Algorithm for solutions of nonlinear equations of strongly monotone type and applications to convex minimization and variational inequality problems
RResearch Article
Algorithm for Solutions of Nonlinear Equations of StronglyMonotone Type and Applications to Convex Minimization andVariational Inequality Problems
Mathew O. Aibinu , Surendra C. Thakur, and Sibusiso Moyo Institute for Systems Science & KZN CoLab, Durban University of Technology, Durban 4000, South Africa KZN CoLab, Durban University of Technology, Durban 4000, South Africa Institute for Systems Science & O ffi ce of the DVC Research, Innovation & Engagement Milena Court, Durban Universityof Technology, Durban 4000, South Africa Correspondence should be addressed to Mathew O. Aibinu; [email protected]
Received 5 March 2020; Revised 29 April 2020; Accepted 19 May 2020; Published 1 August 2020
Academic Editor: Jewgeni DshalalowCopyright © 2020 Mathew O. Aibinu et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.Real-life problems are governed by equations which are nonlinear in nature. Nonlinear equations occur in modeling problems, suchas minimizing costs in industries and minimizing risks in businesses. A technique which does not involve the assumption ofexistence of a real constant whose calculation is unclear is used to obtain a strong convergence result for nonlinear equations of ð p , η Þ - strongly monotone type, where η > 0, p > 1 . An example is presented for the nonlinear equations of ð p , η Þ - stronglymonotone type. As a consequence of the main result, the solutions of convex minimization and variational inequality problemsare obtained. This solution has applications in other fi elds such as engineering, physics, biology, chemistry, economics, andgame theory.
1. Introduction
Let ν : ½ ∞ Þ ⟶ ½ ∞ Þ be a continuous, strictly increasingfunction such that ν ð t Þ ⟶ ∞ as t → ∞ and ν ð Þ = 0 forany t ∈ ½ ∞ Þ . Such a function ν is called a gauge function .Let B be a Banach space, and B ∗ denotes it is dual. A dualitymapping associated with the gauge function ν is a mapping J B ν : B ⟶ B ∗ de fi ned by J B ν x ð Þ = f ∈ B ∗ : x , f h i = ∥ x ∥ ν ∥ x ∥ ð Þ , ∥ f ∥ = ν ∥ x ∥ ð Þf g , ð Þ where h : , : i denotes the duality pairing. For p > 1 , let ν ð t Þ = t p − be a gauge function. J B : B ⟶ B ∗ is called a generalizedduality mapping from B into B ∗ and is given by J Bp x ð Þ = f ∈ B ∗ : x , f h i = ∥ x ∥ p , ∥ f ∥ = ∥ x ∥ p − (cid:1) (cid:3) : ð Þ For p = 2 , the mapping J B is called the normalizedduality mapping . In a Hilbert space, the normalized dualitymapping is the identity map. For x , y ∈ B and η > 0, a map-ping T : B ⟶ B ∗ is said to be(1) monotone if h x − y , Tx − Ty i ≥ (2) strongly monotone (see, e.g., Alber and Ryazantseva[1], p. 25), if h x − y , Tx − Ty i ≥ η ∥ x − y ∥ (3) η -strongly pseudomonotone if h x − y , Ty i ≥ ⟹ h x − y , Tx i ≥ η ∥ x − y ∥ (4) ð p , η Þ - strongly monotone if h x − y , Tx − Ty i ≥ η ∥ x − y ∥ p (see, e.g., Chidume and Djitté [2], Chi-dume and Shehu [3], and Aibinu and Mewomo[4, 5]). HindawiAbstract and Applied AnalysisVolume 2020, Article ID 6579720, 11 pageshttps://doi.org/10.1155/2020/6579720 emark 1.
According to the de fi nition of Chidume and Djitté[2] and Chidume and Shehu [3], a strongly monotone map-ping is referred to as ð η Þ -strongly monotone mapping.A monotone mapping T is called maximal monotone if itis a monotone and its graph is not properly contained in thegraph of any other monotone mapping. As a result ofRockafellar [6], T is said to be a maximum monotone if itis a monotone and the range of ð J B + tT Þ is all of B ∗ for some t > 0 . The set of zeros of a maximum monotone mapping T , T − ð Þ ≔ f x ∈ B : Tx = 0 g is closed and convex. A function φ : B ⟶ ð −∞ ,+ ∞ (cid:2) is said to be proper if the set f x ∈ ℝ : F ð x Þ ∈ ℝ g is nonempty. A proper function φ : B ⟶ ð −∞ ,+ ∞ (cid:2) issaid to be convex if for all x , y ∈ B and τ ∈ ½
0, 1 (cid:2) , we have φ τ x + 1 − τ ð Þ y ð Þ ≤ τφ x ð Þ + 1 − τ ð Þ φ y ð Þ : ð Þ If the set of f x ∈ ℝ : φ ð x Þ ≤ r g is closed in B for all r ∈ ℝ , φ issaid to be lower semicontinuous . For a proper lower semicontin-uous function φ : B ⟶ ð −∞ ,+ ∞ (cid:2) , the subdi ff erential mapping ∂ φ : B ⟶ B ∗ , de fi ned by ∂ φ x ð Þ = x ∗ ∈ B ∗ : φ y ð Þ − φ x ð Þ ≥ y − x , x ∗ h i ∀ y ∈ B f g , ð Þ is a maximal monotone (Rockafellar [7]). Consider a problem of fi nding a solution of the equation Tu = 0, where T is a maximalmonotone mapping. Such a problem is associated with the con-vex minimization problem . Indeed, for a proper lower semicon-tinuous convex function φ : B ⟶ ð −∞ ,+ ∞ (cid:2) , solving theequation Tu = 0 is equivalent to fi nding φ ð u Þ = min x ∈ B φ ð x Þ by setting ∂ φ ≡ T : For a re fl exive smooth strictly convex space B , we let T bea mapping such that the range of ð J Bp + tT Þ is all of B ∗ forsome t > 0 and let x ∈ B be fi xed. Then, for every t > 0, therecorresponds a unique element x t ∈ D ð T Þ such that J Bp x = J Bp x t + tTx t : ð Þ Therefore, the resolvent of T is de fi ned by J Tt x = x t : Inother words, J Tt = ð J Bp + tT Þ − J Bp and T − F ð J Tt Þ for all t > 0, where F ð J Tt Þ denotes the set of all fi xed points of J Tt : The resolvent J Tt is a single-valued mapping from B into D ð T Þ (Kohsaka and Takahashi [8]). J Tt is nonexpansive if E isa Hilbert space (Takahashi [9]). Some existing results proveda strong convergence theorem for nonlinear equations of themonotone type, with the assumption of existence of a realconstant whose calculation is unclear (see, e.g., Aibinu andMewomo [4], Chidume et al. [10], and Diop et al. [11]).Monotone-type mappings occur in many functional equa-tions, and the research on monotone type mappings hasrecently attracted much attention (see, e.g., Shehu [12, 13],Chidume et al. [14], Djitte et al. [15], Tang [16], Uddinet al. [17], Chidume and Idu [18], and Aibinu andMewomo [19]).In this paper, we consider nonlinear equations of ð p , η Þ -strongly monotone type, p > 1 and η ∈ ð ∞ Þ : This isa wider class than the class of strongly monotone mappings. An example is presented for nonlinear equations of ð p , η Þ - strongly monotone type. Under suitable conditionswhich do not involve the assumption of existence of a real con-stant whose calculation is unclear, a sequence of iteration isshown to converge strongly to the zero of a nonlinear equationof ð p , η Þ -strongly monotone type. As a consequence of the mainresult, the solution of convex minimization and variationalinequality problems is obtained, which has applications in sev-eral fi elds such as economics, game theory, and the sciences.
