetC be a nonempty closed convex subset of real Hilbert space ℋ. Let F:C⟶ℝ be a convex, Freschet differentiable mapping, and ∇F is a-inverse strongly monotone mapping. Suppose that the convex minimization problem (47) has a solution and condition (17) holds. Then, the sequence un generated by (50) converges strongly to u∗ which solves convex minimization problem (47) with the following error estimates:(51)un+1−u∗≤ζ3n+1∏k=0n1−ak1−ζu0−u∗,∀n∈ℕ,where(52)ζ=a1−ρa1.Proof.
The desired conclusion is accomplished by takingf=∇F and T,g=I in Theorem 1.Example 2.
Letℋ=L20,1=G:0,1⟶ℝ:∫01G2udu<∞. Then, ℋ,⋅2 is a Hilbert space given by(53)Gu22=Gu,Gu=∫01G2udu.
Consider a closed convex subsetC=G∈L20,1:Gu22≤1 of ℋ. Define F:C⟶ℝ by FG=Gu22. Then, Gu=0 is a unique minimum of a convex function f, and f is the Frechet differentiable at G. The gradient ∇F:C⟶ℋ is evaluated as ∇FG=2G. Then, for all G1,G2∈C, we get(54)∇FG1−∇FG2,G1−G2=∫012G1u−2G2uG1−G2du=2∫01G1u−G2u2du≥−14∫012G1u−2G2u2du=−14∇FG1−∇FG222,i.e., ∇F is 1/4 inverse strongly monotone. Also, ζ<1 for ρ=1/4. Thus, all the assumptions of Theorem 4 are satisfied, and for an=1/n+1, the sequence un generated by (50) is given as(55)u0∈C,wn=pC12un,rn=1−1n+1wn+1n+1pC12wn,vn=pC12rn,un+1=pC12vn,where pC=G,G∈C,G/G,G∉C. Then, the sequence un generated by (50) converges to 0 function.


### 3.2. Split Feasibility Problem

This subsection is devoted to utilization of Theorem1 to examine a split feasibility problem SFP. Let C1 and C2 be nonempty closed convex subsets of real Hilbert spaces ℋ1 and ℋ2, respectively. Let A:ℋ1⟶ℋ2 be a bounded linear operator. The SFP is to locate a point u∗, such that(56)u∗∈C1:Au∗∈C2.LetΓ denotes the solution set of SFP (56); then,(57)Γ≕u∗∈C1:Au∗∈C2=C1∩A−1C2.A class of inverse problems has been solved by usingSFP, for example, [6]. In [32], Xu established the relationship between SFP (56) and the fixed point of problem pC1I−ρA∗I−pC2A. More precisely, for ρ>0, u∗∈C1 solves SFP (56) if and only if pC1I−ρA∗I−pC2Au∗=u∗. Byrne [5] posed the following iterative algorithm for solving SFP (56) as follows:(58)un+1=pC1I−ρA∗I−pC2Aun,∀n≥0,where 0<ρ<2/A2, A∗ is the adjoint of operator A, and pC1 and pC2 are the projections onto C1 and C2, respectively.Note that the operatorpC1I−ρA∗I−pC2A with 0<ρ<2/A2 is nonexpansive. Now, we propose following iterative algorithm to solve SFP (56):(59)u1∈C1,wn=pC1I−ρA∗I−pC2Aun,rn=1−anwn+anpC1I−ρA∗I−pC2Awn,vn=pC1I−ρA∗I−pC2Arn,un+1=pC1I−ρA∗I−pC2Avn,where an is a sequence in 0,1 and 0<ρ<2/A2.Theorem 5.
Suppose thatΓ≠ϕ and condition (17) holds. Then, the sequence un initiated in (59) converges weakly to u∗, which solves SFP (56) with following error estimates:(60)un+1−u∗≤ζ3n+1∏k=0n1−ak1−ζu0−u∗,∀n∈ℕ,where(61)ζ=a1−ρa1.Proof.
The desired conclusion follows by taking∇F=A∗I−pC2A and T,g=I in Theorem 1.


