# Extended Perturbed Mixed Variational-Like Inequalities for Fuzzy Mappings

Authors: Muhammad Bilal Khan; Muhammad Aslam Noor; Khalida Inayat Noor; Ahmad Termimi Ab Ghani; Lazim Abdullah
Journal: Journal of Mathematics
 (2021)

---

## Abstract

In this article, our aim is to consider a class of fuzzy mixed variational-like inequalities (FMVLIs) for fuzzy mapping known as extended perturbed fuzzy mixed variational-like inequalities (EPFMVLIs). As exceptional cases, some new and classically defined “FMVLIs” are also attained. We have also studied the auxiliary principle technique of auxiliary “EPFMVLI” for “EPFMVLI.” By using this technique and some new analytic results, some existence results and efficient numerical techniques of “EPFMVLI” are established. Some advanced and innovative iterative algorithms are also obtained, and the convergence criterion of iterative sequences generated by algorithms is also proven. In the end, some new and previously known existence results and algorithms are also studied. Results secured in this paper can be regarded as purification and development of previously familiar results.


---

## Body

## 1. Introduction

The boundless research work of fuzzy set and systems [1] has been devoted in advancement of different fields. It contributes to a vast class knowledge and appears in pure mathematics and applied sciences as well as operation research, computer science, managements sciences, artificial intelligence, control engineering, and decision sciences [2]. As a part of these knowledge developments, Chang and Zhu [3] initiated to introduce a new type of variational inequality for fuzzy mapping, which is known as fuzzy variational inequality. In fuzzy optimization, Noor [4–6] studied the characterization of minimum of convex fuzzy mapping through fuzzy variational inequality and fuzzy mixed variational inequality and obtained some advanced and effective iterative algorithms. Moreover, they showed the parallel correlation by linking fuzzy variational inequalities and fuzzy Wiener–Hopf equations. Similarly, they established parallel correspondence between fuzzy variational inequalities and the resolvent equations for fuzzy mappings and encouraged some important and novel new iterative algorithms and discussed their convergence criteria. They also introduced fuzzy mixed variational inequalities, and by using the classical auxiliary principle technique, some new existence theorems and iterative algorithms for fuzzy mixed variational inequalities are attained. It is worthy to mention here that one of the most considered generalizations of convex fuzzy mappings is preinvex fuzzy mapping. The idea of fuzzy preinvex mapping on the invex set was introduced and studied by Noor [7]. Moreover, any local minimum of a preinvex fuzzy mapping is a global minimum on invex set, and necessary and sufficient condition for fuzzy mapping is to be preinvex if its epigraph is an invex set.Furthermore, it has been verified that fuzzy optimality conditions of differentiable fuzzy preinvex mappings can be distinguished by variational-like inequalities. Motivated and inspired by the ongoing research work, many authors discussed fuzzy variational inequalities and its important generalizations and its applications in different fields [8, 9]. In the subsequent text, we will review the applications and generalizations of variational inequalities for fuzzy mappings. First, we give some literature survey of variational inequality and its generalizations. In early 1960s, the idea of variational inequality was initiated by Hartman and Stampacchia [10]. The useful generalizations of variational inequality are variational-like inequality (in short, VLI) and generalized mixed variational-like inequality (in short, GMVLI). It is well known that in classical variational inequality theory, the projection method fails to discuss the existence of solutions of variational-like inequalities. This familiarity has attracted many authors to propose the auxiliary principle technique to acknowledge existence results for different variational inequalities. Glowinski et al. [11] suggested the auxiliary principle technique. For further information about the auxiliary principle technique, we refer the readers to [12–17] and the references therein. Similarly, the projection type methods can be used to recommend iterative methods for “VLI” and “GMVLI” for fuzzy mappings. To overcome this challenging task and with the help of classical auxiliary principle techniques, Chang [8], Chang et al. [18], and Kumam and Petrot [19] studied the idea of “GMVLI” and complementarity problems for ordinary set-valued mapping and fuzzy mappings in different contexts with compact and noncompact values. For applications of “GMVLI,” see [3–7, 9, 12–14, 16, 19–26] and the references therein.In this article, we shall introduce and discuss a new type of “GMVLI” for fuzzy mappings, which is known as extended perturbed fuzzy mixed variational-like inequality (EPFMVLI) for fuzzy mapping because by using this technique, we can easily handle the functional which is the sum of differentiable preinvex fuzzy mapping and strongly preinvex fuzzy mapping and their special cases like the single-valued functional which is the sum of the differentiable preinvex functions and strongly preinvex functions. By using this technique and some new analytic results, some existence results and efficient numerical techniques of “PFMVLI” are established. As a result of this technique, some advanced and innovative iterative algorithms can be obtained. Moreover, the convergence criteria of iterative sequences generated by algorithms can also be proven. At the end, we shall discuss some particular cases of “PFMVLI” for ordinary set-valued mappings and fuzzy mappings (see [3–7, 9, 12–14, 16, 19–22] and the references therein).


