A Halpern-type algorithm for a common solution of nonlinear problems in Banach spaces

Habtu Zegeye; Oganeditse A. Boikanyo

doi:10.1515/taa-2022-0133

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A Halpern-type algorithm for a common solution of nonlinear problems in Banach spaces

Habtu Zegeye and Oganeditse A. Boikanyo

Published/Copyright: March 13, 2023

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From the journal Topological Algebra and its Applications Volume 11 Issue 1

Abstract

In this article, we propose a Halpern-type subgradient extragradient algorithm for solving a common element of the set of solutions of variational inequality problems for continuous monotone mappings and the set of f-fixed points of continuous f-pseudocontractive mappings in reflexive real Banach spaces. In addition, we prove a strong convergence theorem for the sequence generated by the algorithm. As a consequence, we obtain a scheme that converges strongly to a common f-fixed point of continuous f-pseudocontractive mappings and a scheme that converges strongly to a common zero of continuous monotone mappings in Banach spaces. Furthermore, we provide a numerical example to illustrate the implementability of our algorithm.

Keywords: f-fixed point; f-pseudocontractive mapping; monotone mapping; semi-pseudocontractive mapping; variational inequality; strong convergence; zero points

MSC 2010: 47H05; 47H10; 47J25; 47J05; 47J20; 47J26

1 Introduction

Let E be a real normed linear space with dual E ∗ , and C be a nonempty, closed, and convex subset of E . Let A : C → E ∗ be a given mapping. The classical variational inequality problem (VIP) associated with A and C is the following:

(1) find x ∈ C such that ⟨ A x , y − x ⟩ ≥ 0 for all y ∈ C .

The solution set of the VIP is denoted by V I ( C , A ) .

The variational inequality theory was introduced independently by Fichera [15] in 1963 and Stampacchia [36] in 1964. This theory has been widely researched because of its applications in many areas of pure and applied mathematics such as partial differential equations, optimal control, optimization, fixed point theory, equilibrium problems, engineering mechanics, computer sciences, and so on, see, for example, [2,9,13,14,18,39] and the references therein. Several researchers have proposed and analyzed various iterative methods for approximating solutions of variational inequalities, see, for example, [1,4,9,17,18,21,25,35,43,44].

A mapping A : C → E ∗ is said to be monotone, if

(2) ⟨ A x − A y , x − y ⟩ ≥ 0 , for every x , y ∈ C ,

and it is said to be γ -inverse strongly monotone, if there exists γ > 0 such that

(3) ⟨ A x − A y , x − y ⟩ ≥ γ ∣ ∣ A x − A y ∣ ∣ 2 , for all x , y ∈ C .

An example of a monotone mapping is the subdifferential mapping ∂ f : E → 2 E ∗ of a proper lower semicontinuous convex function f : E → ( − ∞ , + ∞ ] , defined by

∂ f ( x ) ≔ { x ∗ ∈ E ∗ : f ( x ) + ⟨ y − x , x ∗ ⟩ ≤ f ( y ) , for all y ∈ E } ,

for all x ∈ E (see, Rockafellar [33]).

If, in (1), we consider C = E and A : E → 2 E ∗ is monotone, then the problem reduces to the following zero-point problem:

(4) find x ∈ E such that x ∈ A − 1 ( 0 ) ,

where A − 1 ( 0 ) ≔ { p ∈ E : A p = 0 } .

Motivated by the need to develop techniques for approximating solutions of the inclusion (4), when A is monotone, a new notion of f-pseudocontractive mappings with the notion of f-fixed points has recently been introduced and studied by Zegeye and Wega [45].

A mapping T : C → E ∗ is called f-pseudocontractive, if

⟨ T x − T y , x − y ⟩ ≤ ⟨ ∇ f x − ∇ f y , x − y ⟩ , for all x , y ∈ C ,

where ∇ f represents the gradient of f, and it is called γ -strictly f-pseudocontractive, if there exists γ > 0 such that

⟨ T x − T y , x − y ⟩ ≤ ⟨ ∇ f x − ∇ f y , x − y ⟩ − γ ∣ ∣ ( ∇ f x − ∇ f y ) − ( T x − T y ) ∣ ∣ 2 , for all x , y ∈ C .

