# Graph Convergence and Iterative Approximation of Solution of a Set-Valued Variational Inclusions

Authors: Faizan Ahmad Khan; Sanjeev Gupta
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this article, first, we introduce a class of proximal-point mapping associated with generalizedαiβj-Hp.,.,…-accretive mapping. Further, we discuss the graph convergence of generalized αiβj-Hp.,.,…-accretive mapping. As an application, we consider a set-valued variational inclusion problem (SVIP) in real Banach spaces. Furthermore, we propose an iterative scheme involving the above class of proximal-point mapping to find a solution of SVIP and discuss its convergence under some convenient assumptions. An example is constructed and demonstrated few graphics in support of our main results.


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## Body

## 1. Introduction and Preliminaries

Variational inclusions (a generalization of variational inequalities) have become very interesting concern for researchers due to its important applications in differential equations, operation research, contact problem in elasticity, general equilibrium problem, economics and optimization, etc. It is observed that the accretive property of proximal-point mapping has a remarkable role in the area of variational inequalities theory. The concept ofm-accretive mappings was established by Huang and Fang [1] in 2001. For many important generalizations of m-accretive mappings, we refer to see [2–25].Recently, many researchers have studied and introducedH.,.-accretive mappings and generalized αβ-H.,.-accretive mappings [13, 24, 25]. Very recently, Nazemi [26] and Guan and Hu [27] investigated and introduced Cn-monotone mappings and Cn-η-monotone mappings in Banach spaces to study some classes of variational inclusions.The presentation of graphical convergence related toH.,.-accretive mappings and the equivalence between the proximal-point mapping and graphical convergence of a sequence of H.,.-accretive mapping studied and analyzed by Li and Huang [15]. Recently, graphical convergence related to A-maximal relaxed monotone and A^-maximal m-relaxed η-accretive and the equivalence between the proximal-point mapping and graphical convergence of the sequences of their investigated mappings were studied and analyzed by Verma [28] and Balooee et al. [29]. Since then, many researchers have studied graph convergence in the context of their proximal-point mappings and extensive graph convergence results obtained by Attouch [30]. For the related works, see [2, 3, 10, 15, 31].Inspired and motivated by the work discussed above, we consider and study a class of proximal-point mapping associated with generalizedαiβj-Hp.,.,…-accretive mapping. This class of accretive mapping is the generalization of generalized αβ-H.,.-accretive mappings [13]. Further, we discuss graph convergence of the generalized αiβj-Hp.,.,…-accretive mappings. An iterative algorithm involving the above class of proximal-point mapping is constructed for the SVIP in real Banach space. Furthermore, we discuss the existence of a solution of SVIP and discuss the convergence analysis of the proposed iterative algorithm. Some illustrations are constructed and shown some graphics in support of our main results.Throughout this paper, letY be a real Banach space endowed with a norm ., and let .,. be the duality pairing between Y and Y∗. Let CBY be the family of all nonempty, closed, and bounded subsets of Y and 2Y be the power set of Y. D˜.,. is the Hausdorff metric on CBY. Set Yp=Y×Y×⋯×Y︸ptimes.(1)D˜B1,B2=maxsupy∈B1infz∈B2d˜y,zsupz∈B1infy∈B2d˜y,z,B1,B2∈CBY.Definition 1.
“The normalized duality mappingJ:Y⟶2Y∗ is defined by(2)Jz=z˜∈Y∗:z,z˜=z2,z˜=z,∀z∈Y.
IfY is equivalent to real Hilbert space X, then J become identity mapping on X. Let J be a selection of normalized duality mapping,” see [32].Lemma 1.
“LetY be a real Banach space and J:Y⟶2Y be the normalized duality mapping. Then, for any y˜,z˜∈Y,(3)y˜+z˜2≤y˜2+y˜,Jy˜+z˜,∀Jy˜+z˜∈Jy˜+z˜″,25.Definition 2.
A set-valued mappingU:Y⟶CBY is said to be D˜-Lipschitz continuous with ζ>0, if(4)D˜Uy˜,Uz˜≤ζy˜−z˜,∀y˜,z˜∈Y.Lemma 2.
“Let two non-negative real sequencescn and en, satisfying cn+1≤lcn+en with en⟶0 and 0<l<1. Then, limn⟶∞cn=0,” see [33].Definition 3.
LetT:Y⟶Y be a single-valued mapping. Then, T is called(i)
Accretive if(5)Tw˜−Tu˜,Jw˜−u˜≥0,∀w˜,u˜∈Y;(ii)
ξ-strongly accretive if there exists non-negativeξ such that(6)Tw˜−Tu˜,Jw˜−u˜≥ξw˜−u˜2,∀w˜,u˜∈Y;(iii)
λT-Lipschitz continuous if there exists non-negative λT such that(7)Tw˜−Tu˜≤λTw˜−u˜,∀w˜,u˜∈Y;(iv)
λ-expansive if there exists non-negative λ such that(8)Tw˜−Tu˜≥λw˜−u˜,∀w˜,u˜∈Y;(v)
T becomes expansive if λ=1.Throughout this paper, we consideri∈1,2,…,p, where p≥3, otherwise specified.Definition 4.