## 2. Preliminaries

LetJ be a real Hilbert space and ∅≠ℭ⊂J be a convex set. We denote the collection CBJ of all nonempty bounded and closed subsets of J, and D.,.is the Hausdorff metric on CBJ defined by(1)DK,A=maxsupu∈Adu,K,supϑ∈Kdϑ,A,A,K∈CBJ.A mappingψ:J⟶0,1 is called fuzzy set. If a∈0,1, then the set ψa=u∈J|ψu≥a is known as a-level set of ψ. If a=0, then suppψ=u∈J|ψu>0 is called support of ψ. By ψ0, we define the closure of suppψ.In what follows,ℱJ=A:J⟶I=0,1 denotes the family of all fuzzy sets on J. A mapping P from J to ℱJ is called a fuzzy mapping. If P:J⟶ℱJis a fuzzy mapping, then the set Pu for u∈J is a fuzzy set in ℱJ (in the sequel we denote Pu by Pu) and Puϑϑ∈J is the degree of membership of ϑin Pu.Definition 1.
(see [18]). If for each u∈J, the function ϑ⟶Puϑ is upper semicontinuous, then fuzzy mapping P is called closed. If ϑr⊂J is a net satisfying ϑr⟶ϑ0∈J, then Pu has the following property:(2)limtsuprPuϑu≤Puϑ0.Definition 2.
(see [18]). A closed fuzzy mapping P:J⟶ℱJ is said to satisfy the condition (B), if there exists a function P:J⟶0,1 such that for each u∈J, the set(3)Puru=ϑ∈J:Puϑ≥ru≠∅,is a bounded subset of J.Remark 1.
(see [18]). From Definitions 1 and 2, it can be easily seen that Puru∈CBJ. Indeed, let ϑrr∈Γ⊂Puru be a net and ϑr⟶ϑ0∈J. Then, from (2) and (3), we have limtsuprPuϑr≤Puϑ0 and Puϑr≥ru for each r∈Γ. Since P is closed with condition (B), we have(4)Puϑ0≥limtsuprPuϑr≥ru.
This implies thatϑ0∈Puru, and so Puru∈CBJ.Problem 1.
LetJ be a real Hilbert space. Then, for given nonlinear mappings ξ.,.:J×J⟶J, ℳ.,.:J×J⟶J, we consider the problem of finding u∈J, p,q∈J and constant ω>0, such that(5)ℳp,q,ξϑ,u+Jϑ,u−Ju,u+ωξϑ,u2≥0,∀ϑ∈J,where P,V:ℭ⟶ℱJare two closed fuzzy mappings satisfying condition (B) with function r,s:J⟶0,1, respectively, such that(6)Pup≥ru,Vuq≥su,i.e.∈Puru,q∈Vusu,and J.,.:J×J⟶Ris a nondifferentiable function fulfilling the following assumption.Assumption 1.
(a)
For anyu,ϑ,p∈J, there exists constant ∂>0, such that(i)
Ju,ϑ≤∂uϑ(ii)
Ju,ϑ−Ju,p≤∂uu−p(b)
Ju,ϑis linear in respect of u(i.e., for any u,ϑ∈JJ−u,ϑ=−Ju,ϑ)(c)
Ju,ϑis convex in respect of ϑ
The inequality (5) is called “EPFMVLI.” It is important to mention that the generalized fuzzy mixed variational-like inequality (in short, GFMVLI), classical mixed variational-like inequality (in short, MVLI), and associated fuzzy optimizations problems are particular cases of inequality (5). For applications, see [3, 8–10, 13] and the references therein.
Now, we study some certain cases of problem (5).(1)
Ifω=0, then problem (5) is analogous to finding u∈Jp∈Puruq∈Vusu, such that(7)ℳp,q,ξϑ,u+Jϑ,u−Ju,u≥0,∀ϑ∈J.This type of problem is called “GFMVLI” and was studied and established by Chang et al. [18] and, under some added condition, improved by Kumam and Petrot [19].(2) LetP,V:J⟶ℱJbe an ordinary multivalued mapping and J,ξ be the mapping in problem (5).Now, we define a fuzzy mappingP˜.,V˜.:J⟶ℱJas follows:(8)P˜u=XPu,V˜u=XVu,whereXPuXVuare two characteristic functions of Pu, Vu. From (8), it can straightforwardly be noticed that P˜, V˜ are two closed fuzzy mapping fulfilling condition (B) with constant function ru=1,su=1, respectively, for all u∈J. Also,(9)P˜uru=XPu1=ϑ∈J:Puϑ=1=Pu,V˜usu=XVu1=ϑ∈J:Vuϑ=1=Vu.Then, problem (5) is parallel to detecting u∈J, such that(10)P˜up=1,V˜uq=1,i.e.p∈Pu,q∈Vu,ℳp,q,ξϑ,u+Jϑ,u−Ju,u+ωξϑ,u2≥0,∀ϑ∈J.This kind of problem is called the set-valued “EPMVLI.” This mixed variational-like inequality is also new one.(3)
Ifω=0, then problem (10), is analogous to finding u∈J, p∈Pu, q∈Vu, such that(11)ℳp,q,ξϑ,u+Jϑ,u−Ju,u≥0,∀ϑ∈J.This type of problem is called “GMVLI” and was studied and established by Noor [18] and, under some added condition, improved by Zeng [17].(4) Ifℳp,q=ℳpand P:J⟶FJ, then problem (5) is analogous to finding u∈J, p∈Puru, such that(12)ℳp,ξϑ,u+Jϑ−Ju+ωξϑ,u2≥0,∀ϑ∈J.Then, problem (12) is called “PFMVLI.” This class of fuzzy variational-like inequality is also new one.Note that problem (12) is a particular case of problem (5); therefore, we discuss some other special cases of problem (5) using problem (12).(5)
Ifℳ=I(identity mapping), p∈J, and P:J⟶J is single valued, then problem (10) is parallel to finding u∈Jsuch that(13)Pu,ξϑ,u+Jϑ−Ju+ωξϑ,u2≥0,∀ϑ∈J.This is known as “PMVLI” and was studied by Noor et al. [12].(6)
Ifξϑ,u=ϑ−u, then problem (12) is called strongly fuzzy mixed variational inequality and is parallel to finding u∈J, p∈Purusuch that(14)ℳp,ϑ−u+Jϑ−Ju+ωϑ−u2≥0,∀ϑ∈J.This class of “FMVI” is also new one. The classical FMVLI and associated fuzzy optimizations problems are special cases of inequality (15) [21].(7)
Ifω=0, then problem (12) is parallel to finding u∈J, p∈Puru such that(15)ℳp,ξϑ,u+Jϑ−Ju≥0,∀ϑ∈J,which is known as “FMVLI.” The FMVLI is a generalised form of Chang et al. [18] and Kumam and Petrot [19]. It is also an ordinary set-valued mapping of Noor [22]. For the development of numerical methods and applications of (15), see [9, 15, 16, 27] and the references therein.(8)