An element p ∈ C is called an f-fixed point of T, if T p = ∇ f p . The set of f-fixed points of T is denoted by F f ( T ) , that is, F f ( T ) ≔ { p ∈ C : T p = ∇ f p } . We note that T is f-pseudocontractive if and only if A ≔ ∇ f − T is monotone and T is γ -strictly f-pseudocontractive if and only if A = ∇ f − T is γ -inverse strongly monotone and hence, the zeros of A correspond to f-fixed points of T . Furthermore, if we consider f ( x ) = ∣ ∣ x ∣ ∣ 2 2 , then the notion of f-pseudocontractive mappings coincides with the definition of semi-pseudocontractive mappings and the notion of f-fixed points coincides with the definition of semi-fixed points.

A nonlinear mapping T : C → E ∗ is said to be semi-pseudocontractive if

(5) ⟨ T x − T y , x − y ⟩ ≤ ⟨ J x − J y , x − y ⟩ , for all x , y ∈ C ,

and it is called a γ -strictly semi-pseudocontractive if there exists γ > 0 such that

⟨ T x − T y , x − y ⟩ ≤ ⟨ J x − J y , x − y ⟩ − γ ∣ ∣ ( J x − J y ) − ( T x − T y ) ∣ ∣ 2 , for all x , y ∈ C ,

where J = ∇ f for f ( x ) = ∣ ∣ x ∣ ∣ 2 2 , which is usually called the normalized duality mapping from E into E ∗ defined by J x = { x ∗ ∈ E ∗ : ⟨ x , x ∗ ⟩ = ∣ ∣ x ∣ ∣ ∣ ∣ x ∗ ∣ ∣ , ∣ ∣ x ∣ ∣ = ∣ ∣ x ∗ ∣ ∣ } .

A point p ∈ C is said to be a semi-fixed point of T if T p = J p . The set of all semi-fixed points of T is denoted by F s ( T ) , that is, F s ( T ) ≔ { p ∈ C : T p = J p } .

We remark that if E = H , then J in (5) reduces to the identity mapping I from H onto H , that is, J = I . Thus, in this case, T is called a pseudocontractive mapping, and the notion of semi-fixed point becomes the classical definition of a fixed point (see, for example, [6,11,12,20,21,42]).

We recall that the definition of semi-pseudocontractive was introduced by Zegeye [42] in 2008 and studied by Chidume and Idu [12] in 2016 and called it J -pseudocontractive mapping. This definition of mappings turns out to be very useful and applicable (see, e.g., [11,12,20,37]).

Many authors have studied different algorithms for finding a common element of the set of fixed points of nonexpansive mapping and the set of solutions of VIP for Lipschitz monotone mapping (see, e.g., [3,17,19,26–28,38,44] and the references therein). Recall that a mapping T : C → E is called L-Lipschitz if there exits L ≥ 0 such that

(6) ∣ ∣ T x − T y ∣ ∣ ≤ L ∣ ∣ x − y ∣ ∣ , for all x , y ∈ C ,

where C is a nonempty subset of E . If in (6), we take L = 1 , then T is called nonexpansive.

In 2003, Takahashi and Toyoda [38] employed the following method for finding a common point of the set of fixed points of nonexpansive mapping and the set of solutions of the VIP in a Hilbert space setting: for arbitrary x 0 ∈ C , let the sequence { x n } be generated by

(7) x n + 1 = α n x n + ( 1 − α n ) T P C ( x n − γ n A x n ) , for all n ≥ 0 ,

where A : C → H is an α -inverse strongly monotone mapping, T : C → H is a nonexpansive mapping, { α n } is a sequence in ( 0 , 1 ) , and { γ n } is a sequence in ( 0 , 2 α ) . They proved that the sequence generated by (7)converges weakly to some x ∗ ∈ V I ( C , A ) ∩ F ( T ) .

In 2005, Iiduka et al. [17] proposed the so-called Halpern-type subgradient method for a common point of the set of fixed points of nonexpansive mapping and the set of solutions of VIP. For arbitrary x 0 , x ∈ C , let the sequence { x n } be generated by

(8) x n + 1 = α n x + ( 1 − α n ) T P C ( x n − γ n A x n ) , for all n ≥ 0 ,

where A : C → H is an α -inverse strongly monotone mapping and T : C → H is a nonexpansive mapping. They proved that the sequence generated by (8) converges strongly to x ∗ = P V I ( C , A ) ∩ F ( T ) x provided that the control sequences { γ n } and { α n } satisfy appropriate conditions.