LetAi:Y⟶Y,Hp:Yp⟶Y be single-valued mappings, then(i)
Hp is αi-strongly accretive with respect to Ai if there exists non-negative αi such that(9)Hp…,Aiw˜,…−Hp…,Aiv˜,…,Jw˜−v˜≥αiw˜−v˜2,∀w˜,v˜∈Y;(ii)
Hp is βi-relaxed accretive with respect to mapping Ai if there exists non-negative βi such that(10)Hp…,Aiw˜,…−Hp…,Aiv˜,…,Jw˜−v˜≥−βiw˜−v˜2,∀w˜,v˜∈Y;(iii)
Hp is λi-Lipschitz continuous with respect to mapping Ai if there exists non-negative λi such that(11)Hp…,Aiw˜,…−Hp…,Aiv˜,….≤λiw˜−v˜,∀w˜,v˜∈Y;(iv)
Hp is α1β2α3β4…αp−1βp-symmetric accretive with respect to mappings A1,A2,…,Ap iff for i∈1,3,…,p−1; Hp is αi-strongly accretive with respect to mapping Ai and for j∈2,4,…,p; Hp is βj-relaxed accretive with respect to mapping Aj, where p is even, satisfying ∑βj≤∑αi and ∑βj=∑αi iff w˜=v˜;(v)
Hp is α1β2α3β4,…βp−1,αp-symmetric accretive with respect to mappings A1,A2,…,Ap iff for i∈1,3,…,p; Hp is αi-strongly accretive with respect to mapping Ai and for j∈2,4,…,p−1; Hp is βj-relaxed accretive with respect to mapping Aj, where p is odd, satisfying ∑βj≤∑αi and ∑βj=∑αi iff w˜=v˜.Definition 5.
Lethi:Y⟶Y be a single-valued mapping and N:Yp⟶2Y be a set-valued mapping, then(i)
N is μi-strongly accretive with respect to mapping hi if there exists non-negative μisuch that(12)w˜i−v˜i,Jw˜−v˜≥μiw˜−v˜2,∀w˜,v˜∈Y,w˜i∈N…,hiw˜,…,v˜i∈N…,hiv˜,…;(ii)
N is γi-relaxed accretive with respect to mapping hi if there exists non-negative γi such that(13)w˜i−v˜i,Jw˜−v˜≥−γiw˜−v˜2,∀w˜,v˜∈Y,w˜i∈N…,hiw˜,…,v˜i∈N…,hiv˜,…;(iii)
μ1γ2μ3γ4…μp−1γp-symmetric accretive with respect to mappings h1,h2,…,hp iff for i∈1,3,…,p−1; N is μi-strongly accretive with respect to mapping hi and for j∈2,4,…,p; N is γj-relaxed accretive with respect to mapping hj, where p is even, satisfying ∑γj≤∑μi and ∑γj=∑μi iff w˜=v˜;(iv)
μ1γ2μ3rγ4,…μp,γp−1-symmetric accretive with respect to mappings h1,h2,…,hp iff for i∈1,3,…,p; N is μi-strongly accretive with respect to mapping hi and for j∈2,4,…,p−1; N is γj-relaxed accretive with respect to mapping hj, where p is odd, satisfying ∑γj≤∑μi and ∑γj=∑μi iff w˜=v˜.Definition 6.
LetQ:Yp⟶Y be a mapping and U:Y⟶2Y, N:Yp⟶2Y be set-valued mappings, then(i)
Q is β¯i-strongly accretive w.r.t U in the ith-argument if there exists non-negative β¯i such that(14)Q…,w∗,…−Q…,v∗,…,Jz−y≥β¯iz−y2,∀y,z∈Y,w∗∈Uz,v∗∈Uy;(ii)
Q is ζi-Lipschitz continuous in the ith-argument if there exists non-negative ζi such that(15)Q…,w∗,…−Q…,v∗,…≤ζiz−y,∀y,z∈Y,w∗∈Uz,v∗∈Uy.


## 2. Generalizedαiβj-Hp.,.,…,. Accretive Mappings

LetHp:Yp⟶Y, hi,Ai:Y⟶Y be the single-valued mappings, and let N:Yp⟶2Y be a set-valued mapping. We consider the following assumptions (P1–P4) to define generalized αiβj-Hp.,.,…,.-accretive mappings:P1: Hp is α1β2α3β4…αp−1βp-symmetric accretive with respect to mappings A1,A2,…,Ap if p is even;P2: Hp is α1β2α3β4…βp−1αp-symmetric accretive with respect to mappings A1,A2,…,Ap if p is odd;P3: N is μ1γ2μ3γ4…μp−1γp-symmetric accretive mapping with respect to mappings h1,h2,…,hp if p is even;P4: N is μ1γ2μ3γ4…γp−1μp-symmetric accretive mapping with respect to mappings h1,h2,…,hp if p is odd.Definition 7.
Letp≥3, then N is called a generalized αiβj-Hp.,.,…,.-accretive mapping with respect to mappings h1,h2,…,hp and A1,A2,…,Ap:(i)
iffN is μ1γ2μ3γ4…μp−1γp-symmetric accretive with respect to h1,h2,…,hp (assumption P3) and HpA1,A2,…,Ap+ρNh1,h2,…,hpY=Y, when p is even andρ>0(ii)
iffN is μ1γ2μ3γ4…γp−1μp-symmetric accretive with respect to h1,h2,…,hp (assumption P4) and HpA1,A2,…,Ap+ρNh1,h2,…,hpY=Y, when p is odd and ρ>0Proposition 1.
LetP1 and P2 be satisfied and let N:Yp⟶2Y be a generalized αiβj-Hp.,.,…-accretive mapping with respect to mappings h1,h2,…,hp and A1,A2,…,Ap, and ∑βj<∑αi,∑γj<∑μi. If the following inequality is satisfied(16)y−z,Jv−w≥0,for every w,z∈GrNh1,h2,…,hp, y∈Nh1,h2,…,hpv, where GrNh1,h2,…,hp=v,y:y∈Nh1,h2,…,hpv.Proof.
Assume that there existsv0,y0∈GrNh1,h2,…,hp such that(17)y0−z,Jv0−w≥0,∀w,z∈GrNh1,h2,…,hp.Case 1.
Letp is an even number. Since N is a generalized αiβj-Hp.,.,…-accretive with respect to mappings h1,h2,…,hp and A1,A2,…,Ap, then N is μ1γ2μ3γ4…μp−1γp-symmetric accretive with respect to mappings h1,h2,…,hp and HpA1,A2,…,Ap+ρNh1,h2,…,hpY=Y, where ρ>0, then there exists v1,y1∈GrNh1,h2,…,hp such that(18)HpA1v0,A2v0,…,Apv0+ρy0=HpA1v1,A2v1,…,Apv1+ρy1∈Y.
