Algorithm for solutions of nonlinear equations of strongly monotone type and applications to convex minimization and variational inequality problems

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 of (p, {\eta})-strongly monotone type, where {\eta}>0, p>1. An example is presented for the nonlinear equations of (p, {\eta})-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.


Remark 1.
According to the definition of Chidume and Djitté [2] and Chidume and Shehu [3], a strongly monotone mapping is referred to as ð2, ηÞ-strongly monotone mapping.
A monotone mapping T 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 ðJ B + tTÞ is all of B * for some t > 0. The set of zeros of a maximum monotone mapping T, T −1 ð0Þ ≔ fx ∈ B : Tx = 0g is closed and convex. A function φ : B ⟶ ð−∞,+∞ is said to be proper if the set fx ∈ ℝ : Fðx Þ ∈ ℝg is nonempty. A proper function φ : B ⟶ ð−∞,+∞ is said to be convex if for all x, y ∈ B and τ ∈ ½0, 1,we have If the set of fx ∈ ℝ : φðxÞ ≤ rg is closed in B for all r ∈ ℝ,φ is said to be lower semicontinuous. For a proper lower semicontinuous function φ : B ⟶ ð−∞,+∞, the subdifferential mapping ∂φ : B ⟶ 2 B * , defined by 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 ⟶ ð−∞,+∞, solving the equation Tu = 0 is equivalent to finding φðuÞ = min x∈B φðxÞ by setting ∂φ ≡ T: For a reflexive smooth strictly convex space B, we let T be a mapping such that the range of ðJ B p + tTÞ is all of B * for some t > 0 and let x ∈ B be fixed. Then, for every t > 0, there corresponds a unique element x t ∈ DðTÞ such that Therefore, the resolvent of T is defined by J T t x = x t : In other words, The resolvent J T t is a single-valued mapping from B into DðTÞ (Kohsaka and Takahashi [8]). J T t 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 of ðp, ηÞ-strongly monotone type, p > 1 and η ∈ ð1,∞Þ: This is a wider class than the class of strongly monotone mappings.
An example is presented for nonlinear equations of ðp, ηÞ-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 ðp, ηÞ-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.

Preliminaries
Let B be a real Banach space and S ≔ fx ∈ B : ∥x∥ = 1g. B is said to have a Gateaux differentiable norm if the limit 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 hx, x * i = ∥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, δ B : ð0, 2 ⟶ ½0, 1 is defined by B is uniformly convex if and only if δ B ðεÞ > 0 for every ε ∈ ð0, 2. A normed linear space B is said to be strictly convex if It is well known that a space B is uniformly smooth if and only if B * is uniformly convex.
A mapping T : B ⟶ B * is locally bounded at v ∈ D, if there exist r v > 0 and m > 0 such that In particular, ∥Tv∥≤m: Therefore, hv, Tvi ≤ 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 Let ψðtÞ be a function on the set ℝ + of nonnegative real numbers such that T is said to be uniformly continuous if it admits the modulus of continuity ψ such that Abstract and Applied Analysis The modulus of continuity ψ has some useful properties (for instance, see Altomare and Campiti [20], pp. 266-269; Forster [21] and references therein).

