# Strong Convergence of an Inertial Iterative Algorithm for Generalized Mixed Variational-like Inequality Problem and Bregman Relatively Nonexpansive Mapping in Reflexive Banach Space

Authors: Saud Fahad Aldosary; Watcharaporn Cholamjiak; Rehan Ali; Mohammad Farid
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we consider a generalized mixed variational-like inequality problem and prove a Minty-type lemma for its related auxiliary problems in a real Banach space. We prove the existence of a solution of these auxiliary problems and also prove some properties for the solution set of generalized mixed variational-like inequality problem. Furthermore, we introduce and study an inertial hybrid iterative method for solving the generalized mixed variational-like inequality problem involving Bregman relatively nonexpansive mapping in Banach space. We study the strong convergence for the proposed algorithm. Finally, we list some consequences and computational examples to emphasize the efficiency and relevancy of the main result.


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## Body

## 1. Introduction

Throughout the paper, unless otherwise stated, letX be a reflexive Banach space with X∗ as its dual and C≠∅ be the closed convex subset of X. In this paper, we consider the generalized mixed variational-like inequality problem (in brief, GMVLIP): find u∈C such that(1)Gv,u;u+bu,v−bu,u≥0,∀v∈C,where b:C×C⟶ℝ and G:C×C×C⟶ℝ, be bifunction and trifunction, respectively, and ℝ be the set of real numbers. Sol (GMVLIP equation (1)) stands for the solution of equation (1). If b≡0, GMVLIP equation (1) is reduced to GVLIP: find u∈C such that(2)Gv,u;u≥0,∀v∈C,which is introduced by Preda et al. [1] (see, for instance, [2, 3]).If we setGv,u;u=Du+Au,ηv,u, where D,A:C⟶X and η:C×C⟶X, GMVLIP equation (1) is reduced to MVLIP (see for details [4]).Further, if we setGv,u;u=Du,ηv,u and b≡0, GMVLIP equation (1) is reduced to VLIP: find u∈C such that(3)Du,ηv,u≥0,∀v∈C,which is presented by Parida et al. [5].Moreover, ifηv,u=v−u, VLIP is reduced to VIP: find u∈C such that(4)Du,v−u≥0,∀v∈C,which is introduced by Hartmann and Stampacchia [6].Ifb≡0, X=ℝn, and Gv,u;u=▽Du,ηv,u, where η is continuous and D is differentiable and η-convex, GMVLIP equation (1) is reduced to mathematical programming problem as [5](5)minu∈CDu.Korpelevich [7] proposed the iterative method for VIP in 1976 on Hilbert space H as(6)u0∈C⊆H,vn=projCun−σDun,un+1=projCun−σDvn,n≥0,where σ>0, projC denotes projection of H onto C, and D is monotone and Lipschitz continuous mapping. This method is called the extragradient iterative method.Nadezkhina and Takahashi [8] proposed a hybrid extragradient algorithm involving nonexpansive mapping T on C and studied the convergence analysis in 2006 as(7)u0∈C⊆H,xn=projCun−σnDun,vn=αnun+1−αnTprojCun−σnDxn,Cn=z∈C:vn−z2≤un−z2,Dn=z∈C:un−z,u0−un≥0,un+1=projCn∩Dnu0,n≥0.The idea considered in [8] has been generalized in [9] from Hilbert to Banach space X as(8)u0∈C⊆X,vn=J−1αnJun+1−αnJTun,Cn=z∈K:ϕz,vn≤ϕz,un,Dn=z∈K:un−z,Ju0−Jun≥0,un+1=∏Cn∩Dnu0,where ΠC denotes generalized projection of X onto C, ϕ is the Lyapunov function such that ϕu,v=v2−2v,Ju+u2,∀u,v∈X, and J:X⟶2X∗ is the normalized duality mapping with J−1 being its inverse. For further work, see [10–17].In 1967, an important technique was discovered by Bregman [18] in the light of Bregman distance function. This technique is very useful not only in design and interpretation of the iterative method but also to solve optimization and feasibility problems and to approximate equilibria, fixed point, variational inequalities, etc. (for details [19–22]).In 2010, Reich and Sabach [23] introduced iterative algorithm on Banach space involving maximal monotone operators. In the light of Bregman projection, there were various iterative algorithms studied by researchers in this field (see, for instance, [19, 24–28]).In 2008, Maingé [29] developed and studied an inertial Krasnosel’skiǐ–Mann algorithm as(9)tn=un+θnun−un−1,un+1=1−αntn+αnTtn.For further work, see [30–39].Inspired by the work in [2, 27, 29], we establish an inertial hybrid iterative algorithm involving Bregman relatively nonexpansive mapping to find a common solution of GMVLIP equation (1) and a fixed-point problem in Banach space. Moreover, we study the convergence analysis for the main result. At last, we list some consequences and computational example to emphasize the efficiency and relevancy of the main result.


