# Algorithm for Solutions of Nonlinear Equations of Strongly Monotone Type and Applications to Convex Minimization and Variational Inequality Problems

Authors: Mathew O. Aibinu; Surendra C. Thakur; Sibusiso Moyo
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

Real-life problems are governed by equations which are nonlinear in nature. Nonlinear equations occur in modeling problems, such as minimizing costs in industries and minimizing risks in businesses. A technique which does not involve the assumption of existence of a real constant whose calculation is unclear is used to obtain a strong convergence result for nonlinear equations ofp,η-strongly monotone type, where η>0,p>1. An example is presented for the nonlinear equations of p,η-strongly monotone type. As a consequence of the main result, the solutions of convex minimization and variational inequality problems are obtained. This solution has applications in other fields such as engineering, physics, biology, chemistry, economics, and game theory.


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## Body

## 1. Introduction

Letν:0,∞⟶0,∞ be a continuous, strictly increasing function such that νt⟶∞ as t→∞ and ν0=0 for any t∈0,∞. Such a function ν is called a gauge function. Let B be a Banach space, and B∗ denotes it is dual. A duality mapping associated with the gauge function ν is a mapping JνB:B⟶2B∗ defined by
(1)JνBx=f∈B∗:x,f=∥x∥ν∥x∥,∥f∥=ν∥x∥,where .,. denotes the duality pairing. For p>1, let νt=tp−1 be a gauge function. JB:B⟶2B∗ is called a generalized duality mapping from B into 2B∗ and is given by
(2)JpBx=f∈B∗:x,f=∥x∥p,∥f∥=∥x∥p−1.Forp=2, the mapping J2B is called the normalized duality mapping. In a Hilbert space, the normalized duality mapping is the identity map. For x,y∈B and η>0, a mapping T:B⟶B∗ is said to be
(1)
monotone if x−y,Tx−Ty≥0(2)
strongly monotone (see, e.g., Alber and Ryazantseva [1], p. 25), if x−y,Tx−Ty≥η∥x−y∥2(3)
η-strongly pseudomonotone if x−y,Ty≥0⟹x−y,Tx≥η∥x−y∥2(4)
p,η-strongly monotone if x−y,Tx−Ty≥η∥x−y∥p (see, e.g., Chidume and Djitté [2], Chidume and Shehu [3], and Aibinu and Mewomo [4, 5]).Remark 1.
According to the definition of Chidume and Djitté [2] and Chidume and Shehu [3], a strongly monotone mapping is referred to as 2,η-strongly monotone mapping.A monotone mappingT is called maximal monotone if it is a monotone and its graph is not properly contained in the graph of any other monotone mapping. As a result of Rockafellar [6], T is said to be a maximum monotone if it is a monotone and the range of JB+tT is all of B∗ for some t>0. The set of zeros of a maximum monotone mapping T,T−10≔x∈B:Tx=0 is closed and convex. A function φ:B⟶−∞,+∞ is said to be proper if the set x∈ℝ:Fx∈ℝ is nonempty. A proper function φ:B⟶−∞,+∞ is said to be convex if for all x,y∈B and τ∈0,1,we have
(3)φτx+1−τy≤τφx+1−τφy.If the set ofx∈ℝ:φx≤r is closed in B for all r∈ℝ,φ is said to be lower semicontinuous. For a proper lower semicontinuous function φ:B⟶−∞,+∞, the subdifferential mapping∂φ:B⟶2B∗, defined by
(4)∂φx=x∗∈B∗:φy−φx≥y−x,x∗∀y∈B,is a maximal monotone (Rockafellar [7]). Consider a problem of finding a solution of the equation Tu=0, where T is a maximal monotone mapping. Such a problem is associated with the convex minimization problem. Indeed, for a proper lower semicontinuous convex function φ:B⟶−∞,+∞, solving the equation Tu=0 is equivalent to finding φu=minx∈Bφx by setting ∂φ≡T.For a reflexive smooth strictly convex spaceB, we let T be a mapping such that the range of JpB+tT is all of B∗ for some t>0 and let x∈B be fixed. Then, for every t>0, there corresponds a unique element xt∈DT such that
(5)JpBx=JpBxt+tTxt.Therefore, theresolvent of T is defined by JtTx=xt. In other words, JtT=JpB+tT−1JpB and T−10=FJtT for all t>0, where FJtT denotes the set of all fixed points of JtT. The resolvent JtT is a single-valued mapping from B into DT (Kohsaka and Takahashi [8]). JtT is nonexpansive if E is a Hilbert space (Takahashi [9]). Some existing results proved a strong convergence theorem for nonlinear equations of the monotone type, with the assumption of existence of a real constant whose calculation is unclear (see, e.g., Aibinu and Mewomo [4], Chidume et al. [10], and Diop et al. [11]). Monotone-type mappings occur in many functional equations, and the research on monotone type mappings has recently attracted much attention (see, e.g., Shehu [12, 13], Chidume et al. [14], Djitte et al. [15], Tang [16], Uddin et al. [17], Chidume and Idu [18], and Aibinu and Mewomo [19]).In this paper, we consider nonlinear equations ofp,η-strongly monotone type, p>1 and η∈1,∞. This is a wider class than the class of strongly monotone mappings. An example is presented for nonlinear equations of p,η-strongly monotonetype. Under suitable conditions which do not involve the assumption of existence of a real constant whose calculation is unclear, a sequence of iteration is shown to converge strongly to the zero of a nonlinear equation of p,η-strongly monotone type. As a consequence of the main result, the solution of convex minimization and variational inequality problems is obtained, which has applications in several fields such as economics, game theory, and the sciences.


