Startseite Mathematik Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
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Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems

  • Victor Amarachi Uzor , Timilehin Opeyemi Alakoya und Oluwatosin Temitope Mewomo EMAIL logo
Veröffentlicht/Copyright: 13. April 2022
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 20 Heft 1

Abstract

In this paper, we study the problem of finding a common solution of the pseudomonotone variational inequality problem and fixed point problem for demicontractive mappings. We introduce a new inertial iterative scheme that combines Tseng’s extragradient method with the viscosity method together with the adaptive step size technique for finding a common solution of the investigated problem. We prove a strong convergence result for our proposed algorithm under mild conditions and without prior knowledge of the Lipschitz constant of the pseudomonotone operator in Hilbert spaces. Finally, we present some numerical experiments to show the efficiency of our method in comparison with some of the existing methods in the literature.

Keywords: Tseng’s extragradient method; pseudomonotone; demicontractive; variational inequalities; fixed point; strong convergence; adaptive step size; inertial technique
MSC 2010: 65K15; 47J25; 65J15; 90C33

1 Introduction

Let H be a real Hilbert space with inner product ⟨ ⋅ , ⋅ ⟩ and induced norm ‖ ⋅ ‖ . In this paper, we consider the variational inequality problem (VIP) of finding a point p ∈ C such that

(1) ⟨ A p , x − p ⟩ ≥ 0 , ∀ x ∈ C ,

where C is a nonempty closed convex subset of H , and A : H → H is a nonlinear operator. We denote by V I ( C , A ) the solution set of the VIP (1).

Variational inequality theory, which was first introduced independently by Fichera [1] and Stampacchia [2], is a vital tool in mathematical analysis, and has a vast application across several fields of study, such as optimisation theory, engineering, physics, operator theory, economics, and many others (see [3,4, 5,6] and references therein). Over the years, several iterative methods have been formulated and adopted in solving VIP (1) (see [7,8,9, 10,11] and references therein). There are two common approaches to solving the VIP, namely, the regularised methods and the projection methods. These approaches usually require that the nonlinear operator A in VIP (1) has certain monotonicity. In this study, we adopt the projection method and consider the case in which the associated nonlinear operator is pseudomonotone (see definition below) – a larger class than monotone mappings.

Now, we review some nonlinear operators in nonlinear analysis.

Definition 1.1

A mapping A : H → H is said to be

  1. γ -strongly monotone on H if there exists a constant γ > 0 such that

    (2) ⟨ A x − A y , x − y ⟩ ≥ γ ‖ x − y ‖ 2 , ∀ x , y ∈ H .

  2. γ -inverse strongly monotone on H if there exists a constant γ > 0 such that

    ⟨ A x − A y , x − y ⟩ ≥ γ ‖ A x − A y ‖ 2 , ∀ x , y ∈ H .

  3. Monotone on H , if

    (3) ⟨ A x − A y , x − y ⟩ ≥ 0 , ∀ x , y ∈ H .

  4. γ -strongly pseudomonotone on H , if there exists a constant γ > 0 such that

    (4) ⟨ A y , x − y ⟩ ≥ 0 ⇒ ⟨ A x , x − y ⟩ ≥ γ ‖ x − y ‖ 2 , ∀ x , y ∈ H .

  5. Pseudomonotone on H , if

    (5) ⟨ A y , x − y ⟩ ≥ 0 ⇒ ⟨ A x , x − y ⟩ ≥ 0 , ∀ x , y ∈ H .

  6. Lipschitz-continuous on H , if there exists a constant L > 0 such that

    (6) ‖ A x − A y ‖ ≤ L ‖ x − y ‖ , ∀ x , y ∈ H .

    If L ∈ [ 0 , 1 ) , then A is said to be a contraction mapping.

  7. Sequentially weakly continuous on H , if for each sequence { x n } ,

    x n ⇀ x implies T x n ⇀ T x , x ∈ H .

From the above definitions, we observe that ( 1 ) ⇒ ( 3 ) ⇒ ( 5 ) and ( 1 ) ⇒ ( 4 ) ⇒ ( 5 ) . However, the converses are not generally true. Moreover, if A is γ - strongly monotone and L - Lipschitz continuous, then A is γ L 2 - inverse strongly monotone (see [12,13]).

The simplest known projection method for solving VIP is the gradient method (GM), which involves a single projection onto the feasible set C per iteration. However, the algorithm only converges weakly under some strict conditions that the operator is either strongly monotone or inverse strongly monotone, but fails to converge if A is monotone. The classical gradient projection algorithm proposed by Sibony [14] is given as follows:

(7) x n + 1 = P C ( x n − λ A x n ) ,

where A is strongly monotone and L -Lipschitz continuous, with step size λ ∈ 0 , 2 L .

Korpelevich [15] and Antipin [16] proposed the extragradient method (EGM) for solving VIP (1), thereby relaxing the conditions placed in (7). The initial algorithm proposed by Korpelevich was employed in solving saddle point problems, but was later extended to VIPs in both Euclidean space and infinite dimensional Hilbert spaces. The EGM method is given as follows:

(8) x 0 ∈ C y n = P C ( x n − λ A x n ) x n + 1 = P C ( x n − λ A y n ) ,

where λ ∈ 0 , 1 L , A is monotone and L -Lipschitz continuous, and P C denotes the metric projection from H onto C . If the set V I ( C , A ) is nonempty, then the algorithm only converges weakly to an element in V I ( C , A ) .

Over the years, EGM has been of interest to several researchers. Also, many results and variants have been developed from this method, using the assumptions of Lipschitz continuity, monotonicity, and pseudomonotonicity, see [17,18, 19,20] and references therein.

Due to the extensive amount of time required in executing the EGM method, as a result of calculating two projections onto the closed convex set C in each iteration, Censor et al. [8] proposed the subgradient extragradient method (SEGM) in which they replaced the second projection onto C by a projection onto a half-space, thus, making computation easier and convergence rate faster. The SEGM is presented as follows:

(9) y n = P C ( x n − λ A x n ) T n = { w ∈ H : ⟨ x n − λ A x n − y n , w − y n ⟩ ≤ 0 } x n + 1 = P T n ( x n − λ A y n ) , ∀ n ≥ 0 ,

where λ ∈ 0 , 2 L . The authors only obtained a weak convergence result for the proposed method. However, they later introduced a hybrid SEGM in [7] and obtained a strong convergence result. Likewise, Tseng [21], in the bid to improve on the EGM, proposed Tseng’s extragradient method (TEGM), which only requires one projection per iteration, as follows:

(10) y n = P C ( x n − λ A x n ) x n + 1 = y n + λ ( A x n − A y n ) , ∀ n ≥ 0 ,

where A is monotone, L -Lipschitz continuous, and λ ∈ 0 , 2 L . The TEGM (10) converges to a weak solution of the VIP with the assumption that V I ( C , A ) is nonempty. The TEGM is also known as the forward-backward method. Recently, some authors have carried out some interesting works on the TEGM (see [22,23] and references therein).

In this work, we consider the inertial algorithm, which is a two-step iteration process and a technique for accelerating the speed of convergence of iterative schemes. The inertial extrapolation technique was derived by Polyak [24] from a dynamic system called the heavy ball with friction. Due to its efficiency, the inertial technique has become a centre of attraction and interest to many researchers in this field. Over the years, researchers have studied the inertial algorithm and applied it to solve different optimisation problems, see [25,26, 27,28] and references therein.

Very recently, Tan and Qin [29] proposed the following Tseng’s extragradient algorithm for solving pseudomonotone VIP:

(11) s n = x n + δ n ( x n − x n − 1 ) y n = P C ( s n − ψ n A s n ) z n = y n − ψ n ( A y n − A s n ) x n + 1 = α n f ( z n ) + ( 1 − α n ) z n ,

δ n = min ε n ‖ x n − x n − 1 ‖ , δ if x n ≠ x n − 1 δ , otherwise .

ψ n + 1 = min ϕ ‖ s n − y n ‖ ‖ A s n − A y n ‖ , ψ n if A s n − A y n ≠ 0 ψ n , otherwise ,

where f is a contraction and A is a pseudomonotone, Lipschitz continuous, and sequentially weakly continuous mapping. The authors proved a strong convergence result for the proposed method under mild conditions on the control parameters.

Another area of interest in this study is the fixed point theory. Let U : H → H be a nonlinear map. The fixed point problem (FPP) is to find a point p ∈ H (called the fixed point of U ) such that

(12) U p = p .