Forω=0,problem (14) reduces to(16)u∈J,p∈Puru,ℳp,ϑ−u+Jϑ−Ju≥0,∀ϑ∈J,which is known as “FMVI.” Chang et al. [18] and Kumam and Petrot [19] studied it as a special case. In case of ordinary set-valued mapping as developed by Lions and Stampacchia [20], for applications, see [11, 15, 22, 27] and the references therein.(9)
Whenℳ=Iand P:J⟶ℱJis a fuzzy mapping, then problem (16) is parallel to finding u∈J, p∈Puru.(17)p,ϑ−u+Jϑ−Ju≥0,∀ϑ∈J.This is known as “FMVI” (see [5]).(10)
IfJ. is an indicator mapping of a closed convex set ℭin J, that is,(18)IJu=0,u∈ℭ,∞,otherwise,then problem (15) is analogous to finding p∈J, p∈Purusuch that(19)ℳp,ξϑ,u≥0,∀ϑ∈J,which is known as “FVLI.”(11)
Whenℳ=I, then problem (19) is analogous to finding u∈J, p∈Puru such that(20)p,ξϑ,u≥0,∀ϑ∈J,which is also known as “FVLI.” (see [7]). For the applications of “FVLI” and fuzzy optimization problem, see [9, 16, 22, 27] and the references therein.(12)
Ifξϑ,u=ϑ−u, then problem (19) is analogous to finding u∈J, p∈Puru such that(21)ℳp,ϑ−u≥0,∀ϑ∈J.This is known as “FVI.”(13)
Ifℳ=I, then problem (21) is analogous to finding u∈J, p∈Puru such that(22)p,ϑ−u≥0,∀ϑ∈J.
This is also known as “FVI” (see [3, 4]). The “VI” is mainly due to Stampacchia and Guido [28] and Lions and Stampacchia [20], as developed and studied by Noor [4]. For the applications of problem (22), see [9, 11, 20, 21] and the references therein.
From the above discussion, it can be easily seen that problems (7)–(22) are particular cases of “EPFMVLI” (5). In fact, “EPFMVLI” is more generalized and unifying one, which is main motivation of our work. For a proper and suitable choice of P, ξ, and J, we can choose a number of known and unknown “PFVLI” and complementary problems.
Next, we will use mathematical terminologies S-mixed monotone and L-continuous for strongly mixed monotone and Lipschitz continuous, respectively.Definition 3.
(see [18]). If fuzzy mappings P,V:J⟶ℱJ are closed and fulfil the condition (B) with functions r,s:J⟶0,1, then nonlinear mapping ℳ.,.:J×J⟶J is said to be(i)
γ-L-continuous in respect of first argument if for any u1,u2∈Jand p1∈Pu1ru1, u2∈Pu2ru2, there exists a constant γ>0 such that(23)ℳp1,.−ℳp2,.≤γp1−p2,for allp1∈Pu1ru1,p2∈Pu2ru2.(ii)
β-strongly mixed monotone in respect of P and V if there exists a constant β>0 such that(24)ℳp1,q1−ℳp2,q2,u1−u2≥βu1−u22,for allp1∈Pu1ru1, p2∈Pu2ru2, q1∈Vu1su1, q2∈Vu2su2.(iii)
α-L-continuous in respect of second argument if for any ϑ1,ϑ2∈J and q1∈Vϑ1sϑ1, q2∈Vϑ2sϑ2, there exists a constant α>0, such that(25)ℳ.,q1−ℳ.,q2≤αq1−q2.Definition 4.
(see [18]). Let P,V:J⟶ℱJ be two fuzzy mappings. Then,(iv)
P is said to be P-L-continuous if for any u,ϑ∈J, there exists a function r:ℭ⟶0,1and constant λ>0, such that(26)DPuru,Pϑrϑ≤λϑ−u.(v)
V is said to be V-L-continuous: if there exists a function s:ℭ⟶0,1and constant ξ>0, such that(27)DVusu,Vϑsϑ≤ξϑ−u,whereD.,. is the Hausdorff metric on ℱJ.
In particular, from (i) and (ii), we have(28)βu1−u22≤ℳp1,q1−ℳp2,q2,u1−u2≤ℳp1,q1−ℳp2,q2u1−u2≤γp1−p2u1−u2≤γDPu1ru1,Pu2ru2u1−u2.
From (iv), we have(29)≺¯γλu1−u22,which implies that β≺¯γλ. Similarly, from (ii), (iii), and (v), we can observe that β≺¯αξ.Definition 5.
(see [12]). The bifunction ξ.,.:J×J⟶J is said to be(vi)
S-monotone: if there exists constantμ>0, such that(30)ξϑ,u,ϑ−u≥μϑ−u2,for allu,ϑ∈J.(vii)
L-continuous: if there exists constantδ>0, such that(31)ξϑ,u≺¯δϑ−u,for allu,ϑ∈J.
From (vi) and (vii), we can observe thatμ≺¯δ.
For anyK⊆J, we denote the conv(K), the convex hull of K. A set-valued mapping T:J⟶2Jis called a KKM mapping if, for every finite subset ϑ1,ϑ2,ϑ3,ϑ4,…,ϑnof J,(32)convϑ1,ϑ2,ϑ3,ϑ4…,ϑn⊆∪i=1nTϑi.Lemma 1.
(see [29]). Let K be arbitrary nonempty in a topological vector space G and let T:K⟶2J be a KKM mapping. If Tϑ is closed for all ϑ∈K and is compact for at least one ϑ∈K, then(33)∩ϑ∈KTϑ≠∅.Theorem 1.
(see [8]). Let G be a locally convex Hausdorff topologically vector space and g:G⟶R∪+∞ be a properly convex functional (i.e., a functional g. is called proper, if gu>−∞ for all u∈G and Ju≡+∞). Then, g is lower semicontinuous on G if and only if, g is weakly lower semicontinuous on G.Assumption 2.
Letℳ:J×J⟶J and ξ.,.:J×J⟶J be two mappings satisfying the following condition:(a)
ξϑ,u+ξu,ϑ=0 (and so ξu,u=0, for allu∈J), for all u,ϑ∈J.(b)
For any givenu∈J, the mapping ϑ⟼ℳp,q,ξϑ,uis concave, where p∈Puru,q∈Vusu.(c)
For any givenu∈J, the mapping ϑ↦ℳp,q,ξϑ,uis lower semicontinuous, where p∈Puru,q∈Vusu.(34)un⟶u,pn⟶p,qn⟶qimplyℳp,q,ξϑ,u≤liminfn⟶∞ℳpn,qn,ξϑ,un.Lemma 2.
(see [30]). Let X,d be a complete metric space and K1,K2∈CB(X) and r≥1 be any real number. Then, for every k1∈K1, there existk2∈K2 such that dk1,k2≤rDK1,K2.
In the next sections, we will use the above preliminaries.


