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

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 of -strongly monotone type, where . An example is presented for the nonlinear equations of -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.

1. Introduction

Let be a continuous, strictly increasing function such that as and for any . Such a function is called a gauge function. Let be a Banach space, and denotes it is dual. A duality mapping associated with the gauge function is a mapping defined by where denotes the duality pairing. For , let be a gauge function. is called a generalized duality mapping from into and is given by

For , the mapping is called the normalized duality mapping. In a Hilbert space, the normalized duality mapping is the identity map. For and a mapping is said to be (1)monotone if (2)strongly monotone (see, e.g., Alber and Ryazantseva [1], p. 25), if (3)-strongly pseudomonotone if (4)-strongly monotone if (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 -strongly monotone mapping.

A monotone mapping 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], is said to be a maximum monotone if it is a monotone and the range of is all of for some . The set of zeros of a maximum monotone mapping is closed and convex. A function is said to be proper if the set is nonempty. A proper function is said to be convex if for all and we have

If the set of is closed in for all is said to be lower semicontinuous. For a proper lower semicontinuous function the subdifferential mapping defined by is a maximal monotone (Rockafellar [7]). Consider a problem of finding a solution of the equation where is a maximal monotone mapping. Such a problem is associated with the convex minimization problem. Indeed, for a proper lower semicontinuous convex function solving the equation is equivalent to finding by setting

For a reflexive smooth strictly convex space we let be a mapping such that the range of is all of for some and let be fixed. Then, for every there corresponds a unique element such that

Therefore, the resolvent of is defined by In other words, and for all where denotes the set of all fixed points of The resolvent is a single-valued mapping from into (Kohsaka and Takahashi [8]). is nonexpansive if 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 -strongly monotone type, and This is a wider class than the class of strongly monotone mappings. An example is presented for nonlinear equations of -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 -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

Let be a real Banach space and . is said to have a Gateaux differentiable norm if the limit exists for each A Banach space is said to be smooth if for every in there is a unique such that and where denotes the dual of is said to be uniformly smooth if it is smooth and the limit (6) is attained uniformly for each The modulus of convexity of a Banach space , is defined by

is uniformly convex if and only if for every . A normed linear space is said to be strictly convex if

It is well known that a space is uniformly smooth if and only if is uniformly convex.

A mapping is locally bounded at if there exist and such that

In particular, Therefore, Let and be real Banach spaces and let be a mapping. is uniformly continuous if for each there exists such that

Let be a function on the set of nonnegative real numbers such that (i) is nondecreasing and continuous(ii) if and only if

is said to be uniformly continuous if it admits the modulus of continuity such that

The modulus of continuity 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

Let and be real Banach spaces and let be a map which admits the modulus of continuity (a)Modulus of continuity is subadditive: for all real numbers we have(b)Modulus of continuity is monotonically increasing: if holds for some real numbers then(c)Modulus of continuity is continuous: the modulus of continuity is continuous on the set positive real numbers; in particular, the limit of at from above is

Let be a nonempty subset of a Banach space and be a mapping from into itself. (i) is nonexpansive provided for all (ii) is firmly nonexpansive type (see, e.g., [22]) if for all and

The 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 be a Banach space. (i) is smooth if and only if is single-valued(ii)Ifis reflexive, thenis onto(iii)If has uniform Gateaux differentiable norm, then is norm-to-weak uniformly continuous on bounded sets(iv) is uniformly smooth if and only if is single-valued and uniformly continuous on any bounded subset of (v)If is strictly convex, then is one-to-one, that is, (vi)If and are strictly convex and reflexive, then is the generalized duality mapping from to , and is the inverse of (vii)If is uniformly smooth and uniformly convex, the generalized duality mapping is uniformly continuous on any bounded subset of (viii)If and are strictly convex and reflexive, for all and the equalities and hold

Definition 2. Alber [26] introduced the functions defined by where is the normalized duality mapping from to Let be a smooth real Banach space and with Aibinu and Mewomo [4] introduced the functions defined by and defined as where is the generalized duality mapping from to

Remark 3. These remarks follow from Definition 2: (i)For which is the definition of Alber [26]. It is easy to see from the definition of the function that

Indeed,

By similar analysis, it can verified that for each (ii)It is obvious that

Let be a topological real vector space and a multivalued mapping from into Cauchy-Schwartz’s inequality is given by for any and in and any choice of and (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. Letbe a strictly convex and uniformly smooth real Banach space andThen,for all and

Lemma 5. Letbe a smooth uniformly convex real Banach space andbe an arbitrarily real number. For, let. Then, for arbitrary,

Lemma 6. Letbe a reflexive strictly convex and smooth real Banach space andThen,

Lemma 7. Letbe a real uniformly convex Banach space. For arbitrary, let. Then, there exists a continuous strictly increasing convex functionsuch that for every , we have (see Xu [28]).

Lemma 8. Letbe a sequence of nonnegative real numbers satisfying the following relations:where (i), (ii)(iii), Then, as (see Xu [29]).

Lemma 9. Letbe a smooth uniformly convex real Banach space and letandbe two sequences fromIf eitheroris bounded andas, thenas (see Kamimura and Takahashi [30]).