From (17) and (18), we have(19)ρy0−ρy1=HpA1v1,A2v1,…,Apv1−HpA1v0,A2v0,…,Apv0∈Y,ρy0−ρy1,Jv0−v1=HpA1v1,A2v1,…,Apv1−HpA1v0,A2v0,…,Apv0,Jv0−v1.
Setw,z=v1,y1 in (17) and using P3 in (18), we obtain(20)μ1−γ2+μ3−γ4+⋯+μp−1−γpw0−w12≤ρy0−y1,Jv0−v1≤−HpA1v0,A2v0,…,Apv0−HpA1v1,A2v1,…,Apv1,Jv0−v1=−HpA1v0,A2v0,…,Apv0−HpA1v1,A2v0,…,Apv0,Jv0−v1−HpA1v1,A2v0,…,Apv0−HpA1v1,A2v1,…,Apv0,Jv0−v1⋮−HpA1v1,A2v1,…,Apv0−HpA1v1,A2v1,…,Apv1,Jv0−v1≤−α1v0−v12−β2v0−v12−α3v0−v12−β4v0−v12⋮−αp−1v0−v12−βpv0−v12≤−α1−β2+α3−β4+⋯+αp−1−βpv0−v12.
Then, we have(21)∑αi−∑βj+ρ∑μi−∑γjv0−v12≤0,where(22)∑αi=α1+α3+⋯+αp−1,∑μi=μ1+μ3+⋯+μp−1,∑βj=β2+β4+⋯+βp,∑γj=γ2+γ4−⋯+γp.
Thus, we havev0=v1 due to ∑γj<∑μj, ∑βj<∑αi and ρ>0. From (17), we have y0=y1. Hence, v1,y1=v0,y0∈GrNh1,h2,…,hp. In case of p is odd, we can prove the result by the above similar process. This completes the proof.Theorem 1.
LetP1 and P2 be satisfied and let N:Yp⟶2Y be a generalized αiβj-Hp.,.,…-accretive mapping with respect to mappings h1,h2,…,hp and A1,A2,…,Ap, and ∑γj<∑μi,∑βj<∑αi, then HpA1,A2,…,Ap+ρNh1,h2,…,hp−1 is single-valued.Proof.
Let anyv∈Y, and t,t′∈HpA1,A2,…,Ap+ρNh1,h2,…,hp−1v. It follows that(23)−HpA1t,A2t,…,Apt+v∈ρNh1,h2,…,hpt,−HpA1t′,A2t′,…,Apt′+v∈ρNh1,h2,…,hpt′.Case 2.
Letp is an even number. Since N is μ1γ2μ3γ4…μp−1γp-symmetric accretive with respect to mappings h1,h2,…,hp, then(24)μ1−γ2+μ3−γ4+⋯+μp−1−γpt−t′2≤1ρ−HpA1t,A2t,…,Apt+v−−HpA1t′,A2t′,…,Apt′+v,Jt−t′⇒ρμ1+μ3+⋯+μp−1−γ2−γ4−⋯−γpt−t′2≤−HpA1t,A2t,…,Apt−HpA1t′,A2t′,…,Apt′,Jt−t′=−HpA1t,A2t,…,Apt−HpA1t′,A2t,…,Apt,Jt−t′−HpA1t′,A2t,…,Apt−HpA1t′,A2t′,…,Apt,Jt−t′⋮−HpA1t′,A2t′,…,Apt−HpA1t′,A2t′,…,Apt′,Jt−t′.
Using (21), then(25)∑αi−∑βj+ρ∑μi−∑γjx−y2≤0.
Thus, we havet−t′≤0, which gives t=t′ due to ∑γj<∑μj, ∑βj<∑αi and ρ>0. Hence, HpA1,A2,…,Ap+ρNh1,h2,…,hp−1 is single-valued. In case of p is odd, we can prove the result by the above similar process. This completes the proof.Definition 8.
LetP1 and P2 be satisfied and let N:Yp⟶2Y be a generalized αiβj-Hp.,.,…-accretive mapping with respect to mappings h1,h2,…,hp andA1,A2,…,Ap, and ∑γj<∑μi,∑βj<∑αi. The proximal-point mapping Rρ,N.,.,..,.Hp.,.,..,.:Y⟶Y is defined by(26)Rρ,N.,.,…Hp.,.,…z=HpA1,A2,…,Ap+ρNh1,h2,…,hp−1z,∀z∈Yandρ>0.Next, we have shown the Lipschitz continuity of the proximal-point mapping.Theorem 2.
LetP1 and P2 be satisfied and let N:Yp⟶2Y be a generalized αiβj-Hp.,.,…-accretive mapping with respect to mappings h1,h2,…,hp and A1,A2,…,Ap, and∑γj<∑μi,∑βj<∑αi. Then, the proximal-point mapping Rρ,N.,.,..,.Hp.,.,..,.:Y⟶Y is 1/∑αi−∑βj+ρ∑μi−∑γj-Lipschitz continuous.Proof.
Lett,t′∈Y and using (26), then(27)Rρ,N.,.,..,.Hp.,.,..,.t=HpA1,A2,…,Ap+ρNh1,h2,…,hp−1t,Rρ,N.,.,..,.Hp.,.,..,.t′=HpA1,A2,…,Ap+ρNh1,h2,…,hp−1t′.
It follows that(28)1ρt−HpA1Rρ,N.,.,..,.Hp.,.,..,.t,A2Rρ,N.,.,..,.Hp.,.,..,.t,…,ApRρ,N.,.,..,.Hp.,.,..,.t∈NRρ,N.,.,..,.Hp.,.,..,.t,1ρt′−HpA1Rρ,N.,.,..,.Hp.,.,..,.t′,A2Rρ,N.,.,..,.Hp.,.,..,.t′,…,ApRρ,N.,.,…Hp.,.,…t′∈NRρ,N.,.,..,.Hp.,.,..,.t′.