## 2. Preliminaries

LetB be a real Banach space and S≔x∈B:∥x∥=1. B is said to have a Gateaux differentiable norm if the limit
(6)limt→0∥x+ty∥−∥x∥t,exists for each x,y∈S. A Banach space B is said to be smooth if for every x≠0 in B, there is a unique x∗∈B∗ such that ∥x∗∥=1 and x,x∗=∥x∥, where B∗ denotes the dual of B.B is said to be uniformly smooth if it is smooth and the limit (6) is attained uniformly for each x,y∈S. The modulus of convexity of a Banach space B, δB:0,2⟶0,1 is defined by
(7)δBε=inf1−∥x+y∥2:∥x∥=∥y∥=1,∥x−y∥>ε.B is uniformly convex if and only if δBε>0 for every ε∈0,2. A normed linear space B is said to be strictly convex if
(8)∥x∥=∥y∥=1,x≠y⟹∥x+y∥2<1.It is well known that a spaceB is uniformly smooth if and only if B∗ is uniformly convex.A mappingT:B⟶B∗ is locally bounded at v∈D, if there exist rv>0 and m>0 such that
(9)∥Tx∥≤m,∀x∈Drvv.In particular,∥Tv∥≤m. Therefore, v,Tv≤m∥v∥. Let X and Y be real Banach spaces and let T:X⟶Y be a mapping. T is uniformly continuous if for each ε>0, there exists δ>0 such that
(10)∥Tx−Ty∥<ε∀x,y∈Xwith∥x−y∥<δ.Letψt be a function on the set ℝ+ of nonnegative real numbers such that
(i)
ψ is nondecreasing and continuous(ii)
ψt=0 if and only if t=0T is said to be uniformly continuous if it admits the modulus of continuity ψ such that
(11)∥Tx−Ty∥≤ψ∥x−y∥∀x,y∈X.The modulus of continuityψ has some useful properties (for instance, see Altomare and Campiti [20], pp. 266–269; Forster [21] and references therein).


### 2.1. Properties of Modulus of Continuity

LetX and Y be real Banach spaces and let T:X⟶Y be a map which admits the modulus of continuity ψ.(a)
Modulus of continuity is subadditive: for all real numbers t1≥0,t2≥0, we have(12)ψt1+t2≤ψt1+ψt2(b)
Modulus of continuity is monotonically increasing: if 0≤t1≤t2 holds for some real numbers t1,t2, then(13)0≤ψt1≤ψt2(c)
Modulus of continuity is continuous: the modulus of continuity ψ:ℝ+⟶ℝ+ is continuous on the set positive real numbers; in particular, the limit of ψ at 0 from above is(14)limt→0ψt=0LetC be a nonempty subset of a Banach space B and T be a mapping from C into itself.
(i)
T is nonexpansive provided ∥Tx−Ty∥≤∥x−y∥ for all x,y∈C(ii)
T is firmly nonexpansive type (see, e.g., [22]) if Tx−Ty,jpBTx−jpBTy≤Tx−Ty,jpBx−jpBy for all x,y∈C and jpB∈JpBThe following results about the generalized duality mappings are well known which are established in, e.g., Alber and Ryazantseva [1] (p. 36), Cioranescu [23] (pp. 25–77), Xu and Roach [24], and Zalinescu [25]. Let B be a Banach space.
(i)
B is smooth if and only if JpB is single-valued(ii)
IfBis reflexive, thenJpBis onto(iii)
IfB has uniform Gateaux differentiable norm, then JpB is norm-to-weak∗ uniformly continuous on bounded sets(iv)
B is uniformly smooth if and only if JpB is single-valued and uniformly continuous on any bounded subset of B(v)
IfB is strictly convex, then JpB is one-to-one, that is, ∀x,y∈B,x≠y⟹JpBx∩JpBy=∅(vi)
IfB and B∗ are strictly convex and reflexive, then JpB∗ is the generalized duality mapping from B∗ to B, and JpB∗ is the inverse of JpB(vii)
IfB is uniformly smooth and uniformly convex, the generalized duality mapping JpB∗ is uniformly continuous on any bounded subset of B∗(viii)
IfB and B∗ are strictly convex and reflexive, for all x∈B and f∈B∗, the equalities JpBJpB∗f=f and JpB∗JpBx=x holdDefinition 2.
Alber [26] introduced the functions ϕ:B×B→ℝ, defined by
(15)ϕx,y=∥x∥2−2x,J2By+∥y∥2,forallx,y∈B,where J2B is the normalized duality mapping from B to B∗. Let B be a smooth real Banach space and p,q>1 with 1/p+1/q=1. Aibinu and Mewomo [4] introduced the functions ϕp:B×B⟶ℝ, defined by
(16)ϕpx,y=pq∥x∥q−px,JpBy+∥y∥p,forallx,y∈B,and Vp:B×B∗→ℝ, defined as
(17)Vpx,x∗=pq∥x∥q−px,x∗+∥x∗∥p∀x∈B,x∗∈B∗,where JpB is the generalized duality mapping from B to B∗.Remark 3.
These remarks follow from Definition2:
(i)
Forp=2,ϕ2x,y=ϕx,y, which is the definition of Alber [26]. It is easy to see from the definition of the function ϕ that(18)∥x∥−∥y∥2≤ϕx,y≤∥x∥+∥y∥2forallx,y∈B.Indeed,(19)∥x∥−∥y∥2=∥x∥2−2∥x∥∥y∥+∥y∥2≤∥x∥2−2x,JpBy+∥y∥2=ϕx,y≤∥x∥2+2∥x∥∥y∥+∥y∥2=∥x∥+∥y∥2.By similar analysis, it can verified that for eachp≥2,(20)∥x∥−∥y∥p≤ϕpx,y≤∥x∥+∥y∥pforallx,y∈B.(ii)
It is obvious that(21)Vpx,x∗=ϕpx,JpB∗x∗∀x∈B,x∗∈B∗.LetB be a topological real vector space and T a multivalued mapping from B into 2B∗. Cauchy-Schwartz’s inequality is given by
(22)∣x,y∗∣≤x,x∗1/2y,y∗1/2,for any x and y in B and any choice of x∗∈Tx and y∗∈Ty (Zarantonello [27]).In the sequel, we shall need the lemmas whose proofs have been established (see, e.g., Alber [26] and Aibinu and Mewomo [4]).Lemma 4.LetBbe a strictly convex and uniformly smooth real Banach space andp>1.Then,(23)Vpx,x∗+pJpB∗x∗−x,y∗≤Vpx,x∗+y∗,for all x∈B and x∗,y∗∈B∗.Lemma 5.
LetBbe a smooth uniformly convex real Banach space andp>1be an arbitrarily real number. Ford>0, letBd0≔x∈B:∥x∥≤d. Then, for arbitraryx,y∈Bd0,
(24)∥x−y∥p≥ϕpx,y−pq∥x∥q,where1p+1q=1.Lemma 6.
LetBbe a reflexive strictly convex and smooth real Banach space andp>1.Then,(25)ϕpy,x−ϕpy,z≥pz−y,JpBx−JpBz=py−z,JpBz−JpBxforallx,y,z∈B.Lemma 7.
LetBbe a real uniformly convex Banach space. For arbitraryr>0, letBr0≔x∈B:∥x∥≤r. Then, there exists a continuous strictly increasing convex function(26)g:0,∞⟶0,∞,g0=0,such that for every x,y∈Br0,jpBx∈JpBx,jpBy∈JpBy, we have x−y,jpBx−jpBy≥g∥x−y∥ (see Xu [28]).Lemma 8.
Letanbe a sequence of nonnegative real numbers satisfying the following relations:(27)an+1≤1−αnan+αnσn+γn,n∈ℕ,where
(i)
αn⊂0,1, ∑n=1∞αn=∞(ii)
limsupσn≤0(iii)
γn≥0, ∑n=1∞γn<∞
Then,an⟶0 as n⟶∞ (see Xu [29]).Lemma 9.
LetBbe a smooth uniformly convex real Banach space and letxnandynbe two sequences fromB.If eitherxnorynis bounded andϕxn,yn⟶0asn⟶∞, then∥xn−yn∥⟶0asn⟶∞ (see Kamimura and Takahashi [30]).Lemma 10.
A monotone mapT:B→B∗ is locally bounded at the interior points of its domain (see, e.g., Rockafellar [31] and Pascali and Sburlan [32]).Lemma 11.
If a functionalϕon the open convex setM⊂domϕhas a subdifferential, thenϕis convex and lower semicontinuous on the set (see Alber and Ryazantseva [1], p. 17).Lemma 12.
LetXandYbe real normed linear spaces and letT:X⟶Ybe a uniformly continuous map. For arbitraryr>0and fixedx∗∈X, let(28)BXx∗,r:x∈X:∥x−x∗∥X≤r.Then,TBx∗,r is bounded (see, e.g., Chidume and Djitte [33]).