In this work, we denote the set of fixed points of U by F ( U ) . Our interest in this study is to find a common element of the fixed point set, F ( U ) , and the solution set of the variational inequality, V I ( C , A ) . That is, the problem of finding a point x ∗ ∈ H such that

(13) x ∗ ∈ V I ( C , A ) ∩ F ( U ) .

Many algorithms have been proposed over the years and in recent times for solving the common solution problem (13) (see [30,31,32, 33,34,35, 36,37,38, 39,40] and references therein). Common solution problem of this type has drawn the attention of researchers because of its potential application to mathematical models whose constraints can be expressed as FPP and VIP. This arises in areas like signal processing, image recovery, and network resource allocation. An instance of this is in network bandwidth allocation problem for two services in a heterogeneous wireless access networks in which the bandwidth of the services is mathematically related (see [37,41,42] and references therein).

Recently, Cai et al. [22] proposed the following inertial Tseng’s extragradient algorithm for approximating the common solution of pseudomonotone VIP and FPP for nonexpansive mappings in real Hilbert spaces:

(14) x 0 , x 1 ∈ H w n = x n + θ n ( x n − x n − 1 ) y n = P C ( w n − ψ A w n ) z n = y n − ψ ( A y n − A w n ) x n + 1 = α n f ( x n ) + ( 1 − α n ) [ β n T z n + ( 1 − β n ) z n ] ,

where f is a contraction, T is a nonexpansive mapping, A is pseudomonotone, L -Lipschitz and sequentially weakly continuous, and ψ ∈ 0 , 1 L . They proved a strong convergence result for the proposed algorithm under some suitable conditions.

One of the major drawbacks of Algorithm (14) is the fact that the step size ψ of the algorithm depends on the Lipschitz constant of the cost operator. In many cases, this Lipschitz constant is unknown or even difficult to estimate. This makes it difficult to implement algorithms of this nature.

Very recently, Thong and Hieu [23] proposed an iterative scheme for finding a common element of the solution set of monotone variational inequality and set of fixed points of demicontractive mappings as follows:

(15) y n = P C ( x n − ψ n A x n ) z n = y n − ψ n ( A y n − A x n ) x n + 1 = α n f ( x n ) + ( 1 − α n ) [ β n U z n + ( 1 − β n ) z n ] ,

ψ n + 1 = min μ ‖ x n − y n ‖ ‖ A x n − A y n ‖ , ψ n if A x n − A y n ≠ 0 ψ n , otherwise ,

where A is monotone and L -Lipschitz continuous, U is a demicontractive mapping such that I − U is demiclosed at zero, and f is a contraction. The authors proved a strong convergence result under suitable conditions for the proposed method.

Motivated by the above results and the ongoing research activities in this direction, in this paper our aim is to introduce an effective iterative technique, which employs the efficient combination of the inertial technique, TEGM together with the viscosity method for finding a common solution of FPP of demicontractive mappings and pseudomonotone VIP with Lipschitz continuous and sequentially weakly continuous operator in Hilbert spaces. In line with this goal, we construct an algorithm with the following features:

  1. Our algorithm approximates the solution of a more general class of VIP and FPP.

  2. The proposed method only requires one projection per iteration onto the feasible set, which guarantees the minimal cost of computation.

  3. Moreover, our method is computationally efficient. It employs an efficient self-adaptive step size technique which makes the algorithm independent of the Lipschitz constant of the cost operator.

  4. We employ the combination of the inertial technique together with the viscosity method, which are two of the efficient techniques for accelerating the rate of convergence of iterative schemes.

  5. We prove a strong convergence theorem for the proposed algorithm without following the conventional “two-cases” approach often employed by researchers (e.g. see [22,23,29,43,44,45]). This makes our results in this paper to be more concise and precise.

Furthermore, by several numerical experiments, we demonstrate the efficiency of our proposed method over many other existing methods in related literature.

The remainder of this paper is organised as follows. In Section 2, useful definitions and lemmas employed in the study are presented. In Section 3, we present the proposed algorithm and highlight some of its notable features. Section 4 presents the convergence analysis of the proposed method. In Section 5, we carry out some numerical experiments to illustrate the computational advantage of our method over some of the existing methods in the literature. Finally, in Section 6 we give a concluding remark.

2 Preliminaries

Let H be a real Hilbert space and C be a nonempty closed convex subset of H . We denote the weak and strong convergence of sequence { x n } n = 1 ∞ to x by x n ⇀ x , as n → ∞ and x n → x , as n → ∞ .

The metric projection [46,47], P C : H → C is defined, for each x ∈ H , as the unique element P C x ∈ C such that

‖ x − P C x ‖ = inf { ‖ x − z ‖ : z ∈ C } .

It is a known fact that P C is nonexpansive, i.e. ‖ P C x − P C y ‖ ≤ ‖ x − y ‖ ∀ x , y ∈ C . Also, the mapping P C is firmly nonexpansive, i.e.

‖ P C x − P C y ‖ 2 ≤ ⟨ P C x − P C y , x − y ⟩ ,

for all x , y ∈ H . Some results on the metric projection map are given below.

Lemma 2.1

[48] Let C be a nonempty closed convex subset of a real Hilbert space H. For any x ∈ H and z ∈ C , Then,

z = P C x ⇔ ⟨ x − z , z − y ⟩ ≥ 0 , for all y ∈ C .

Lemma 2.2

[48,49] Let C be a nonempty, closed, and convex subset of a real Hilbert space H, x ∈ H . Then:

  1. ∣ ∣ P C x − P C y ∣ ∣ 2 ≤ ⟨ x − y , P C x − P C y ⟩ , ∀ y ∈ C .

  2. ∣ ∣ x − P C x ∣ ∣ 2 + ∣ ∣ y − P C x ∣ ∣ 2 ≤ ∣ ∣ x − y ∣ ∣ 2 , ∀ y ∈ C .

  3. ∣ ∣ ( I − P C ) x − ( I − P C ) y ∣ ∣ 2 ≤ ⟨ x − y , ( I − P C ) x − ( I − P C ) y ⟩ , ∀ y ∈ C .

Definition 2.3

A mapping T : H → H is said to be

  1. Nonexpansive on H , if there exists a constant L > 0 such that

    ‖ T x − T y ‖ ≤ ‖ x − y ‖ , ∀ x , y ∈ H .

  2. Quasi-nonexpansive on H , if F ( T ) ≠ ∅ and

    ‖ T x − p ‖ ≤ ‖ x − p ‖ , ∀ p ∈ F ( T ) , x ∈ H .

  3. λ -strictly pseudocontractive on H with 0 ≤ λ < 1 , if

    ‖ T x − T y ‖ 2 ≤ ‖ x − y ‖ 2 + λ ‖ ( I − T ) x − ( I − T ) y ‖ 2 , ∀ x , y ∈ H .

  4. β -demicontractive with 0 ≤ β < 1 if

    ‖ T x − p ‖ 2 ≤ ‖ x − p ‖ 2 + β ‖ ( I − T ) x ‖ 2 , ∀ p ∈ F ( T ) , x ∈ H ,

    or equivalently

    ⟨ T x − x , x − p ⟩ ≤ β − 1 2 ‖ x − T x ‖ 2 , ∀ p ∈ F ( T ) , x ∈ H ,

    or equivalently

    ⟨ T x − p , x − p ⟩ ≤ ‖ x − p ‖ 2 + β − 1 2 ‖ x − T x ‖ 2 , ∀ p ∈ F ( T ) , x ∈ H .

Remark 2.4

It is known that every strictly pseudocontractive mapping with a nonempty fixed point set is demicontractive. The class of demicontractive mappings includes all the other classes of mappings defined above (see [23]).

Next, we give some examples of the class of demicontractive mappings, as shown in [23,50].

Example 2.5

  1. Let H be the real line and C = [ − 1 , 1 ] . Define T on C by:

    T x = 2 3 x sin 1 x , x ≠ 0 0 if x = 0 .

    Then T is demicontractive.

  2. Consider a mapping T : [ − 2 , 1 ] → [ − 2 , 1 ] defined such that,

    T x = − x 2 − x .

Then T is a demicontractive map that is neither quasi-nonexpansive nor strictly pseudocontractive.

We have the following lemmas which will be employed in our convergence analysis.