## 2. Preliminaries

Assumeg:X⟶−∞,+∞ is a proper, convex, and lower semicontinuous mapping and g∗:X∗⟶−∞,+∞ is a Fenchel conjugate of g, defined as(10)g∗u0=supu0,u−gu:u∈Y,u0∈Y∗.And, for any w∈intdomg, interior of the domain of g and u∈X, the right-hand derivative of g at w in the direction u is(11)g0w,u=limλ⟶0+gw+λu−gwλ.A mappingg is called Gateaux differentiable at w if the above limit exists. So, g0w,u agrees with ∇gw, the value of the gradient of g at w. It is called Frechet differentiable at w, if the limit is attained uniformly in u=1. It is called uniformly Frechet differentiable on C⊆X, if the above limit is attained uniformly for w∈C and u=1.The mappingg is called Legendre if the following holds [19]:(i)
intdomg≠∅, g is Gateaux differentiable on intdomg, and dom∇g=intdomg(ii)
intdomg∗≠∅, g∗ is Gateaux differentiable on intdomg∗, and dom∇g∗=intdomg∗We have the following [19]:(i)
g be Legendre iff g∗ be Legendre mapping(ii)
∂g−1=∂g∗(iii)
∇g=∇g∗−1, ran∇g=dom∇g∗=intdomg∗, ran∇g∗=dom∇g=intdomg(iv)
The mappingsg and g∗ are strictly convex on intdomg and intdomg∗Definition 1 (see [18]).
Letg:Y⟶−∞,+∞ be Gateaux differentiable and convex and Dg:domg×intdomg⟶0,+∞ such that(12)Dgu,w=gu−gw−∇gw,u−w,w∈intdomg,u∈domg,is known as Bregman distance with respect to g.
We notice that the Bregman distance is not a distance in the usual sense of term. Obviously,Dgw,w=0, but Dgw,u=0 may not imply w=u. It holds if g is the Legendre function. However, Dg is neither symmetric nor satisfy the triangle inequality. We have the following important properties of Dg [40] for u,u1,u2∈domg and w1,w2∈intdomg.(i)
Two-point identity:(13)Dgw1,w2+Dgw2,w1=∇gw1−∇gw2,w1−w2.(ii)
Three-point identity:(14)Dgu,w1+Dgw1,w2−Dgu,w2=∇gw2−∇gw1,u−w1.(iii)
Four-point identity:(15)Dgu1,w1−Dgu1,w2−Dgu2,w1+Dgu2,w2=∇gw2−∇gw1,u1−u2.Definition 2 (see [23, 25]).
LetT:C⟶intdomg be a mapping and FT=u∈C:Tu=u, where FT is the set of fixed points of T. Then, we have the following:(i)
A pointu0∈C is called an asymptotic fixed point if C contains a sequence un with un⇀u0 such that limn⟶∞Tun−un=0. We represent F^T as the set of asymptotic fixed points of T.(ii)
T is called Bregman quasi-nonexpansive if(16)FT≠∅;Dgu0,Tu≤Dgu0,u,∀u∈C,u0∈FT.(iii)
T is called Bregman relatively nonexpansive if(17)FT=F^T≠∅;Dgu0,Tu≤Dgu0,u,∀u∈C,u0∈FT.(iv)
T is called Bregman firmly nonexpansive if ∀u1,u2∈C,(18)∇gTu1−∇gTu2,Tu1−Tu2≤∇gu1−∇gu2,Tu1−Tu2,or, correspondingly,(19)DgTu1,Tu2+DgTu2,Tu1+DgTu1,u1+DgTu2,u2≤DgTu1,u2+DgTu2,u1.Example 1 (see [26]).
LetA:X⟶2X∗ be a maximal monotone mapping. If A−10≠∅ and the Legendre function g:X⟶−∞,+∞ is bounded on bounded subsets of X and uniformly Frechet differentiable, then the resolvent with respect to A,(20)resAgu=∇g+A−1°∇gu,is a single-valued, closed, and Bregman relatively nonexpansive mapping from X onto DA and FresAg=A−10.Definition 3 (see [18]).
Letg:X⟶−∞,+∞ be a Gateaux differentiable and convex function. The Bregman projection of w∈intdomg onto C⊂intdomg is a unique vector projCgw∈C with(21)DgprojCgw,w=infDgu,w:u∈C.Remark 1 (see [24]).
(i) IfX is a smooth Banach space and gu=1/2u2, ∀u∈X, then the Bregman projection projCgu reduces to ΠCu, generalized projection (see [41]), and it is defined as(22)ϕΠCu,u=minv∈Cϕv,u,where ϕ is a Lyapunov function. (ii) If X is a Hilbert space and gu=1/2u2, ∀u∈X, then projCgu reduces to the metric projection of u onto C.
For allr>0, assume Br:=z∈X:z≤r. Then, a map g:X⟶ℝ is said to be uniformly convex on bounded subsets of X, if ρrt>0, ∀t>0, where ρr:0,+∞⟶0,+∞ is defined as(23)ρrt=infw,v∈Br,w−v=t,α∈0,1αgw+1−αgv−gαw+1−αvα1−α,∀t≥0. The function ρr is known as the gauge of uniform convexity of g. The function g is also said to be uniformly smooth on bounded subsets of X if limt⟶0σrt/t=0, for all r>0, where σr:0,+∞⟶0,+∞ is defined by(24)σrt=supw∈Br,v∈SX,α∈0,1αgw+1−αtv+1−αgw−αtv−gwα1−α,∀t≥0. The function g is said to be uniformly convex if the function δg:0,+∞⟶0,+∞, defined by(25)δgt≔sup12gw+12gv−gw+v2:v−w=t,satisfies that limt⟶0σrt/t=0.Remark 2.
LetX be a Banach space, r>0 be a constant, and g:X⟶ℝ be a convex function which is uniformly convex on bounded subsets. Then,(26)gαw+1−αv≤αgw+1−αgv−α1−αρrw−v,for all w,v∈Br and α∈0,1, where ρr is the gauge of uniform convexity of g.Definition 4 (see [20]).
Letg:X⟶−∞,+∞ be a Gateaux differentiable and convex function. Then, g is called the following:(i)
Totally convex atw∈intdomg if its modulus of total convexity at u, i.e., the mapping vg:intdomg×0,+∞⟶0,+∞ such that(27)vgw,s=infDgv,w:v∈domg,v−w=s,is positive, fors>0(ii)
Totally convex if it is totally convex at each point ofw∈intdomg(iii)
Totally convex on bounded sets ifvg:intdomg×0,+∞⟶0,+∞ such that(28)vgB,s=infvgw,s:w∈B∩domg.
By [20] (Section 1.3, p.30), we notice that any uniformly convex function is totally convex but the converse is not true. Also, by [21] (Theorem 2.10, p.9), g is totally convex on bounded sets if and only if g is uniformly convex on bounded sets.Definition 5 (see [20, 23]).
A mappingg:X⟶−∞,+∞ is called the following:(i)
Coercive iflimu⟶+∞gu/u=+∞(ii)
Sequentially consistent if for anyun,vn⊆X with un bounded,(29)limn⟶∞Dgvn,un=0⇒limn⟶∞vn−un=0.Lemma 1 (see [21]).
Letg:X⟶−∞,+∞ be a convex function with domain at least two points. Then, g is sequentially consistent iff it is totally convex on bounded sets.Lemma 2 (see [42]).
Letg:X⟶−∞,+∞ be uniformly Frechet differentiable and bounded on C⊆X, a bounded set. Then, g is uniformly continuous on C and ∇g is uniformly continuous on C from the strong topology of X to the strong topology of X∗.Lemma 3 (see [23]).
Letg:X⟶−∞,+∞ be a Gateaux differentiable and totally convex function. If u0∈X and Dgun,u0 are bounded, then un is also bounded.Lemma 4 (see [21]).
Letg:X⟶−∞,+∞ be a Gateaux differentiable and totally convex function on intdomg. Let w∈intdomg and C⊆intdomg, a nonempty closed convex set. If v∈C, then the following statements are equivalent:(i)
v∈C is the Bregman projection of w onto C with respect to g, i.e., v=projCgw(ii)
The vectorv is the unique solution of the variational inequality:(30)∇gw−∇gv,v−u≥0,∀u∈C(iii)
The vectorv is the unique solution of the inequality:(31)Dgu,v+Dgv,w≤Dgu,w,∀u∈CLemma 5 (see [25]).
Letg:X⟶−∞,+∞ be Legendre and T:C⟶C be Bregman quasi nonexpansive mapping with respect to g. Then, FT is closed and convex.Lemma 6 (see [23]).
Letg:X⟶−∞,+∞ be Gateaux differentiable and totally convex function, u0∈X, and C⊆X, a nonempty closed convex set. Suppose that un is bounded and any weak subsequential limit of un belongs to C. If Dgun,u0≤DgprojCgu0,u0, then un strongly converges to projCgu0.Lemma 7 (see [43]).
LetC be a nonempty subset of a Hausdroff topological vector space X∗ and let f:C⟶2X be a KKM mapping. If fv is closed in X∗ for all v∈C and compact for some v∈C, then ∩v∈Cfv≠∅.Definition 6 (see [1]).
A functionG:C×C×C⟶ℝ is said to be generalized relaxed α-monotone if for any u,v∈C, we have(32)Gv,u;v−Gv,u;u≥αu,v,where(33)limt⟶0αu,tv+1−tut=0.Remark 3.
(i)
IfGv,u;w=Aw,ηv,u, where η:C×C⟶X, we say that the mapping A is a generalized η-α monotone(ii)
In Definition6, let Gv,u;w=Aw,ηv,u and αu,v=βv−u, where β:C⟶R with βtw=tpβw, for t>0 and p>1, then we say that A is called a relaxed η-α monotone mapping(iii)
In case (ii), ifηv,u=v−u for all u,v∈C, then Definition 6 reduces to Av−Au,v−u≥βv−u for all u,v∈C and A is called a relaxed α-monotone mapping(iv)
In case (iii), ifβw=kwp,wherek>0 is a constant, then Definition 6 reduces to Av−Au,v−u≥kv−up for all u,v∈C and A is called a p-monotone mapping(v)
Ifα≡0, then (iii) reduces to Av−Au,v−u≥0 for all u,v∈C and A is called a monotone mapping
We construct an example for generalized relaxedα-monotone mapping as follows.Example 2.
ConsiderX=X∗, C=−∞,∞, and(34)Gv,u;w=−cwv−u,v<u,cwv−u,v≥u,where c>0 is a constant. Thus, G is generalized relaxed α-monotone with(35)αu,v=−cv−u2,v<u,cv−u2,v≥u.Assumption 1.
Letb:C×C⟶ℝ satisfy the following:(i)
b is skew-symmetric, i.e., bu,u−bu,v−bv,u+bv,v≥0,∀u,v∈C(ii)
b is convex in the second argument(iii)
b is continuous