## 2.1. Properties of Modulus of Continuity

LetX and Y be real Banach spaces and let T:X⟶Y be a map which admits the modulus of continuity ψ.(a)
Modulus of continuity is subadditive: for all real numbers t1≥0,t2≥0, we have(12)ψt1+t2≤ψt1+ψt2(b)
Modulus of continuity is monotonically increasing: if 0≤t1≤t2 holds for some real numbers t1,t2, then(13)0≤ψt1≤ψt2(c)
Modulus of continuity is continuous: the modulus of continuity ψ:ℝ+⟶ℝ+ is continuous on the set positive real numbers; in particular, the limit of ψ at 0 from above is(14)limt→0ψt=0LetC be a nonempty subset of a Banach space B and T be a mapping from C into itself.
(i)
T is nonexpansive provided ∥Tx−Ty∥≤∥x−y∥ for all x,y∈C(ii)
T is firmly nonexpansive type (see, e.g., [22]) if Tx−Ty,jpBTx−jpBTy≤Tx−Ty,jpBx−jpBy for all x,y∈C and jpB∈JpBThe following results about the generalized duality mappings are well known which are established in, e.g., Alber and Ryazantseva [1] (p. 36), Cioranescu [23] (pp. 25–77), Xu and Roach [24], and Zalinescu [25]. Let B be a Banach space.
(i)
B is smooth if and only if JpB is single-valued(ii)
IfBis reflexive, thenJpBis onto(iii)
IfB has uniform Gateaux differentiable norm, then JpB is norm-to-weak∗ uniformly continuous on bounded sets(iv)
B is uniformly smooth if and only if JpB is single-valued and uniformly continuous on any bounded subset of B(v)
IfB is strictly convex, then JpB is one-to-one, that is, ∀x,y∈B,x≠y⟹JpBx∩JpBy=∅(vi)
IfB and B∗ are strictly convex and reflexive, then JpB∗ is the generalized duality mapping from B∗ to B, and JpB∗ is the inverse of JpB(vii)
IfB is uniformly smooth and uniformly convex, the generalized duality mapping JpB∗ is uniformly continuous on any bounded subset of B∗(viii)
IfB and B∗ are strictly convex and reflexive, for all x∈B and f∈B∗, the equalities JpBJpB∗f=f and JpB∗JpBx=x holdDefinition 2.
Alber [26] introduced the functions ϕ:B×B→ℝ, defined by
(15)ϕx,y=∥x∥2−2x,J2By+∥y∥2,forallx,y∈B,where J2B is the normalized duality mapping from B to B∗. Let B be a smooth real Banach space and p,q>1 with 1/p+1/q=1. Aibinu and Mewomo [4] introduced the functions ϕp:B×B⟶ℝ, defined by
(16)ϕpx,y=pq∥x∥q−px,JpBy+∥y∥p,forallx,y∈B,and Vp:B×B∗→ℝ, defined as
(17)Vpx,x∗=pq∥x∥q−px,x∗+∥x∗∥p∀x∈B,x∗∈B∗,where JpB is the generalized duality mapping from B to B∗.Remark 3.
These remarks follow from Definition2:
(i)
Forp=2,ϕ2x,y=ϕx,y, which is the definition of Alber [26]. It is easy to see from the definition of the function ϕ that(18)∥x∥−∥y∥2≤ϕx,y≤∥x∥+∥y∥2forallx,y∈B.Indeed,(19)∥x∥−∥y∥2=∥x∥2−2∥x∥∥y∥+∥y∥2≤∥x∥2−2x,JpBy+∥y∥2=ϕx,y≤∥x∥2+2∥x∥∥y∥+∥y∥2=∥x∥+∥y∥2.By similar analysis, it can verified that for eachp≥2,(20)∥x∥−∥y∥p≤ϕpx,y≤∥x∥+∥y∥pforallx,y∈B.(ii)
It is obvious that(21)Vpx,x∗=ϕpx,JpB∗x∗∀x∈B,x∗∈B∗.LetB be a topological real vector space and T a multivalued mapping from B into 2B∗. Cauchy-Schwartz’s inequality is given by
(22)∣x,y∗∣≤x,x∗1/2y,y∗1/2,for any x and y in B and any choice of x∗∈Tx and y∗∈Ty (Zarantonello [27]).In the sequel, we shall need the lemmas whose proofs have been established (see, e.g., Alber [26] and Aibinu and Mewomo [4]).Lemma 4.LetBbe a strictly convex and uniformly smooth real Banach space andp>1.Then,(23)Vpx,x∗+pJpB∗x∗−x,y∗≤Vpx,x∗+y∗,for all x∈B and x∗,y∗∈B∗.Lemma 5.
LetBbe a smooth uniformly convex real Banach space andp>1be an arbitrarily real number. Ford>0, letBd0≔x∈B:∥x∥≤d. Then, for arbitraryx,y∈Bd0,
(24)∥x−y∥p≥ϕpx,y−pq∥x∥q,where1p+1q=1.Lemma 6.
LetBbe a reflexive strictly convex and smooth real Banach space andp>1.Then,(25)ϕpy,x−ϕpy,z≥pz−y,JpBx−JpBz=py−z,JpBz−JpBxforallx,y,z∈B.Lemma 7.
LetBbe a real uniformly convex Banach space. For arbitraryr>0, letBr0≔x∈B:∥x∥≤r. Then, there exists a continuous strictly increasing convex function(26)g:0,∞⟶0,∞,g0=0,such that for every x,y∈Br0,jpBx∈JpBx,jpBy∈JpBy, we have x−y,jpBx−jpBy≥g∥x−y∥ (see Xu [28]).Lemma 8.
Letanbe a sequence of nonnegative real numbers satisfying the following relations:(27)an+1≤1−αnan+αnσn+γn,n∈ℕ,where
(i)
αn⊂0,1, ∑n=1∞αn=∞(ii)
limsupσn≤0(iii)
γn≥0, ∑n=1∞γn<∞
Then,an⟶0 as n⟶∞ (see Xu [29]).Lemma 9.
LetBbe a smooth uniformly convex real Banach space and letxnandynbe two sequences fromB.If eitherxnorynis bounded andϕxn,yn⟶0asn⟶∞, then∥xn−yn∥⟶0asn⟶∞ (see Kamimura and Takahashi [30]).Lemma 10.
A monotone mapT:B→B∗ is locally bounded at the interior points of its domain (see, e.g., Rockafellar [31] and Pascali and Sburlan [32]).Lemma 11.
If a functionalϕon the open convex setM⊂domϕhas a subdifferential, thenϕis convex and lower semicontinuous on the set (see Alber and Ryazantseva [1], p. 17).Lemma 12.
LetXandYbe real normed linear spaces and letT:X⟶Ybe a uniformly continuous map. For arbitraryr>0and fixedx∗∈X, let(28)BXx∗,r:x∈X:∥x−x∗∥X≤r.Then,TBx∗,r is bounded (see, e.g., Chidume and Djitte [33]).