## 3.1. Convex Minimization Problem

Now, we solve convex minimization problem as an application of Theorem1.LetC be a closed convex subset of a real Hilbert space ℋ, ρC:ℋ⟶C be a projection, and F:C⟶ℝ be a convex, Frechet differentiable mapping. We consider the following convex minimization problem:(47)minu∗∈CFu∗.Clearly,u∗∈C is a solution of pCI−ρ∇F if and only if(48)∇Fu∗,u−u∗≥0,∀u∈C.More precisely,u∗∈C solves problem (47) if and only if u∗ is a fixed point of the projection mapping pCI−ρ∇F, i e.,(49)u∗=ρCu∗−ρ∇Fu∗,where ∇F is the gradient of mapping F. This formulation is known as gradient projection, which plays a key role in solving problem (47). So far, several iterative methods have been employed to solve minimization problems [7, 26, 32]. By considering f:=∇F and assuming T=g=I, the identity mapping, we propose the following modified gradient projection algorithm for solving pCI−ρ∇F as follows:(50)u1∈C,wn=pCun−ρ∇Fun,rn=1−anwn+anpCwn−ρ∇Fwn,vn=pCrn−ρ∇Frn,un+1=pCvn−ρ∇Fvn,where an is a sequence in 0,1. Now, we approximate the proposed algorithm (50) to estimate the solution of (47).Theorem 4.
LetC be a nonempty closed convex subset of real Hilbert space ℋ. Let F:C⟶ℝ be a convex, Freschet differentiable mapping, and ∇F is a-inverse strongly monotone mapping. Suppose that the convex minimization problem (47) has a solution and condition (17) holds. Then, the sequence un generated by (50) converges strongly to u∗ which solves convex minimization problem (47) with the following error estimates:(51)un+1−u∗≤ζ3n+1∏k=0n1−ak1−ζu0−u∗,∀n∈ℕ,where(52)ζ=a1−ρa1.Proof.
The desired conclusion is accomplished by takingf=∇F and T,g=I in Theorem 1.Example 2.
Letℋ=L20,1=G:0,1⟶ℝ:∫01G2udu<∞. Then, ℋ,⋅2 is a Hilbert space given by(53)Gu22=Gu,Gu=∫01G2udu.
Consider a closed convex subsetC=G∈L20,1:Gu22≤1 of ℋ. Define F:C⟶ℝ by FG=Gu22. Then, Gu=0 is a unique minimum of a convex function f, and f is the Frechet differentiable at G. The gradient ∇F:C⟶ℋ is evaluated as ∇FG=2G. Then, for all G1,G2∈C, we get(54)∇FG1−∇FG2,G1−G2=∫012G1u−2G2uG1−G2du=2∫01G1u−G2u2du≥−14∫012G1u−2G2u2du=−14∇FG1−∇FG222,i.e., ∇F is 1/4 inverse strongly monotone. Also, ζ<1 for ρ=1/4. Thus, all the assumptions of Theorem 4 are satisfied, and for an=1/n+1, the sequence un generated by (50) is given as(55)u0∈C,wn=pC12un,rn=1−1n+1wn+1n+1pC12wn,vn=pC12rn,un+1=pC12vn,where pC=G,G∈C,G/G,G∉C. Then, the sequence un generated by (50) converges to 0 function.


## 3.2. Split Feasibility Problem

This subsection is devoted to utilization of Theorem1 to examine a split feasibility problem SFP. Let C1 and C2 be nonempty closed convex subsets of real Hilbert spaces ℋ1 and ℋ2, respectively. Let A:ℋ1⟶ℋ2 be a bounded linear operator. The SFP is to locate a point u∗, such that(56)u∗∈C1:Au∗∈C2.LetΓ denotes the solution set of SFP (56); then,(57)Γ≕u∗∈C1:Au∗∈C2=C1∩A−1C2.A class of inverse problems has been solved by usingSFP, for example, [6]. In [32], Xu established the relationship between SFP (56) and the fixed point of problem pC1I−ρA∗I−pC2A. More precisely, for ρ>0, u∗∈C1 solves SFP (56) if and only if pC1I−ρA∗I−pC2Au∗=u∗. Byrne [5] posed the following iterative algorithm for solving SFP (56) as follows:(58)un+1=pC1I−ρA∗I−pC2Aun,∀n≥0,where 0<ρ<2/A2, A∗ is the adjoint of operator A, and pC1 and pC2 are the projections onto C1 and C2, respectively.Note that the operatorpC1I−ρA∗I−pC2A with 0<ρ<2/A2 is nonexpansive. Now, we propose following iterative algorithm to solve SFP (56):(59)u1∈C1,wn=pC1I−ρA∗I−pC2Aun,rn=1−anwn+anpC1I−ρA∗I−pC2Awn,vn=pC1I−ρA∗I−pC2Arn,un+1=pC1I−ρA∗I−pC2Avn,where an is a sequence in 0,1 and 0<ρ<2/A2.Theorem 5.
Suppose thatΓ≠ϕ and condition (17) holds. Then, the sequence un initiated in (59) converges weakly to u∗, which solves SFP (56) with following error estimates:(60)un+1−u∗≤ζ3n+1∏k=0n1−ak1−ζu0−u∗,∀n∈ℕ,where(61)ζ=a1−ρa1.Proof.
The desired conclusion follows by taking∇F=A∗I−pC2A and T,g=I in Theorem 1.


## 4. Conclusion

In this study, a new iterative algorithm (16) has been proposed and employed to explore convergence analysis. Using this newly constructed iterative procedure, a common solution of the generalized variational inequality problem and fixed points of nonexpansive mapping is investigated, and theoretical findings are verified by a numerical example. Furthermore, we have shown that our iteration algorithm converges faster than the normal S-iteration process for contraction mapping. Finally, we applied our newly constructed iterative algorithm to investigate the convex optimization problem and split feasibility problem.



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