## 3. Existence of Solutions and Resolvent Operator

Forw∈C, assume the auxiliary problems (in short, AP) related to GMVLIP equation (1): find u∈C such that(36)Gv,u;u+∇gu−∇gw,v−u+bu,v−bu,u≥0,∀v∈C,and find u∈C such that(37)Gv,u;v+∇gu−∇gw,v−u+bu,v−bu,u≥αu,v,∀v∈C.We have the Minty-type lemma as follows.Lemma 8.
Letg:X⟶−∞,+∞ be Gateaux differentiable and coercive function, and let b:C×C⟶ℝ satisfy Assumption 1 (ii). Assume G:C×C×C⟶ℝ with the following cases:(i)
Gv,u;. is hemicontinuous(ii)
G.,u;w is convex(iii)
Gu,u;w=0(iv)
G is a generalized relaxed α-monotone
Then, AP equation (36) and AP equation (37) are equivalent.Proof.
Letu∈C be a solution of AP equation (36) and by the concept of G, we obtain(38)Gv,u;v+∇gu−∇gw,v−u+bu,v−bu,u≥Gv,u;u+∇gu−∇gw,v−u+bu,v−bu,u+αu,v≥αu,v,which shows that u∈C is a solution of AP equation (37).
Conversely, letu∈C be a solution of AP equation (37). Then,(39)Gv,u;v+∇gu−∇gw,v−u+bu,v−bu,u≥αu,v,∀v∈C.
For anyv∈C, let vt=tv+1−tu,t∈0,1, and we get vt∈C. By equation (39), we have(40)Gvt,u;vt+∇gu−∇gw,vt−u+bu,vt−bu,u≥αu,vt.
Using conditions (ii) and (iii), we obtain(41)Gvt,u;vt≤tψv,u;vt+1−tψu,u;vt=tψv,u;vt.
By Assumption1 (ii), we have(42)∇gu−∇gw,vt−u=t∇gu−∇gw,v−uand(43)bu,vt≤tbu,v+1−tbu,u.
Using equations (40)–(43), we have(44)tGv,u;vt+t∇gu−∇gw,v−u+tbu,v+1−tbu,u−bu,u≥Gvt,u;vt+∇gu−∇gw,vt−u+bu,vt−bu,utGv,u;vt+t∇gu−∇gw,v−u+tbu,v−tbu,u≥αu,vt.
Hence,(45)Gv,u;vt+∇gu−∇gw,v−u+bu,v−bu,u≥αu,vtt.
Lett⟶0, and by condition (i), we obtain(46)Gv,u;u+∇gu−∇gw,v−u+bu,v−bu,u≥0.
Thus,u∈C be a solution of AP equation (95).Theorem 1.
Letg:X⟶−∞,+∞ be a Gateaux differentiable and coercive function, b:C×C⟶ℝ satisfy Assumption 1 (ii)-(iii), and α:C×C⟶ℝ be a bifunction. Consider G:C×C×C⟶ℝ and for any u,v,w∈C, assume the following:(i)
Gv,u;. is hemicontinuous(ii)
G.,u;w is convex and lower semicontinuous(iii)
Gu,v;w+Gv,u;w=0(iv)
G is a generalized relaxed α-monotone(v)
α.,v is lower semicontinuous
Then, AP equation (36) has solution.Proof.
LetFw,Gw:C⟶2C, for any w∈C, be two set-valued mappings with(47)Fwv=u∈C:Gv,u;u+∇gu−∇gw,v−u+bu,v−bu,u≥0,∀v∈C,and(48)Gwv=u∈C:Gv,u;v+∇gu−∇gw,v−u+bu,v−bu,u≥αu,v,∀v∈C.
Obviously,u¯∈C solves AP equation (36) if and only if u¯∈∩v∈CFwv. Hence, ∩v∈CFwv≠∅. Next, we prove that Fw is a KKM mapping. On the contrary, let Fw be not a KKM mapping; then, ∃v1,v2,…,vm⊂C such that cov1,v2,…,vm⊈∪i=1mFwvi; this means there exists a u0∈cov1,v2,…,vm,u0=∑i=1mtiviwhereti≥0,i=1,2,…m,∑i=1mti=1, but u0∉∪i=1mFwvi. Then,(49)Gvi,u0;u0+∇gu0−∇gw,vi−u0+bu0,vi−bu0,u0<0.
By Theorem1 (ii)-(iii), we get(50)0=Gu0,u0;u0+∇gu0−∇gw,u0−u0+bu0,u0−bu0,u0≤∑i=1mtiGvi,u0;u0+∑i=1mti∇gu0−∇gw,vi−u0+∑i=1mtibu0,vi−∑i=1mtibu0,u0=∑i=1mtiGvi,u0,u0+∇gu0−∇gw,vi−u0+bu0,vi−bu0,u0<0,which is a contradiction. Thus, Fw is a KKM mapping.
Next, we prove thatFwv⊂Gwv,∀v∈C. Let u∈Fwv, for any v∈C; then,(51)Gv,u;u+∇gu−∇gw,v−u+bu,v−bu,u≥0.
Using the concept ofG, we obtain(52)Gv,u;v+∇gu−∇gw,v−u+bu,v−bu,u≥Gv,u;u+∇gu−∇gw,v−u+bu,v−bu,u+αu,v≥αu,v.
Thus,Fwv⊂Gwv, ∀v∈C, which yields that Gwv is a KKM mapping.
Letun be any sequence in Gwv with un⟶u as n⟶∞. Then,(53)Gv,un;v+∇gun−∇gw,v−un+bun,v−bun,un≥αun,v.
Sinceg is Gateaux differentiable function, ∇g is norm-to-weak ∗ continuous. By (ii) and (iii) and lower semicontinuity of α, we have(54)αu,v+Gu,v;v≤limn⟶∞infαun,v+limn⟶∞infGun,v;v≤limn⟶∞infαun,v+Gun,v;v≤limn⟶∞supαun,v+Gun,v;v=limn⟶∞supαun,v−Gv,un,v≤∇gu−∇gw,v−u+bu,v−bu,u,which yields that Gv,u;v+∇gu−∇gw,v−u+bu,v−bu,u≥αu,v. Thus, u∈Gwv and Gwv are the closed subset of C, ∀v∈C. As C is closed convex and bounded subset in X, it is weakly compact. Thus, Gwv is also compact. By Lemmas 7 and 10, we have ∩v∈CFwv=∩v∈CGwv≠∅. Therefore, AP equation (36) has a solution. □
The resolvent ofG:C×C×C⟶ℝ with respect to b is the operator resG,bf:X⟶2C, defined as follows:(55)resG,bgu=w∈C:Gv,w;w+∇gw−∇gu,v−w+bw,v−bw,w≥0,∀v∈C,∀u∈X.
We obtain some properties of the resolvent operatorresG,bg. First, we show that resG,bgu≠∅ for u∈X and dom resG,bg=X under some suitable conditions.Lemma 9.
Letg:X⟶−∞,+∞ be a coercive and Gateaux differentiable function. If G:C×C×C⟶ℝ satisfies all conditions of Theorem 1 and b:C×C⟶ℝ satisfies Assumption 1, then dom resG,bg=X.Proof.
First, we prove that for anyξ∈X∗∃u∈C such that(56)Gv,u;u+bu,v−bu,u+gv−gu−ξ,v−u≥0,for any v∈C. As g is coercive, the function h:X×X⟶−∞,+∞ defined by(57)hu,v=gv−gu−ξ,v−usatisfies(58)limu−v⟶+∞hu,vu−v=−∞,for each fixed v∈C. By Theorem 1 in [44], equation (56) holds. Now, we show that equation (56) yields(59)Gv,u;u+bu,v−bu,u+∇gu,v−u−ξ,v−u≥0,for any v∈C. Assume vt=tv+1−tu and t∈0,1; we get vt∈C. By equation (59) and the concept of G, we get(60)Gvt,u;vt+bu,vt−bu,u+∇gu,vt−u−ξ,vt−u≥αu,vt,(61)Gtv+1−tu,u;vt+bu,tv+1−tu−bu,u+gtv+1−tu−gv−ξ,tv+1−tu−u≥αu,vt,∀v∈C.