2. Preliminaries
Let B be a real Banach space and S ≔ f x ∈ B : ∥ x ∥ = 1 g . B issaid to have a Gateaux di ff erentiable norm if the limit lim t → ∥ x + ty ∥−∥ x ∥ t , ð Þ exists for each x , y ∈ S : A Banach space B is said to be smooth if for every x ≠ in B , there is a unique x ∗ ∈ B ∗ such that ∥ x ∗ ∥ = 1 and h x , x ∗ i = ∥ x ∥ , where B ∗ denotes the dual of B : B is said to be uniformly smooth if it is smooth and thelimit (6) is attained uniformly for each x , y ∈ S : The mod-ulus of convexity of a Banach space B , δ B : ð
0, 2 (cid:2) ⟶ ½
0, 1 (cid:2) is de fi ned by δ B ε ð Þ = inf 1 − ∥ x + y ∥ : ∥ x ∥ = ∥ y ∥ = 1, ∥ x − y ∥ > ε (cid:4) (cid:5) : ð Þ B is uniformly convex if and only if δ B ð ε Þ > 0 for every ε ∈ ð
0, 2 (cid:2) . A normed linear space B is said to be strictly convex if ∥ x ∥ = ∥ y ∥ = 1, x ≠ y ⟹ ∥ x + y ∥ : ð Þ It is well known that a space B is uniformly smooth if andonly if B ∗ is uniformly convex.A mapping T : B ⟶ B ∗ is locally bounded at v ∈ D , ifthere exist r v > 0 and m > 0 such that ∥ Tx ∥≤ m , ∀ x ∈ D r v v ð Þ : ð Þ In particular, ∥ Tv ∥≤ m : Therefore, h v , Tv i ≤ m ∥ v ∥ : Let X and Y be real Banach spaces and let T : X ⟶ Y be a map-ping. T is uniformly continuous if for each ε > 0, there exists δ > 0 such that ∥ Tx − Ty ∥ < ε ∀ x , y ∈ X with ∥ x − y ∥ < δ : ð Þ Let ψ ð t Þ be a function on the set ℝ + of nonnegative realnumbers such that(i) ψ is nondecreasing and continuous(ii) ψ ð t Þ = 0 if and only if t = 0 T is said to be uniformly continuous if it admits themodulus of continuity ψ such that ∥ T x ð Þ − T y ð Þ ∥≤ ψ ∥ x − y ∥ ð Þ ∀ x , y ∈ X : ð Þ ψ has some useful properties(for instance, see Altomare and Campiti [20], pp. 266 – Let X and Y be realBanach spaces and let T : X ⟶ Y be a map which admitsthe modulus of continuity ψ : (a) Modulus of continuity is subadditive : for all real num-bers t ≥ t ≥ we have ψ t + t ð Þ ≤ ψ t ð Þ + ψ t ð Þ ð Þ (b) Modulus of continuity is monotonically increasing : if ≤ t ≤ t holds for some real numbers t , t , then ≤ ψ t ð Þ ≤ ψ t ð Þ ð Þ (c) Modulus of continuity is continuous : the modulus ofcontinuity ψ : ℝ + ⟶ ℝ + is continuous on the setpositive real numbers; in particular, the limit of ψ at from above is lim t → ψ t ð Þ = 0 ð Þ Let C be a nonempty subset of a Banach space B and T bea mapping from C into itself.(i) T is nonexpansive provided ∥ Tx − Ty ∥≤∥ x − y ∥ for all x , y ∈ C (ii) T is fi rmly nonexpansive type (see, e.g., [22]) if h Tx − Ty , j Bp Tx − j Bp Ty i ≤ h Tx − Ty , j Bp x − j Bp y i for all x , y ∈ C and j Bp ∈ J Bp The following results about the generalized duality map-pings are well known which are established in, e.g., Alber andRyazantseva [1] (p. 36), Cioranescu [23] (pp. 25 – B be a Banach space.(i) B is smooth if and only if J Bp is single-valued(ii) If B is re fl exive, then J Bp is onto(iii) If B has uniform Gateaux di ff erentiable norm, then J Bp is norm-to-weak ∗ uniformly continuous onbounded sets(iv) B is uniformly smooth if and only if J Bp is single-valued and uniformly continuous on any boundedsubset of B (v) If B is strictly convex, then J Bp is one-to-one, that is, ∀ x , y ∈ B , x ≠ y ⟹ J Bp ð x Þ ∩ J Bp ð y Þ = ∅ (vi) If B and B ∗ are strictly convex and re fl exive, then J B ∗ p is the generalized duality mapping from B ∗ to B , and J B ∗ p is the inverse of J Bp (vii) If B is uniformly smooth and uniformly convex, thegeneralized duality mapping J B ∗ p is uniformly con-tinuous on any bounded subset of B ∗ (viii) If B and B ∗ are strictly convex and re fl exive, forall x ∈ B and f ∈ B ∗ , the equalities J Bp J B ∗ p f = f and J B ∗ p J Bp x = x hold De fi nition 2. Alber [26] introduced the functions ϕ : B × B → ℝ , de fi ned by ϕ x , y ð Þ = ∥ x ∥ − x , J B y (cid:6) (cid:7) + ∥ y ∥ , for all x , y ∈ B , ð Þ where J B is the normalized duality mapping from B to B ∗ : Let B be a smooth real Banach space and p , q > 1 with p + 1/ q = 1 : Aibinu and Mewomo [4] introducedthe functions ϕ p : B × B ⟶ ℝ , de fi ned by ϕ p x , y ð Þ = pq ∥ x ∥ q − p x , J Bp y D E + ∥ y ∥ p , for all x , y ∈ B , ð Þ and V p : B × B ∗ → ℝ , de fi ned as V p x , x ∗ ð Þ = pq ∥ x ∥ q − p x , x ∗ h i + ∥ x ∗ ∥ p ∀ x ∈ B , x ∗ ∈ B ∗ , ð Þ where J Bp is the generalized duality mapping from B to B ∗ : Remark 3.