## 3. Main Results

Theorem 13.
LetBbe a uniformly smooth and uniformly convex real Banach space. Letp>1,η∈1,∞; supposeT:B⟶B∗is a continuousp,η-strongly monotone mapping such that the range ofJpB+tTis all ofB∗for allt>0andT−10≠∅.Letλnn=1∞⊂0,1andθnn=1∞in0,1/2be real sequences such that(i)
limn⟶∞θn=0 and θnn=1∞ is decreasing(ii)
∑n=1∞λnθn=∞(iii)
limn⟶∞θn−1/θn−1/λnθn=0,∑n=1∞λn<∞∀n∈ℕ
For arbitraryx1∈B, define xnn=1∞ iteratively by:
(29)xn+1=JpB∗JpBxn−λnTxn+θnJpBxn−JpBx1,n∈ℕ,where JpB is the generalized duality mapping from B into B∗. Then, the sequence xnn=1∞ converges strongly to the solution of Tx=0.Proof.
Observe that there is no need for constructing a convergence sequence ifx=0 because it is a zero of T (since T is strongly monotone, which is one to one). Consequently, we are looking for a unique nonzero solution of Tx=0. The proof is divided into two parts.
Part 1: the sequence xnn=1∞ is shown to be bounded.
Letq>1 with 1/p+1/q=1 and x∈B be a solution of the equation Tx=0. It suffices to show that ϕpx,xn≤r,∀n∈ℕ. The induction method will be adopted. Let r>0 be sufficiently large such that
(30)r≥maxϕpx,x1,4M0M,4pq∥x∥q,where M0>0 and M>0 are arbitrary but fixed. For n=1, by construction, we have that ϕpx,x1≤r for real p>1. Assume that ϕpx,xn≤r for some n≥1. From inequality (20), we have ∥xn∥≤r1/p+∥x∥. Let D≔z∈B:ϕpx,z≤r. Next is to show that ϕpx,xn+1≤r. It is known that T is locally bounded (Lemma 10) and JpB is uniformly continuous on bounded subsets of B. Define
(31)M0≔sup∥Txn+θnJpBxn−JpBx1∥:θn∈0,12,xn∈D+1.Letψ denotes the modulus of continuity of JpB∗. Then,
(32)∥xn−xn+1∥=∥xn−JpB∗JpBxn−λnTxn+θnJpBxn−JpBx1∥=∥JpB∗JpBxn−JpB∗JpBxn−λnTxn+θnJpBxn−JpBx1∥≤ψ∣λn∣∥Txn+θnJpBxn−JpBx1∥≤ψ∣λn∣M0≤ψsup∣λn∣M0:λn∈0,1.SinceT is locally bounded and the duality mapping JpB is uniformly continuous on bounded subsets of B, the sup∣λn∣M0 exists and it is a real number different from infinity. Choose M≕ψsup∣λn∣M0. Applying Lemma 4 with y∗≔λnTxn+θnJpBxn−JpBx1 and by using the definition of xn+1, we compute as follows:
(33)ϕpx,xn+1=ϕpx,JB∗JpBxn−λnTxn+θnJpBxn−JpBx1=Vpx,JpBxn−λnTxn+θnJpBxn−JpBx1by21≤Vpx,JpBxn−pλnJpB∗JpBxn−λnTxn+θnJpBxn−JpBx1−x,Txn+θnJpBxn−JpBx1=ϕpx,xn−pλnxn−x,Txn+θnJpBxn−JpBx1−pλnJpB∗JpBxn−λnTxn+θnJpBxn−JpBx1−xn,Txn+θnJpBxn−JpBx1.By Schwartz inequality and by applying inequality (32), we obtain
(34)ϕpx,xn+1≤ϕpx,xn−pλnxn−x,Txn+θnJpBxn−JpBx1+pλnM0M≤ϕpx,xn−pλnxn−x,Txn−Txsincex∈T−10−pλnθnxn−x,JpBxn−JpBx1+pλnM0M.By Lemma6, px−xn,JpBxn−JpBx1≤ϕpx,xn−ϕpx,x1. Consequently, px−xn,JpBxn−JpBx1≤ϕpx,xn. Therefore, using p,η-strongly monotonicity property of T, we have
(35)ϕpx,xn+1≤ϕpx,xn−pηλn∥xn−x∥p−pλnθnxn−x,JpBxn−JpBx1+pλnM0M≤ϕpx,xn−pλn∥xn−x∥p+pλnθnx−xn,JpBxn−JpBx1+pλnM0M≤ϕpx,xn−pλnϕpx,xn−pq∥x∥q+pλnθnϕpx,xn+pλnM0M=1−pλnϕpx,xn+pλnpq∥x∥q+pλnθnϕpx,xn+pλnM0M≤1−pλnr+pλnr4+pλnr2+pλn4r=1−pλn+pλn14+pλn12+pλn14r=r.Hence,ϕpx,xn+1≤r. By induction, ϕpx,xn≤r∀n∈ℕ. Thus, from inequality (20), xnn=1∞ is bounded.Part 2: we now show that xnn=1∞ converges strongly to a solution of Tx=0.p,η-strongly monotone implies a monotone and the range of JpB+tT is all of B∗ for all t>0. By Kohsaka and Takahashi [8], since B is a reflexive smooth strictly convex space, we obtain for every t>0 and x∈B, there exists a unique xt∈B such that