Letv1=Rρ,N.,.,..,.Hp.,.,..,.t and w1=Rρ,N.,.,..,.Hp.,.,..,.t′.Case 3.
Letp is an even number. Since N is μ1γ2…μp−1γp-symmetric accretive with respect to mappings h1,h2,…,hp, then(29)t−HpA1v1,A2v1,…,Apv1−t′−HpA1w1,A2w1,…,Apw1,Jv1−w1≥ρμ1−γ2+μ3−γ4+⋯+μp−1−γpv1−w12,t−t′−HpA1v1,A2v1,…,Apv1−HpA1w1,A2w1,…,Apw1,Jv1−w1≥ρμ1+μ3+⋯+μp−1−γ2+γ4+⋯+γpv1−w12.
It implies that(30)t−t′v1−w1≥t−t′,Jv1−w1≥HpA1v1,A2v1,…,Apv1−HpA1w1,A2w1,…,Apw1,Jv1−w1+ρ∑μi−∑γjv1−w12≥α1v1−w12−β2v1−w12+α3v1−w12−⋯βpv1−w12+ρ∑μi−∑γjv1−w12=∑αi−∑βj+ρ∑μi−∑γjv1−w12.
We have(31)t−t′v1−w1≥∑αi−∑βj+ρ∑μi−∑γjv1−w12,that is,(32)Rρ,N.,.,..,.Hp.,.,..,.t−Rρ,N.,.,..,.Hp.,.,..,.t′≤1∑αi−∑βj+ρ∑μi−∑γjt−t′,∀t,t′∈Y.
In case ofp is odd, we can prove the result by the above similar process. This completes the proof.


## 3. Graph Convergence forαiβj-Hp.,.,…-Accretive Mappings

Definition 9.
LetN:Yp⟶2Y be a set-valued mapping, then the graph of N is given as(33)GrNu˜1,u˜2,…,u˜p=u˜1,u˜2,…,u˜p,x∗:x∗∈Nu˜1,u˜2,…,u˜p.
Now, we discuss graph convergence of generalizedαiβj-Hp.,.,…-accretive mapping.Definition 10.
Forn=0,1,2,…., let Nn,N:Y⟶2Y be the generalized αiβj-Hp.,.,…-accretive mappings with respect to mappings h1,h2,…,hp and A1,A2,…,Ap. Graph convergence of a sequence Nn to N expressed as Nn⟶GN, if for each h1x,h2x,…,hpx,y∈GrNh1,h2,…,hp, there exists a sequence h1xn,h2xn,…,hpxn,yn∈GrNnh1,h2,…,hp such that(34)h1xn⟶h1x,h2xn⟶h2x,…,hpxn⟶hpx,yn⟶yasn⟶∞.Theorem 3.
LetP1 and P2 be satisfied. For n=0,1,2,…., Nn,N:Yp⟶2Y be the generalized αiβj-Hp.,.,…-accretive mappings with respect to mappings h1,h2,…,hp and A1,A2,…,Ap and ∑γj<∑μi,∑βj<∑αi. We assume that Hp is λi-Lipschitz continuous with respect to mapping Ai, and hi is bi-expansive in the ith-argument. Then, Nn⟶GN if and only if(35)Rρ,N.,.,…Hp.,.,…x⟶Rρ,N.,.,…Hp.,.,…x,∀x∈Y,ρ>0,where Rρ,N.,.,…Hp.,.,…x=HpA1,A2,…,Ap+ρNh1,h2,…,hp−1x and Rρ,Nn.,.,…Hp.,.,…x=HpA1,A2,…,Ap+ρNnh1,h2,…,hp−1x.Proof.
By Theorem2, we know that Rρ,N.,.,…Hp.,.,… and Rρ,N.,.,…Hp.,.,… are 1/∑αi−∑βj+ρ∑μi−∑γj-Lipschitz continuous.
If part: letNn⟶GN. Given for any x∈Y, let zn=Rρ,Nn.,.,…Hp.,.,…x, z=Rρ,N.,.,…Hp.,.,…x. Then, 1/ρx−HpA1z,A2z,…,Apz∈Nh1,h2,…,hpz, which implies that(36)z,1ρx−HpA1z,A2z,…,Apz∈GrNh1,h2,…,hp.
By definition ofGrNh1,h2,…,hp, then ∃ is a sequence h1z˜n,h2z˜n,…,hpz˜n,y˜n such that(37)h1z˜n⟶h1z,h2zn˜⟶h2z,…,hpz˜n⟶hpz,y˜n⟶1ρx−HpA1z,A2z,…,Apz,as n⟶∞. Since y˜n∈Nnh1z˜n,h2z˜n,…,hpz˜n, we have(38)HpA1z˜n,A2z˜n,…,Apz˜n+ρy˜n∈HpA1,A2,…,Ap+ρNnh1,h2,…,hpz˜n.
Therefore,(39)zn˜=Rρ,Nn.,.,…Hp.,.,…HpA1z˜n,A2z˜n,…,Apz˜n+ρy˜n.
Since proximal-point mappingRρ,Nn.,.,…Hp.,.,… is 1/∑αi−∑βj+ρ∑μi−∑γj-Lipschitz continuous, we have(40)zn−z≤zn−z˜n+z˜n−z=Rρ,Nn.,.,…Hp.,.,…x−Rρ,Nn.,.,…Hp.,.,…HpA1z˜n,A2z˜n,…,Apz˜n+ρy˜n+zn′−z≤1∑αi−∑βj+ρ∑μi−∑γjx−HpA1z˜n,A2z˜n,…,Apz˜n−ρy˜n+zn′−z≤1∑αi−∑βj+ρ∑μi−∑γjx−HpA1z,A2z,…,Apz−ρy˜n+HpA1z,A2z,…,Apz−HpA1z˜n,A2z˜n,…,Apz˜n+z˜n−z.