Lemma 2.6

[25] For each x , y ∈ H , and δ ∈ R , we have the following results:

  1. ∣ ∣ x + y ∣ ∣ 2 ≤ ∣ ∣ x ∣ ∣ 2 + 2 ⟨ y , x + y ⟩ ;

  2. ∣ ∣ x + y ∣ ∣ 2 = ∣ ∣ x ∣ ∣ 2 + 2 ⟨ x , y ⟩ + ∣ ∣ y ∣ ∣ 2 ;

  3. ∣ ∣ δ x + ( 1 − δ ) y ∣ ∣ 2 = δ ∣ ∣ x ∣ ∣ 2 + ( 1 − δ ) ∣ ∣ y ∣ ∣ 2 − δ ( 1 − δ ) ∣ ∣ x − y ∣ ∣ 2 .

Lemma 2.7

[51] Let { a n } be a sequence of nonnegative real numbers, { α n } be a sequence in ( 0 , 1 ) with ∑ n = 1 ∞ α n = ∞ , and { b n } be a sequence of real numbers. Assume that

a n + 1 ≤ ( 1 − α n ) a n + α n b n , for all n ≥ 1 ,

if lim sup k → ∞ b n k ≤ 0 for every subsequence { a n k } of { a n } satisfying lim inf k → ∞ ( a n k + 1 − a n k ) ≥ 0 , then lim n → ∞ a n = 0 .

Lemma 2.8

[52] Assume that T : H → H is a nonlinear operator with F ( T ) ≠ 0 . Then, I − T is said to be demiclosed at zero if for any { x n } in H, the following implication holds: x n ⇀ x and ( I − T ) x n → 0 ⇒ x ∈ F ( T ) .

Lemma 2.9

[53] Assume that D is a strongly positive bounded linear operator on a Hilbert space H with coefficient γ ¯ > 0 and 0 < ρ ≤ ∣ ∣ D ∣ ∣ − 1 . Then ∣ ∣ I − ρ D ∣ ∣ ≤ 1 − ρ γ ¯ .

Lemma 2.10

[54] Let U : H → H be β -demicontractive with F ( U ) ≠ ∅ and set U λ = ( 1 − λ ) + λ U , λ ∈ ( 0 , 1 − β ) . Then,

  1. F ( U ) = F i x ( U λ ) .

  2. ‖ U λ x − z ‖ 2 ≤ ‖ x − z ‖ 2 − 1 λ ( 1 − β − λ ) ‖ ( I − U λ ) x ‖ 2 , ∀ x ∈ H , z ∈ F ( U ) .

  3. F ( U ) is a closed convex subset of H.

Lemma 2.11

[55] Consider the problem with C being a nonempty, closed, convex subset of a real Hilbert space H and A : C → H being pseudomonotone and continuous. Then p is a solution of VIP (1) if and only if

⟨ A x , x − p ⟩ ≥ 0 , ∀ x ∈ C .

3 Proposed algorithm

In this section, we propose an inertial viscosity-type Tseng’s extragradient algorithm with self adaptive step size and highlight some of its important features. We establish the convergence of the algorithm under the following conditions:

Condition A

  1. The feasible set C is closed, convex, and nonempty.

  2. The solution set denoted by Ω = V I ( C , A ) ∩ F ( U ) is nonempty.

  3. The mapping A is pseudomonotone, L -Lipschitz continuous on H , and sequentially weakly continuous on C .

  4. The mapping U : H → H is a τ -demicontractive map such that I − U is demiclosed at zero.

  5. D : H → H is a strongly positive bounded linear operator with coefficient γ ¯ .

  6. f : H → H is a contraction with coefficient ρ ∈ ( 0 , 1 ) such that 0 < γ < γ ¯ ρ .

Condition B

  1. { α n } ⊂ ( 0 , 1 ) such that lim n → ∞ α n = 0 and ∑ n = 1 ∞ α n = ∞ .

  2. The positive sequence { ε n } satisfies lim n → ∞ ε n α n = 0 , { β n } ⊂ ( a , 1 − τ ) for some a > 0 .

Now, the algorithm is presented as follows:

Algorithm 3.1

Inertial TEGM with self-adaptive stepsize

  1. Given δ > 0 , ψ 1 > 0 , ϕ ∈ ( 0 , 1 ) . Select initial data x 0 , x 1 ∈ H , and set n = 1 .

  2. Given the ( n − 1 )th and nth iterates, choose δ n such that 0 ≤ δ n ≤ δ ˆ n , ∀ n ∈ N with δ ˆ n defined by

    (16) δ ˆ n = min ε n ∣ ∣ x n − x n − 1 ∣ ∣ , δ , if x n ≠ x n − 1 , δ , otherwise .

  3. Compute

    r n = x n + δ n ( x n − x n − 1 ) .

  4. Compute

    y n = P C ( r n − ψ n A r n ) .

    If y n = r n , then set z n = r n and go to Step 5. Else go to Step 4.

  5. Compute

    z n = y n − ψ n ( A y n − A r n ) .

  6. Compute

    x n + 1 = α n γ f ( r n ) + ( I − α n D ) [ ( 1 − β n ) z n + β n U z n ] .

  7. Compute

    (17) ψ n + 1 = min ϕ ∣ ∣ r n − y n ∣ ∣ ∣ ∣ A r n − A y n ∣ ∣ , ψ n , if A r n − A y n ≠ 0 , ψ n , otherwise .

    Set n ≔ n + 1 and return to Step 1.

Below are some of the interesting features of our proposed algorithm.

Remark 3.2

  1. Observe that Algorithm 3.1 involves only one projection onto the feasible set C per iteration, which makes the algorithm computationally efficient.

  2. The step size ψ n in (17) is self-adaptive and supports easy and simple computations, which makes it possible to implement our algorithm without prior knowledge of the Lipschitz constant of the cost operator.

  3. We also point out that in Step 1 of the algorithm, the inertial technique employed can easily be implemented in numerical computation, since the value of ∣ ∣ x n − x n − 1 ∣ ∣ is known prior to choosing δ n .

Remark 3.3

It can easily be seen from (16) and condition (B1) that

lim n → ∞ δ n ∣ ∣ x n − x n − 1 ∣ ∣ = 0 and lim n → ∞ δ n α n ∣ ∣ x n − x n − 1 ∣ ∣ = 0 .

4 Convergence analysis

First, we establish some lemmas which will be employed in the convergence analysis of our proposed algorithm.

Lemma 4.1

The sequence { ψ n } generated by (17) is a nonincreasing sequence and lim n → ∞ ψ n = ψ ≥ min ψ 1 , ϕ L .

Proof

It follows from (17) that ψ n + 1 ≤ ψ n , ∀ n ∈ N . Hence, { ψ n } is nonincreasing. Also, since A is Lipschitz continuous, we have

‖ A r n − A y n ‖ ≤ L ‖ r n − y n ‖ ,

which implies that

‖ r n − y n ‖ ‖ A r n − A y n ‖ ≥ 1 L .

Consequently, we obtain

ϕ ‖ r n − y n ‖ ‖ A r n − A y n ‖ ≥ ϕ L , when A r n − A y n ≠ 0 .

Combining this together with (17), we obtain

ψ n ≥ min ψ 1 , ϕ L .

Since { ψ n } is nonincreasing and bounded below, we can conclude that

lim n → ∞ ψ n = ψ ≥ min ψ 1 , ϕ L . □

Lemma 4.2

Let { r n } and { y n } be two sequences generated by Algorithm 3.1, and suppose that conditions (A1)–(A3) hold. If there exists a subsequence { r n k } of { r n } convergent weakly to z ∈ H and lim n → ∞ ‖ r n k − y n k ‖ = 0 , then z ∈ V I ( C , A )

Proof

Using the property of the projection map and y n = P C ( r n − ψ n A r n ) , we obtain

⟨ r n k − ψ n k A r n k − y n k , x − y n k ⟩ ≤ 0 ∀ x ∈ C ,

which implies that

1 ψ n k ⟨ r n k − y n k , x − y n k ⟩ ≤ ⟨ A r n k , x − y n k ⟩ ∀ x ∈ C .

From this we obtain

(18) 1 ψ n k ⟨ r n k − y n k , x − y n k ⟩ + ⟨ A r n k , y n k − r n k ⟩ ≤ ⟨ A r n k , x − r n k ⟩ ∀ x ∈ C .

Since { r n k } converges weakly to z ∈ H , we have that { r n k } is bounded. Then, from the Lipschitz continuity of A and ‖ r n k − y n k ‖ → 0 , we obtain that { A r n k } and { y n k } are also bounded. Since ψ n k ≥ ψ 1 , ϕ L , from (18) it follows that

(19) lim inf k → ∞ ⟨ A r n k , x − r n k ⟩ ≥ 0 ∀ x ∈ C .