In 2014, Zegeye and Shahzad [44] introduced the following algorithm: for arbitrary x 0 , x ∈ C , let the sequence { x n } be generated by

(9) y n = ( 1 − β n ) x n + β n S x n , x n + 1 = P C [ α n x + ( 1 − α n ) ( δ n x n + θ n S y n + γ n P C ( x n − γ n A x n ) ) ] ,

where A : C → H is an α -inverse strongly monotone mapping, S : C → H is a Lipschitz continuous mapping, and { δ n } , { γ n } , { θ n } , { α n } , and { β n } are sequences in ( 0 , 1 ) satisfying appropriate conditions. Under some suitable conditions, they proved that the sequence generated by (9) converges strongly to the point x ∗ such that x ∗ = P V I ( C , A ) ∩ F ( T ) x .

In 2015, Alghamdi et al. [3] studied the following Halpern-type subgradient extragradient algorithm given for approximating a common point of the set of fixed points of a continuous pseudocontractive mapping T from H to H and set of solutions of a VIP for a Lipschitz monotone mapping A from C into H with Ω ≔ V I ( C , A ) ∩ F ( T ) ≠ ∅ . For arbitrary x 0 , u ∈ C , let the sequence { x n } be generated by

(10) z n = P C ( x n − λ n A x n ) , x n + 1 = α n u + ( 1 − α n ) ( a n x n + b n K r n S x n + c n P C [ x n − γ n A z n ] ) ,

where P C is the metric projection from H onto C , K r S x = { z ∈ E : ⟨ S z , y − z ⟩ − 1 r ⟨ ( 1 + r ) z − x , y − z ⟩ ≤ 0 , ∀ y ∈ E } , { γ n } ⊂ [ a , b ] ⊂ ( 0 , 1 L ) , and { a n } , { b n } , { c n } ⊂ [ a , b ] ⊂ ( 0 , 1 ) , { α n } ⊂ ( 0 , c ] ⊂ ( 0 , 1 ) , for some a , b , c > 0 . Under suitable conditions, they proved that the sequence { x n } generated by (10) converges strongly to the common solution x ∗ of Ω nearest to u .

Recently, Bello and Nnakwe [6] extended the results of Alghamdi et al. [3] to Banach spaces. They proved the following convergence theorem for a common point of the set of semi-fixed points of a continuous semi-pseudocontractive mapping and the set of solutions of VIP in Banach spaces.

Theorem 1.1

[6] Let E be a 2-uniformly convex and uniformly smooth real Banach space with dual E ∗ and C be a nonempty, closed and convex subset of E. Let S : E → E ∗ be a continuous semi-pseudocontractive mapping and A : C → E ∗ be a Lipschitz monotone mapping with Ω ≔ F s ( S ) ∩ V I ( C , A ) ≠ ∅ . For arbitrary x 0 ∈ C , let the sequence { x n } be generated by

(11) z n = Π C J − 1 ( J x n − τ A x n ) , T n = { w ∈ E : ⟨ J x n − τ A x n − J z n , w − z n ⟩ ≤ 0 } , x n + 1 = J − 1 ( α n J x 0 + ( 1 − α n ) [ β J v n + ( 1 − β ) J w n ] ) ,

where v n = T r n S x n , and T r S x = { z ∈ E : ⟨ S z , y − z ⟩ − 1 r ⟨ ( 1 + r ) J z − J x , y − z ⟩ ≤ 0 , ∀ y ∈ E } , for all x ∈ E , w n = Π T n J − 1 ( J x n − τ A x n ) with τ , β > 0 , { r n } and { α n } are real sequences satisfying certain conditions. They proved that the sequence generated by (11) converges strongly to the point x ∗ in Ω nearest to u .

This brings us to the following question.

Question: Can we obtain an iterative scheme that converges strongly to a common point of the set of f-fixed points of a continuous f-pseudocontractive mapping T and the set of solutions of a VIP for a continuous monotone mapping A in Banach spaces?