## 3. Auxiliary Principle and Algorithm

Now, we review the Auxiliary problem associated with “EPFMVLI” (5) and prove an existence theorem for the auxiliary problem. Furthermore, based on this existence result, we suggest an iterative algorithm for the “PFMVLI” (5).Auxiliary problem: for a givenu∈ℭ,p∈Puru,q∈Vususatisfying problem (5), we consider the problem of finding u∈J, such that(35)z,ϑ−z≥u,ϑ−z−ρℳp,q,ξϑ,z+ρJu,z−ρJu,ϑ−ωρξϑ,z2,for ∀ϑ∈ℭ, where ρ>0 is a constant.The inequality (35) is also called auxiliary “EPFMVLI.”Theorem 2.
LetJ be real Hilbert space and ℭ≠∅ be a closed bounded subset of J. Let P,V:J×J⟶ℱJ be two closed continuous fuzzy mappings satisfying condition (B) with function r,s:J⟶0,1. If Assumptions 1 and 2 are satisfied and bifunction ξ.,.:ℭ×ℭ⟶J is L-continuous with constant δ>0, then auxiliary problem (35) has a unique solution.Proof.
Letu∈ℭ, p∈Puru,q∈Vusu; we consider the mapping T:J⟶2J defined by(36)Tϑ=w∈ℭ:w−u,ϑ−w≥−ρℳp,q,ξϑ,w+ρJu,w−ρJu,ϑ−ωρξϑ,w2,for all ϑ∈J.
From (36), it can be easily seen that for each ϑ∈ℭ, Tϑ≠∅, since ϑ∈Tϑ.To obtain the solution, firstly we show that T:J⟶2Jis a KKM mapping. Suppose the contrary, that is, T:J⟶2Jis not KKM mapping, then there exists a finite subset ϑ1,ϑ2,ϑ3,ϑ4…,ϑnof ℭ and constant τi≥0, i=1,2,3,4,…,k with ∑i=1kτi=1such that(37)w∗=∑i=1kτiϑi∉∪i=1kTϑi,for all1,2,3,4,…,k.
Then, we have(38)w∗−u,ϑi−w∗+ρℳp,q,ξϑi,w∗+ρJu,ϑi−ρJu,w∗+ωρξϑi,w∗2<0.
From Assumptions1 and 2, the above inequality yields(39)0>∑i=1kτiw∗−u,ϑi−w∗+ρ∑i=1kτiℳp,q,ξϑi,w∗+ρ∑i=1kτiJu,ϑi−ρJw∗+ωρ∑i=1kτiξϑi,w∗2≥w∗−u,w∗−w∗+ρℳp,q,ξw∗,w∗−ρJu,w∗+ρJu,w∗+ωρξw∗,w∗2=0.
This is a contradiction. Hence,T:J⟶2J is a KKM mapping.
SinceTϑ¯p, the weak closure of Tϑ is weakly closed subset of a bounded set ℭ⊆J, so it is weakly compact. Hence, by Lemma 1,(40)∩ϑ∈ℭTϑp¯≠∅.
Let(41)z∈∩ϑ∈ℭTϑp¯.
Then, there exists a sequenceznin Tϑ(fix ϑ), such that zn⟶z. Then,(42)zn−u,ϑ−zn+ρℳp,q,ξϑ,zn+ρJu,ϑ−ρJu,zn+ωρξϑ,zn2≥0.
Now, by using the property of inner product,(43)zn−u,ϑ−zn=zn−u,ϑ+u,zn−zn,zn=zn−u,ϑ+u,zn−zn2.
Since. is weakly lower semicontinuous, we have(44)limsupn⟶∞zn−u,ϑ−zn=limsupn⟶∞zn−u,ϑ+u,zn−zn2=limsupn⟶∞zn−u,ϑ+limsupn⟶∞u,zn−liminfn⟶∞zn2≤z−u,ϑ−z.
Again, by Assumptions1 and 2, we get(45)0≤limsupn⟶∞zn−u,ϑ−zn+ρℳp,q,ξϑ,zn+ρJu,ϑ−ρJu,zn+ωρξϑ,zn2=limsupn⟶∞zn−u,ϑ−zn+ρlimsupn⟶∞ℳp,q,ξϑ,zn+ρlimsupn⟶∞Ju,ϑ−ρliminfn⟶∞Ju,zn+ωρlimsupn⟶∞ξϑ,zn2≤z−u,ϑ−z+ρℳp,q,ξϑ,z+ρJu,ϑ+ρJu,z+ωρξϑ,z2.
This implies that(46)z,ϑ−z≥u,ϑ−z−ρℳp,q,ξϑ,z+ρJϑ−ρJz−ωρξϑ,z2,for all ϑ∈J.
Hence,z is solution of auxiliary problem (35).
Uniqueness: Letz1∈ℭ be also a solution of Auxiliary problem. Then, we have(47)z1,ϑ−z1≥u,ϑ−z1−ρℳp,q,ξϑ,z1−ρJu,ϑ+ρJu,z1−ρωξϑ,z12,for all ϑ∈J.
Replacingϑ by z1in (46) and ϑ by zin (47), we have(48)z,z1−z≥u,z1−z−ρℳp,q,ξz1,z+ρJu,z1−ρJu,z−ωρξz1,z2,(49)z1,z−z1≥u,z−z1−ρℳp,q,ξz,z1−ρJu,z+ρJu,z1−ρωξz,z12.
Using Assumption2 (ξu1,u2=−ξu2,u1) and then adding (48) and (49), we have(50)z1−z,z−z1≥0,which implies that z1=z is the uniqueness of the solution of auxiliary problem (35).
This completes the proof of Theorem2.Algorithm 1.
Atn=0, start with initial value u0∈J, p0∈Pu0ru0,q0∈Vu0su0; from Theorem 2, auxiliary problem (35) has a unique solution u1∈J, such that(51)u1,ϑ−u1≥u0,ϑ−u1−ρℳp0,q0,ξϑ,u1−ρJu0,ϑ+ρJu0,u1−ρωξϑ,u12,∀ϑ∈ℭ.
Sincep0∈Pu0ru0,q0∈Vu0su0, then by Nadler’s Lemma 2, there exist p1∈Pu1ru1, q1∈Vu1su1such that(52)p0−p1≤1+1DPu0ru0,Pu1ru1,q0−q1≤1+1DVu0su0,Vu1su1.
Forn=1, u1∈J, p1∈Pu1ru1,q1∈Vu1su1, again from Theorem 2, auxiliary problem (35) has a unique solution u2∈J, such that(53)u2,ϑ−u2≥u1,ϑ−u2−ρℳp1,q1,ξϑ,u2−ρJu1,ϑ+ρJu1,u2−ρωξϑ,u22,∀ϑ∈ℭ.
Sincep1∈Pu1ru1, q1∈Vu1su1, then by Nadler’s Lemma 2, there exist p2∈Pu2ru2, q2∈Vu2su2 such that(54)p1−p2≤1+12DPu1ru1,Pu2ru2,q1−q2≤1+12DVu1su1,Vu2su2,⋅⋅
At stepn, we can obtain sequences un∈J,pn∈Punrun,qn∈Vunsun∈CBJ, such that(55)ipn−pn+1≤1+11+nDPunrun,Pun+1run+1,iiqn−qn+1≤1+11+nDVunsun,Vun+1sun+1,iiiun+1,ϑ−un+1≥un,ϑ−un+1−ρℳpn,qn,ξϑ,un+1−ρJun,ϑ+ρJun,un+1−ρωξϑ,un+12,∀ϑ∈ℭ,∀n≥0.