Moreover, we have that

(20) ⟨ A y n k , x − y n k ⟩ = ⟨ A y n k − A r n k , x − r n k ⟩ + ⟨ A r n k , x − r n k ⟩ + ⟨ A y n k , r n k − y n k ⟩ .

Since lim k → ∞ ‖ r n k − y n k ‖ = 0 , then by the Lipschitz continuity of A we have lim k → ∞ ‖ A r n k − A y n k ‖ = 0 . This together with (19) and (20) gives

lim inf k → ∞ ⟨ A y n k , x − y n k ⟩ ≥ 0 .

Now, choose a decreasing sequence { θ k } of positive numbers such that θ k → 0 as k → ∞ . For any k , we represent the smallest positive integer with N k such that:

(21) ⟨ A y n j , x − y n j ⟩ + θ k ≥ 0 ∀ j ≥ N k .

It is clear that the sequence { N k } is increasing since θ k is decreasing. Furthermore, for any k , from { y N k } ⊂ C , we can assume A y N k ≠ 0 (otherwise, y N k is a solution) and set:

υ N k = A y N k ‖ A y N k ‖ 2 .

Consequently, we have ⟨ A y N k , υ N k ⟩ = 1 , for each k . From (21), one can easily deduce that

⟨ A y N k , x + θ k υ N k − y N k ⟩ ≥ 0 , ∀ k .

By the pseudomonotonicity of A , we have

⟨ A ( x + θ k υ N k ) , x + θ k υ N k − y N k ⟩ ≥ 0 ,

which implies that

(22) ⟨ A x , x − y N k ⟩ ≥ ⟨ A x − A ( x + θ k υ N k ) , x + θ k υ N k − y N k ⟩ − θ k ⟨ A x , υ N k ⟩ .

Next, we show that lim k → ∞ θ k υ N k = 0 . Indeed, since r n k ⇀ z and lim k → ∞ ‖ r n k − y n k ‖ = 0 , we obtain y N k ⇀ z , k → ∞ . Since { y n } ⊂ C , we obtain z ∈ C . By the sequentially weakly continuity of A on C , we have { A y n k } ⇀ A z . We can assume that A z ≠ 0 (otherwise, z is a solution). Since the norm mapping is sequentially weakly lower semicontinuous, we have

0 < ‖ A z ‖ ≤ lim k → ∞ ‖ A y n k ‖ .

By the fact that { y N k } ⊂ { y n k } and θ k → 0 as k → ∞ , we obtain

0 ≤ lim sup k → ∞ ‖ θ k υ N k ‖ = lim sup k → ∞ θ k ‖ A y N k ‖ ≤ lim sup k → ∞ θ k lim inf k → ∞ ‖ A y n k ‖ = 0 ,

and this implies that lim sup k → ∞ θ k υ N k = 0 . Now, by the facts that A is Lipschitz continuous, sequences { y N k } , { υ N k } are bounded and lim k → ∞ θ k υ N k = 0 , we conclude from (22) that

lim inf k → ∞ ⟨ A x , x − y N k ⟩ ≥ 0 .

Consequently, we have

⟨ A x , x − z ⟩ = lim k → ∞ ⟨ A x , x − y N k ⟩ = lim inf k → ∞ ⟨ A x , x − y N k ⟩ ≥ 0 , ∀ x ∈ C .

Thus, by Lemma 2.11, z ∈ V I ( C , A ) as required.□

Lemma 4.3

Let sequences { z n } and { y n } be two sequences generated by Algorithm 3.1  such that conditions (A1)–(A3) hold. Then, for all p ∈ Ω we have

(23) ‖ z n − p ‖ 2 ≤ ‖ r n − p ‖ 2 − 1 − ϕ 2 ψ n 2 ψ n + 1 2 ‖ r n − y n ‖ 2 ,

and

(24) ‖ z n − y n ‖ ≤ ϕ ψ n ψ n + 1 ‖ r n − y n ‖ .

Proof

By applying the definition of { ψ n } , we have

(25) ‖ A r n − A y n ‖ ≤ ϕ ψ n + 1 ‖ r n − y n ‖ , ∀ n ∈ N .

Clearly, if A r n = A y n , then inequality (25) holds. Otherwise, from (17) we have

ψ n + 1 = min ψ ‖ r n − y n ‖ ‖ A r n − A y n ‖ , ψ n ≤ ϕ ‖ r n − y n ‖ ‖ A r n − A y n ‖ .

It then follows that

‖ A r n − A y n ‖ ≤ ϕ ψ n + 1 ‖ r n − y n ‖ .

Thus, the inequality (25) is valid both when A r n = A y n and A r n ≠ A y n . Now, from the definition of z n and applying Lemma 2.6 we have

(26) ‖ z n − p ‖ 2 = ‖ y n − ψ n ( A y n − A r n ) − p ‖ 2 = ‖ y n − p ‖ 2 + ψ n 2 ‖ A y n − A r n ‖ 2 − 2 ψ n ⟨ y n − p , A y n − A r n ⟩ = ‖ r n − p ‖ 2 + ‖ y n − r n ‖ 2 + 2 ⟨ y n − r n , r n − p ⟩ + ψ n 2 ‖ A y n − A r n ‖ 2 − 2 ψ n ⟨ y n − p , A y n − A r n ⟩ = ‖ r n − p ‖ 2 + ‖ y n − r n ‖ 2 − 2 ⟨ y n − r n , y n − r n ⟩ + 2 ⟨ y n − r n , y n − p ⟩ + ψ n 2 ‖ A y n − A r n ‖ 2 − 2 ψ n ⟨ y n − p , A y n − A r n ⟩ = ‖ r n − p ‖ − ‖ y n − r n ‖ + 2 ⟨ y n − r n , y n − p ⟩ + ψ n 2 ‖ A y n − A r n ‖ 2 − 2 ψ n ⟨ y n − p , A y n − A r n ⟩ .

Since y n = P C ( r n − ψ n A r n ) , then by the projection property, we obtain

⟨ y n − r n + ψ n A r n , y n − p ⟩ ≤ 0 ,

or equivalently,

(27) ⟨ y n − r n , y n − p ⟩ ≤ − ψ n ⟨ A r n , y n − p ⟩ .

So, from (25), (26), and (27), we have

(28) ‖ z n − p ‖ 2 ≤ ‖ r n − p ‖ 2 − ‖ y n − r n ‖ 2 − 2 ψ n ⟨ A r n , y n − p ⟩ + ϕ 2 ψ n 2 ψ n + 1 2 ‖ r n − y n ‖ 2 − 2 ψ n ⟨ y n − p , A y n − A r n ⟩ = ‖ r n − p ‖ 2 − 1 − ϕ 2 ψ n 2 ψ n + 1 2 ‖ r n − y n ‖ 2 − 2 ψ n ⟨ y n − p , A y n ⟩ .

Now, from p ∈ V I ( C , A ) , we have that

⟨ A p , y n − p ⟩ ≥ 0 , y n ∈ C .

Then, by the pseudomonotonicity of A , we obtain

(29) ⟨ A y n , y n − p ⟩ ≥ 0 .

Combining (28) and (29), we have that

‖ z n − p ‖ 2 ≤ ‖ r n − p ‖ 2 − 1 − ϕ 2 ψ n 2 ψ n + 1 2 ‖ r n − y n ‖ 2 .

Moreover, from the definition of z n and (25), we obtain

‖ z n − y n ‖ ≤ ϕ ψ n ψ n + 1 ‖ r n − y n ‖ ,

which completes the proof.□

Theorem 4.4

Assume conditions ( A ) and ( B ) hold. Then, the sequence { x n } generated by Algorithm 3.1  converges strongly to an element p ∈ Ω , where p = P Ω ( I − D + γ f ) ( p ) is a solution of the variational inequality

⟨ ( D − γ f ) p , p − q ⟩ ≤ 0 , ∀ q ∈ Ω .

Proof

We divide the proof of Theorem 4.4 as follows:

Claim 1. The sequence { x n } generated by Algorithm 3.1 is bounded.

First, we show that P Ω ( I − D + γ f ) is a contraction of H . For all x , y ∈ H , we have

‖ P Ω ( I − D + γ f ) ( x ) − P Ω ( I − D + γ f ) ( y ) ‖ ≤ ‖ ( I − D + γ f ) ( x ) − ( I − D + γ f ) ( y ) ‖ ≤ ‖ ( I − D ) x − ( I − D ) y ‖ + γ ‖ f x − f y ‖ ≤ ( 1 − γ ¯ ) ‖ x − y ‖ + γ ρ ‖ x − y ‖ = ( 1 − ( γ ¯ − γ ρ ) ) ‖ x − y ‖ .