SinceHp is λi-Lipschitz continuous with respect to mapping Ai, we have(41)HpA1,A2,…,Apz−HpA1,A2,…,Apz˜n≤HpA1z,A2z,…,Apz−HpA1z˜n,A2z,…,Apz+HpA1z˜n,A2z,…,Apz−HpA1z˜n,A2z˜n,…,Apz⋮+HpA1z˜n,A2z˜n,…,Apz−HpA1z˜n,A2z˜n,…,Apz˜n≤λ1z−z˜n+λ2z−z˜n+⋯+λpz−z˜n=λ1+λ2+⋯+λpz−z˜n=λz−z˜n,whereλ=λ1+λ2+⋯+λp.
Using (40) and (41), we have(42)zn−z≤1∑αi−∑βj+ρ∑μi−∑γjx−HpA1z,A2z,…,Apz−ρy˜n+1+λ1+λ2+⋯+λp∑αi−∑βj+ρ∑μi−∑γjz−z˜n.
Sincehi is bi-expansive, we have(43)hiz˜n−hiz≥biz˜n−z≥0.
Sincelimn⟶∞hiz˜n⟶hiz. From (40), (43), we get z˜n⟶z and(44)1ρx−HpA1z,A2z,…,Apz−y˜n⟶0.
From (42), we have limn⟶∞zn−z⟶0; that is,(45)Rρ,Nn.,.,…Hp.,.,…x⟶Rρ,N.,.,…Hp.,.,…x.
Only if part: let us considerRρ,Nn.,.,…Hp.,.,…⟶Rρ,N.,.,…Hp.,.,…,∀u∈Y. For any given h1x,h2x,…,hpx,y∈GrNh1,h2,…,hp, we have y∈Nh1,h2,…,hp and(46)HpA1x,A2x,…,Apx+ρy∈HpA1,A2,…,Ap+ρNh1,h2,…,hpx.
Therefore,x=Rρ,N.,.,…Hp.,.,…HpA1x,A2x,…,Apx+ρy.
Letxn=Rρ,Nn.,.,…Hp.,.,…HpA1x,A2x,…,Apx+ρy. Then, we have(47)1ρHpA1x,A2x,…,Apx−HpA1xn,A2xn,…,Apxn+ρy∈Nnh1xn,h2xn,…,hpxn.
Letyn=1/ρHpA1x,A2x,…,Apx−HpA1xn,A2xn,…,Apxn+ρy. Now, we evaluate(48)yn−y=1ρHpA1x,A2x,…,Apx−HpA1xn,A2xn,…,Apxn+ρy−y=1ρHpA1x,A2x,…,Apx−HpA1xn,A2xn,…,Apxn≤λρxn−x,whereλ=λ1+λ2+⋯+λp.
AsRρ,Nn.,.,…Hp.,.,…⟶Rρ,N.,.,…Hp.,.,… for given any x∈Y, we have xn−x⟶0. Let n⟶∞, equation (48) gives yn⟶y. Therefore, Nn⟶GN.
Now, we are giving an example to show thatN is a αiβj-Hp.,.,.,..-accretive mapping with respect to mappings h1,h2,…,hp and A1,A2,…,Ap, and Nn⟶GN, Rρ,Nn.,.,…Hp.,.,…⟶Rρ,N.,.,…Hp.,.,…. Through the MATLAB programming, we demonstrate some graphics for the convergence of the proximal-point mappings.Example 1.
For everyi=1,2,…,p, assume that p be even and Y=ℝ. Let Ai,hi:ℝ⟶ℝ and Hp:ℝp⟶ℝ be the mappings given as(49)A1z=z38+2z3,A3z=z38+2z3,….,Ap−1z=z38+2z3,A2z=z2,A4z=z2,…,=Apz=z2,h1z=z2,h3z=z4,….,hp−1x=z4,h2z=5z12,h4z=5z12,…,=hpz=5z12,HpA1z,A2z,…,Apz=A1z−A2z+⋯+Ap−1z−Apz,such that the inequality zz′+z2+z′2>0 is satisfied for each z∈ℝ2.
LetNp,N:ℝp⟶2ℝ be the set-valued mappings given as(50)Nnh1z,h2z,…,hp−1z,hpz=1n2+h1z−h2z+⋯+hp−1z−hpz,Nh1z,h2z,…,hp−1z,hpz=h1z−h2z+⋯+hp−1z−hpz.Assume that for any givent2,t3,….tp∈Y,(51)HpA1z,t2,…,tp−1−HpA1z′,t2,…,tp−1,z−z′=A1z−A1z′,z−z′=z38+2z3−z′38+2z′3,z−z′=18z−z′2z2+z′2+zz′+23z−z′2≥23z−z′2.Therefore,Hp is 2/3-strongly accretive with respect to mapping A1. Similarly, we can show that for each i=1,3,…,p−1, Hp is 2/3-strongly accretive with respect to mapping Ai.Assume that for any givent1,t3,….tp∈Y,(52)Hpt1,A2z,…,tp−1−Hpt1,A2z′,…,tp−1,z−z′=−A2z−A2z,z−z′=−z2−z′2,z−z′=−12z−z′2≥−32z−z′2.Therefore,Hp is 3/2-relaxed accretive with respect to mapping A2. Similarly, we can show that for each i=2,4,…,p, Hp is 3/2-relaxed accretive with respect to mapping Ai.Assume that for any givent˜2,t˜3,….t˜p∈Y,(53)Nnh1z˜,t˜2,…,t˜p−1−Nnh1z˜′,t˜2,…,t˜p−1,z˜−z˜′=h1z˜+1n2−h1z˜′−1n2,z˜−z˜′=z˜4−z˜′4,z˜−z˜′=14z˜−z˜′2≥14z˜−z˜′2.Therefore,Nn is 1/4-strongly accretive with respect to mapping h1. Similarly, we can show that for each i=1,3,…,p−1, Nn is 1/4-strongly accretive with respect to mapping hi.Assume that for any givent˜1,t˜3,….t˜p∈Y,(54)Nnt˜1,h2z˜,…,t˜p−1−Nnt˜1h2z˜′,…,t˜p−1,z˜−z˜′=−h2z˜+1n2−h2z˜′−1n2,z˜−z˜′=−5z˜12−5z˜′12,z˜−z˜′=−512z˜−z˜′2≥−1712z˜−z˜′2.Therefore,Nn is 17/12-relaxed accretive with respect to mapping h2. Similarly, we can show that for each i=2,4,…,p, Nn is 17/12-relaxed accretive with respect to mapping hi.Forρ=1, it is easily obtained that(55)HpA1,A2,…,Ap+ρNh1,h2,…,hpℝ=ℝ.Figure1presents the convergence of proximal-point mapping Rρ,Nn.,.,…Hp.,.,… to Rρ,N.,.,…Hp.,.,… as Nn graphical convergent to N, where Nz=p/2z/4−5z/12 and Nnz=p/2z/4−5z/12+1/n2.Figure 1