Properties of Modulus of Continuity.
Let X 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 t 1 ≥ 0, t 2 ≥ 0, we have (b) Modulus of continuity is monotonically increasing: if 0 ≤ t 1 ≤ t 2 holds for some real numbers t 1 , t 2 , then (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 Let C be a nonempty subset of a Banach space B and T be a mapping from C into itself.
where J B 2 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 ϕ p : and V p : where J B p is the generalized duality mapping from B to B * : Remark 3. These remarks follow from Definition 2: (i) For p = 2, ϕ 2 ðx, yÞ = ϕðx, yÞ, which is the definition of Alber [26]. It is easy to see from the definition of the function ϕ that Indeed, 3 Abstract and Applied Analysis By similar analysis, it can verified that for each p ≥ 2, (ii) It is obvious that Let B be a topological real vector space and T a multivalued mapping from B into 2 B * : Cauchy-Schwartz's inequality is given by for any x and y in B and any choice of x * ∈ Tx and y * ∈ Ty (Zarantonello [27]).
Lemma 4. Let B be a strictly convex and uniformly smooth real Banach space and p > 1: Then, for all x ∈ B and x * , y * ∈ B * : Lemma 5. Let B be a smooth uniformly convex real Banach space and p > 1 be an arbitrarily real number. For d > 0 , let B d ð0Þ ≔ fx ∈ B : ∥x∥≤dg . Then, for arbitrary x, y ∈ B d ð0Þ, Lemma 6. Let B be a reflexive strictly convex and smooth real Banach space and p > 1: Then, such that for every x, y ∈ B r ð0Þ, j B p ðxÞ ∈ J B p ðxÞ, j B p ðyÞ ∈ J B p ðyÞ, [28]).

Lemma 8.
Let fa n g be a sequence of nonnegative real numbers satisfying the following relations: where Then, a n ⟶ 0 as n ⟶ ∞ (see Xu [29]).

Lemma 9.
Let B be a smooth uniformly convex real Banach space and let fx n g and fy n g be two sequences from B: If either fx n g or fy n g is bounded and ϕðx n , y n Þ ⟶ 0 as n ⟶ ∞ , then ∥x n − y n ∥⟶0 as n ⟶ ∞ (see Kamimura and Takahashi [30]).
Lemma 11. If a functional ϕ on the open convex set M ⊂ dom ϕ has a subdifferential, then ϕ is convex and lower semicontinuous 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 ⟶ Ybe a uniformly continuous map. For arbitrary r > 0 and fixed x * ∈ X , let Then, TðBðx * , rÞÞ is bounded (see, e.g., Chidume and Djitte [33]).