It shows that P Ω ( I − D + γ f ) is a contraction. Thus, by the Banach contraction principle there exists an element p ∈ Ω such that p = P Ω ( I − D + γ f ) ( p ) . Next, setting g n = ( 1 − β n ) z n + β n U z n and applying (23) we have

(30) ‖ g n − p ‖ 2 = ‖ ( 1 − β n ) z n + β n U z n − p ‖ 2 = ‖ ( 1 − β n ) ( z n − p ) + β n ( U z n − p ) ‖ 2 = ( 1 − β n ) 2 ‖ z n − p ‖ 2 + β n 2 ‖ U z n − p ‖ 2 + 2 ( 1 − β n ) β n ⟨ U z n − p , z n − p ⟩

≤ ( 1 − β n ) 2 ‖ z n − p ‖ 2 + β n 2 [ ‖ z n − p ‖ 2 + τ ‖ z n − U z n ‖ 2 ] + 2 ( 1 − β n ) β n ‖ z n − p ‖ 2 − 1 − τ 2 ‖ z n − U z n ‖ 2 = ‖ z n − p ‖ 2 + β n ( β n τ − ( 1 − β n ) ( 1 − τ ) ‖ z n − U z n ‖ 2 = ‖ z n − p ‖ 2 − β n ( 1 − τ − β n ) ‖ z n − U z n ‖ 2 ≤ ‖ r n − p ‖ 2 − 1 − ϕ 2 ψ n 2 ψ n + 1 2 ‖ r n − y n ‖ 2 − β n ( 1 − τ − β n ) ‖ U z n − z n ‖ 2 .

By the condition on β n , from this we obtain

(31) ‖ g n − p ‖ 2 ≤ ‖ r n − p ‖ 2 − 1 − ϕ 2 ψ n 2 ψ n + 1 2 ‖ r n − y n ‖ 2 .

From Lemma 4.1, we have that

lim n → ∞ 1 − ϕ 2 ψ n 2 ψ n + 1 2 = 1 − ϕ 2 > 0 .

This implies that there exists n 0 ∈ N such that 1 − ϕ 2 ψ n 2 ψ n + 1 2 > 0 for all n ≥ n 0 . Hence, from (31) we obtain

(32) ‖ g n − p ‖ 2 ≤ ‖ r n − p ‖ 2 ∀ n ≥ n 0 .

Also, by definition of r n and triangle inequality,

(33) ‖ r n − p ‖ = ‖ x n + δ n ( x n − x n − 1 − p ) ‖ ≤ ‖ x n − p ‖ + δ n ‖ x n − x n − 1 ‖ = ‖ x n − p ‖ + α n δ n α n ‖ x n − x n − 1 ‖ .

From Remark 3.3, we have δ n α n ‖ x n − x n − 1 ‖ → 0 as n → ∞ . Thus, there exists a constant G 1 > 0 that satisfies:

(34) δ n α n ‖ x n − x n − 1 ‖ ≤ G 1 , ∀ n ≥ 1 .

So, from (32), (33), and (34) we obtain

(35) ‖ g n − p ‖ ≤ ‖ r n − p ‖ ≤ ‖ x n − p ‖ + α n G 1 , ∀ n ≥ n 0 .

Now, by applying Lemma 2.6 and (35), ∀ n ≥ n 0 we have

‖ x n + 1 − p ‖ = ‖ α n γ f ( r n ) + ( I − α n D ) g n − p ‖ = ‖ α n ( γ f ( r n ) − D p ) + ( I − α n D ) ( g n − p ) ‖ ≤ α n ‖ γ f ( r n ) − D p ‖ + ( 1 − α n γ ¯ ) ‖ g n − p ‖ ≤ α n ‖ γ f ( r n ) − γ f ( p ) ‖ + α n ‖ γ f ( p ) − D p ‖ + ( 1 − α n γ ¯ ) ( ‖ x n − p ‖ + α n G 1 ) ≤ α n γ ρ ‖ r n − p ‖ + α n ‖ γ f ( p ) − D p ‖ + ( 1 − α n γ ¯ ) ( ‖ x n − p ‖ + α n G 1 ) ≤ α n γ ρ ( ‖ x n − p ‖ + α n G 1 ) + α n ‖ γ f ( p ) − D p ‖ + ( 1 − α n γ ¯ ) ( ‖ x n − p ‖ + α n G 1 ) = ( 1 − α n ( γ ¯ − γ ρ ) ) ‖ x n − p ‖ + α n ‖ γ f ( p ) − D p ‖ + ( 1 − α n ( γ ¯ − γ ρ ) ) α n G 1 ≤ ( 1 − α n ( γ ¯ − γ ρ ) ) ‖ x n − p ‖ + α n ( γ ¯ − γ ρ ) ‖ γ f ( p ) − D p ‖ γ ¯ − γ ρ + G 1 γ ¯ − γ ρ ≤ max ‖ x n − p ‖ , ‖ γ f ( p ) − D p ‖ γ ¯ − γ ρ + G 1 γ ¯ − γ ρ ⋮ ≤ max ‖ x n 0 − p ‖ , ‖ γ f ( p ) − D p ‖ γ ¯ − γ ρ + G 1 γ ¯ − γ ρ .

Hence, the sequence { x n } is bounded, and so { r n } , { y n } , { z n } are also bounded.

Claim 2. The following inequality holds for all p ∈ Ω and n ∈ N

∣ ∣ x n + 1 − p ∣ ∣ 2 ≤ 1 − 2 α n ( γ ¯ − γ ρ ) ( 1 − α n γ ρ ) ∣ ∣ x n − p ∣ ∣ 2 + 2 α n ( γ ¯ − γ ρ ) ( 1 − α n γ ρ ) α n γ ¯ 2 2 ( γ ¯ − γ ρ ) G 3 + 3 G 2 ( ( 1 − α n γ ¯ ) 2 + α n γ ρ ) 2 ( γ ¯ − γ ρ ) δ n α n ∣ ∣ x n − x n − 1 ∣ ∣ + 1 ( γ ¯ − γ ρ ) ⟨ γ f ( p ) − D p , x n + 1 − p ⟩ − ( 1 − α n γ ¯ ) 2 ( 1 − α n γ ρ ) 1 − ϕ 2 ψ n 2 ψ n + 1 2 ‖ r n − y n ‖ 2 + β n ( 1 − τ − β n ) ‖ U z n − z n ‖ 2 .

Using the Cauchy-Schwartz inequality and Lemma 2.6, we obtain

(36) ∣ ∣ r n − p ∣ ∣ 2 = ∣ ∣ x n + δ n ( x n − x n − 1 ) − p ∣ ∣ 2 = ∣ ∣ x n − p ∣ ∣ 2 + δ n 2 ∣ ∣ x n − x n − 1 ∣ ∣ 2 + 2 δ n ⟨ x n − p , x n − x n − 1 ⟩ ≤ ∣ ∣ x n − p ∣ ∣ 2 + δ n 2 ∣ ∣ x n − x n − 1 ∣ ∣ 2 + 2 δ n ∣ ∣ x n − x n − 1 ∣ ∣ ∣ ∣ x n − p ∣ ∣ = ∣ ∣ x n − p ∣ ∣ 2 + δ n ∣ ∣ x n − x n − 1 ∣ ∣ ( δ n ∣ ∣ x n − x n − 1 ∣ ∣ + 2 ∣ ∣ x n − p ∣ ∣ ) ≤ ∣ ∣ x n − p ∣ ∣ 2 + 3 G 2 δ n ∣ ∣ x n − x n − 1 ∣ ∣ = ∣ ∣ x n − p ∣ ∣ 2 + 3 G 2 α n δ n α n ∣ ∣ x n − x n − 1 ∣ ∣ ,

where G 2 ≔ sup n ∈ N { ∣ ∣ x n − p ∣ ∣ , θ n ∣ ∣ x n − x n − 1 ∣ ∣ } > 0 .