Convergence of the proximal-point mappingRρ,Nn.,.,…Hp.,.,… to Rρ,N.,.,…Hp.,.,….Figure2 presents the graph of proximal-point mapping Rρ,N.,.,…Hp.,.,…, where Nz=p/2z/4−5z/12.Figure 2
Graph of proximal-point mappingRρ,N.,.,…Hp.,.,….Now, the aim is to show thatNn is graphically convergent to N.Since, for everyh1z,h2z,…,hpz,y∈GrNh1,h2,…,hp, there exists h1zn,h2zn,…hpzn,yn∈GrNnh1,h2,…,hp, such that h1zn⟶h1z,h2zn⟶h2z,…,hzn⟶hpz,yn⟶ywithn⟶∞. Let us consider(56)zn=1+1nz,h1zn=h3zn=⋯=hp−1zn=zn4,h2zn=h4zn=⋯=hpzn=512zn,n∈ℕ.Sincelimnzn=limn1+1/nz=z. Thus, we have limn⟶∞zn⟶z.(57)Nowlimnh1zn⟶h1z,limnh3zn⟶h3z,….,limnhp−1zn⟶hp−1z,limnh2zn⟶h2z,limnh4zn⟶h4z,….,limnhpzn⟶hpz,Sinceyn=Nh1zn,h2zn,…,hpzn=h1zn−h2zn+⋯+hp−1zn−hpzn+1n2=p2zn4−5zn4+1n2.Now, we compute(58)limnyn=limnp2zn4−5zn12+1n2=p2z4−5z12=z4−5z12+z4−5z12+⋯+z4−5z12︸pterms=h1z−h2z+⋯+hp−1x−hpz=Nz=y.Thus,limn⟶∞yn⟶y, and hence, Nn graphically convergent to N. Next, we will show that Rρ,Nn.,.,…Hp.,.,…⟶Rρ,N.,.,…Hp.,.,… as Nn graphically convergent to N. Since(59)Rρ,Nn.,.,…Hp.,.,…z=HpA1,A2,…,Ap+ρNnh1,h2,…,hpz=2z−n−2−3,thenRρ,N.,.,…Hp.,.,…z=HpA1,A2,…,Ap+ρNh1,h2,…,hpz=2z−3.We computeRρ,Nn.,.,…Hp.,.,…z−Rρ,N.,.,…Hp.,.,…z=2z−n−2−3−2z−3, which shows that as n⟶∞Rρ,Nn.,.,…Hp.,.,…−Rρ,N.,.,…Hp.,.,…⟶0. i.e.Rρ,Nn.,.,…Hp.,.,…⟶Rρ,N.,.,…Hp.,.,…withNn graphically convergent to N.


## 4. Set-Valued Variational Inclusions

LetY be a real Banach space. Let Ai,hi:Y⟶Y,Hp,Q:Yp⟶Y be the single-valued mappings and Ui:Y⟶2Y be a set-valued mapping. Let set-valued mapping N:Yp⟶2Y be a generalized αiβj-Hp.,.,…,.-accretive mapping.Now, problem is to findz∈Y,w1∈U1z,w2∈U2z,…,wp∈Upz such that(60)Θ∈Qw1,w2,…,wp+Nh1z,h2z,…,hpz.Problem (60) is called a set-valued variational inclusion problem (for short, SVIP).Special cases are as follows:(i)
IfQw1,w2,…,wp=Qw1,w2 and Nh1z,h2z,…,hpz=Nh1z,h2z, then SVIP (60) reduced to find z∈Y,w1∈U1z,w2∈U2z such that(61)Θ∈Qw1,w2+Nh1z,h2z.(ii)
Ifh1=h2=h, U1=U2=U, and N.,.=N., then SVIP (61) reduced to find z∈Y,w∈Uz such that(62)Θ∈w+Nhz.SVIP (62) has been studied by Huang [34] when N is a m-accretive mapping.(iii)
IfQw1,w2=Qw, U1=U2=U, and U is single-valued mapping, then SVIP (40) reduced to find z∈Y such that(63)Θ∈Uz+Nz.VIP (63) has been studied by Zou and Huang [24] when N is an H.,.-accretive mapping. For the generalized m-accretive mapping, VIP (63) was studied by Bi et al. [35].Theorem 4.
For any givenz,w1,w2,…,wp∈Y×Yp, z,w1,w2,…,wp is a solution of SVIP (60) if and only if z,w1,w2,…,wp satisfies(64)z=Rρ,N.,.,…Hp.,.,…HpA1,A2,…,Apz−ρQw1,w2,…,wp,where ρ is non-negative constant.Proof.
Letz,w1,w2,…,wp be the solution of SVIP (60), we have(65)Θ∈Qw1,w2,…,wp+Nh1z,h2z,…,hpz⇔HpA1z,A2z,…,Apz∈HpA1z,A2z,…,Apz+ρQw1,w2,…,wp+Nh1z,h2z,…,hpz⇔HpA1z,A2z,…,Apz−ρQw1,w2,…,wp∈HpA1z,A2z,…,Apz+ρNh1z,h2z,…,hpz⇔z=Rρ,N.,.,…Hp.,.,…HpA1z,A2z,…,Apz−ρQw1,w2,…,wp.Algorithm 1.