Since(62)gtv+1−tu−gv≤∇gtv+1−tu,tv+1−tu−u,we get from equation (61), Theorem 1 (ii), and Assumption 1 (ii) that(63)tGv,u;vt+1−tGu,u;vt+tbu,v+1−tbu,u−bu,u+∇gtv+1−tu,tv+1−tu−u−ξ,tv+1−tu−u≥αu,vt,∀v¯∈C.
From Lemma10 (iii), we have(64)tGv,u;vt+tbu,v−tbu,u+∇gtv+1−tu,tv−u−ξ,tv−u≥αu,vtand(65)tGv,u;vt+bu,v−bu,u+∇gtv+1−tu,v−u−ξ,v−u≥αu,vt.
Therefore,(66)Gv,u;vt+bu,v−bu,u+∇gtv+1−tu,v−u−ξ,v−u≥αu,vtt,∀v¯∈C.
Asg is a Gateaux differentiable function, ∇g is norm-to-weak ∗ continuous. Taking t⟶0, we have(67)Gv,u;u+bu,v−bu,u+∇gu,v−u−ξ,v−u≥0,∀v¯∈C.
Thus, for anyu∈X, let ξ=∇gu¯; we have u¯∈C such that(68)Gv,u;u+bu,v−bu,u+∇gu,v−u−∇gu¯,v−u≥0,∀v¯∈C,i.e.,(69)Gv,u;u+bu,v−bu,u+∇gu−∇gu¯,v−u≥0,∀v¯∈C,that is, u∈resG,bgu. Hence, dom resG,bg=X. □Lemma 10.
LetG:C×C×C⟶ℝ satisfy all conditions of Theorem 1, and let b:C×C⟶ℝ satisfy Assumption 1. Let g:X⟶−∞,+∞ be a coercive Legendre function and the resolvent operator resG,bg:X⟶2C be defined by equation (55). Then, the following holds:(i)
resG,bg is single-valued(ii)
resG,bg is Bregman firmly nonexpansive type mapping, that is,(70)∇gresG,bgu−∇gresG,bgv,resG,bgu−resG,bgv≤∇gu−∇gv,resG,bgu−resG,bgy,∀u,v∈X(iii)
FresG,bg=SolGMVLIP1 is closed and convex(iv)
(71)Dgq,resG,bgu+DgresG,bgu,u≤Dgq,u,∀q∈FresG,bg.(v)
resG,bg is Bregman quasi-nonexpansiveProof.
(i)
Foru∈X, let w1,w2∈FresG,bg. Then, w1,w2∈C and hence(72)Gw2,w1;w1+∇gw1−∇gu,w2−w1+bw1,w2−bw1,w1≥0.and(73)Gw1,w2;w2+∇gw2−∇gu,w1−w2+bw2,w1−bw2,w2.Adding the above two inequalities, we get(74)Gw2,w1;w1+Gw1,w2;w2+∇gw1−∇gw2,w2−w1+bw1,w2−bw1,w1+bw2,w1−bw2,w2≥0.By condition (iii) of Theorem1, we get(75)−Gw1,w2;w1+Gw1,w2;w2+∇gw1−∇gw2,w2−w1+bw1,w2−bw1,w1+bw2,w1−bw2,w2≥0.Asb is skew symmetric and G is a generalized relaxed α-monotone,(76)αw2,w1−∇gw1−∇gw2,w2−w1≤0∇gw1−∇gw2,w2−w1≥αw2,w1.By interchanging the position ofw1 and w2 in equation (76), we get(77)∇gw2−∇gw1,w1−w2≥αw2,w1.Adding equations (76) and (77), we have(78)2∇gw1−∇gw2,w2−w1≥αw1,w2+αw2,w1.Asαu,v+αv,u≥0,∀v∈C,(79)∇gw1−∇gw2,w2−w1≥0.This implies that(80)∇gw2−∇gw1,w2−w1≤0.Asg is convex and Gateaux differentiable,(81)∇gw2−∇gw1,w2−w1≥0.By equations (80) and (81), we have(82)∇gw2−∇gw1,w2−w1=0.Sinceg is a Legendre function, w1=w2. Hence, resG,bg is single-valued.(ii)
Foru,v∈C, we obtain(83)GresG,bgv,resG,bgu;resG,bgu+∇gresG,bgu−∇gu,resG,bgv−resG,bgu+bresG,bgu,resG,bgv−bresG,bgu,resG,bgu≥0and(84)GresG,bgu,resG,bgv;resG,bgv+∇gresG,bgv−∇gv,resG,bgu−resG,bgv+bresG,bgv,resG,bgu−bresG,bgv,resG,bgv≥0.Adding the above two inequalities, we have(85)GresG,bgv,resG,bgu;resG,bgu+GresG,bgu,resG,bgv;resG,bgv+∇gresG,bgu−∇gu−∇gresG,bgv+∇gv,resG,bgv−resG,bgu+bresG,bgu,resG,bgv−bresG,bgu,resG,bgu+bresG,bgv,resG,bgu−bresG,bgv,resG,bgv≥0,which yields by applying the concept ofb and G,(86)∇gresG,bgu−∇gu−∇gresG,bgv+∇gv,resG,bgv−resG,bgu≥−GresG,bgv,resG,bgu;resG,bgu+GresG,bgu,resG,bgv;resG,bgv=GresG,bgv,resG,bgu;resG,bgv−GresG,bgv,resG,bgu;resG,bgu≥αresG,bgu,resG,bgv.In equation (86), interchanging the position of resG,bgu and resG,bgv, we get(87)∇gresG,bgv−∇gv−∇gresG,bgu+∇gu,resG,bgv−resG,bgu≥αresG,bgv,resG,bgu.Adding equations (86) and (87) and using αu,v+αv,u≥0,∀v∈C, we get(88)2∇gresG,bgu−∇gu−∇gresG,bgv+∇gv,resG,bgv−resG,bgu≥αresG,bgu,resG,bgv+αresG,bgv,resG,bgu≥0.This implies that(89)∇gresG,bgu−∇gresG,bgv,resG,bgu−resG,bgv≤∇gu−∇gv,resG,bgu−resG,bgv.This means thatresG,bg is a Bregman firmly nonexpansive type mapping.(iii)
Letu∈FresG,bg; then,(90)u∈FresG,bg⇔u=resG,bgu⇔Gv,u;u+∇gu−∇gu,v−u+bu,v−bu,u≥0,∀v∈C⇔Gv,u;u+bu,v−bu,u,∀v∈C⇔u∈SolGMVLIP1.Furthermore, SinceresG,bg is a Bregman firmly nonexpansive type mapping, in ([42], Lemma 1.3.1), FresG,bg is a closed and convex subset of C. Therefore, by equation (90), we get that SolGMVLIP1=FresG,bg is closed and convex.(iv)
Now, we show thatresG,bg is Bregman quasi-nonexpansive mapping.Foru,v∈C, from (b), we have(91)∇gresG,bgu−∇gresG,bgv,resG,bgu−resG,bgv≤∇gu−∇gv,resG,bgu−resG,bgv.Moreover, we have(92)DgresG,bgu,resG,bgv+DgresG,bgv,resG,bgu≤DgresG,bgu,v−DgresG,bgu,u+DgresG,bgv,u−DgresG,bgv,v.Takingv=w∈FresG,bg, we see that(93)DgresG,bgu,w+Dgw,resG,bgu≤DgresG,bgu,w−DgresG,bgu,u+Dgw,u−Dgw,w.Hence,(94)Dgw,resG,bgu+DgresG,bgu,u≤Dgw,u.(v)
Equation (94) implies that resG,bg is Bregman quasi-nonexpansive mapping.