It is our purpose in this article to propose a Halpern-type subgradient extragradient method for finding a common point of the set of f-fixed points of continuous f-pseudocontractive mappings and the set of solutions of VIPs for continuous monotone mappings in reflexive real Banach spaces. As a consequence, we obtain a scheme that converges strongly to a common point of set of f-fixed points of continuous f-pseudocontractive mappings and a scheme that converge strongly to a common zero of continuous monotone mappings in Banach spaces. Moreover, we provide a numerical example to illustrate the implementability of our scheme. Our results, complement, improve and unify some related recent results in the literature.

2 Preliminaries

Throughout this section, we assume that E is a real reflexive Banach space with E ∗ as its dual and f : E → ( − ∞ , + ∞ ] is a proper, lower semicontinuous and convex function. The domain of f, denoted by dom f , is the set { x ∈ E : f ( x ) < + ∞ } . The function f is coercive if lim ∣ ∣ x ∣ ∣ → + ∞ f ( x ) = + ∞ and strongly coercive if lim ∣ ∣ x ∣ ∣ → + ∞ f ( x ) ∣ ∣ x ∣ ∣ = + ∞ . The Fenchel conjugate of f is a function f ∗ : E ∗ → ( − ∞ , + ∞ ] defined by f ∗ ( x ∗ ) = sup { ⟨ x , x ∗ ⟩ − f ( x ) : x ∈ E } , for x ∗ ∈ E ∗ .

The function f is said to be Gâteaux differentiable at x if

(12) f o ( x , y ) = lim t → 0 + f ( x + t y ) − f ( x ) t ,

exists for any y ∈ E . In this case, f o ( x , y ) coincides with ∇ f ( x ) , the value of the gradient ∇ f of f at x . Moreover, f is said to be uniformly Fréchet differentiable on a subset C of E , if the limit in (12) is attained uniformly for x ∈ C and ∣ ∣ y ∣ ∣ = 1 .

Lemma 1

[30] If f : E → R is bounded on bounded subsets of E and uniformly Fréchet differentiable, then f is uniformly continuous on bounded subsets of E and ∇ f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E ∗ .

The function f : E → ( − ∞ , + ∞ ] is said to be Legendre if it satisfies the following two conditions:

The interior of the domain of f denoted by int ( dom f ) , is nonempty, f is Gâteaux differentiable on int ( dom f ) , and dom ∇ f = int ( dom f ) .
The interior of the domain f ∗ denoted by int ( dom f ∗ ) , is nonempty, f ∗ is Gâteaux differentiable on int ( dom f ∗ ) , and dom ∇ f ∗ = int ( dom f ∗ ) .

We remark that f is Legendre if and only if f ∗ is Legendre (see [5]) and that the functions f and f ∗ are both strictly convex and Gâteaux differentiable. Furthermore, we note that if f is a Legendre function, then ( ∇ f ) − 1 = ∇ f ∗ (see [7]). An example of a Legendre function is the function given by f ( x ) = 1 p ∣ ∣ x ∣ ∣ p ( 1 < p < + ∞ ) , when E is a smooth and strictly convex Banach space, whose conjugate is the function f ∗ ( x ∗ ) = 1 q ∣ ∣ x ∗ ∣ ∣ q ( 1 < q < + ∞ ) , where 1 p + 1 q = 1 , (see [16]). In this case, ∇ f coincides with the generalized duality mapping, J p of E , that is, ∇ f = J p , where J p : E → 2 E ∗ is defined by

(13) J p ( x ) = { y ∗ ∈ E ∗ : ⟨ x , y ∗ ⟩ = ∣ ∣ x ∣ ∣ p , ∣ ∣ y ∗ ∣ ∣ = ∣ ∣ x ∣ ∣ p − 1 } .

We remark that J p x = ∣ ∣ x ∣ ∣ p − 2 J x and if p = 2 , then we write J 2 = J , which is the normalized duality mapping.

Let f : E → ( − ∞ , + ∞ ] be a convex and Gâteaux differentiable function. The function D f : dom f × int ( dom f ) → [ 0 , + ∞ ) is defined by

(14) D f ( y , x ) = f ( y ) − f ( x ) − ⟨ ∇ f ( x ) , y − x ⟩ , for all x , y ∈ E ,

which is called the Bregman distance with respect to f (see Censor and Lent [10]).