(36)JpBx=JpBxt+tTxt.DefineJtTx≔xt; in other words, define a single-valued mapping JtT:B⟶B by JtT=JpB+tT−1JpB. Such a JtT is called the resolvent of T. Setting t≔1/θn and by the result of Aoyama et al. [34] and Reich [35], for some x1∈B, there exists in B a unique
(37)yn≔JpB+1θnT−1JpBx1with yn⟶x∈T−10. Obviously, one can obtain that
(38)Tyn=θnJpBx1−JpByn,and ynn=1∞ is known to be bounded. Also it can be obtained that
(39)θnJpByn−JpBx1+Tyn=0.From (39), we have that
(40)θnJpByn−JpBx1+Tyn−Tx1=−Tx1,which is equivalent to
(41)Tyn−Tx1=−θnJpByn−JpBx1−Tx1.Consequently,(42)η∥yn−x1∥p≤yn−x1,Tyn−Tx1byp,η‐stronglymonotonicityofT=−θnyn−x1,JpByn−JpBx1−yn−x1,Tx1≤∥Tx1∥∥yn−x1∥,which shows that the sequence ynn=1∞ is bounded. Moreover, xnn=1∞ is bounded, and hence, Txnn=1∞ is bounded. Following the same arguments as in part 1, we get
(43)ϕpyn,xn+1≤ϕpyn,xn−pλnxn−yn,Txn+θnJpBxn−JpBx1+pλnM0M.By thep,η-strongly monotonicity property of T and using Lemma 7 and Equation (39), we obtain
(44)xn−yn,Txn+θnJpBxn−JpBx1=xn−yn,Txn+θnJpBxn−JpByn+JpByn−JpBx1=θnxn−yn,JpBxn−JpByn+xn−yn,Txn+θnJpByn−JpBx1=θnxn−yn,JpBxn−JpByn+xn−yn,Txn−Tyn≥θng∥xn−yn∥+η∥xn−yn∥p≥1pθnϕpyn,xn.Therefore, the inequality (43) becomes
(45)ϕpyn,xn+1≤1−λnθnϕpyn,xn+pλnM0M.Observe that by Lemma6, we have
(46)ϕpyn,xn≤ϕpyn−1,xn−pyn−xn,JpByn−1−JpByn=ϕpyn−1,xn+pxn−yn,JpByn−1−JpByn≤ϕpyn−1,xn+∥JpByn−1−JpByn∥∥xn−yn∥.LetR>0 such that ∥x1∥≤R,∥yn∥≤R for all n∈ℕ. We obtain from Equation (39) that
(47)JpByn−1−JpByn+1θnTyn−1−Tyn=θn−1−θnθnJpBx1−JpByn−1.By taking the duality pairing of each side of this equation with respect toyn−1−yn and by the strong monotonicity of T, we have
(48)JpByn−1−JpByn,yn−1−yn≤θn−1−θnθn∥JpBx1−JpByn−1∥∥yn−1−yn∥.Sinceθnn=1∞ is a decreasing sequence, it is known that θn−1≥θn. Therefore,
(49)θn−1−θnθn=θn−1θn−1≥0.Consequently,(50)∥JpByn−1−JpByn∥≤θn−1θn−1∥JpByn−1−JpBx1∥.Using (46) and (50), the inequality (45) becomes
(51)ϕpyn,xn+1≤1−λnθnϕpyn−1,xn+Cθn−1θn−1+pλnM0M,for some constant C>0. By Lemma 8, ϕpyn−1,xn⟶0 as n⟶∞ and using Lemma 9, we have that xn−yn−1⟶0as n⟶∞. Since yn⟶x∈T−10, we obtain that xn⟶x as n⟶∞.Corollary 14.
LetHbe a Hilbert space,p>1,η∈1,∞and supposeT:H⟶His a continuous,p,η-strongly monotone mapping such thatDT⊆rangeI+tTfor allt>0.For arbitraryx1∈H, define the sequencexnn=1∞iteratively by(52)xn+1≔xn−λnTxn−λnθnxn−x1,n∈ℕ,where λnn=1∞⊂0,1 and θnn=1∞ in 0,1/2 are real sequences satisfying the conditions:
(i)
limn→∞θn=0 and θnn=1∞ is decreasing(ii)
∑n=1∞λnθn=∞(iii)
limn⟶∞θn−1/θn−1/λnθn=0,∑n=1∞λn<∞∀n∈ℕ
Suppose that the equationTx=0 has a solution. Then, the sequence xnn=1∞ converges strongly to the solution of the equation Tx=0.Proof.
The result follows from Theorem13 since uniformly smooth and uniformly convex spaces are more general than the Hilbert spaces.
Examples are given for nonlinear mappings of the monotone type which satisfies the conditions stated in the main theorem.Example 15.
LetT:ℝn→ℝnwithTx=xp−2andp≥2.Then,(53)x−y,xp−2x−yp−2y=12xp−2+yp−2x−y2+12xp−2−yp−2x2−y2≥2−1xp−2+yp−2x−y2≥22−px−yp.Thus,T is p,η-strongly monotone with η≔22−p.Example 16.
LetE=ℝ2with the usual norm. Consider the functionT:ℝ2⟶ℝ2defined byTx=gx,where(54)g=8−5513,x=x1x2.Then,Tisp,η-strongly monotone withp≔2andη≔8.Indeed,x,Tx≥8∥x∥2.