## 4. Existence and Convergence Analysis

Theorem 3.
Let statement of Theorem2 hold and let P,V be P-L-continuous and V-L-continuous with constants λ>0 and ξ>0, respectively. Let nonlinear continuous mapping ℳ:J×J⟶J be β-S-mixed monotone with constant β>0 and γ-L-continuous in respect of first argument and α-L-continuous in respect of second argument with constants γ>0 and α>0, respectively. If J. is nondifferentiable and the bifunction ξ.,.:J×J⟶J is S-monotone with constant μ>0, respectively, then for constant ρ>0,(56)0<ρ<2β−η+2ωδ2γ2λ2−η+2ωδ22,ρ<1η+2ωδ2,β>η+2ωδ2,where(57)η=γλ1−2μ+δ2+αξδ+∂.
Then, there exist∈ℭ, p∈Puru, q∈Vusu satisfying “EPFMVLI” (5), and the sequences un, pn, and qngenerated by (55) converge strongly to u, p, and q, respectively.Proof.
From Algorithm1 and auxiliary problem (35), for any ϑ∈J, we have(58)un,ϑ−un≥un−1,ϑ−un−ρℳpn−1,qn−1,ξϑ,un−ρJun−1,ϑ+ρJun−1,un−ρωξϑ,un2,(59)un+1,ϑ−un+1≥un,ϑ−un+1−ρℳpn,qn,ξϑ,un+1−ρJun,ϑ+ρJun,un+1−ρωξϑ,un+12.
By takingϑ=un+1in (58) and ϑ=unin (59), we get(60)un,un+1−un≥un−1,un+1−un−ρℳpn−1,qn−1,ξun+1,un−ρJun−1,un+1+ρJun−1,un−ρωξun+1,un2,(61)un+1,un−un+1≥un,un−un+1−ρℳpn,qn,ξun,un+1−ρJun,un+ρJun,un+1−ρωξun,un+12.
Adding (60) and (61) and from Assumption 2 (ξϑ,u+ξu,ϑ=0), we have(62)un−un+1,un−un+1≤un−1−un,un−un+1−ρℳpn,qn−ℳpn−1,qn−1,ξun,un+1+ρJun−un−1,un−ρJun−1−un,un+1+2ρωξun,un+12≤un−1−un,un−un+1+ρℳpn,qn−ℳpn−1,qn−1,ξun,un+1+ρJun−un−1,un−un+1+2ρωξun,un+12,and so(63)un−un+1,un−un+1≤un−1−un−ρℳpn−1,qn−1−ℳpn,qn−1,un−un+1+ρℳpn−1,qn−1−ℳpn,qn−1,un−un+1−ξun,un+1+ρℳpn,qn−ℳpn,qn−1,ξun,un+1+ρJun−un−1,un−un+1+2ρωξun,un+12.
It follows that(64)un−un+12≤un−1−un−ρℳpn−1,qn−1−ℳpn,qn−1un−un+1+ρℳpn−1,qn−1−ℳpnun−un+1−ξun,un+1ρℳpn,qn−ℳpn,qn−1ξun,un+1+ρ∂un−1−unun−un+1+2ωρξun,un+12.
By theβ-S-mixed monotonicity, γ-L-continuity in respect of fist argument of ℳ, and P-L-continuity of P, we have(65)un−1−un−ρℳpn−1,qn−1−ℳpn,qn−12≤un−1−un2−2ρℳpn−1,qn−1−ℳpn,qn−1,un−1−un+ρ2ℳpn−1,qn−1−ℳpn,qn−12≤un−1−un2−2ρβun−1−un2+ρ2γ2pn−1−pn2≤un−1−un2−2ρβun−1−un2+ρ2γ2DPun−1run,Punrun2≤1−2ρβ+ρ2γ2λ21+1n2un−1−un2,(66)ℳpn−1,qn−1−ℳpn,qn−1≤γpn−1−pn≤γ1+1nDPun−1run−1,Punrun≤γλ1+1nun−1−un.
Similarly, by the S-monotonicity and L-continuity of bifunctionξ, we have(67)un−un+1−ξun,un+12≤un−un+12−2un−un+1,ξun,un+1+ξun,un+12≤un−un+12−2μun−un+12+δ2un−un+12=1−2μ+δ2un−un+12.
By theα-L-continuity of ℳ in respect of second argument and V-L-continuity of V, we get(68)ℳpn,qn−ℳpn,qn−1≤αpn−1−pn≤α1+1nDPun−1run−1,Punrun≤αξ1+1nun−1−un.
Combining (64)–(68) and by the L-continuity of ξ, we have(69)un−un+12≤1−2ρβ+ρ2γλ21+1n2+ργλ1+1n1−2μ+δ2un−1−unun−un+1ραξδ1+1nun−1−unun−un+1+ρ∂un−1−unun−un+1+2ωρδ2un−un+12,which implies that(70)un−un+1≤1−2ρβ+ρ2γλ21+1n2+ργλ1+1n1−2μ+δ2+ραξδ1+1n+ρ∂un−1−un+2ωρδ2un−un+1.
It follows that(71)un−un+1≤1−2ρβ+ρ2γλ21+1/n2+ργλ1+1/n1−2μ+δ2+ραξδ1+1/n+ρ∂1−2ωρδ2un−1−un=φnun−1−un,where(72)φn=1−2ρβ+ρ2γλ21+1/n2+ργλ1+1/n1−2μ+δ2+ραξδ1+1/n+ρ∂1−2ωρδ2,φ=1−2ρβ+ρ2γλ2+ργλ1−2μ+δ2+ραξδ+ρ∂1−2ωρδ2.
Clearly,(73)φn⟶φ=kρ+ρη1−2ωρδ2,where(74)kρ=1−2ρβ+ρ2γλ2,η=γλ1−2μ+δ2+ραξδ+ρ∂.
From (56), it follows that φ<1. Hence, it follows from (71) and (74) that unis Cauchy sequence in ℭ. Hence, it converges to some point. Since ℭis closed convex set in ℭ, there exists uin ℭsuch that un⟶u, which satisfies “EPFMVLI” (5).
On the other hand, from Algorithm1, we have(75)pn−pn+1≤1+11+nDPunrun,Pun+1run+1≤1+11+nγun−un+12qn−1−qn≤1+1nDVun−1sun−1,Vunsun≤1+1nξun−1−un.
This implies thatpn, qnboth are Cauchy sequences in J, since un is convergence sequence. Then, we consider that pn⟶p and qn⟶q whenn⟶∞. Since pn∈Punrunand qn∈Vqnsun, we have(76)dp,Punrun≤p−pn+dpn,Punrun+DPunrun,Puru≤p−pn+0+γun−u2dq,Vunsun≤q−qn+dqn,Vunsun+DVunsun,Vusu≤q−qn+0+ξun−u2.
Whenn⟶∞, we have(77)dp,Puru≤p−pn+0+γun−u2⟶0,dq,Puru≤p−pn+0+ξun−u2⟶0.
Hence,p∈Puruand q∈Vusu.
Finally, we show that(78)ℳp,ξϑ,u+Jϑ−Ju+ωξϑ,u2≥0,∀ϑ∈J.
We again study (55), as follows:(79)un+1,ϑ−un+1≥un,ϑ−un+1−ρℳpn,qn,ξϑ,un+1−ρJun,ϑ+ρJun,un+1−ρωξϑ,un+12,∀ϑ∈J,∀n≥0.
Now from Assumptions1 and 2, we get(80)0≤limsupn⟶∞un+1,ϑ−un+1−un,ϑ−un+1+ρℳpn,qn,ξϑ,un+1+ρJun,ϑ−ρJun,un+1+ρωξϑ,un+12,≤ρℳp,q,ξϑ,u+Ju,ϑ−Ju,u+ωξϑ,u2.
This implies that(81)0≤ℳp,q,ξϑ,u+Ju,ϑ−Ju,u+ωξϑ,u2.
This completes the proof.