Main Results
Theorem 13. Let B be a uniformly smooth and uniformly convex real Banach space. Let p > 1, η ∈ ð1,∞Þ ; suppose T : B ⟶ B * is a continuous ðp, ηÞ -strongly monotone mapping such that the range of ðJ B p + tTÞ is all of B * for all t > 0 and T −1 ð0Þ ≠ ∅: Let fλ n g ∞ n=1 ⊂ ð0, 1Þ and fθ n g ∞ n=1 in ð0, 1/2Þ be real sequences such that (i) lim n⟶∞ θ n = 0 and fθ n g ∞ n=1 is decreasing (ii) ∑ ∞ n=1 λ n θ n = ∞ (iii) lim n⟶∞ ððθ n−1 /θ n Þ − 1Þ/λ n θ n = 0, ∑ ∞ n=1 λ n < ∞∀n ∈ ℕ For arbitrary x 1 ∈ B, define fx n g ∞ n=1 iteratively by: Abstract and Applied Analysis where J B p is the generalized duality mapping from B into B * : Then, the sequence fx n g ∞ n=1 converges strongly to the solution of Tx = 0: Proof. Observe that there is no need for constructing a convergence 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 fx n g ∞ n=1 is shown to be bounded. Let q > 1 with 1/p + 1/q = 1 and x ∈ B be a solution of the equation Tx = 0: It suffices to show that ϕ p ðx, x n Þ ≤ r, ∀n ∈ ℕ : The induction method will be adopted. Let r > 0 be sufficiently large such that where M 0 > 0 and M > 0 are arbitrary but fixed. For n = 1, by construction, we have that ϕ p ðx, x 1 Þ ≤ r for real p > 1: Assume that ϕ p ðx, x n Þ ≤ r for some n ≥ 1: From inequality (20), we have ∥x n ∥≤r 1/p + ∥x∥: Let D ≔ fz ∈ B : ϕ p ðx, zÞ ≤ rg: Next is to show that ϕ p ðx, x n+1 Þ ≤ r: It is known that T is locally bounded (Lemma 10) and J B p is uniformly continuous on bounded subsets of B: Define Let ψ denotes the modulus of continuity of J B * p : Then, Since T is locally bounded and the duality mapping J B p is uniformly continuous on bounded subsets of B, the sup f|λ n | M 0 g exists and it is a real number different from infinity. Choose M ≕ ψðsup f|λ n | M 0 gÞ: Applying Lemma 4 with y * ≔ λ n ðTx n + θ n ðJ B p x n − J B p x 1 ÞÞ and by using the definition of x n+1 , we compute as follows: By Schwartz inequality and by applying inequality (32), we obtain Therefore, using ðp, ηÞ-strongly monotonicity property of T, we have Hence, ϕ p ðx, x n+1 Þ ≤ r: By induction, ϕ p ðx, x n Þ ≤ r∀n ∈ ℕ: Thus, from inequality (20), fx n g ∞ n=1 is bounded. Part 2: we now show that fx n g ∞ n=1 converges strongly to a solution of Tx = 0:ðp, ηÞ-strongly monotone implies a monotone and the range of ðJ B p + tTÞ 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 x t ∈ B such that Define J T t x ≔ x t ; in other words, define a single-valued mapping J T t : B ⟶ B by J T t = ðJ B p + tTÞ −1 J B p : Such a J T t is called the resolvent of T: Setting t ≔ 1/θ n and by the result of Aoyama et al. [34] and Reich [35], for some x 1 ∈ B, there exists in B a unique 5 Abstract and Applied Analysis with y n ⟶ x ∈ T −1 ð0Þ: Obviously, one can obtain that and fy n g ∞ n=1 is known to be bounded. Also it can be obtained that From (39), we have that which is equivalent to Consequently, which shows that the sequence fy n g ∞ n=1 is bounded. Moreover, fx n g ∞ n=1 is bounded, and hence, fTx 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− pλ n x n − y n , Tx n + θ n J B p x n − J B p x 1 D E By the ðp, ηÞ-strongly monotonicity property of T and using Lemma 7 and Equation (39), we obtain Therefore, the inequality (43) becomes Observe that by Lemma 6, we have Let R > 0 such that ∥x 1 ∥≤R, ∥y n ∥≤R for all n ∈ ℕ. We obtain from Equation (39) that By taking the duality pairing of each side of this equation with respect to y n−1 − y n and by the strong monotonicity of T, we have Since fθ n g ∞ n=1 is a decreasing sequence, it is known that θ n−1 ≥ θ n : Therefore, Consequently, Using (46) and (50), the inequality (45) becomes for some constant C > 0. By Lemma 8, ϕ p ðy n−1 , x n Þ ⟶ 0 as n ⟶ ∞ and using Lemma 9, we have that x n − y n−1 ⟶ 0 as n ⟶ ∞: Since y n ⟶ x ∈ T −1 ð0Þ, we obtain that x n ⟶ x as n ⟶ ∞: Let H be a Hilbert space, p > 1, η ∈ ð1,∞Þ and suppose T : H ⟶ H is a continuous, ðp, ηÞ -strongly monotone mapping such that DðTÞ ⊆ range ðI + tTÞ for all t > 0: For arbitrary x 1 ∈ H , define the sequence fx n g ∞ n=1 iteratively by where fλ n g ∞ n=1 ⊂ ð0, 1Þ and fθ n g ∞ n=1 in ð0, 1/2Þ are real sequences satisfying the conditions: (i) lim n→∞ θ n = 0 and fθ n g ∞ n=1 is decreasing 6 Abstract and Applied Analysis (ii) ∑ ∞ n=1 λ n θ n = ∞ (iii) lim n⟶∞ ððθ n−1 /θ n Þ − 1Þ/λ n θ n = 0, ∑ ∞ n=1 λ n < ∞∀n ∈ ℕ Suppose that the equation Tx = 0 has a solution. Then, the sequence fx n g ∞ n=1 converges strongly to the solution of the equation Tx = 0: Proof. The result follows from Theorem 13 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.