## 4. Main Result

We developed the strong convergence algorithm for the inertial iterative method to find the common solution of GMVLIP equation (1) and fixed-point problem of a Bregman relatively nonexpansive mapping in reflexive Banach space.Iterative Algorithm 1.
Let the sequencesxn and zn be generated by the iterative algorithm:(95)x0,x−1∈C,un=xn+θnxn−xn−1,vn=∇g∗αn∇gun+1−αn∇gTun,wn=∇g∗βn∇gTun+1−βn∇gvn,zn=resG,bgwn,Cn=z∈C:Dgz,zn≤Dgz,un,Qn=z∈C:∇gx0−∇gxn,z−xn≤0,xn+1=projCn∩Qngx0,foralln≥0,where θn⊆0,1 and αn,βn⊆0,1.Theorem 2.
LetC⊆X with C⊆intdomg, where g:X⟶−∞,+∞ be a coercive Legendre function which is bounded, uniformly Frechet differentiable, and totally convex on bounded subsets of X. Let G:C×C×C⟶ℝ satisfy all conditions of Theorem 1 with continuous Gy,⋅;y, and b:C×C⟶ℝ satisfies Assumption 1, respectively. Let T:C⟶C be a Bregman relatively nonexpansive mapping. Let Ω=SolGMVLIP1∩FT≠∅. Let xn,zn be generated by Iterative 1 and θn⊆0,1, αn,βn⊆0,1 with limn⟶∞αn=0. Then, xn converges strongly to projΩgx0.Proof.
For convenience, we divide its proof into several steps as in the following.Step 1.
Ω and Cn∩Qn are closed and convex, ∀n≥0.
By Lemmas5 and 9, Ω is a closed and convex, and therefore, projΩgx0 is well defined.
Obviously,Qn is closed and convex. Furthermore, we prove that Cn is closed and convex, ∀n≥0. We can easily show that Cn is closed and convex, ∀n. Thus, Cn∩Qn is closed and convex, ∀n≥0.Step 2.
Ω⊂Cn∩Qn, ∀n≥0, and xn is well defined.
Letp∈Ω; then,(96)Dgp,zn=Dgp,resG,bgwn≤Dgp,wn=Dgp,∇g∗βn∇gTun+1−βn∇gvn≤βnDgp,un+1−βnDgp,vn,and(97)Dgp,vn=Dgp,∇g∗αn∇gun+1−αn∇gTun≤αnDgp,un+1−αnDgp,un=Dgp,un.
Substituting equation (97) into equation (96), we have(98)Dgp,zn≤Dgp,un.
Thus,p∈Cn. Therefore, Ω⊂Cn, ∀n≥0. Furthermore, by induction, we show that Ω⊂Cn∩Qn, n≥0. As Q0=C, Ω⊂C0∩Q0. Suppose that Ω⊂Cm∩Qm, for some m>0. Then, ∃xm+1∈Cm∩Qm such that xm+1=projCm∩Qmgx0. From the definition of xm+1, we get ∇gx0−∇gxm+1,xm+1−z≥0,∀z∈Ck∩Qm. Since Ω⊂Cm∩Qm, we have(99)∇gx0−∇gxm+1,p−xm+1≤0,∀p∈Ω,which implies p∈Qm+1. Hence, Ω⊂Cm+1∩Qm+1 implies Ω⊂Cn∩Qn, ∀n≥0, and thus, xn+1=projCn∩Qngx0 is well defined, ∀n≥0. Hence, xn is well defined.Step 3.
The sequencesxn, un, vn, zn, and wn are bounded.
Using the concept ofQn, we get xn=projQngx0. By xn=projQngx0 and Lemma 10 (iii), we obtain(100)Dgxn,x0=DgprojQngx0,x0≤Dgu,x0−Dgu,projQngx0≤Dgu,x0,∀u∈Ω⊂Qn.
This implies thatDgxn,x0 is bounded, and hence, xn is bounded by Lemma 3.
Now,(101)Dgp,xn=Dgp,projCn−1∩Qn−1gx0≤Dgp,x0−Dgxn,x0,which implies that Dgp,xn is bounded. Using Dgp,Txn≤Dgp,xn,∀p∈Ω, Txn is bounded. Therefore, un, vn, wn, and zn are bounded.Step 4.
limn⟶∞xn+1−xn=0;limn⟶∞xn−un=0;limn⟶∞zn−un=0;limn⟶∞zn−wn=0; limn⟶∞un−wn=0, and limn⟶∞un−Tun=0.
Sincexn+1=projCn∩Qngx0∈Qn and xn∈projQngx0, we get(102)Dgxn,x0≤Dgxn+1,x0,∀n≥0,which implies Dgxn,x0 is nondecreasing. By boundedness of Dgxn,x0, limn⟶∞Dgxn,x0 exists and is finite. Furthermore,(103)Dgxn+1,xn=Dgxn+1,projQngx0≤Dgxn+1,x0−DgprojQngx0,x0=Dgxn+1,x0−Dgxn,x0,which yields(104)limn⟶∞Dgxn+1,xn=0.
Using Lemma1,(105)limn⟶∞xn+1−xn=0.
From the definition ofun, un−xn=θnxn−xn−1≤xn−xn−1, which implies by equation (105) that(106)limn⟶∞un−xn=0.
Since(107)un−xn+1≤un−xn+xn−xn+1,it follows from equations (105) and (106) that(108)limn⟶∞un−xn+1=0.