Now, by applying Lemma 2.6, (30), and (36) we have

∣ ∣ x n + 1 − p ∣ ∣ 2 = ∣ ∣ α n γ f ( r n ) + ( I − α n D ) g n − p ∣ ∣ 2 = ∣ ∣ α n ( γ f ( r n ) − D p ) + ( I − α n D ) ( g n − p ) ∣ ∣ 2 ≤ ( 1 − α n γ ¯ ) 2 ∣ ∣ g n − p ∣ ∣ 2 + 2 α n ⟨ γ f ( r n ) − D p , x n + 1 − p ⟩ ≤ ( 1 − α n γ ¯ ) 2 ‖ r n − p ‖ 2 − 1 − ϕ 2 ψ n 2 ψ n + 1 2 ‖ r n − y n ‖ 2 − β n ( 1 − τ − β n ) ‖ U z n − z n ‖ 2 + 2 α n γ ⟨ f ( r n ) − f ( p ) , x n + 1 − p ⟩ + 2 α n ⟨ γ f ( p ) − D p , x n + 1 − p ⟩ ≤ ( 1 − α n γ ¯ ) 2 ∣ ∣ r n − p ∣ ∣ 2 − 1 − ϕ 2 ψ n 2 ψ n + 1 2 ‖ r n − y n ‖ 2 − β n ( 1 − τ − β n ) ‖ U z n − z n ‖ 2 + α n γ ρ ( ‖ r n − p ‖ 2 + ‖ x n + 1 − p ‖ 2 ) + 2 α n ⟨ γ f ( p ) − D p , x n + 1 − p ⟩ ≤ ( 1 − α n γ ¯ ) 2 ∣ ∣ x n − p ∣ ∣ 2 + 3 G 2 α n δ n α n ∣ ∣ x n − x n − 1 ∣ ∣ − 1 − ϕ 2 ψ n 2 ψ n + 1 2 ‖ r n − y n ‖ 2 − β n ( 1 − τ − β n ) ‖ U z n − z n ‖ 2 ) + α n γ ρ ∣ ∣ x n − p ∣ ∣ 2 + 3 G 2 α n δ n α n ∣ ∣ x n − x n − 1 ∣ ∣ + ‖ x n + 1 − p ‖ 2 + 2 α n ⟨ γ f ( p ) − D p , x n + 1 − p ⟩ .

Consequently, we obtain

∣ ∣ x n + 1 − p ∣ ∣ 2 ≤ ( 1 − 2 α n γ ¯ + ( α n γ ¯ ) 2 + α n γ ρ ) ( 1 − α n γ ρ ) ∣ ∣ x n − p ∣ ∣ 2 + 3 G 2 ( ( 1 − α n γ ¯ ) 2 + α n γ ρ ) ( 1 − α n γ ρ ) α n δ n α n ∣ ∣ x n − x n − 1 ∣ ∣ + 2 α n ( 1 − α n γ ρ ) ⟨ γ f ( p ) − D p , x n + 1 − p ⟩ − ( 1 − α n γ ¯ ) 2 ( 1 − α n γ ρ ) 1 − ϕ 2 ψ n 2 ψ n + 1 2 ‖ r n − y n ‖ 2 + β n ( 1 − τ − β n ) ‖ U z n − z n ‖ 2 = ( 1 − 2 α n γ ¯ + α n γ ρ ) ( 1 − α n γ ρ ) ∣ ∣ x n − p ∣ ∣ 2 + ( α n γ ¯ ) 2 ( 1 − α n γ ρ ) ∣ ∣ x n − p ∣ ∣ 2 + 3 G 2 ( ( 1 − α n γ ¯ ) 2 + α n γ ρ ) ( 1 − α n γ ρ ) α n δ n α n ∣ ∣ x n − x n − 1 ∣ ∣ + 2 α n ( 1 − α n γ ρ ) ⟨ γ f ( p ) − D p , x n + 1 − p ⟩ − ( 1 − α n γ ¯ ) 2 ( 1 − α n γ ρ ) 1 − ϕ 2 ψ n 2 ψ n + 1 2 ‖ r n − y n ‖ 2 + β n ( 1 − τ − β n ) ‖ U z n − z n ‖ 2

≤ ( 1 − 2 α n ( γ ¯ − γ ρ ) ( 1 − α n γ ρ ) ) ∣ ∣ x n − p ∣ ∣ 2 + 2 α n ( γ ¯ − γ ρ ) ( 1 − α n γ ρ ) α n γ ¯ 2 2 ( γ ¯ − γ ρ ) G 3 + 3 G 2 ( ( 1 − α n γ ¯ ) 2 + α n γ ρ ) 2 ( γ ¯ − γ ρ ) δ n α n ∣ ∣ x n − x n − 1 ∣ ∣ + 1 ( γ ¯ − γ ρ ) ⟨ γ f ( p ) − D p , x n + 1 − p ⟩ − ( 1 − α n γ ¯ ) 2 ( 1 − α n γ ρ ) 1 − ϕ 2 ψ n 2 ψ n + 1 2 ‖ r n − y n ‖ 2 + β n ( 1 − τ − β n ) ‖ U z n − z n ‖ 2 ,

where G 3 ≔ sup { ∣ ∣ x n − p ∣ ∣ 2 : n ∈ N } . This gives the required inequality.

Claim 3. The sequence { ‖ x n − p ‖ 2 } converges to zero.

Let p = P Ω ( I − D + γ f ) ( p ) . From Claim 2, we obtain

(37) ∣ ∣ x n + 1 − p ∣ ∣ 2 ≤ 1 − 2 α n ( γ ¯ − γ ρ ) ( 1 − α n γ ρ ) ∣ ∣ x n − p ∣ ∣ 2 + 2 α n ( γ ¯ − γ ρ ) ( 1 − α n γ ρ ) α n γ ¯ 2 2 ( γ ¯ − γ ρ ) G 3 + 3 G 2 ( ( 1 − α n γ ¯ ) 2 + α n γ ρ ) 2 ( γ ¯ − γ ρ ) δ n α n ∣ ∣ x n − x n − 1 ∣ ∣ + 1 ( γ ¯ − γ ρ ) ⟨ γ f ( p ) − D p , x n + 1 − p ⟩ .

To establish Claim 3, in view of Lemma 2.7, Remark 3.3, and the fact that lim n → ∞ α n = 0 , it suffices to show that lim sup k → ∞ ⟨ γ f ( p ) − D p , x n k + 1 − p ⟩ ≤ 0 for every subsequence { ‖ x n k − p ‖ } of { ‖ x n − p ‖ } satisfying

lim inf k → ∞ ( ‖ x n k + 1 − p ‖ − ‖ x n k − p ‖ ) ≥ 0 .

Suppose that { ‖ x n k − p ‖ } is a subsequence of { ‖ x n − p ‖ } such that

(38) lim inf k → ∞ ( ‖ x n k + 1 − p ‖ − ‖ x n k − p ‖ ) ≥ 0 .

Again, from Claim 2 we obtain

( 1 − α n k γ ¯ ) 2 ( 1 − α n k γ ρ ) 1 − ϕ 2 ψ n k 2 ψ n k + 1 2 ‖ r n k − y n k ‖ 2 ≤ 1 − 2 α n k ( γ ¯ − γ ρ ) ( 1 − α n k γ ρ ) ∣ ∣ x n k − p ∣ ∣ 2 − ∣ ∣ x n k + 1 − p ∣ ∣ 2 + 2 α n k ( γ ¯ − γ ρ ) ( 1 − α n k γ ρ ) α n k γ ¯ 2 2 ( γ ¯ − γ ρ ) G 3 + 3 G 2 ( ( 1 − α n k γ ¯ ) 2 + α n k γ ρ ) 2 ( γ ¯ − γ ρ ) δ n k α n k ∣ ∣ x n k − x n k − 1 ∣ ∣ + 1 ( γ ¯ − γ ρ ) ⟨ γ f ( p ) − D p , x n k + 1 − p ⟩ .

Applying (38) and the fact that lim k → ∞ α n k = 0 , we have

( 1 − α n k γ ¯ ) 2 ( 1 − α n k γ ρ ) 1 − ϕ 2 ψ n k 2 ψ n k + 1 2 ‖ r n k − y n k ‖ 2 → 0 , k → ∞ .

By the conditions on the control parameters, we obtain

(39) ‖ r n k − y n k ‖ → 0 , k → ∞ .

Following similar argument, from Claim 2 we have

(40) ‖ U z n k − z n k ‖ → 0 , k → ∞ .

From (24) and (39), we obtain

(41) ‖ z n k − y n k ‖ → 0 , k → ∞ .

Combining (39) and (41), we have

(42) ‖ r n k − z n k ‖ ≤ ‖ r n k − y n k ‖ + ‖ y n k − z n k ‖ → 0 , k → ∞ .