Solution of Convex Minimization Problems
The result of Theorem 13 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, The following well-known basic results will be used.

Lemma 17.
Let φ : B → ℝ be a real-valued differentiable convex function and u ∈ B: Let dφ : B ⟶ B * denote the differential map associated to φ: Then, the following hold: (1) The point u is a minimizer of φ on B if and only if d φðuÞ = 0 (2) If φ is bounded, then φ is locally Lipschitzian, i.e., for every x 0 ∈ B and r > 0, there exists L > 0 such that φ is L-Lipschitzian on Bðx 0 , rÞ, i.e., The main result in this section is given below.
Theorem 18. Let B be a uniformly smooth and uniformly convex real Banach space. Let φ : B ⟶ ℝ be a differentiable, convex, bounded, and coercive function. Let fλ n g ∞ n=1 ⊂ ð0, 1Þ and fθ n g ∞ n=1 in ð0, 1/2Þ be real sequences such that, For arbitrary x 1 ∈ B, define fx n g ∞ n=1 iteratively by where J B is the generalized duality mapping from B into B * : Then, φ has a minimizer x * ∈ B and the sequence fx n g ∞ n=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 dφðx * Þ = 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 x 0 ∈ B and r > 0: By Lemma 17, there exists L > 0 such that Let v * ∈ dφðBðx 0 , rÞÞ and x * ∈ Bðx 0 , rÞ such that v * = dφ ðx * Þ:Since Bðx 0 , rÞ is open, for all u ∈ B, there exists σ > 0 such that x * + σu ∈ ðBðx 0 , rÞ: From the fact that v * = dφðx * Þ and inequality (58), it is obtained that such that Consequently, ∥v * ∥≤L, which implies that dφðBðx 0 , rÞÞ is bounded. Thus, dφ is bounded. Hence, it can be deduced from Theorem 13 that the sequence fx n g ∞ n=1 converges strongly to x * , a minimizer of φ: 7 Abstract and Applied Analysis Example 19. An example of a function which is coercive is a real valued function f : ℝ 2 ⟶ ℝ which is defined by f ðu, vÞ = u 4 − 7uv + v 3 : Constructively, f ðu, vÞ = ðu 3 + v 4 Þð1 − ð7uv/ðu 3 + v 4 ÞÞÞ: As ∥ðu, vÞ∥⟶∞, 7uv/ðu 3 Hence, f is coercive.

Solutions of Variational Inequality Problems
Let K 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 for some j p ðx − yÞ ∈ J p ðx − yÞ: The set of solutions of a variational inequality problem is denoted by VIðT, KÞ: If B ≔ H, a Hilbert space, the variational inequality problem reduces to 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). Let K be a closed convex subset of H: The projection into K is defined to be the mapping, P K : H → K, which is given by Gradient projection method is an orthodox way for solving (63). The projection algorithm is given by 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 form VIðT, CÞ, where T : B ⟶ B * is a ðp, ηÞ-strongly monotone mapping, C ≔ ∩ N i=1 Fðφ i Þ ≠ ∅, φ i : K ⟶ B, i = 1, 2, ⋯, N is a finite family of quasi-ϕ p -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 FðφÞ: A mapping φ is said to be quasiϕ p -nonexpansive if FðφÞ ≠ ∅ and ϕ p ðu, φxÞ ≤ ϕ p ðu, xÞ, ∀x ∈ K and u ∈ FðφÞ: The proof of the following theorem is given.

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.

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 of ðp, ηÞ-strongly monotone type, where η ∈ ð1,∞Þ, 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.

Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Disclosure
The abstract of this manuscript is submitted for presentation in the "9th International Conference on Mathematical Modeling in Physical Sciences," September 7-10, 2020, Tinos island, Greece.