These remarks follow from De fi nition 2:(i) For p = 2, ϕ ð x , y Þ = ϕ ð x , y Þ , which is the de fi nition ofAlber [26]. It is easy to see from the de fi nition of thefunction ϕ that ∥ x ∥−∥ y ∥ ð Þ ≤ ϕ x , y ð Þ ≤ ∥ x ∥ + ∥ y ∥ ð Þ for all x , y ∈ B : ð Þ Indeed, ∥ x ∥−∥ y ∥ ð Þ = ∥ x ∥ − ∥ x ∥∥ y ∥ + ∥ y ∥ ≤ ∥ x ∥ − x , J Bp y D E + ∥ y ∥ = ϕ x , y ð Þ ≤ ∥ x ∥ + 2 ∥ x ∥∥ y ∥ + ∥ y ∥ = ∥ x ∥ + ∥ y ∥ ð Þ : ð Þ fi ed that for each p ≥ ∥ x ∥−∥ y ∥ ð Þ p ≤ ϕ p x , y ð Þ ≤ ∥ x ∥ + ∥ y ∥ ð Þ p for all x , y ∈ B : ð Þ (ii) It is obvious that V p x , x ∗ ð Þ = ϕ p x , J B ∗ p x ∗ (cid:8) (cid:9) ∀ x ∈ B , x ∗ ∈ B ∗ : ð Þ Let B be a topological real vector space and T a multiva-lued mapping from B into B ∗ : Cauchy-Schwartz ’ s inequalityis given by ∣ x , y ∗ h i ∣ ≤ x , x ∗ h i y , y ∗ h i , ð Þ 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 proofshave been established (see, e.g., Alber [26] and Aibinu andMewomo [4]).Lemma 4. Let B be a strictly convex and uniformly smoothreal Banach space and p > 1 : Then, V p x , x ∗ ð Þ + p J B ∗ p x ∗ − x , y ∗ D E ≤ V p x , x ∗ + y ∗ ð Þ , ð Þ for all x ∈ B and x ∗ , y ∗ ∈ B ∗ : Lemma 5.
Let B be a smooth uniformly convex real Banachspace and p > be an arbitrarily real number. For d > , let B d ð Þ ≔ f x ∈ B : ∥ x ∥≤ d g . Then, for arbitrary x , y ∈ B d ð Þ , ∥ x − y ∥ p ≥ ϕ p x , y ð Þ − pq ∥ x ∥ q , where 1p + = : ð Þ Lemma 6.
Let B be a re fl exive strictly convex and smooth realBanach space and p > : Then, ϕ p y , x ð Þ − ϕ p y , z ð Þ ≥ p z − y , J Bp x − J Bp z D E = p y − z , J Bp z − J Bp x D E for all x , y , z ∈ B : ð Þ Lemma 7.
Let B be a real uniformly convex Banach space. Forarbitrary r > , let B r ð Þ ≔ f x ∈ B : ∥ x ∥≤ r g . Then, there existsa continuous strictly increasing convex function g : , ∞ ½ Þ ⟶ , ∞ ½ Þ , g 0 ð Þ = , ð Þ such that for every x , y ∈ B r ð Þ , j Bp ð x Þ ∈ J Bp ð x Þ , j Bp ð y Þ ∈ J Bp ð y Þ ,we have h x − y , j Bp ð x Þ − j Bp ð y Þi ≥ g ð ∥ x − y ∥ Þ (see Xu [28]). Lemma 8.
Let f a n g be a sequence of nonnegative real num-bers satisfying the following relations: a n + ≤ − α n ð Þ a n + α n σ n + γ n , n ∈ ℕ , ð Þ where(i) f α g n ⊂ ð , Þ , ∑ ∞ n = α n = ∞ (ii) limsup f σ g n ≤ (iii) γ n ≥ , ∑ ∞ n = γ n < ∞ Then, a n ⟶ as n ⟶ ∞ (see Xu [29]). Lemma 9.
Let B be a smooth uniformly convex real Banachspace and let f x n g and f y n g be two sequences from B : If either f x n g or f y n g is bounded and ϕ ð x n , y n Þ ⟶ as n ⟶ ∞ ,then ∥ x n − y n ∥ ⟶ as n ⟶ ∞ (see Kamimura andTakahashi [30]). Lemma 10.
A monotone map T : B → B ∗ is locally boundedat the interior points of its domain (see, e.g., Rockafellar [31]and Pascali and Sburlan [32]). Lemma 11.
If a functional ϕ on the open convex set M ⊂ dom ϕ has a subdi ff erential, then ϕ is convex and lower semicontin-uous on the set (see Alber and Ryazantseva [1], p. 17). Lemma 12.
Let X and Y be real normed linear spaces and let T : X ⟶ Y be a uniformly continuous map. For arbitrary r > and fi xed x ∗ ∈ X , let B X x ∗ , r ð Þ : x ∈ X : ∥ x − x ∗ ∥ X ≤ r f g : ð Þ Then, T ð B ð x ∗ , r ÞÞ is bounded (see, e.g., Chidume andDjitte [33]).
3. Main Results
Theorem 13.
Let B be a uniformly smooth and uniformly con-vex real Banach space. Let p > , η ∈ ð , ∞ Þ ; suppose T : B ⟶ B ∗ is a continuous ð p , η Þ -strongly monotone mappingsuch that the range of ð J Bp + tT Þ is all of B ∗ for all t > and T − ð Þ ≠ ∅ : Let f λ n g ∞ n = ⊂ ð , Þ and f θ n g ∞ n = in ð , / Þ bereal sequences such that(i) lim n ⟶ ∞ θ n = and f θ n g ∞ n = is decreasing(ii) ∑ ∞ n = λ n θ n = ∞ (iii) lim n ⟶ ∞ ðð θ n − / θ n Þ − Þ / λ n θ n = , ∑ ∞ n = λ n < ∞∀ n ∈ ℕ For arbitrary x ∈ B , de fi ne f x n g ∞ n = iteratively by: x n + = J B ∗ p J Bp x n − λ n Tx n + θ n J Bp x n − J Bp x (cid:8) (cid:9)(cid:8) (cid:9)(cid:8) (cid:9) , n ∈ ℕ , ð Þ here J Bp is the generalized duality mapping from B into B ∗ : Then, the sequence f x n g ∞ n = converges strongly to the solutionof Tx = : Proof.