Using Lemma2 because g is uniformly Frechet differentiable, we get(109)limn⟶∞gun−gxn+1=0and(110)limn⟶∞∇gun−∇gxn+1=0.
By the concept ofDg, we get(111)Dgxn+1,un=gxn+1−gun−∇gun,xn+1−un.∇g is bounded on the bounded subset of X because g is bounded on X. Since g is uniformly Frechet differentiable, it is uniformly continuous on bounded subsets. Hence, by equations (108), (109), and (111),(112)limn⟶∞Dgxn+1,un=0.
Asxn+1=projCn∩Qngx0∈Cn, we have(113)Dgxn+1,zn≤Dgxn+1,un,and hence, by equations (112) and (113),(114)limn⟶∞Dgxn+1,zn=0.
Thanks to Lemma1,(115)limn⟶∞xn+1−zn=0.
Taking into account(116)zn−un≤zn−xn+1+xn+1−un,by equations (108) and (115), we get(117)limn⟶∞zn−un=0.
By Lemma2,(118)limn⟶∞gzn−gun=0and(119)limn⟶∞∇gzn−∇gun=0.
Next, we estimate(120)Dgp,un−Dgp,zn=gp−gun−∇gun,p−un−gp+gzn+∇gzn,p−zn=gzn−gun+∇gzn,p−zn−∇gun,p−un=gzn−gun+∇gzn,un−zn+∇gzn−∇gun,p−un.
Sincezn, un, ∇gzn, and ∇gun are bounded and by equations (117)–(120), we get(121)limn⟶∞Dgp,un−Dgp,zn=0.
Furthermore, it follows from Lemma9 (v) that(122)Dgzn,wn≤Dgp,wn−Dgp,zn≤Dgp,∇g∗βn∇gTun+1−βn∇gvn−Dgp,zn≤βnDgp,Tun+1−βnDgp,un−Dgp,zn≤Dgp,un−Dgp,zn.
SinceDgp,un and Dgp,zn are bounded, by equations (121) and (122),(123)limn⟶∞Dgzn,wn=0,and hence,(124)limn⟶∞zn−wn=0.
From equations (117) and (124), we get(125)limn⟶∞un−wn=0.
By uniform Frechet differentiable ofg, Lemma 2, and equations (124) and (125), we have(126)limn⟶∞∇gzn−∇gwn=0,(127)limn⟶∞∇gun−∇gwn=0.
Note that(128)∇gun−∇gwn=∇gun−∇g∇g∗βn∇gTun+1−βn∇gvn=∇gun−βn∇gTun−1−βn∇gvn=βn∇gun−∇gTun+1−βn∇gun−∇gvn=βn∇gun−∇gTun+1−βn∇gun−∇g∇g∗αn∇gun+1−αn∇gTun=βn∇gun−∇gTun+1−βn1−αn∇gun−∇gTun=1−αn1−βn∇gun−∇gTun.
By equations (127) and (128) and limn⟶∞αn=0, we get(129)limn⟶∞∇gun−∇gTun=0.
Moreover, we have from equation (129) that(130)limn⟶∞un−Tun=0.Step 5.
x¯∈Ω.
First, we prove thatx¯∈FT. As xn is bounded, ∃ a subsequence xnk⊆xn such that xnk⇀x¯∈C as k⟶∞. By equations (106), (117), (124), and (125), xn, un, wn, and zn have the same asymptotic behaviour and thus ∃ subsequences unk of un, wnk of wn, and znk of zn such that unk⇀x¯, wnk⇀x¯, and znk⇀x¯ as k⟶∞. Using unk⇀x¯ and equation (130), we get(131)limk⟶∞unk−Tunk=0.
By the concept ofT, x¯∈F^T=FT.
Next, prove thatx¯∈SolGMVLIP1. As zn=resG,bgwn, we have(132)Gv,znk;znk+∇gznk−∇gwnk,v−znk+bv,znk−bznk,znk≥0,∀v∈C.
Using generalized relaxedα-monotonicity of G, we have(133)∇gznk−∇gwnk,v−znk≥−Gv,znk;znk−bv,znk+bznk,znk,∀v∈C,≥αznk,v−Gv,znk;v−bv,znk+bznk,znk.
Using the concept ofG, b, equation (126), and k⟶∞ in equation (133), we obtain(134)αx¯,v−Gv,x¯;v+bx¯,x¯−bx¯,v≤0,for allv∈C.
Fort∈0,1 and v∈C, let vt=tv+1−tx¯. Since vt∈C, we have(135)αix¯,vt−Gvt,x¯;vt+bx¯,x¯−bx¯,vt≤0,which implies that(136)αx¯,vt≤Gvt,x¯;vt−bx¯,x¯+bx¯,vt≤tGv,x¯;vt+1−tGx¯,x¯;vt−bx¯,x¯+tbx¯,v+1−tbx¯,x¯≤tGv,x¯;vt+bx¯,v−bx¯,x¯.
SinceGv,x¯;⋅ is hemicontinuous, we have(137)limt⟶0Gv,x¯;vt+bx¯,v−bx¯,x¯≥limt⟶0αx¯,vtt,which implies(138)Gv,x¯;x¯+bx¯,v−bx¯,x¯≥0.
Hence,x¯∈SolGMVLIP1. Thus, x¯∈Ω.Step 6.
We prove thatxn⟶x¯=projΩgx0.Proof of Step6.
Letu˜=projΩgx0. As xn is weakly convergent, xn+1=projΩgx0 and projΩgx0∈Ω⊂Cn∩Qn. By equation (100), we have(139)Dgxn+1,x0≤DgprojΩgx0,x0.
Using Lemma6, xn is strongly convergent to u˜=projΩgx0. Hence, by the uniqueness of the limit, xn converges strongly to x¯=projΩgx0.