For any givenz01∈Y, select w01∈U1z01,w02∈U2z01,…,w0p∈Upz01 and obtain zn1, wn1, wn2, …, wnp, by an iterative scheme(66)zn+11=Rρ,N.,.,…Hp.,.,…HpA1,A2,…,Apzn1−ρQwn1,wn2,…,wnp,wn1∈U1zn1:wn+11−wn1≤D˜U1zn+11,U1zn1,wn2∈U2zn1:wn+12−wn2≤D˜U2zn+11,U2zn1,⋮wnp∈Upzn1:wn+1p−wnp≤D˜Upzn+11,Upzn1,where n=0,1,2,… and ρ is non-negative constant.Now, we establish the result in the context of the existence and uniqueness of the solution of SVIP (60).Theorem 5.
Let SVIP (60) satisfies the assumptions P1-P4 and Nn,N:Yp⟶2Y be the generalized αiβj-Hp.,.,…-accretive mappings with respect to mappings h1,h2,…,hp and A1,A2,…,Ap and ∑γj<∑μi,∑βj<∑αi. For each i∈1,2,…,p, Hp is λi-Lipschitz continuous with respect to mapping Ai, Ui is ξi-D-Lipschitz, and Q is β¯i-strongly accretive and ζi-Lipschitz continuous in the ith-argument. In addition, the following condition is satisfied:(67)λ2+2ρβ¯−kλ+ρk+1∑αi−∑βj+ρ∑μi−∑γj2<1,whereλ=λ1+λ2+⋯+λp,β¯=β¯1+β¯2+⋯+β¯p,k=ξ1ζ1+ξ2ζ2+⋯+ξpζp.Then,(1)
SVIP (60) has a unique solution z1,w1,w2,…,wp in Y(2)
Iterative sequenceszn1,wn1,wn2,…wnp developed by Algorithm 1 converge strongly to z1,w1,w2,…,wp a solution of SVIP (60)Proof.
LetP:Y⟶Y be given as(68)Px=Rρ,N.,.,…Hp.,.,…HpA1x,A2x,…,Apx−ρQw1,w2,…,wp,∀x,w1,w2,..,wp∈Y.
Letz1,w1,w2,…,wp,y1,w1′,w2′,…,wp′∈Y, we have(69)Pz1=Rρ,N.,.,…Hp.,.,…HpA1z1,A2z1,…,Apz1−ρQw1,w2,…,wp,Py1=Rρ,N.,.,…Hp.,.,…HpA1y1,A1y1,…,Apy1−ρQw1′,w2′,…,wp′.
Using (64) and Lipschitz continuity of the proximal-point mapping, then(70)Pz1−Py1=Rρ,N.,.,…Hp.,.,…HpA1,A2,…,Apz1−ρQw1,w2,…,wp−Rρ,N.,.,…Hp.,.,…HpA1,A2,…,Apy1−ρQw1′,w2′,…,wp′≤1∑αi−∑βj+ρ∑μi−∑γj×HpA1,A2,…,Apz1−ρQw1,w2,…,wp−HpA1,A2,…,Apy1−ρQw1′,w2′,…,wp′≤1∑αi−∑βj+ρ∑μi−∑γjHpA1,A2,…,Apz1−HpA1,A2,…,Apy1−ρQw1,w2,…,wp−Qw1′,w2′,…,wp′.
From Lemma1, we have(71)HpA1,A2,...,Apz1−HpA1,A2,...,Apy1−ρQw1,w2,...,wp−Qw1′,w2′,...,wp′2≤HpA1,A2,...,Apz1−HpA1,A2,...,Apy12−2ρQw1,w2,...,wp−Qw1′,w2′,...,wp′,JHpA1,A2,...,Apz1−HpA1,A2,...,Apy1−ρQw1,w2,...,wp−Qw1′,w2′,...,wp′≤HpA1,A2,...,Apz1−HpA1,A2,...,Apy12−2ρQw1,w2,...,wp−Qw1′,w2′,...,wp′,JHpA1,A2,...,Apz1−HpA1,A2,...,Apy1−ρQw1,w2,...,wp−Qw1′,w2′,...,wp′+Jz1−y1+2ρQw1,w2,...,wp−Qw1′,w2′,...,wp′,Jz1−y1≤HpA1,A2,...,Apz1−HpA1,A2,...,Apy12−2ρQw1,w2,...,wp−Qw1′,w2′,...,wp′HpA1,A2,...,Apz1−HpA1,A2,...,Apy1+ρQw1,w2,...,wp−Qw1′,w2′,...,wp′+z1−y1+2ρQw1,w2,...,wp−Qw1′,w2′,...,wp′,Jz1−y1.
SinceHP is λi-Lipschitz continuous with respect to mapping Ai, then(72)HpA1,A2,…,Apz1−HpA1,A2,…,Apy1≤HpA1z1,A2z1,…,Apz1−HpA1y1,A2z1,…,Apz1+HpA1y1,A2z1,…,Apz1−HpA1y1,A2y1,…,Apz1⋮+HpA1y1,A2y1,…,Apz1−HpA1y1,A2y1,…,Apy1≤λ1z1−y1+λ2z1−y1+⋯+λpz1−y1=λ1+λ2+⋯+λpz1−y1=λz1−y1,whereλ=λ1+λ2+⋯+λp.
Now, we compute(73)Qw1,w2,…,wp−Qw1′,w2′,…,wp′,Jz1−y1=Qw1,w2,…,wp−Qw1′,w2,…,wp,Jz1−y1+Qw1′,w2,…,wp−Qw1′,w2′,…,wp,Jz1−y1⋮+Qw1′,w2′,…,wp−Qw1′,w2′,…,wp′,Jz1−y1≥β¯1z1−y12+β¯2z1−y12+⋯+β¯pz1−y12≥β¯1+β¯2+⋯+β¯pz1−y12=β¯z1−y12,whereβ¯=β¯1+β¯2+⋯+β¯p.