### 4.1. Special Cases

Theorem 4.
LetJ be real Hilbert space and ℭ≠∅ be a closed bounded subset of J. Let P:J⟶ℱJ be a closed continuous fuzzy mapping satisfying condition (B) with function α:J⟶0,1 and P-L-continuous with constant λ>0. Let nonlinear continuous mapping ℳ:J⟶J be β-S mixed monotone and γ-L-continuous with constants β>0 and γ>0, respectively. Let J:ℭ⟶R be a nondifferentiable convex functional and ∂-L-continuous with constant ∂. Let the nonlinear mapping ξ.,.:J×J⟶J is S-monotone and L-continuous with constants μ>0 and δ>0 respectively If Assumption 2 holds, then for constantρ>0,(82)0<ρ<2β−η+2ωδ2γ2λ2−η+2ωδ22,ρ<1η+2ωδ2,β>η+2ωδ2,where(83)η=γλ1−2μ+δ2.
Then, there exist uniqueu∈ℭ, u∈Puαusatisfying the set-valued “PFMVLI” (12), and the sequences unand pn generated by (55) converge strongly to uand p, respectively.Proof.
Demonstration of Theorem4 is similar to demonstration of Theorems 2 and 3.Algorithm 2.
Atn=0, start with initial value u0∈J, p0∈Pu0αu0; from Theorem 4, the auxiliary problem (35) has a unique solution u1∈J, such that(84)u1,ϑ−u1≥u0,ϑ−u1−ρℳp0,ξϑ,u1−ρJϑ+ρJu1−ρωξϑ,u12,∀ϑ∈J.
Sincep0∈Pu0αu0, then by Nadler’s Lemma 2, there exists p1∈Pu1αu1, such that(85)p0−p1≤1+1DPu0αu0,Pu1αu1.
Forn=1, u1∈J, p1∈Pu1αu1, again from Theorem 2, auxiliary problem (35) has a unique solution u2∈J, such that(86)u2,ϑ−u2≥u1,ϑ−u2−ρℳp1,ξϑ,p2−ρJϑ+ρJu2−ρωξϑ,u22,∀ϑ∈J.
Sincep1∈Pu1αu1, then by Nadler’s Lemma 2, there exists p2∈Pu2αu2, such that(87)p1−p2≤1+12DPu1αu1,Pu2αu2,⋅⋅
At stepn, we can obtain sequences un∈J,pn∈Punαun∈CBJ, such that(88)ipn−pn+1≤1+11+nDPunαun,Pun+1αun+1iiun+1,ϑ−un+1≥un,ϑ−un+1−ρℳpn,ξϑ,un+1−ρJϑ+ρJun+1−ρωξϑ,un+12,∀ϑ∈J,∀n≥0.Theorem 5.
Let the operatorP,V:J⟶CBJ be P-L-continuous and V-L-continuous with constants λ>0 and ξ>0, respectively, and nonlinear continuous mapping ℳ:J⟶J be β-S-mixed monotone, γ-L-continuous in respect of first argument, and α-L-continuous in respect of second argument with constants β>0, γ>0, and α>0, respectively, and J. is nondifferentiable. Let bifunction ξ.,. be S-monotone with constant μ>0 and L-continuous with constant δ>0, respectively. If Assumptions 1 and 2 hold, then for constant ρ>0,(89)0<ρ<2β−η+2ωδ2γ2λ2−η+2ωδ22,ρ<1η+2ωδ2,β>η+2ωδ2,where(90)η=γλ1−2μ+δ2+αξδ+∂.
Then, there existu∈ℭ, p∈Pu,q∈Vusatisfying the set-valued “EPMVLI” (10), and the sequences un, pn, and qngenerated by (55) converge strongly to u, p, and q, respectively.Proof.
By using the set-valued mappingF:J⟶CBJ, we define the fuzzy mapping P˜:J⟶ℱJ, as follows:(91)P˜u=XPu,V˜u=XVu,where XPuand XVuare characteristic functions of the sets Pu. It is easy to see that P˜,V˜are closed fuzzy mappings satisfying condition (B) with constant functions ru=1,su=1 for all u∈J. Also,(92)P˜uru=XPu1=ϑ∈J:Puϑ=1=Pu,V˜usu=XVu1=ϑ∈J:Vuϑ=1=Vu.(1)
Then, problem (5) is analogous to finding u∈J, such that(93)P˜up=1,i.e.p∈Pu,V˜uq=1,i.e.q∈Vu,ℳp,q,ξϑ,u+Ju,ϑ−Ju,u+ωξϑ,u2≥0,∀ϑ∈J.
Hence, from Theorems2 and 3, we can obtain the conclusion of Theorem 5 immediately.Algorithm 3.
Atn=0, start with initial value u0∈J, p0∈Pu0, q0∈Vu0; from Theorem 5, auxiliary problem (35) has a unique solution u1∈J, such that(94)u1,ϑ−u1≥u0,ϑ−u1−ρℳp0,q0,ξϑ,u1−ρJu0,ϑ+ρJu0,u1−ρωξϑ,u12,∀ϑ∈J.
Sincep0∈Pu0, q0∈Vu0, then by Nadler’s Lemma 2, there exist p1∈Pu1, q1∈Vu1 such that(95)p0−p1≤1+1DPu0,Pu1,q0−q1≤1+1DVu0,Vu1.
Forn=1, u1∈J,p1∈Pu1, q1∈Vu1again from Theorem 5, the Auxiliary problem (35) has a unique solution u2∈J, such that(96)u2,ϑ−u2≥u1,ϑ−u2−ρℳp1,q1,ξϑ,u2−ρJu1,ϑ+ρJu1,u2−ρωξϑ,u22,∀ϑ∈J.
Sincep1∈Pu1, q1∈Vu1, then by Nadler’s Lemma 2, there exist p2∈Pu2, q2∈Vu2 such that(97)p1−p2≤1+12DPu1,Pu2,q1−q2≤1+12DVu1,Vu2,⋅⋅
At stepn, we can obtain sequences un∈J,pn∈Pun,qn∈Vun∈CBJ, such that(98)pn−pn+1≤1+11+nDPun,Pun+1,qn−qn+1≤1+11+nDVun,Vun+1,un+1,ϑ−un+1≥un,ϑ−un+1−ρℳpn,qn,ξϑ,un+1−ρJun,ϑ+ρJun,un+1−ρωξϑ,un+12,∀ϑ∈ℭ,∀n≥0.
Ifω=0, then Theorem 3 reduces to the following result (see [18]).Corollary 1.
Let statement of Theorem2 hold and let P,V be P-L-continuous and V-L-continuous with constants λ>0 and ξ>0, respectively. Let nonlinear continuous mapping ℳ:J×J⟶J be β-S-mixed monotone with constant β>0 and γ-L-continuous in respect of first argument and α-L-continuous in respect of second argument with constants γ>0 and α>0, respectively. If J. is nondifferentiable and the bifunction ξ.,.:J×J⟶J is S-monotone with constant μ>0, respectively, then for constant ρ>0,(99)0<ρ<2β−ηγ2λ2−η2,ρ<1η,β>η,where(100)η=γλ1−2μ+δ2+αξδ+∂.
Then, there exist∈ℭ, p∈Puru, q∈Vusu satisfying “EPFMVLI” (5), and the sequences un, pn, and qn generated by (55) converge strongly to u, p, and q, respectively.
Ifω=0, then Theorem 5 reduces to the following result (see [13]).Corollary 2.
Let the operatorP,V:J⟶CBJ be P-L-continuous and V-L-continuous with constants λ>0 and ξ>0, respectively, and nonlinear continuous mapping ℳ:J⟶J be β-S-mixed monotone, γ-L-continuous in respect of first argument, and α-L-continuous in respect of second argument with constants β>0, γ>0, and α>0, respectively, and J. is nondifferentiable. Let bifunction ξ.,. be S-monotone with constant μ>0 and L-continuous with constant δ>0, respectively. If Assumptions 1 and 2 hold, then for constant ρ>0,(101)0<ρ<2β−ηγ2λ2−η2,ρ<1η,β>η,where(102)η=γλ1−2μ+δ2+αξδ+∂.
Then, there existu∈ℭ, p∈Pu,q∈Vu satisfying the set-valued “EPMVLI” (10), and the sequences un, pn, and qn generated by (55) converge strongly to u, p, and q, respectively.
Ifℳ=I(identity mapping), p∈J, and P:J⟶J is single valued, then Theorem 5 reduces to the following result (see [12]).Corollary 3.
LetP:J⟶J be P-S-monotone with constant β>0 and P-L-continuous with constant λ>0, respectively. Let J:J⟶R be a nondifferentiable mapping and the bifunction ξ.,. be S-monotone with constant μ>0 and L-continuous with constant δ>0, respectively. If ξ.,. satisfies following condition:(103)ξϑ,u+ξu,ϑ=0and soξu,u=0,for allu∈J,for allu,ϑ∈J,then for constant ρ>0,(104)0<ρ<2β−η+2ωδ2γ2−η+2ωδ22,ρ<1η+2ωδ2,β>η+2ωδ2,where(105)η=γ1−2μ+δ2.
Then, there existsu∈ℭ, satisfying “PMVLI” (13), and the sequence un generated by (55) converges strongly to u.
Ifℳ=Iandω=0, p∈J, and P:J⟶J is single valued, then Theorem 5 reduces to the following result (see [22]).Corollary 4.
LetP:J⟶J be P-S-monotone with constant β>0 and P-L-continuous with constant λ>0, respectively. Let J:J⟶R be a nondifferentiable mapping and the bifunction ξ.,. be S-monotone with constant μ>0 and L-continuous with constant δ>0, respectively. If ξ.,. satisfies following condition:(106)ξϑ,u+ξu,ϑ=0and soξu,u=0,for allu∈J,for allu,ϑ∈J,then for constant ρ>0,(107)0<ρ<2β−ηγ2−η+2ωδ22,ρη<1,β>η,where(108)η=γ1−2μ+δ2.
Then, there existsu∈ℭ, satisfying “MVLI” (15), and the sequence ungenerated by (55) converges strongly to u.