## 5. Consequences

Finally, we get the following consequences of Theorem2.Corollary 1.
LetC⊆X with C⊆intdomg, where g:X⟶−∞,+∞ be a coercive Legendre function which is bounded, uniformly Frechet differentiable, and totally convex on bounded subsets of X. Let G:C×C×C⟶ℝ satisfy conditions (i), (ii), and (iii) of Theorem 1 and G be monotone, i.e.,(140)Gy,x;y−Gy,x;x≥0,for anyx,y∈C.
Letb:C×C⟶ℝ satisfy Assumption 1, and Let T:C⟶C be a Bregman relatively nonexpansive mapping. Let Ω=SolGMVLIP1∩FT≠∅. Let xn,zn be generated by Iterative 1 and θn⊆0,1 and αn,βn⊆0,1 with limn⟶∞αn=0. Then, xn converges strongly to projΩgx0.
Moreover, if GMVLIP equation (1)=C and by the concept of Example 1 for A:X⟶2X∗, we have the maximal monotone operator.Corollary 2.
LetC⊆X with C⊆intdomg, where g:X⟶−∞,+∞ be a coercive Legendre function which is bounded, uniformly Frechet differentiable, and totally convex on bounded subsets of X. Let A:X⟶2X∗ be a maximal monotone operator with A−10≠∅. Let xn,zn⊆C generated by(141)x0,x−1∈C,un=xn+αnxn−xn−1,vn=∇g∗αn∇gun+1−αn∇gresAgun,zn=∇g∗βn∇gresAgun+1−βn∇gvn,Cn=z∈C:Dgz,zn≤Dgz,un,Qn=z∈C:∇gx0−∇gxn,z−xn≤0,xn+1=projCn∩Qngx0,∀n≥0,where θn⊆0,1 and αn,βn⊆0,1 with limn⟶∞αn=0. Then, xn converges strongly to projA−10x0.Remark 4.
Ifgx=1/2x2, ∀x∈X, then Theorem 2 is reduced to the strong convergence theorem for finding the common solution of GMVLIP equation (1) and fixed-point problem of a relatively nonexpansive mapping in reflexive Banach space.