## 4. Solution of Convex Minimization Problems

The result of Theorem13 is applied in this section for solving a problem of finding a minimizer of a convex function φ defined from a real Banach space B to ℝ. Recall that a mapping T:B⟶B∗ is said to be coercive if for any x∈B,(55)x,Tx∥x∥⟶∞as∥x∥⟶∞.The following well-known basic results will be used.Lemma 17.
Letφ:B→ℝbe a real-valued differentiable convex function andu∈B.Letdφ:B⟶B∗denote the differential map associated toφ.Then, the following hold:(1)
The pointu is a minimizer of φ on B if and only if dφu=0(2)
Ifφ is bounded, then φ is locally Lipschitzian, i.e., for every x0∈B and r>0, there exists L>0 such that φ is L-Lipschitzian on Bx0,r, i.e.,(56)∥φx−φy∣≤L∥x−y∥∀x,y∈Bx0,r.The main result in this section is given below.Theorem 18.
LetBbe a uniformly smooth and uniformly convex real Banach space. Letφ:B⟶ℝbe a differentiable, convex, bounded, and coercive function. Letλnn=1∞⊂0,1andθnn=1∞in0,1/2be real sequences such that,(i)
limn⟶∞θn=0 and θnn=1∞ is decreasing(ii)
∑n=1∞λnθn=∞(iii)
limn⟶∞θn−1/θn−1/λnθn=0,∑n=1∞λn<∞∀n∈ℕ
For arbitraryx1∈B, define xnn=1∞ iteratively by
(57)xn+1=JpB∗JpBxn−λndφxn+θnJpBxn−JpBx1,n∈ℕ,where JB is the generalized duality mapping from B into B∗. Then, φ has a minimizer x∗∈B and the sequence xnn=1∞ converges strongly to x∗.Proof.
φ has a minimizer because it is a function which is lower semicontinuous, convex, and coercive. Moreover, x∗∈B minimizes φ if and only if dφx∗=0. It can be inferred that dφ is a maximal monotone due to the convexity, the differentiability, and the boundedness of φ (see, e.g., Minty [36] and Moreau [37]). The next task is to show that dφ is bounded. Indeed, let x0∈B and r>0. By Lemma 17, there exists L>0 such that
(58)∥φx−φy∣≤L∥x−y∥∀x,y∈Bx0,r.Letv∗∈dφBx0,r and x∗∈Bx0,r such that v∗=dφx∗.Since Bx0,r is open, for all u∈B, there exists σ>0 such that x∗+σu∈(Bx0,r. From the fact that v∗=dφx∗ and inequality (58), it is obtained that
(59)v∗,σu≤φx∗+σu−φx∗≤σL∥u∥,such that
(60)v∗,u≤L∥u∥∀u∈B.Consequently,∥v∗∥≤L, which implies that dφBx0,r is bounded. Thus, dφ is bounded. Hence, it can be deduced from Theorem 13 that the sequence xnn=1∞ converges strongly to x∗, a minimizer of φ.Example 19.
An example of a function which is coercive is a real valued functionf:ℝ2⟶ℝwhich is defined byfu,v=u4−7uv+v3.Constructively,fu,v=u3+v41−7uv/u3+v4. As ∥u,v∥⟶∞,7uv/u3+v4⟶0 while u3+v4⟶∞. It follows that
(61)lim∥u,v∥→∞fu,v=lim∥u,v∥⟶∞u3+v41−0=+∞.Hence,f is coercive.