By Remark 3.3 and the definition of r n , we obtain

(43) ‖ x n k − r n k ‖ = δ n k ‖ x n k − x n k − 1 ‖ → 0 , k → ∞ .

From (39), (42), and (43), we obtain

(44) ‖ x n k − y n k ‖ → 0 , k → ∞ , ‖ x n k − z n k ‖ → 0 , k → ∞ .

Also, from (40) and (44), we obtain

(45) ‖ x n k − U z n k ‖ → 0 , k → ∞ .

Using (44) and (45), we have

(46) ‖ x n k − g n k ‖ ≤ ( 1 − β n k ) ‖ x n k − z n k ‖ + β n k ‖ x n k − U z n k ‖ → 0 , k → ∞ .

Combining this together with the fact that lim k → ∞ α n k = 0 , we obtain

(47) ‖ x n k + 1 − x n k ‖ ≤ α n k ‖ γ f ( r n k ) − x n k ‖ + ( 1 − α n k γ ¯ ) ‖ g n k − x n k ‖ → 0 , k → ∞ .

To complete the proof, we need to show that w ω ( x n ) ⊂ Ω . Since { x n } is bounded, then w ω ( x n ) is nonempty. Let x ∗ ∈ w ω ( x n ) be an arbitrary element. Then there exists a subsequence { x n k } of { x n } such that x n k ⇀ x ∗ as k → ∞ . By Lemma 4.2 and (39), it follows that x ∗ ∈ V I ( C , A ) . Consequently, we have w ω ( x n ) ⊂ V I ( C , A ) . From (44), we have that z n k ⇀ x ∗ as k → ∞ . Since I − U is demiclosed at zero, then it follows from (40) that x ∗ ∈ F ( U ) . That is, w ω ( x n ) ⊂ F ( U ) . Therefore, we have w ω ( x n ) ⊂ Ω .

Moreover, from (44) it follows that w ω { y n } = w ω { x n } = w ω { z n } . By the boundedness of { x n k } , there exists a subsequence { x n k j } of { x n k } such that x n k j ⇀ x † and

(48) lim j → ∞ ⟨ γ f ( p ) − D p , x n k j − p ⟩ = lim sup k → ∞ ⟨ γ f ( p ) − D p , x n k − p ⟩ = lim sup k → ∞ ⟨ γ f ( p ) − D p , z n k − p ⟩ .

Since p = P Ω ( I − D + γ f ) ( p ) , it follows from (48) that

(49) lim sup k → ∞ ⟨ γ f ( p ) − D p , x n k − p ⟩ = lim j → ∞ ⟨ γ f ( p ) − D p , x n k j − p ⟩ = ⟨ γ f ( p ) − D p , x † − p ⟩ ≤ 0 .

Hence, from (47) and (49), we obtain

(50) lim sup k → ∞ ⟨ γ f ( p ) − D p , x n k + 1 − p ⟩ = lim sup k → ∞ ⟨ γ f ( p ) − D p , x n k + 1 − x n k ⟩ + lim sup k → ∞ ⟨ γ f ( p ) − D p , x n k − p ⟩ = ⟨ γ f ( p ) − D p , x † − p ⟩ ≤ 0 .

Applying Lemma 2.7 to (37), and using (50) together with the fact that lim n → ∞ θ n α n ∣ ∣ x n − x n − 1 ∣ ∣ = 0 and lim n → ∞ α n = 0 , we deduce that lim n → ∞ ∣ ∣ x n − p ∣ ∣ = 0 as required.□

Taking γ = 1 and D = I in Theorem 4.4, where I is the identity mapping, then we have the following corollary.

Corollary 4.5

Let H be a Hilbert space and suppose U : H → H is a τ -demicontractive map. Let { x n } be a sequence generated as follows:

Algorithm 4.6

  1. Given δ > 0 , ϕ ∈ ( 0 , 1 ) , select initial data x 0 , x 1 ∈ H , λ 0 > 0 , and set n = 1 .

  2. Given the ( n − 1 )th and nth iterates, choose δ n such that 0 ≤ δ ≤ δ n , ∀ n ∈ N with δ n defined by:

    (51) δ n = min ε n ∣ ∣ x n − x n − 1 ∣ ∣ , δ , if x n ≠ x n − 1 , δ , otherwise .

  3. Compute

    (52) r n = x n + δ n ( x n − x n − 1 ) .

  4. Compute the projection:

    (53) y n = P C ( r n − ψ n A r n ) ,

    If y n = r n , then set y n = r n and go to Step 5. Else go to Step 4.

  5. Compute

    (54) z n = y n − ψ n ( A y n − A r n ) .

  6. Compute

    (55) ψ n + 1 = min ϕ ∣ ∣ r n − y n ∣ ∣ ∣ ∣ A r n − A y n ∣ ∣ , ψ n , if A r n − A y n ≠ 0 , ψ n , otherwise .

  7. Compute

    (56) x n + 1 = α n f ( r n ) + ( 1 − α n ) [ ( 1 − β n ) z n + β n U z n ] .

    Set n ≔ n + 1 and return to Step 1.

Assume that Ω = V I ( C , A ) ∩ F ( U ) ≠ 0 and other assumptions in conditions A and B are satisfied. Then the sequence { x n } generated by Algorithm 4.6 converges strongly to a point p ∈ Ω where p = P Ω ∘ f ( p ) is a solution of the variational inequalities.

⟨ ( I − f ) p , p − z ⟩ ≤ 0 for all z ∈ Ω .

Remark 4.7

The result in Corollary 4.5 complements the result of Tan and Qin [29], Gang et al. [22] and Thong and Hieu [23] in the following ways:

  1. Our result in Corollary 4.5 extends the result of Tan and Qin [29] from pseudomonotone VIP to common solution problem of pseudomonotone variational inequality and FPPs of demicontractive maps.

  2. Corollary 4.5 result extends the result of Cai et al. [22] from FPP of nonexpansive maps to FPP of demicontractive maps.

  3. The result of Cai et al. [22] requires the knowledge of the Lipschitz constant of the cost operator while our result in Corollary 4.5 does not require any knowledge of the Lipschitz constant of the cost operator.

  4. The result of Corollary 4.5 extends the result of Thong and Hieu [23] from monotone VIP to pseudomonotone VIP.

  5. Unlike the result of Thong and Hieu [23], our result in Corollary 4.5 employs inertial technique to speed up the rate of convergence of the algorithm.

  6. As shown in our convergence analysis, we did not adopt the conventional “two cases” approach employed in several papers to prove strong convergence. Our procedure is more concise and easy to comprehend.

5 Numerical examples

In this section, we proceed to perform two numerical experiments to show the computational efficiency of our Algorithm 3.1 in comparison with some other algorithms in the literature. The graph of errors is plotted against the number of iterations in each case. All numerical computations were carried out using Matlab 2019(b). We use ‖ x n + 1 − x n ‖ ≤ 1 0 − 2 as the stopping criterion. The parameters are chosen as follows:

  • Let f ( x ) = 1 5 x , then ρ = 1 5 is the Lipschitz constant for f . Let D ( x ) = x 3 with constant γ ¯ = 1 3 , then we take γ = 1 , which satisfies 0 < γ < γ ¯ ρ . Let U x = − 3 2 x . Choose δ = 0.8 , ψ 1 = 0.6 , ϕ = 0.7 , α n = 1 n + 3 , ε n = 1 ( n + 3 ) 3 , β n = 3 n + 1 5 n + 3 in our Algorithm 3.1.

  • Take T x = x 2 , ψ = 0.8 L , θ n = 1 ( n + 3 ) 2 in Algorithm (14).

  • Let G x = x − x 1 , γ n = 1 n + 1 , ω = 0.09 , ρ n = n 2 n + 1 in Appendix 6.1.

  • Take T n x = − 2 n mod 5 x , λ = m = μ = 1 2 , σ n = 1 n + 3 , τ n = 1 3 , γ n = 1 6 , μ n = 1 2 , in Appendices 6.2 and 6.3.

Example 5.1

Consider the linear operator A : R m → R m ( m = 5 , 10 , 15 , 20 ) as follows: A ( x ) = F x + g , where g ∈ R m and F = B B T + M + E , matrix B ∈ R m × m , matrix M ∈ R m × m , is skew symmetric, and matrix E ∈ R m × m is a diagonal matrix whose diagonal terms are nonnegative (which implies that F is positive symmetric definite). We choose the feasible set as C = { x ∈ R m : − 2 ≤ x i ≤ 5 , i = 1 , … , m } . It can easily be verified that the mapping A is strongly pseudomonotone and Lipschitz continuous with L = ‖ F ‖ . In this example, both B and M entries are generated randomly in [ − 2 , 2 ] , E is generated randomly in [ 0 , 2 ] , and g = 0 . The initial values x 0 = x 1 are generated randomly by rand ( m , 1 ) .