## 6. Numerical Example

Finally, to support our main theorem, we now give an example in infinitely dimensional spacesL20,1 such that ⋅ is L2-norm defined by x=∫01xt2dt where xt∈L20,1.Example 3.
LetX=L20,1 and C=xt∈L20,1:∫01txtdt≤2. Define mappings as follows:(i)
Coercive Legendre functiong:X⟶−∞,+∞ by gx=1/2x2, ∀x∈X(ii)
∀x,y,z∈C, FunctionG:C×C×C⟶ℝ by Gx,y,z=1/2y2−x2, with α:C×C⟶ℝ such that αx,y=0, ∀x,y∈C(iii)
Bifunctionb:C×C⟶ℝ by bx,y=−x,y, ∀x,y∈C(iv)
Bregman relatively nonexpansive mappingT:C⟶C with respect to g by Tx=x/2, ∀x∈C
It is obvious thatG:C×C×C⟶ℝ satisfies all conditions of Theorem 1 with continuous Gy,⋅;y and b:C×C⟶ℝ satisfies Assumption 1, respectively. On the other hand, we consider(142)u∈resG,bgw⟺Gu,y,y+∇gu−∇gw,y−u+bu,y−bu,u≥0,∀y∈C⟺12y2−u2+u−w,y−u−u,y+u,u≥0,∀y∈C⟺12y2−u2−w,y−u≥0,∀y∈C⟺12y2−u2−w,y−w+w,u−w≥0,∀y∈C⟺12u2−w2−w,u−w≤12y2−w2−w,y−w,∀y∈C⟺Dgu,w≤Dgy,w,∀y∈C⟺u=ProjCgw.
For the experiments in this section, we use the Cauchy errorxn+1−xn2<10−5 for the stopping criterion. We will start with the initialization x−1 and x0 in two cases. We split considering all of the performances of our algorithm in four cases by considering all of the parameters that have an effect on the convergence of the algorithm.Case 1.
We start computation by comparison of the algorithm with different parametersϵn where(143)θn=minϵnxn−xn−1,θ,ifn≤N,ϵn,otherwise,where N is the number of iterations that we want to stop, limn⟶∞εn=0, and θ∈0,1. We choose θ=0.3, αn=1/2n+1, and βn=αn. Then, the results are presented in Table 1.Table 1
Numerical results of the differenceεn.
εn1/n+11/2n+11/n2+11/2n2+11/n3+1x−1=sint/2,x0=sintNo. of iter.915192020CPU time (s)7.5993212.2274814.7102415.5730615.66219x−1=2sint−t/2,x0=2sint−tNo. of iter.1015202021CPU time (s)8.7582012.2497115.6506815.7860717.30471x−1=t+log2t+1,x0=logt+1No. of iter.1115212022CPU time (s)9.0621710.1557416.5308415.8173817.65972Case 2.
We compare the performance of the algorithm with different parametersθ by setting ϵn=1/n+1, αn=1/2n+1, and βn=αn. Then, the results are presented in Table 2.Table 2
Numerical results of the differenceθ.
θ0.10.30.50.70.9x−1=sint/2,x0=sintNo. of iter.99999CPU time (s)7.758787.507407.679077.598647.60107x−1=2sint−t/2,x0=2sint−tNo. of iter.1010101010CPU time (s)8.533628.821508.622028.820758.64536x−1=t+log2t+1,x0=logt+1No. of iter.1111111111CPU time (s)9.619679.062179.562749.475709.10264Case 3.
We compare the performance of the algorithm with different parametersαn by setting εn=1/n+1, βn=αn, and θ=0.3 for the initialization x−1=sint/2,x0=sint and x−1=t+log2t+1,x0=logt+1 and θ=0.1 for the initialization x−1=2sint−t/2,x0=2sint−t. Then, the results are presented in Table 3.Table 3
Numerical results of the differenceαn.
αn1/2n+11/10n+11/100n+11/2n2+11/10n2+1x−1=sint/2,x0=sintNo. of iter.96575CPU time (s)7.538285.630664.784616.192904.80899x−1=2sint−t/2,x0=2sint−tNo. of iter.106576CPU time (s)8.511655.842075.108836.473655.94383x−1=t+log2t+1,x0=logt+1No. of iter.116576CPU time (s)8.878435.522234.951056.232865.59795Case 4.
We compare the performance of the algorithm with different parametersβn by setting εn=1/n+1, αn=1/100n+1, and θ=0.3 for the initialization x−1=sint/2,x0=sint and x−1=t+log2t+1,x0=logt+1 and θ=0.1 for the initialization x−1=2sint−t/2,x0=2sint−t. Then, the results are presented in Table 4.
From Tables1–4 and Figures 1–12, we noticed that in all the above 4 cases, choosing θ=0.3, εn=1/n+1, αn=1/100n+1, and βn=1/2n2+1 yields the best results for the initialization x−1=sint/2,x0=sint. Choosing θ=0.1, εn=1/n+1, αn=1/100n+1, and βn=1/2n+1 yields the best results for the initialization x−1=2sint−t/2,x0=2sint−t, and choosing θ=0.3, εn=1/n+1, αn=1/100n+1, and βn=1/100n+1 yields the best results for the initialization x−1=t+log2t+1,x0=logt+1.Table 4
Numerical results of the differenceβn.
βn1/2n+11/10n+11/100n+11/2n2+11/10n2+1x−1=sint/2,x0=sintNo. of iter.55555CPU time (s)4.808894.751284.791564.751094.76763x−1=2sint−t/2,x0=2sint−tNo. of iter.55555CPU time (s)4.973115.096685.031534.993854.98385x−1=t+log2t+1,x0=logt+1No. of iter.55555CPU time (s)5.160314.972004.835505.495815.47782Figure 1
The Cauchy error plotting number of iterations for different parametersεn.Figure 2
The Cauchy error plotting number of iterations for different parametersεn.Figure 3
The Cauchy error plotting number of iterations for different parametersεn.Figure 4
The Cauchy error plotting number of iterations for different parametersθ.Figure 5
The Cauchy error plotting number of iterations for different parametersθ.Figure 6
The Cauchy error plotting number of iterations for different parametersθ.Figure 7
The Cauchy error plotting number of iterations for different parametersαn.Figure 8
The Cauchy error plotting number of iterations for different parametersαn.Figure 9
The Cauchy error plotting number of iterations for different parametersαn.Figure 10
The Cauchy error plotting number of iterations for different parametersβn.Figure 11
The Cauchy error plotting number of iterations for different parametersβn.Figure 12
The Cauchy error plotting number of iterations for different parametersβn.


## 7. Conclusion

In this paper, we established an inertial hybrid iterative algorithm involving Bregman relatively nonexpansive mapping to find a common solution of GMVLIP equation (1) and FPP in Banach space. Moreover, we study the convergence analysis for the main result. At last, we list some consequences and computational example to emphasize the efficiency and relevancy of the main result. From the theoretical and application point of view, the inertial method via Bregman relatively nonexpansive mapping has a great importance on data analysis and some imaging problems. The inertial method has been studied by various researchers due to its importance (see for details [19, 24–28, 30, 31, 33–36, 39]).



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