## 4.1. Special Cases

Theorem 4.
LetJ be real Hilbert space and ℭ≠∅ be a closed bounded subset of J. Let P:J⟶ℱJ be a closed continuous fuzzy mapping satisfying condition (B) with function α:J⟶0,1 and P-L-continuous with constant λ>0. Let nonlinear continuous mapping ℳ:J⟶J be β-S mixed monotone and γ-L-continuous with constants β>0 and γ>0, respectively. Let J:ℭ⟶R be a nondifferentiable convex functional and ∂-L-continuous with constant ∂. Let the nonlinear mapping ξ.,.:J×J⟶J is S-monotone and L-continuous with constants μ>0 and δ>0 respectively If Assumption 2 holds, then for constantρ>0,(82)0<ρ<2β−η+2ωδ2γ2λ2−η+2ωδ22,ρ<1η+2ωδ2,β>η+2ωδ2,where(83)η=γλ1−2μ+δ2.
Then, there exist uniqueu∈ℭ, u∈Puαusatisfying the set-valued “PFMVLI” (12), and the sequences unand pn generated by (55) converge strongly to uand p, respectively.Proof.
Demonstration of Theorem4 is similar to demonstration of Theorems 2 and 3.Algorithm 2.
Atn=0, start with initial value u0∈J, p0∈Pu0αu0; from Theorem 4, the auxiliary problem (35) has a unique solution u1∈J, such that(84)u1,ϑ−u1≥u0,ϑ−u1−ρℳp0,ξϑ,u1−ρJϑ+ρJu1−ρωξϑ,u12,∀ϑ∈J.
Sincep0∈Pu0αu0, then by Nadler’s Lemma 2, there exists p1∈Pu1αu1, such that(85)p0−p1≤1+1DPu0αu0,Pu1αu1.
Forn=1, u1∈J, p1∈Pu1αu1, again from Theorem 2, auxiliary problem (35) has a unique solution u2∈J, such that(86)u2,ϑ−u2≥u1,ϑ−u2−ρℳp1,ξϑ,p2−ρJϑ+ρJu2−ρωξϑ,u22,∀ϑ∈J.
Sincep1∈Pu1αu1, then by Nadler’s Lemma 2, there exists p2∈Pu2αu2, such that(87)p1−p2≤1+12DPu1αu1,Pu2αu2,⋅⋅
At stepn, we can obtain sequences un∈J,pn∈Punαun∈CBJ, such that(88)ipn−pn+1≤1+11+nDPunαun,Pun+1αun+1iiun+1,ϑ−un+1≥un,ϑ−un+1−ρℳpn,ξϑ,un+1−ρJϑ+ρJun+1−ρωξϑ,un+12,∀ϑ∈J,∀n≥0.Theorem 5.
Let the operatorP,V:J⟶CBJ be P-L-continuous and V-L-continuous with constants λ>0 and ξ>0, respectively, and nonlinear continuous mapping ℳ:J⟶J be β-S-mixed monotone, γ-L-continuous in respect of first argument, and α-L-continuous in respect of second argument with constants β>0, γ>0, and α>0, respectively, and J. is nondifferentiable. Let bifunction ξ.,. be S-monotone with constant μ>0 and L-continuous with constant δ>0, respectively. If Assumptions 1 and 2 hold, then for constant ρ>0,(89)0<ρ<2β−η+2ωδ2γ2λ2−η+2ωδ22,ρ<1η+2ωδ2,β>η+2ωδ2,where(90)η=γλ1−2μ+δ2+αξδ+∂.
Then, there existu∈ℭ, p∈Pu,q∈Vusatisfying the set-valued “EPMVLI” (10), and the sequences un, pn, and qngenerated by (55) converge strongly to u, p, and q, respectively.Proof.
By using the set-valued mappingF:J⟶CBJ, we define the fuzzy mapping P˜:J⟶ℱJ, as follows:(91)P˜u=XPu,V˜u=XVu,where XPuand XVuare characteristic functions of the sets Pu. It is easy to see that P˜,V˜are closed fuzzy mappings satisfying condition (B) with constant functions ru=1,su=1 for all u∈J. Also,(92)P˜uru=XPu1=ϑ∈J:Puϑ=1=Pu,V˜usu=XVu1=ϑ∈J:Vuϑ=1=Vu.(1)
Then, problem (5) is analogous to finding u∈J, such that(93)P˜up=1,i.e.p∈Pu,V˜uq=1,i.e.q∈Vu,ℳp,q,ξϑ,u+Ju,ϑ−Ju,u+ωξϑ,u2≥0,∀ϑ∈J.
Hence, from Theorems2 and 3, we can obtain the conclusion of Theorem 5 immediately.Algorithm 3.
Atn=0, start with initial value u0∈J, p0∈Pu0, q0∈Vu0; from Theorem 5, auxiliary problem (35) has a unique solution u1∈J, such that(94)u1,ϑ−u1≥u0,ϑ−u1−ρℳp0,q0,ξϑ,u1−ρJu0,ϑ+ρJu0,u1−ρωξϑ,u12,∀ϑ∈J.
Sincep0∈Pu0, q0∈Vu0, then by Nadler’s Lemma 2, there exist p1∈Pu1, q1∈Vu1 such that(95)p0−p1≤1+1DPu0,Pu1,q0−q1≤1+1DVu0,Vu1.
Forn=1, u1∈J,p1∈Pu1, q1∈Vu1again from Theorem 5, the Auxiliary problem (35) has a unique solution u2∈J, such that(96)u2,ϑ−u2≥u1,ϑ−u2−ρℳp1,q1,ξϑ,u2−ρJu1,ϑ+ρJu1,u2−ρωξϑ,u22,∀ϑ∈J.
Sincep1∈Pu1, q1∈Vu1, then by Nadler’s Lemma 2, there exist p2∈Pu2, q2∈Vu2 such that(97)p1−p2≤1+12DPu1,Pu2,q1−q2≤1+12DVu1,Vu2,⋅⋅
At stepn, we can obtain sequences un∈J,pn∈Pun,qn∈Vun∈CBJ, such that(98)pn−pn+1≤1+11+nDPun,Pun+1,qn−qn+1≤1+11+nDVun,Vun+1,un+1,ϑ−un+1≥un,ϑ−un+1−ρℳpn,qn,ξϑ,un+1−ρJun,ϑ+ρJun,un+1−ρωξϑ,un+12,∀ϑ∈ℭ,∀n≥0.