The stopping criterion used for our computation is ‖ x n + 1 − x n ‖ < 1 0 − 2 . We plot the graphs of errors against the number of iterations in each case. The numerical results are reported in Figure 1 and Table 1.

Figure 1 Top left: m = 5 m=5 ; top right: m = 10 m=10 ; bottom left: m = 15 m=15 ; bottom right: m = 20 m=20 .
Figure 1

Top left: m = 5 ; top right: m = 10 ; bottom left: m = 15 ; bottom right: m = 20 .

Table 1

Numerical results for Example 5.1

Algorithm 14 Appendix 6.1 Appendix 6.2 Appendix 6.3 Algorithm 3.1
m = 5 No. of Iter. 10 11 11 22 6
CPU time (s) 1.7148 0.9880 0.8846 1.9379 0.4792
m = 10 No. of Iter. 11 11 11 22 6
CPU time (s) 1.4375 1.1026 0.9923 1.9712 0.5256
m = 15 No. of Iter. 11 11 11 24 6
CPU time (s) 1.4107 0.9337 1.0554 1.9888 0.6284
m = 20 No. of Iter. 11 11 12 25 6
CPU time (s) 1.2953 0.8184 0.9771 1.6142 0.4390

Example 5.2

We consider the next example in the infinite dimensional Hilbert space H = L 2 ( [ 0 , 1 ] ) with inner product

(57) ⟨ x , y ⟩ ≔ ∫ 0 1 x ( t ) y ( t ) d t for all x , y ∈ H ,

and induced norm

(58) ∣ ∣ x ∣ ∣ ≔ ∫ 0 1 ∣ x ( t ) ∣ 2 d t 1 2 for all x ∈ H .

Now, define A : H → H by A ( x ) ( t ) = max { 0 , x ( t ) } , for all t ∈ [ 0 , 1 ] , x ∈ H . It is easy to see that A is pseudomonotone and 1-Lipschitz continuous on H . It can easily be verified that all the conditions of Theorem 4.4 are satisfied.

We choose four different initial values as follows:

Case I: x 0 = 2 t 3 + 1 3 and x 1 = 3 t 5 + t 2 + 1 ;

Case II: x 0 = exp ( − t ) and x 1 = cos 2 t ;

Case III: x 0 = t 3 + t + 5 and x 1 = exp ( − 2 t ) ;

Case IV: x 0 = 2 t 5 + t 2 + 3 and x 1 = 2 t 3 − t 2 + 3 .

The stopping criterion used for our computation is ‖ x n + 1 − x n ‖ < 1 0 − 2 . We plot the graphs of errors against the number of iterations in each case. The numerical results are reported in Figure 2 and Table 2.

Figure 2 Top left: Case I; top right: Case II; bottom left: Case III; bottom right: Case IV.
Figure 2

Top left: Case I; top right: Case II; bottom left: Case III; bottom right: Case IV.

Table 2

Numerical results for Example 5.2

Algorithm 14 Appendix 6.1 Appendix 6.2 Appendix 6.3 Algorithm 3.1
No. of Iter. 6 8 12 5 4
No. of Iter. 6 8 12 5 4
No. of Iter. 6 8 12 5 4
No. of Iter. 9 11 17 8 5

6 Conclusion

We studied the pseudomonotone VIP with a fixed point constraint. We introduced a new inertial TEGM with an adaptive step size for approximating a solution of the pseudomonotone VIP, which is also a fixed point of demicontractive mappings. We proved strong convergence results for the proposed algorithm without the knowledge of the Lipschitz constant of the cost operator. Finally, we presented several numerical experiments to demonstrate the efficiency of our proposed method in comparison with some of the existing methods in the literature.

Acknowledgements

The authors sincerely thank the reviewers for their careful reading, constructive comments, and fruitful suggestions that improved the manuscript. The research of Timilehin Opeyemi Alakoya is wholly supported by the University of KwaZulu-Natal, Durban, South Africa Postdoctoral Fellowship. He is grateful for the funding and financial support. Oluwatosin Temitope Mewomo is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF.

  1. Funding information: Timilehin Opeyemi Alakoya is supported by University of KwaZulu-Natal Postdoctoral Fellowship. The research of Oluwatosin Temitope Mewomo is supported by the National Research Foundation (NRF) of South Africa (Grant Number 119903).

  2. Author contributions: Conceptualisation of the article was given by VAU and OTM; methodology by VAU and TOA; formal analysis, investigation, and writing-original draft preparation by VAU and TOA; software and validation by OTM, writing-review and editing by VAU, TOA, and OTM; and project administration by OTM. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors declare that they have no competing interests.

Appendix

Appendix 6.1. (Algorithm 3.3 in [56])
Take x 1 ∈ H , ψ 1 > 0 , ω ∈ 0 , 1 − β 2 and ϕ ∈ ( 0 , 1 ) . Choose the sequences { α n } and { γ n } satisfying the assumptions made on the control parameters
Step 1. Compute
y n = P C ( x n − ψ n A ( x n ) ) .
Step 2. Compute
z n = P H n ( x n − ψ n A ( y n ) ) ,
where
H n ≔ { x ∈ H : ⟨ x n − ψ n A ( x n ) − y n , x − y n ⟩ ≤ 0 } ,
and
ψ n + 1 ≔ min ϕ ∣ ∣ x n − y n ∣ ∣ ∣ ∣ A ( x n ) − A ( y n ) ∣ ∣ , ψ n , if A ( x n ) − A ( y n ) ≠ 0 , ψ n , otherwise .
Step 3. Compute
t n ≔ ( 1 − ρ n ) x n + ρ n z n .
Step 4. Compute
v n = t n − γ n G ( t n ) .
Step 5. Compute
x n + 1 = [ ( 1 − ω ) I + ω U ] v n .
Let n = n + 1 and return to Step 1.
Appendix 6.2. (Algorithm 1 in [57])
Initial step: Given x 0 , x 1 ∈ H arbitrarily. Let γ > 0 , m ∈ ( 0 , 1 ) μ ∈ ( 0 , 1 )
Iteration steps: Compute x n + 1 below:
Step 1. Put v n = x n − σ n ( x n − 1 − x n ) and calculate u n = P C ( v n − l n A v n ) , where l n is picked to be the largest l ∈ { λ , λ m , λ m 2 , … } s.t
l ‖ A v n − A u n ‖ ≤ μ ‖ v n − u n ‖ .
Step 2. Calculate
z n = ( 1 − α n ) P C n ( v n − l n A ( u n ) ) + α n f ( x n ) ,
where
C n ≔ { v ∈ H : ⟨ v n − l n A v n − u n , u n − v ⟩ ≥ 0 } .
Step 3. Compute
x n + 1 = γ n P C n ( v n − l n A u n ) + μ n T n z n + τ n x n .
Update n = n + 1 and return to Step 1.
Appendix 6.3. (Algorithm 2 in [57])
Initial step: Given x 0 , x 1 ∈ H arbitrarily. Let γ > 0 , m ∈ ( 0 , 1 ) μ ∈ ( 0 , 1 )
Iteration steps: Compute x n + 1 below:
Step 1. Put v n = x n − σ n ( x n − 1 − x n ) and calculate u n = P c ( v n − l n A v n ) , where l n is picked to be the largest l ∈ { λ , λ m , λ m 2 , … } s.t
l ‖ A v n − A u n ‖ ≤ μ ‖ v n − u n ‖
Step 2. Calculate
z n = ( 1 − α n ) P C n ( v n − l n A ( u n ) ) + α n f ( x n ) ,
where
C n ≔ { v ∈ H : ⟨ v n − l n A v n − u n , u n − v ⟩ ≥ 0 }
Step 3. Compute
x n + 1 = γ n P C n ( v n − l n A u n ) + δ n T n z n + β n v n
Update n = n + 1 and return to Step 1.

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Received: 2021-09-04
Revised: 2022-01-21
Accepted: 2022-02-03
Published Online: 2022-04-13

© 2022 Victor Amarachi Uzor et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2022-0030
Schlagwörter für diesen Artikel
Tseng’s extragradient method; pseudomonotone; demicontractive; variational inequalities; fixed point; strong convergence; adaptive step size; inertial technique
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Artikel in diesem Heft

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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