Observe that there is no need for constructing a con-vergence sequence if x = 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 f x n g ∞ n =1 is shown to be bounded.Let q > 1 with p + 1/ q = 1 and x ∈ B be a solution of theequation Tx = 0 : It su ffi ces to show that ϕ p ð x , x n Þ ≤ r , ∀ n ∈ ℕ : The induction method will be adopted. Let r > 0 be su ffi -ciently large such that r ≥ max ϕ p x , x ð Þ , 4 M M , 4 pq ∥ x ∥ q (cid:4) (cid:5) , ð Þ where M > 0 and M > 0 are arbitrary but fi xed. For n = 1, byconstruction, we have that ϕ p ð x , x Þ ≤ r for real p > 1 : Assumethat ϕ p ð x , x n Þ ≤ r for some n ≥ : From inequality (20), wehave ∥ x n ∥≤ r p + ∥ x ∥ : Let D ≔ f z ∈ B : ϕ p ð x , z Þ ≤ r g : Next isto show that ϕ p ð x , x n +1 Þ ≤ r : It is known that T is locallybounded (Lemma 10) and J Bp is uniformly continuous onbounded subsets of B : De fi ne M ≔ sup ∥ Tx n + θ n J Bp x n − J Bp x (cid:8) (cid:9) ∥ : θ n ∈
0, 12 (cid:10) (cid:11) , x n ∈ D (cid:4) (cid:5) + 1 : ð Þ Let ψ denotes the modulus of continuity of J B ∗ p : Then, ∥ x n − x n +1 ∥ = ∥ x n − J B ∗ p J Bp x n − λ n Tx n + θ n J Bp x n − J Bp x (cid:8) (cid:9)(cid:8) (cid:9)(cid:8) (cid:9) ∥ = ∥ J B ∗ p J Bp x n (cid:8) (cid:9) − J B ∗ p J Bp x n − λ n Tx n + θ n J Bp x n − J Bp x (cid:8) (cid:9)(cid:8) (cid:9)(cid:8) (cid:9) ∥≤ ψ ∣ λ n ∣∥ Tx n + θ n J Bp x n − J Bp x (cid:8) (cid:9) ∥ (cid:8) (cid:9) ≤ ψ ∣ λ n ∣ M ð Þ ≤ ψ sup ∣ λ n ∣ M : λ n ∈
0, 1 ð Þf gð Þ : ð Þ Since T is locally bounded and the duality mapping J Bp is uniformly continuous on bounded subsets of B , the sup f ∣ λ n ∣ M g exists and it is a real number di ff erent fromin fi nity. Choose M ≕ ψ ð sup f ∣ λ n ∣ M gÞ : Applying Lemma4 with y ∗ ≔ λ n ð Tx n + θ n ð J Bp x n − J Bp x ÞÞ and by using thede fi nition of x n +1 , we compute as follows: ϕ p x , x n +1 ð Þ = ϕ p x , J B ∗ J Bp x n − λ n Tx n + θ n J Bp x n − J Bp x (cid:8) (cid:9)(cid:8) (cid:9)(cid:8) (cid:9)(cid:8) (cid:9) = V p x , J Bp x n − λ n Tx n + θ n J Bp x n − J Bp x (cid:8) (cid:9)(cid:8) (cid:9)(cid:8) (cid:9) by 21 ð Þð Þ ≤ V p x , J Bp x n (cid:8) (cid:9) − p λ n J B ∗ p J Bp x n − λ n Tx n + θ n J Bp x n − J Bp x (cid:8) (cid:9)(cid:8) (cid:9)(cid:8) (cid:9)D − x , Tx n + θ n J Bp x n − J Bp x (cid:8) (cid:9)E = ϕ p x , x n ð Þ − p λ n x n − x , Tx n + θ n J Bp x n − J Bp x (cid:8) (cid:9)D E − p λ n J B ∗ p J Bp x n − λ n Tx n + θ n J Bp x n − J Bp x (cid:8) (cid:9)(cid:8) (cid:9)(cid:8) (cid:9)D − x n , Tx n + θ n J Bp x n − J Bp x (cid:8) (cid:9)E : ð Þ By Schwartz inequality and by applying inequality (32),we obtain ϕ p x , x n +1 ð Þ ≤ ϕ p x , x n ð Þ − p λ n x n − x , Tx n + θ n J Bp x n − J Bp x (cid:8) (cid:9)D E + p λ n M M ≤ ϕ p x , x n ð Þ − p λ n x n − x , Tx n h − Tx i since x ∈ T − ð Þ (cid:12) (cid:13) − p λ n θ n x n − x , J Bp x n − J Bp x D E + p λ n M M : ð Þ By Lemma 6, p h x − x n , J Bp x n − J Bp x i ≤ ϕ p ð x , x n Þ − ϕ p ð x , x Þ : Consequently, p h x − x n , J Bp x n − J Bp x i ≤ ϕ p ð x , x n Þ : There-fore, using ð p , η Þ -strongly monotonicity property of T , we have ϕ p x , x n +1 ð Þ ≤ ϕ p x , x n ð Þ − p ηλ n ∥ x n − x ∥ p − p λ n θ n x n − x , J Bp x n − J Bp x D E + p λ n M M ≤ ϕ p x , x n ð Þ − p λ n ∥ x n − x ∥ p + p λ n θ n x − x n , J Bp x n − J Bp x D E + p λ n M M ≤ ϕ p x , x n ð Þ − p λ n ϕ p x , x n ð Þ − pq ∥ x ∥ q (cid:10) (cid:11) + p λ n θ n ϕ p x , x n ð Þ + p λ n M M = 1 − p λ n ð Þ ϕ p x , x n ð Þ + p λ n pq ∥ x ∥ q (cid:10) (cid:11) + p λ n θ n ϕ p x , x n ð Þ + p λ n M M ≤ − p λ n ð Þ r + p λ n r p λ n r p λ n r = 1 − p λ n + p λ n
14 + p λ n
12 + p λ n (cid:10) (cid:11) r = r : ð Þ Hence, ϕ p ð x , x n +1 Þ ≤ r : By induction, ϕ p ð x , x n Þ ≤ r ∀ n ∈ ℕ : Thus, from inequality (20), f x n g ∞ n =1 is bounded. Part 2 : we now show that f x n g ∞ n =1 converges strongly to asolution of Tx = 0 : ð p , η Þ -strongly monotone implies a mono-tone and the range of ð J Bp + tT Þ is all of B ∗ for all t > 0 : ByKohsaka and Takahashi [8], since B is a re fl exive smoothstrictly convex space, we obtain for every t > 0 and x ∈ B , there exists a unique x t ∈ B such that J Bp x = J Bp x t + tTx t : ð Þ De fi ne J Tt x ≔ x t ; in other words, de fi ne a single-valuedmapping J Tt : B ⟶ B by J Tt = ð J Bp + tT Þ − J Bp : Such a J Tt iscalled the resolvent of T : Setting t ≔ θ n and by the resultof Aoyama et al. [34] and Reich [35], for some x ∈ B , thereexists in B a unique y n ≔ J Bp + 1 θ n T (cid:10) (cid:11) − J Bp x ð Þ y n ⟶ x ∈ T − ð Þ : Obviously, one can obtain that Ty n = θ n J Bp x − J Bp y n (cid:8) (cid:9) , ð Þ and f y n g ∞ n =1 is known to be bounded. Also it can be obtainedthat θ n J Bp y n − J Bp x (cid:8) (cid:9) + Ty n = 0 : ð Þ From (39), we have that θ n J Bp y n − J Bp x (cid:8) (cid:9) + Ty n − Tx = − Tx , ð Þ which is equivalent to Ty n − Tx = − θ n J Bp y n − J Bp x (cid:8) (cid:9) − Tx : ð Þ Consequently, η ∥ y n − x ∥ p ≤ y n − x , Ty n − Tx h i by p , η ð Þ ‐ strongly monotonicity of T ð Þ = − θ n y n − x , J Bp y n − J Bp x (cid:8) (cid:9)D E − y n − x , Tx h i ≤ ∥ Tx ∥∥ y n − x ∥ , ð Þ which shows that the sequence f y n g ∞ n =1 is bounded. More-over, f x n g ∞ n =1 is bounded, and hence, f Tx n g ∞ n =1 is bounded.Following the same arguments as in part 1, we get ϕ p y n , x n +1 ð Þ ≤ ϕ p y n , x n ð Þ − p λ n x n − y n , Tx n + θ n J Bp x n − J Bp x (cid:8) (cid:9)D E + p λ n M M : ð Þ By the ð p , η Þ -strongly monotonicity property of T andusing Lemma 7 and Equation (39), we obtain x n − y n , Tx n + θ n J Bp x n − J Bp x (cid:8) (cid:9)D E = x n − y n , Tx n + θ n J Bp x n − J Bp y n + J Bp y n − J Bp x (cid:8) (cid:9)D E = θ n x n − y n , J Bp x n − J Bp y n D E + x n − y n , Tx n + θ n J Bp y n − J Bp x (cid:8) (cid:9)D E = θ n x n − y n , J Bp x n − J Bp y n D E + x n − y n , Tx n − Ty n h i ≥ θ n g ∥ x n − y n ∥ ð Þ + η ∥ x n − y n ∥ p ≥ p θ n ϕ p y n , x n ð Þ : ð Þ Therefore, the inequality (43) becomes ϕ p y n , x n +1 ð Þ ≤ − λ n θ n ð Þ ϕ p y n , x n ð Þ + p λ n M M : ð Þ Observe that by Lemma 6, we have ϕ p y n , x n ð Þ ≤ ϕ p y n − , x n ð Þ − p y n − x n , J Bp y n − − J Bp y n D E = ϕ p y n − , x n ð Þ + p x n − y n , J Bp y n − − J Bp y n D E ≤ ϕ p y n − , x n ð Þ + ∥ J Bp y n − − J Bp y n ∥∥ x n − y n ∥ : ð Þ Let R > 0 such that ∥ x ∥≤ R , ∥ y n ∥≤ R for all n ∈ ℕ . Weobtain from Equation (39) that J Bp y n − − J Bp y n + 1 θ n Ty n − − Ty n ð Þ = θ n − − θ n θ n J Bp x − J Bp y n − (cid:8) (cid:9) : ð Þ By taking the duality pairing of each side of this equationwith respect to y n − − y n and by the strong monotonicity of T , we have J Bp y n − − J Bp y n , y n − − y n D E ≤ θ n − − θ n θ n ∥ J Bp x − J Bp y n − ∥∥ y n − − y n ∥ : ð Þ Since f θ n g ∞ n =1 is a decreasing sequence, it is known that θ n − ≥ θ n : Therefore, θ n − − θ n θ n = θ n − θ n − ≥ : ð Þ Consequently, ∥ J Bp y n − − J Bp y n ∥≤ θ n − θ n − (cid:10) (cid:11) ∥ J Bp y n − − J Bp x ∥ : ð Þ Using (46) and (50), the inequality (45) becomes ϕ p y n , x n +1 ð Þ ≤ − λ n θ n ð Þ ϕ p y n − , x n ð Þ + C θ n − θ n − (cid:10) (cid:11) + p λ n M M , ð Þ for some constant C > 0 . By Lemma 8, ϕ p ð y n − , x n Þ ⟶ as n ⟶ ∞ and using Lemma 9, we have that x n − y n − ⟶ as n ⟶ ∞ : Since y n ⟶ x ∈ T − ð Þ , we obtain that x n ⟶ x as n ⟶ ∞ : Corollary 14.
Let H be a Hilbert space, p > , η ∈ ð , ∞ Þ andsuppose T : H ⟶ H is a continuous, ð p , η Þ -stronglymonotone mapping such that D ð T Þ ⊆ range ð I + tT Þ forall t > : For arbitrary x ∈ H , de fi ne the sequence f x n g ∞ n = iteratively by x n + ≔ x n − λ n Tx n − λ n θ n x n − x ð Þ , n ∈ ℕ , ð Þ where f λ n g ∞ n = ⊂ ð , Þ and f θ n g ∞ n = in ð , / Þ are realsequences satisfying the conditions:(i) lim n → ∞ θ n = and f θ n g ∞ n = is decreasing ii) ∑ ∞ n = λ n θ n = ∞ (iii) lim n ⟶ ∞ ðð θ n − / θ n Þ − Þ / λ n θ n = , ∑ ∞ n = λ n < ∞∀ n ∈ ℕ Suppose that the equation Tx = has a solution. Then, thesequence f x n g ∞ n = converges strongly to the solution of theequation Tx = : Proof.
The result follows from Theorem 13 since uniformlysmooth and uniformly convex spaces are more general thanthe Hilbert spaces.Examples are given for nonlinear mappings of the mono-tone type which satis fi es the conditions stated in the maintheorem. Example 15. Let T : ℝ n → ℝ n with Tx = j x j p − and p ≥ : Then, x − y , x j j p − x − y j j p − y (cid:6) (cid:7) = 12 x j j p − + y j j p − (cid:12) (cid:13) x − y j j + 12 x j j p − − y j j p − (cid:12) (cid:13) x j j − y j j (cid:12) (cid:13) ≥ − x j j p − + y j j p − (cid:12) (cid:13) x − y j j ≥ − p x − y j j p : ð Þ Thus, T is ð p , η Þ -strongly monotone with η ≔ − p : Example 16. Let E = ℝ with the usual norm. Consider thefunction T : ℝ ⟶ ℝ de fi ned by Tx = gx , where g = 8 −
55 13 ! , x = x x ! : ð Þ Then, T is ð p , η Þ -strongly monotone with p ≔ and η ≔ : Indeed, h x , Tx i ≥ ∥ x ∥ :
4. Solution of Convex Minimization Problems
The result of Theorem 13 is applied in this section for solvinga problem of fi nding a minimizer of a convex function φ de fi ned from a real Banach space B to ℝ : Recall that a map-ping T : B ⟶ B ∗ is said to be coercive if for any x ∈ B , x , Tx h i ∥ x ∥ ⟶ ∞ as ∥ x ∥ ⟶ ∞ : ð Þ The following well-known basic results will be used.
Lemma 17.
Let φ : B → ℝ be a real-valued di ff erentiableconvex function and u ∈ B : Let d φ : B ⟶ B ∗ denote thedi ff erential map associated to φ : Then, the following hold:(1) The point u is a minimizer of φ on B if and only if d φ ð u Þ = (2) If φ is bounded, then φ is locally Lipschitzian, i.e., forevery x ∈ B and r > , there exists L > such that φ is L -Lipschitzian on B ð x , r Þ , i.e., ∥ φ x ð Þ − φ y ð Þ ∣ ≤ L ∥ x − y ∥ ∀ x , y ∈ B x , r ð Þ : ð Þ The main result in this section is given below.