Ifω=0, then Theorem 3 reduces to the following result (see [18]).Corollary 1.
Let statement of Theorem2 hold and let P,V be P-L-continuous and V-L-continuous with constants λ>0 and ξ>0, respectively. Let nonlinear continuous mapping ℳ:J×J⟶J be β-S-mixed monotone with constant β>0 and γ-L-continuous in respect of first argument and α-L-continuous in respect of second argument with constants γ>0 and α>0, respectively. If J. is nondifferentiable and the bifunction ξ.,.:J×J⟶J is S-monotone with constant μ>0, respectively, then for constant ρ>0,(99)0<ρ<2β−ηγ2λ2−η2,ρ<1η,β>η,where(100)η=γλ1−2μ+δ2+αξδ+∂.
Then, there exist∈ℭ, p∈Puru, q∈Vusu satisfying “EPFMVLI” (5), and the sequences un, pn, and qn generated by (55) converge strongly to u, p, and q, respectively.
Ifω=0, then Theorem 5 reduces to the following result (see [13]).Corollary 2.
Let the operatorP,V:J⟶CBJ be P-L-continuous and V-L-continuous with constants λ>0 and ξ>0, respectively, and nonlinear continuous mapping ℳ:J⟶J be β-S-mixed monotone, γ-L-continuous in respect of first argument, and α-L-continuous in respect of second argument with constants β>0, γ>0, and α>0, respectively, and J. is nondifferentiable. Let bifunction ξ.,. be S-monotone with constant μ>0 and L-continuous with constant δ>0, respectively. If Assumptions 1 and 2 hold, then for constant ρ>0,(101)0<ρ<2β−ηγ2λ2−η2,ρ<1η,β>η,where(102)η=γλ1−2μ+δ2+αξδ+∂.
Then, there existu∈ℭ, p∈Pu,q∈Vu satisfying the set-valued “EPMVLI” (10), and the sequences un, pn, and qn generated by (55) converge strongly to u, p, and q, respectively.
Ifℳ=I(identity mapping), p∈J, and P:J⟶J is single valued, then Theorem 5 reduces to the following result (see [12]).Corollary 3.
LetP:J⟶J be P-S-monotone with constant β>0 and P-L-continuous with constant λ>0, respectively. Let J:J⟶R be a nondifferentiable mapping and the bifunction ξ.,. be S-monotone with constant μ>0 and L-continuous with constant δ>0, respectively. If ξ.,. satisfies following condition:(103)ξϑ,u+ξu,ϑ=0and soξu,u=0,for allu∈J,for allu,ϑ∈J,then for constant ρ>0,(104)0<ρ<2β−η+2ωδ2γ2−η+2ωδ22,ρ<1η+2ωδ2,β>η+2ωδ2,where(105)η=γ1−2μ+δ2.
Then, there existsu∈ℭ, satisfying “PMVLI” (13), and the sequence un generated by (55) converges strongly to u.
Ifℳ=Iandω=0, p∈J, and P:J⟶J is single valued, then Theorem 5 reduces to the following result (see [22]).Corollary 4.
LetP:J⟶J be P-S-monotone with constant β>0 and P-L-continuous with constant λ>0, respectively. Let J:J⟶R be a nondifferentiable mapping and the bifunction ξ.,. be S-monotone with constant μ>0 and L-continuous with constant δ>0, respectively. If ξ.,. satisfies following condition:(106)ξϑ,u+ξu,ϑ=0and soξu,u=0,for allu∈J,for allu,ϑ∈J,then for constant ρ>0,(107)0<ρ<2β−ηγ2−η+2ωδ22,ρη<1,β>η,where(108)η=γ1−2μ+δ2.
Then, there existsu∈ℭ, satisfying “MVLI” (15), and the sequence ungenerated by (55) converges strongly to u.


## 5. Conclusion

In this paper, we have proposed the idea of “EPFMVLI.” As a particular case of “EPFMVLI,” extended perturbed mixed variational-like inequalities are also introduced. With the help of the extended auxiliary principle technique and some new analytic techniques, some existence theorems of auxiliary “EPFMVLI” are studied for “EPFMVLI,” and some iterative methods are obtained for the solution of “EPFMVLI.” Then, we have obtained some known and new results. We would like to mention that many earlier defined familiar methods as well as decent, projection techniques and its mixed forms such as relaxation and Newton’s methods can be obtained from auxiliary “SPFMVLI.” There are much rooms for exploring of this concept, as the suitable choices of fuzzy mappings can be obtained from EPFMVLI (5), see [3–8, 12–14, 18–22].



---

*Source: 6652930-2021-01-18.xml*