SinceQ is ζi-Lipschitz continuous and Ui is ξi-D-Lipschitz continuous, we have(74)Qw1,w2,…,wp−Qw1′,w2′,…,wp′≤Qw1,w2,…,wp−Qw1′,w2,…,wp+Qw1′,w2,…,wp−Qw1′,w2′,…,wp⋮+Qw1′,w2′,…,wp−Qw1′,w2′,…,wp′≤ζ1w1−w1′+ζ2w2−w2′+⋯+ζpwp−wp′≤ζ1D˜U1z1,U1y1+ζ2D˜U2z1,U2y1+⋯+ζpD˜Upzp,Upyp=ζ1ξ1z1−y1+ζ2ξ2z1−y1+⋯+ζpξpz1−y1=ζ1ξ1+ζ2ξ2+⋯+ζpξpz1−y1=kz1−y1,wherek=ζ1ξ1+ζ2ξ2+⋯+ζpξp.
Using equations (71)–(74) in equation (70), we have(75)Pz1−Py1≤Yz1−y1,whereY=λ2+2ρβ¯−kλ+ρk+1∑αi−∑βj+ρ∑μi−∑γj2.
From condition (67), we have(76)λ2+2ρβ¯−kλ+ρk+1∑αi−∑βj+ρ∑μi−∑γj2<1.
Thus, we have0<Y<1. Therefore, (75) implies that(77)P=Rρ,N.,.,…Hp.,.,…HpA1,A2,…,Ap−ρQ,which is a contraction mapping and has a unique fixed point z1 in Y. Hence, z1 is a unique solution of SVIP (60).
Now, we will prove thatzn1 strongly converges to z1. Using Algorithm 1, we have(78)zn+11−z1=Rρ,Nn.,.,…Hp.,.,…HpA1,A2,…,Apzn1−ρQwn1,wn2,…,wnp−Rρ,N.,.,…Hp.,.,…HpA1,A2,…,Apz1−ρQw1,w2,…,wp≤Rρ,Nn.,.,…Hp.,.,…HpA1,A2,…,Apzn1−ρQwn1,wn2,…,wnp−Rρ,Nn.,.,…Hp.,.,…HpA1,A2,…,Apz1−ρQw1,w2,…,wp+Rρ,Nn.,.,…Hp.,.,…HpA1,A2,…,Apz1−ρQw1,w2,…,wp−Rρ,N.,.,…Hp.,.,…HpA1,A2,…,Apz1−ρQw1,w2,…,wp.
Using Theorem3, we have(79)Rρ,Nn.,.,…Hp.,.,…HpA1,A2,…,Apz1−ρQw1,w2,…,wp⟶Rρ,N.,.,…Hp.,.,…HpA1,A2,…,Apz1−ρQw1,w2,…,wp.
Let(80)θn=Rρ,Nn.,.,…Hp.,.,…HpA1,A2,…,Apz1−ρQw1,w2,…,wp−Rρ,N.,.,…Hp.,.,…HpA1,A2,…,Apz1−ρQw1,w2,…,wp⟶0.
In the light of equations (71)–(73), one can obtain(81)Rρ,N.,.,…Hp.,.,…HpA1,A2,…,Apzn1−ρQwn1,wn2,…,wnp−Rρ,Nn.,.,…Hp.,.,…HpA1,A2,…,Apz1−ρQw1,w2,…,wp≤Λzn1−z1,where Λ=λ2+2ρβ¯−kλ+ρk+1/∑αi−∑βj+ρ∑μi−∑γj2.
Using (79)–(81) in (78), we get(82)‖zn+11−z1‖=Λ‖zn1−z1‖+θn.
From (67), we have λ2+2ρβ¯−kλ+ρk+1/∑αi−∑βj+ρ∑μi−∑γj2<1. From (80), we have limn⟶∞θn⟶0. By using Lemma 2, limn⟶∞zn1⟶z1. By calD-Lipschitz continuity of w1,w2,…,wp and Algorithm 1, we have(83)wn+11−wn1≤D˜U1zn+11,U1zn1≤ζ1zn+11−zn1,wn+12−wn2≤D˜U2zn+11,U2zn1≤ζ2zn+11−zn1,⋮wn+1p−wnp≤D˜Upzn+11,Upzn1≤ζpzn+11−zn1.
Thus,wn1,wn2,….,wnp are the Cauchy sequences, and then, there exists w1,w2,…,wp such that wn1⟶w1,wn2⟶w2,…,wnp⟶wp, with n⟶∞. We will prove that w1∈U1z1. Since wn1∈U1z1, we have(84)d˜w1,U1z1≤w1−wn1+d˜wn1,U1z1≤w1−wn1+D˜U1zn1,U1z1≤w1−wn1+ξ1zn1−z1.
SinceU1z1 is closed, thus w1∈U1z1. Similarly, we can prove w2∈U2z1,w3∈U3z1,…,wp∈Upz1. By the Lipschitz continuity, we know that z1,w1,w2,…,wp satisfy the following relation:(85)z1=Rρ,N.,.,…Hp.,.,…HpA1,A2,…,Apz1−ρQw1,w2,…,wpp.
By Theorem4, SVIP (60) has a solution z1,w1,w2,…,wp.


## 5. Conclusion

In this manuscript, we used proximal-point mapping linked with generalizedαiβj-Hp.,.,…-accretive mapping to discuss the graph convergence of this class of accretive mappings. Furthermore, we have discussed convergence analysis of the proposed iterative algorithm to solve SVIP (60) in real Banach space. As an extension of αβ-H.,.-accretive mapping introduced by Kazmi et al. [13], the generalized αiβj-Hp.,.,…-accretive mapping has some important applications in physics, economics, and physical sciences. In the future, we can apply the technique used in this article to study the Yosida inclusion problems in the framework of Banach spaces.



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