Theorem 18.
Let B be a uniformly smooth and uniformly con-vex real Banach space. Let φ : B ⟶ ℝ be a di ff erentiable,convex, bounded, and coercive function. Let f λ n g ∞ n = ⊂ ð , Þ and f θ n g ∞ n = in ð , / Þ be real sequences such that,(i) lim n ⟶ ∞ θ n = and f θ n g ∞ n = is decreasing(ii) ∑ ∞ n = λ n θ n = ∞ (iii) lim n ⟶ ∞ ðð θ n − / θ n Þ − Þ / λ n θ n = , ∑ ∞ n = λ n < ∞∀ n ∈ ℕ For arbitrary x ∈ B , de fi ne f x n g ∞ n = iteratively by x n + = J B ∗ p J Bp x n − λ n d φ x n ð Þ + θ n J Bp x n − J Bp x (cid:8) (cid:9)(cid:8) (cid:9)(cid:8) (cid:9) , n ∈ ℕ , ð Þ where J B is the generalized duality mapping from B into B ∗ : Then, φ has a minimizer x ∗ ∈ B and the sequence f x n g ∞ n = con-verges strongly to x ∗ : Proof. φ has a minimizer because it is a function which islower semicontinuous, convex, and coercive. Moreover, x ∗ ∈ B minimizes φ if and only if d φ ð x ∗ Þ = 0 : It can be inferredthat d φ is a maximal monotone due to the convexity, the dif-ferentiability, and the boundedness of φ (see, e.g., Minty [36]and Moreau [37]). The next task is to show that d φ isbounded. Indeed, let x ∈ B and r > 0 : By Lemma 17, thereexists L > 0 such that ∥ φ x ð Þ − φ y ð Þ ∣ ≤ L ∥ x − y ∥ ∀ x , y ∈ B x , r ð Þ : ð Þ Let v ∗ ∈ d φ ð B ð x , r ÞÞ and x ∗ ∈ B ð x , r Þ such that v ∗ = d φ ð x ∗ Þ : Since B ð x , r Þ is open, for all u ∈ B , there exists σ > 0 suchthat x ∗ + σ u ∈ ð B ð x , r Þ : From the fact that v ∗ = d φ ð x ∗ Þ andinequality (58), it is obtained that v ∗ , σ u h i ≤ φ x ∗ + σ u ð Þ − φ x ∗ ð Þ ≤ σ L ∥ u ∥ , ð Þ such that v ∗ , u h i ≤ L ∥ u ∥ ∀ u ∈ B : ð Þ Consequently, ∥ v ∗ ∥≤ L , which implies that d φ ð B ð x , r ÞÞ isbounded. Thus, d φ is bounded. Hence, it can be deduced fromTheorem 13 that the sequence f x n g ∞ n =1 converges strongly to x ∗ , a minimizer of φ : xample 19. An example of a function which is coerciveis a real valued function f : ℝ ⟶ ℝ which is de fi nedby f ð u , v Þ = u − uv + v : Constructively, f ð u , v Þ = ð u + v Þð − ð uv / ð u + v ÞÞÞ : As ∥ ð u , v Þ ∥ ⟶ ∞ , 7 uv / ð u + v Þ ⟶ while u + v ⟶ ∞ : It follows that lim ∥ u , v ð Þ ∥ → ∞ f u , v ð Þ = lim ∥ u , v ð Þ ∥ ⟶ ∞ u + v (cid:12) (cid:13) − ð Þ = + ∞ : ð Þ Hence, f is coercive.
5. Solutions of Variational Inequality Problems
Let K be a nonempty, closed, and convex subset of a realnormed linear space B and let T : K → B be a nonlinear map-ping. The variational inequality problem is to find x ∈ K such that j p x − y ð Þ , Tx D E ≥ ∀ y ∈ K , ð Þ for some j p ð x − y Þ ∈ J p ð x − y Þ : The set of solutions of a varia-tional inequality problem is denoted by VI ð T , K Þ : If B ≔ H , aHilbert space, the variational inequality problem reduces to find x ∈ K such that x − y , Tx h i ≥ ∀ y ∈ K , ð Þ which was introduced and studied by Stampacchia [38]. Var-iational inequality theory has emerged as an important toolin studying a wide class of related problems arising in math-ematical, physical, regional, engineering, and nonlinear opti-mization sciences. The theories of variational inequalityproblems have numerous applications in the study of nonlin-ear analysis (see, e.g., Censor et al. [39], Korpelevich [40], Shi[41], and Stampacchia [38] and the references contained inthem). 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 Hartmanand Stampacchia [44] and the references contained in them).Let K be a closed convex subset of H : The projection into K is de fi ned to be the mapping, P K : H → K , which is givenby ∥ P K x ð Þ − x ∥ = min ∥ y − x ∥ : y ∈ K f g : ð Þ Gradient projection method is an orthodox way for solv-ing (63). The projection algorithm is given by x ∈ K , x n +1 = P K x n − η n T x n ð Þð Þ , n ∈ ℕ , ( ð Þ where T is η -strongly pseudomonotone and L -Lipschitz con-tinuous mapping (see, e.g., Khanh and Vuong [45]). A recentreport 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 sub-sets of K : We are interested in the set of solutions of the form VI ð T , C Þ , where T : B ⟶ B ∗ is a ð p , η Þ -strongly monotonemapping, C ≔ ∩ Ni =1 F ð φ i Þ ≠ ∅ , φ i : K ⟶ B , i = 1, 2, ⋯ , N isa fi nite family of quasi- ϕ p -nonexpansive mappings, and B isa uniformly smooth and uniformly convex real Banach space.Recall that a mapping φ : K ⟶ K is called nonexpansive if ∥ φ x − φ y ∥≤∥ x − y ∥ , ∀ x , y ∈ K : The set of fi xed points of the map-ping φ will be denoted by F ð φ Þ : A mapping φ is said to be quasi- ϕ p -nonexpansive if F ð φ Þ ≠ ∅ and ϕ p ð u , φ x Þ ≤ ϕ p ð u , x Þ , ∀ x ∈ K and u ∈ F ð φ Þ : The proof of the following theorem is given.
Theorem 20.