## 5. Solutions of Variational Inequality Problems

LetK be a nonempty, closed, and convex subset of a real normed linear space B and let T:K→B be a nonlinear mapping. The variational inequality problem is to
(62)findx∈Ksuchthatjpx−y,Tx≥0,∀y∈K,for some jpx−y∈Jpx−y. The set of solutions of a variational inequality problem is denoted by VIT,K. If B≔H, a Hilbert space, the variational inequality problem reduces to
(63)findx∈Ksuchthatx−y,Tx≥0,∀y∈K,which was introduced and studied by Stampacchia [38]. Variational inequality theory has emerged as an important tool in studying a wide class of related problems arising in mathematical, physical, regional, engineering, and nonlinear optimization sciences. The theories of variational inequality problems have numerous applications in the study of nonlinear analysis (see, e.g., Censor et al. [39], Korpelevich [40], Shi [41], and Stampacchia [38] and the references contained in them). Several existence results have been established for (62) and (63) when T is a monotone type mapping (see, e.g., Barbu and Precupanu [42], Browder [43], and Hartman and Stampacchia [44] and the references contained in them).LetK be a closed convex subset of H. The projection into K is defined to be the mapping, PK:H→K, which is given by
(64)∥PKx−x∥=min∥y−x∥:y∈K.Gradient projection method is an orthodox way for solving (63). The projection algorithm is given by
(65)x1∈K,xn+1=PKxn−ηnTxn,n∈ℕ,where T is η-strongly pseudomonotone and L-Lipschitz continuous mapping (see, e.g., Khanh and Vuong [45]). A recent report eliminated some drawbacks in the study of algorithm (65) [46]. The report considered a mapping T, which is η-strongly pseudomonotone and bounded on bounded subsets of K.We are interested in the set of solutions of the formVIT,C, where T:B⟶B∗ is a p,η-strongly monotone mapping, C≔∩i=1NFφi≠∅,φi:K⟶B,i=1,2,⋯,N is a finite family of quasi-ϕp-nonexpansive mappings, and B is a uniformly smooth and uniformly convex real Banach space. Recall that a mapping φ:K⟶K is called nonexpansive if ∥φx−φy∥≤∥x−y∥,∀x,y∈K.The set of fixed points of the mapping φ will be denoted by Fφ. A mapping φ is said to be quasi-ϕp-nonexpansive if Fφ≠∅ and ϕpu,φx≤ϕpu,x,∀x∈K and u∈Fφ. The proof of the following theorem is given.Theorem 20.
LetBbe a uniformly smooth and uniformly convex real Banach space andKa nonempty, closed, and convex subset ofB.Letp>1,η∈1,∞,supposeT:B⟶B∗is a continuous,p,η-strongly monotone mapping such that the range ofJp+tTis all ofB∗for allt>0.Letφi:K⟶B,i=1,2,⋯,Nbe a finite family of quasi-ϕp-nonexpansive mappings withC≔∩i=1NFφi≠∅.Letλnn=1∞⊂0,1andθnn=1∞in0,1/2be real sequences such that(i)
limn⟶∞θn=0 and θnn=1∞ is decreasing(ii)
∑n=1∞λnθn=∞(iii)
limn⟶∞θn−1/θn−1/λnθn=0,∑n=1∞λn<∞∀n∈ℕ
For arbitraryx1∈B, define xnn=1∞ iteratively by
(66)xn+1=JpB∗JpBφnxn−λnTφnxn+θnJpBφnxn−JpBφnx1,n∈ℕ,where φn≔φnModN and JpB is the generalized duality mapping from B into B∗. Then, the sequence xnn=1∞ converges strongly to x∈VIT,C.Proof.
Firstly, it is shown that the sequencexnn=1∞ is bounded.
Letq>1 with 1/p+1/q=1 and x∈VIT,C. It suffices to show that ϕpx,xn≤r,∀n∈ℕ. The proof is by induction. Let r>0 be sufficiently large such that