Lemma 10. A monotone map is locally bounded at the interior points of its domain (see, e.g., Rockafellar [31] and Pascali and Sburlan [32]).

Lemma 11. If a functionalon the open convex setdomhas a subdifferential, thenis convex and lower semicontinuous on the set (see Alber and Ryazantseva [1], p. 17).

Lemma 12. Letandbe real normed linear spaces and letbe a uniformly continuous map. For arbitraryand fixed, let

Then, is bounded (see, e.g., Chidume and Djitte [33]).

3. Main Results

Theorem 13. Letbe a uniformly smooth and uniformly convex real Banach space. Let; supposeis a continuous-strongly monotone mapping such that the range ofis all offor allandLetandinbe real sequences such that(i) and is decreasing(ii)(iii)For arbitrary , define iteratively by: where is the generalized duality mapping from into Then, the sequence converges strongly to the solution of

Proof. Observe that there is no need for constructing a convergence sequence if because it is a zero of (since is strongly monotone, which is one to one). Consequently, we are looking for a unique nonzero solution of The proof is divided into two parts.
Part 1: the sequence is shown to be bounded.
Let with and be a solution of the equation It suffices to show that The induction method will be adopted. Let be sufficiently large such that where and are arbitrary but fixed. For by construction, we have that for real Assume that for some From inequality (20), we have Let Next is to show that It is known that is locally bounded (Lemma 10) and is uniformly continuous on bounded subsets of Define

Let denotes the modulus of continuity of Then,

Since is locally bounded and the duality mapping is uniformly continuous on bounded subsets of the exists and it is a real number different from infinity. Choose Applying Lemma 4 with and by using the definition of we compute as follows:

By Schwartz inequality and by applying inequality (32), we obtain

By Lemma 6, Consequently, Therefore, using -strongly monotonicity property of we have

Hence, By induction, Thus, from inequality (20), is bounded.

Part 2: we now show that converges strongly to a solution of -strongly monotone implies a monotone and the range of is all of for all By Kohsaka and Takahashi [8], since is a reflexive smooth strictly convex space, we obtain for every and there exists a unique such that

Define ; in other words, define a single-valued mapping by Such a is called the resolvent of Setting and by the result of Aoyama et al. [34] and Reich [35], for some there exists in a unique with Obviously, one can obtain that and 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 is bounded. Moreover, is bounded, and hence, is bounded. Following the same arguments as in part 1, we get

By the -strongly monotonicity property of and using Lemma 7 and Equation (39), we obtain

Therefore, the inequality (43) becomes

Observe that by Lemma 6, we have

Let such that for all . We obtain from Equation (39) that

By taking the duality pairing of each side of this equation with respect to and by the strong monotonicity of we have

Since is a decreasing sequence, it is known that Therefore,

Consequently,

Using (46) and (50), the inequality (45) becomes for some constant . By Lemma 8, as and using Lemma 9, we have that as Since we obtain that as

Corollary 14. Letbe a Hilbert space,and supposeis a continuous,-strongly monotone mapping such thatrangefor allFor arbitrary, define the sequenceiteratively bywhere and in are real sequences satisfying the conditions: (i) and is decreasing(ii)(iii)Suppose that the equation has a solution. Then, the sequence converges strongly to the solution of the equation

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.

Example 15. LetwithandThen,

Thus, is -strongly monotone with

Example 16. Letwith the usual norm. Consider the functiondefined bywhere

Then,is-strongly monotone withandIndeed,

4. 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 defined from a real Banach space to Recall that a mapping is said to be coercive if for any

The following well-known basic results will be used.

Lemma 17. Letbe a real-valued differentiable convex function andLetdenote the differential map associated toThen, the following hold:(1)The point is a minimizer of on if and only if (2)If is bounded, then is locally Lipschitzian, i.e., for every and there exists such that is -Lipschitzian on i.e.,

The main result in this section is given below.

Theorem 18. Letbe a uniformly smooth and uniformly convex real Banach space. Letbe a differentiable, convex, bounded, and coercive function. Letandinbe real sequences such that,(i) and is decreasing(ii)(iii)For arbitrary , define iteratively by where is the generalized duality mapping from into Then, has a minimizer and the sequence converges strongly to

Proof. has a minimizer because it is a function which is lower semicontinuous, convex, and coercive. Moreover, minimizes if and only if It can be inferred that is a maximal monotone due to the convexity, the differentiability, and the boundedness of (see, e.g., Minty [36] and Moreau [37]). The next task is to show that is bounded. Indeed, let and By Lemma 17, there exists such that

Let and such that Since is open, for all there exists such that From the fact that and inequality (58), it is obtained that such that

Consequently, which implies that is bounded. Thus, is bounded. Hence, it can be deduced from Theorem 13 that the sequence converges strongly to a minimizer of

Example 19. An example of a function which is coercive is a real valued functionwhich is defined by

Constructively, As while It follows that

Hence, is coercive.

5. Solutions of Variational Inequality Problems

Let be a nonempty, closed, and convex subset of a real normed linear space and let be a nonlinear mapping. The variational inequality problem is to for some The set of solutions of a variational inequality problem is denoted by If 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 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 be a closed convex subset of The projection into is defined to be the mapping, which is given by

Gradient projection method is an orthodox way for solving (63). The projection algorithm is given by where is