Let B be a uniformly smooth and uniformly con-vex real Banach space and K a nonempty, closed, and convexsubset of B : Let p > , η ∈ ð , ∞ Þ , suppose T : B ⟶ B ∗ is acontinuous, ð p , η Þ -strongly monotone mapping such that therange of ð J p + tT Þ is all of B ∗ for all t > : Let φ i : K ⟶ B , i = , , ⋯ , N be a fi nite family of quasi- ϕ p -nonexpansivemappings with C ≔ ∩ Ni = F ð φ i Þ ≠ ∅ : Let f λ n g ∞ n = ⊂ ð , Þ and f θ n g ∞ n = in ð , / Þ be real sequences such that(i) lim n ⟶ ∞ θ n = and f θ n g ∞ n = is decreasing(ii) ∑ ∞ n = λ n θ n = ∞ (iii) lim n ⟶ ∞ ðð θ n − / θ n Þ − Þ / λ n θ n = , ∑ ∞ n = λ n < ∞∀ n ∈ ℕ For arbitrary x ∈ B , de fi ne f x n g ∞ n = iteratively by x n + = J B ∗ p J Bp φ n ½ (cid:2) x n (cid:8) (cid:9) − λ n T φ n ½ (cid:2) x n (cid:8) (cid:9)(cid:8)(cid:8) + θ n J Bp φ n ½ (cid:2) x n (cid:8) (cid:9) − J Bp φ n ½ (cid:2) x (cid:8) (cid:9)(cid:8) (cid:9)(cid:9)(cid:9) , n ∈ ℕ , ð Þ where φ ½ n (cid:2) ≔ φ n ModN and J Bp is the generalized duality map-ping from B into B ∗ : Then, the sequence f x n g ∞ n = convergesstrongly to x ∈ VI ð T , C Þ : Proof.
Firstly, it is shown that the sequence f x n g ∞ n =1 isbounded.Let q > 1 with p + 1/ q = 1 and x ∈ VI ð T , C Þ : It su ffi cesto show that ϕ p ð x , x n Þ ≤ r , ∀ n ∈ ℕ : The proof is by induction.Let r > 0 be su ffi ciently large such that r ≥ max ϕ p x , x ð Þ , 4 M M , 4 pq ∥ x ∥ q (cid:4) (cid:5) , ð Þ where M > 0 and M > 0 are arbitrary but fi xed. By con-struction, ϕ p ð x , x Þ ≤ r . Suppose that ϕ p ð x , x n Þ ≤ r for some n ∈ ℕ . From inequality (20), for real p > 1, we have ∥ x n ∥≤ r p + ∥ x ∥ : Let D ≔ f z ∈ B : ϕ p ð x , z Þ ≤ r g : We show that ϕ p ð x , x n +1 Þ ≤ r : It is known that T is locally bounded and J Bp is uniformly continuous on bounded subsets of B : fi ne M ≔ sup ∥ T φ n ½ (cid:2) x n (cid:8) (cid:9) + θ n J Bp φ n ½ (cid:2) x n (cid:8) (cid:9)(cid:8)n − J Bp φ n ½ (cid:2) x (cid:8) (cid:9)(cid:9) ∥ : θ n ∈
0, 12 (cid:10) (cid:11) , x n ∈ D o + 1 : ð Þ Let ψ denotes the modulus of continuity of J B ∗ p : Then, ∥ φ n ½ (cid:2) x n − x n +1 ∥ = ∥ φ n ½ (cid:2) x n − J B ∗ p J p φ n ½ (cid:2) x n (cid:8) (cid:9) − λ n T φ n ½ (cid:2) x n (cid:8) (cid:9)(cid:8)(cid:8) + θ n J Bp φ n ½ (cid:2) x n (cid:8) (cid:9) − J Bp φ n ½ (cid:2) x (cid:8) (cid:9)(cid:8) (cid:9)(cid:9)(cid:9) ∥ = ∥ J B ∗ p J Bp φ n ½ (cid:2) x n (cid:8) (cid:9)(cid:8) (cid:9) − J B ∗ p J Bp φ n ½ (cid:2) x n (cid:8) (cid:9)(cid:8) − λ n T φ n ½ (cid:2) x n (cid:8) (cid:9) + θ n J Bp φ n ½ (cid:2) x n (cid:8) (cid:9)(cid:8)(cid:8) − J Bp φ n ½ (cid:2) x (cid:8) (cid:9)(cid:9)(cid:9)(cid:9) ∥≤ ψ ∣ λ n ∣∥ T φ n ½ (cid:2) x n (cid:8) (cid:9)(cid:8) + θ n J Bp φ n ½ (cid:2) x n (cid:8) (cid:9) − J Bp φ n ½ (cid:2) x (cid:8) (cid:9)(cid:8) (cid:9) ∥ (cid:9) ≤ ψ ∣ λ n ∣ M ð Þ ≤ ψ sup ∣ λ n ∣ M : λ n ∈
0, 1 ð Þf gð Þ : ð Þ Since T is locally bounded and the duality mapping J Bp is uniformly continuous on bounded subsets of B , the sup f ∣ λ n ∣ M g exists, and it is a real number di ff erent fromin fi nity. De fi ne M ≕ ψ ð sup f ∣ λ n ∣ M gÞ : Applying Lemma 4with y ∗ ≔ λ n ð T ð φ ½ n (cid:2) x n Þ + θ n ð J Bp ð φ ½ n (cid:2) x n Þ − J Bp ð φ ½ n (cid:2) x ÞÞÞ and byusing the de fi nition of x n +1 , we compute as follows: ϕ p x , x n +1 ð Þ = ϕ p x , J B ∗ p J Bp φ n ½ (cid:2) x n (cid:8) (cid:9) − λ n T φ n ½ (cid:2) x n (cid:8) (cid:9)(cid:8)(cid:8)(cid:8) + θ n J Bp φ n ½ (cid:2) x n (cid:8) (cid:9) − J Bp φ n ½ (cid:2) x (cid:8) (cid:9)(cid:8) (cid:9)(cid:9)(cid:9)(cid:9) = V p x , J Bp φ n ½ (cid:2) x n (cid:8) (cid:9) − λ n T φ n ½ (cid:2) x n (cid:8) (cid:9)(cid:8)(cid:8) + θ n J Bp φ n ½ (cid:2) x n (cid:8) (cid:9) − J Bp φ n ½ (cid:2) x (cid:8) (cid:9)(cid:8) (cid:9)(cid:9)(cid:9) by 21 ð Þð Þ ≤ V p x , J Bp φ n ½ (cid:2) x n (cid:8) (cid:9)(cid:8) (cid:9) − p λ n x n +1 − x , T φ n ½ (cid:2) x n (cid:8) (cid:9)D + θ n J Bp φ n ½ (cid:2) x n (cid:8) (cid:9) − J Bp φ n ½ (cid:2) x (cid:8) (cid:9)(cid:8) (cid:9)E = ϕ p x , φ n ½ (cid:2) x n (cid:8) (cid:9) − p λ n φ n ½ (cid:2) x n − x , T φ n ½ (cid:2) x n (cid:8) (cid:9)D + θ n J Bp φ n ½ (cid:2) x n (cid:8) (cid:9) − J Bp φ n ½ (cid:2) x (cid:8) (cid:9)(cid:8) (cid:9)E − p λ n x n +1 − φ n ½ (cid:2) x n , T φ n ½ (cid:2) x n (cid:8) (cid:9)D + θ n J Bp φ n ½ (cid:2) x n (cid:8) (cid:9) − J Bp φ n ½ (cid:2) x (cid:8) (cid:9)(cid:8) (cid:9)E : ð Þ By Schwartz inequality and by applying inequality (69), we obtain ϕ p x , x n +1 ð Þ ≤ ϕ p x , φ n ½ (cid:2) x n (cid:8) (cid:9) − p λ n φ n ½ (cid:2) x n − x , T φ n ½ (cid:2) x n (cid:8) (cid:9)D + θ n J Bp φ n ½ (cid:2) x n (cid:8) (cid:9) − J Bp φ n ½ (cid:2) x (cid:8) (cid:9)(cid:8) (cid:9) i + p λ n M M ≤ ϕ p x , φ n ½ (cid:2) x n (cid:8) (cid:9) − p λ n φ n ½ (cid:2) x n − x , T φ n ½ (cid:2) x n (cid:8) (cid:9) − Tx D E − p λ n φ n ½ (cid:2) x n − x , Tx D E − p λ n θ n φ n ½ (cid:2) x n − x , J Bp φ n ½ (cid:2) x n (cid:8) (cid:9)D − J Bp φ n ½ (cid:2) x (cid:8) (cid:9)E + p λ n M M ≤ ϕ p x , φ n ½ (cid:2) x n (cid:8) (cid:9) − p λ n φ n ½ (cid:2) x n − x , T φ n ½ (cid:2) x n (cid:8) (cid:9) − Tx D E since x ∈ VI T , C ð Þð Þ − p λ n θ n φ n ½ (cid:2) x n − x , J Bp φ n ½ (cid:2) x n (cid:8) (cid:9) − J Bp φ n ½ (cid:2) x (cid:8) (cid:9)D E + p λ n M M : ð Þ By Lemma 6, p h x − φ ½ n (cid:2) x n , J Bp ð φ ½ n (cid:2) x n Þ − J Bp ð φ ½ n (cid:2) x Þi ≤ ϕ p ð x , φ ½ n (cid:2) x n Þ − ϕ p ð x , φ ½ n (cid:2) x Þ : Consequently, p h x − φ ½ n (cid:2) x n , J Bp ð φ ½ n (cid:2) x n Þ − J Bp ð φ ½ n (cid:2) x Þi ≤ ϕ p ð x , φ ½ n (cid:2) x n Þ : Therefore, using ð p , η Þ -strongly monotonicity property of T , we have ϕ p x , x n +1 ð Þ ≤ ϕ p x , φ n ½ (cid:2) x n (cid:8) (cid:9) − p ηλ n ∥ T φ n ½ (cid:2) x n (cid:8) (cid:9) − x ∥ p − p λ n θ n φ n ½ (cid:2) x n − x , J Bp φ n ½ (cid:2) x n (cid:8) (cid:9) − J Bp φ n ½ (cid:2) x (cid:8) (cid:9)D E + p λ n M M ≤ ϕ p x , φ n ½ (cid:2) x n (cid:8) (cid:9) − p λ n ϕ p x , T φ n ½ (cid:2) x n (cid:8) (cid:9)(cid:8) (cid:9) − pq ∥ x ∥ q (cid:10) (cid:11) + p λ n θ n φ n ½ (cid:2) x n − x , J Bp φ n ½ (cid:2) x n (cid:8) (cid:9)D − J Bp φ n ½ (cid:2) x (cid:8) (cid:9)E + p λ n M M ≤ ϕ p x , φ n ½ (cid:2) x n (cid:8) (cid:9) − p λ n ϕ p x , φ n ½ (cid:2) x n (cid:8) (cid:9) + p λ n pq ∥ x ∥ q + p λ n θ n ϕ p x , φ n ½ (cid:2) x n (cid:8) (cid:9) + p λ n M M = 1 − p λ n ð Þ ϕ p x , φ n ½ (cid:2) x n (cid:8) (cid:9) + p λ n pq ∥ x ∥ q (cid:10) (cid:11) + p λ n θ n ϕ p x , φ n ½ (cid:2) x n (cid:8) (cid:9) + p λ n M M ≤ − p λ n ð Þ ϕ p x , x n ð Þ + p λ n pq ∥ x ∥ q (cid:10) (cid:11) + p λ n θ n ϕ p x , x n ð Þ + p λ n M M ≤ − p λ n ð Þ r + p λ n r p λ n r p λ n r = 1 − p λ n + p λ n
14 + p λ n
12 + p λ n (cid:10) (cid:11) r = r : ð Þ Hence, ϕ p ð x , x n +1 Þ ≤ r : By induction, ϕ p ð x , x n Þ ≤ r ∀ n ∈ ℕ : Thus, from inequality (20), f x n g ∞ n =1 is bounded. Theremaining part of the proof follows from the proof of The-orem 13. 9Abstract and Applied Analysis emark 21. It well known that uniformly smooth and uni-formly convex spaces are more general than the Hilbertspaces. Therefore, the following corollary is readily obtainable.
Corollary 22.
Let H be a Hilbert space and and K a nonempty,closed, and convex subset of H : Let p > , η ∈ ð , ∞ Þ ; suppose T : H → H is a continuous, ð p , η Þ -strongly monotone mappingsuch that D ð T Þ ⊆ range ð I + tT Þ for all t > : Let φ i : K → H , i = , , ⋯ , N be a fi nite family of quasi- ϕ p -nonexpansivemappings with C ≔ ∩ Ni = F ð φ i Þ ≠ ∅ : Let f λ n g ∞ n = ⊂ ð , Þ and f θ n g ∞ n = in ð , / Þ be real sequences such that(i) lim n → ∞ θ n = and f θ n g ∞ n = is decreasing(ii) ∑ ∞ n = λ n θ n = ∞ (iii) lim n → ∞ ðð θ n − / θ n Þ − Þ / λ n θ n = , ∑ ∞ n = λ n < ∞∀ n ∈ ℕ For arbitrary x ∈ H , de fi ne f x n g ∞ n = iteratively by x n + = φ n ½ (cid:2) x n − λ n T φ n ½ (cid:2) x n (cid:8) (cid:9) + θ n φ n ½ (cid:2) x n − φ n ½ (cid:2) x (cid:8) (cid:9)(cid:8) (cid:9) , n ∈ ℕ , ð Þ where φ ½ n (cid:2) ≔ φ n ModN : Then, the sequence f x n g ∞ n = convergesstrongly to x ∈ VI ð T , C Þ :
6. Conclusion
Real-life problems are usually modeled by nonlinear equa-tions. Nonlinear equations occur in modeling problems, suchas minimizing costs in industries and minimizing risks inbusinesses. Nonlinear equations of ð p , η Þ -strongly monotonetype, where η ∈ ð ∞ Þ , p > 1 , have been studied in this paper.The result was applied to obtain the solution of convex min-imization and variational inequality problems, which haveapplications in several fi elds such as economics, game theory,and the sciences. Data Availability
Data sharing is not applicable to this article as no datasetswere generated or analyzed during the current study.
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The abstract of this manuscript is submitted for presentationin the “ th International Conference on MathematicalModeling in Physical Sciences, ” September 7 –
10, 2020, Tinosisland, Greece.
Conflicts of Interest
The authors declare no con fl icts of interest. References [1] Y. Alber and I. Ryazantseva,
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