(67)r≥maxϕpx,x1,4M0M,4pq∥x∥q,where M0>0 and M>0 are arbitrary but fixed. By construction, ϕpx,x1≤r. Suppose that ϕpx,xn≤r for some n∈ℕ. From inequality (20), for real p>1, we have ∥xn∥≤r1/p+∥x∥. Let D≔z∈B:ϕpx,z≤r. We show that ϕpx,xn+1≤r. It is known that T is locally bounded and JpB is uniformly continuous on bounded subsets of B. Define
(68)M0≔sup∥Tφnxn+θnJpBφnxn−JpBφnx1∥:θn∈0,12,xn∈D+1.Letψ denotes the modulus of continuity of JpB∗. Then,
(69)∥φnxn−xn+1∥=∥φnxn−JpB∗Jpφnxn−λnTφnxn+θnJpBφnxn−JpBφnx1∥=∥JpB∗JpBφnxn−JpB∗JpBφnxn−λnTφnxn+θnJpBφnxn−JpBφnx1∥≤ψ∣λn∣∥Tφnxn+θnJpBφnxn−JpBφnx1∥≤ψ∣λn∣M0≤ψsup∣λn∣M0:λn∈0,1.SinceT is locally bounded and the duality mapping JpB is uniformly continuous on bounded subsets of B, the sup∣λn∣M0exists, and it is a real number different from infinity. Define M≕ψsup∣λn∣M0. Applying Lemma 4 with y∗≔λnTφnxn+θnJpBφnxn−JpBφnx1 and by using the definition of xn+1, we compute as follows:
(70)ϕpx,xn+1=ϕpx,JpB∗JpBφnxn−λnTφnxn+θnJpBφnxn−JpBφnx1=Vpx,JpBφnxn−λnTφnxn+θnJpBφnxn−JpBφnx1by21≤Vpx,JpBφnxn−pλnxn+1−x,Tφnxn+θnJpBφnxn−JpBφnx1=ϕpx,φnxn−pλnφnxn−x,Tφnxn+θnJpBφnxn−JpBφnx1−pλnxn+1−φnxn,Tφnxn+θnJpBφnxn−JpBφnx1.By Schwartz inequality and by applying inequality (69), we obtain
(71)ϕpx,xn+1≤ϕpx,φnxn−pλnφnxn−x,Tφnxn+θnJpBφnxn−JpBφnx1+pλnM0M≤ϕpx,φnxn−pλnφnxn−x,Tφnxn−Tx−pλnφnxn−x,Tx−pλnθnφnxn−x,JpBφnxn−JpBφnx1+pλnM0M≤ϕpx,φnxn−pλnφnxn−x,Tφnxn−Txsincex∈VIT,C−pλnθnφnxn−x,JpBφnxn−JpBφnx1+pλnM0M.By Lemma6, px−φnxn,JpBφnxn−JpBφnx1≤ϕpx,φnxn−ϕpx,φnx1. Consequently, px−φnxn,JpBφnxn−JpBφnx1≤ϕpx,φnxn. Therefore, using p,η-strongly monotonicity property of T, we have
(72)ϕpx,xn+1≤ϕpx,φnxn−pηλn∥Tφnxn−x∥p−pλnθnφnxn−x,JpBφnxn−JpBφnx1+pλnM0M≤ϕpx,φnxn−pλnϕpx,Tφnxn−pq∥x∥q+pλnθnφnxn−x,JpBφnxn−JpBφnx1+pλnM0M≤ϕpx,φnxn−pλnϕpx,φnxn+pλnpq∥x∥q+pλnθnϕpx,φnxn+pλnM0M=1−pλnϕpx,φnxn+pλnpq∥x∥q+pλnθnϕpx,φnxn+pλnM0M≤1−pλnϕpx,xn+pλnpq∥x∥q+pλnθnϕpx,xn+pλnM0M≤1−pλnr+pλnr4+pλnr2+pλn4r=1−pλn+pλn14+pλn12+pλn14r=r.Hence,ϕpx,xn+1≤r. By induction, ϕpx,xn≤r∀n∈ℕ. Thus, from inequality (20), xnn=1∞ is bounded. The remaining part of the proof follows from the proof of Theorem 13.Remark 21.
It well known that uniformly smooth and uniformly convex spaces are more general than the Hilbert spaces. Therefore, the following corollary is readily obtainable.Corollary 22.
LetHbe a Hilbert space and andKa nonempty, closed, and convex subset ofH.Letp>1,η∈1,∞; supposeT:H→His a continuous,p,η-strongly monotone mapping such thatDT⊆rangeI+tTfor allt>0.Letφi:K→H,i=1,2,⋯,Nbe a finite family of quasi-ϕp-nonexpansive mappings withC≔∩i=1NFφi≠∅.Letλnn=1∞⊂0,1andθnn=1∞in0,1/2be real sequences such that(i)
limn→∞θn=0 and θnn=1∞ is decreasing(ii)
∑n=1∞λnθn=∞(iii)
limn→∞θn−1/θn−1/λnθn=0,∑n=1∞λn<∞∀n∈ℕ
For arbitraryx1∈H, define xnn=1∞ iteratively by
(73)xn+1=φnxn−λnTφnxn+θnφnxn−φnx1,n∈ℕ,where φn≔φnModN. Then, the sequence xnn=1∞ converges strongly to x∈VIT,C.


## 6. Conclusion

Real-life problems are usually modeled by nonlinear equations. Nonlinear equations occur in modeling problems, such as minimizing costs in industries and minimizing risks in businesses. Nonlinear equations ofp,η-strongly monotone type, where η∈1,∞,p>1, have been studied in this paper. The result was applied to obtain the solution of convex minimization and variational inequality problems, which have applications in several fields such as economics, game theory, and the sciences.



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