We note that D f ( . , . ) is not a metric since it does not satisfy symmetric and the triangular inequality properties.

Lemma 2

[29] If f is a proper, lower semi-continuous, and convex function, then f ∗ : E ∗ → ( − ∞ , + ∞ ] is a proper, weak ∗ lower semi-continuous, and convex function. In addition, for all z ∈ E , we have

(15) D f z , ∇ f ∗ ∑ i = 1 N t i ∇ f ( x i ) ≤ ∑ i = 1 N t i D f ( z , x i ) , f o r a l l z ∈ E ,

where { x i } i = 1 N ⊂ E and { t i } i = 1 N ⊂ ( 0 , 1 ) such that ∑ i = 1 N t i = 1 .

Let f be a Gâteaux differentiable function. The function f is said to be strongly convex with constant α > 0 if

(16) f ( y ) ≥ f ( x ) + ⟨ ∇ f ( x ) , y − x ⟩ + α 2 ∣ ∣ x − y ∣ ∣ 2 , for all x , y ∈ dom f ,

or equivalently (see, e.g., [24]) if for all x , y ∈ dom f , we have

⟨ ∇ f ( x ) − ∇ f ( y ) , x − y ⟩ ≥ α ∣ ∣ x − y ∣ ∣ 2 .

We note that the function f ( x ) = 1 2 ∣ ∣ x ∣ ∣ 2 is a bounded, strongly coercive, and uniformly Fréchet differentiable Legendre function, which is strongly convex with a constant 0 < α ≤ 1 and conjugate f ∗ ( x ∗ ) = 1 2 ∣ ∣ x ∗ ∣ ∣ 2 in a Banach space E , which is smooth and 2-uniformly convex.

A function f is said to be a uniformly convex with modulus g , if for all ( x , x ∗ ) , ( y , y ∗ ) ∈ G p h ( ∂ f ) = { ( u , u ∗ ) ∈ E × E ∗ : u ∗ ∈ ∂ f ( u ) } , there exists modulus g such that

f ( y ) ≥ f ( x ) + ⟨ y − x , x ∗ ⟩ + g ( ∣ ∣ x − y ∣ ∣ ) ,

where g is a function that is increasing and vanishes only at 0.

We remark that any strongly convex function is uniformly convex function with g ( t ) = α 2 t 2 .

Lemma 3

[23] Let f : E → ( − ∞ , + ∞ ] be a Gâteaux differentiable convex function, which is uniformly convex on bounded subsets of E . If { x n } and { y n } are bounded sequences in E , then D f ( x n , y n ) → 0 as n → + ∞ if and only if ∣ ∣ x n − y n ∣ ∣ → 0 as n → + ∞ .

The function f is called totally convex at a point x ∈ int dom ( f ) if its modulus of total convexity at x , ν f ( . , . ) : E × R + → R defined by

(17) ν f ( x , t ) = inf { y ∈ E : ∣ ∣ x − y ∣ ∣ = t } D f ( y , x ) ,

is positive, whenever t > 0 . The function f is called totally convex when it is totally convex at every point of int dom ( f ) .

We note that the function f is uniformly convex on bounded subsets of E if and only if f is totally convex on bounded subsets of E (see, e.g., [8]).

Lemma 4

[41] If f is a convex function, then the statements below are equivalent.

f is strongly coercive and uniformly convex;
dom f ∗ = E ∗ , f ∗ is Frèchet differentiable and ∇ f ∗ is uniformly continuous on bounded subsets of E .

The Bregman projection of x ∈ int ( dom f ) with respect to f, onto a nonempty closed convex subset C ⊂ int ( dom f ) is defined as the unique point in C , denoted by P C f ( x ) ∈ C , which satisfies

(18) D f ( P C f ( x ) , x ) = inf { D f ( y , x ) : y ∈ C } .

Concerning the Bregman projection, the followings are well known.

Lemma 5

[8] Let f be a totally convex and Gâteaux differentiable function on the int ( dom f ) . If C ⊂ int ( dom f ) is a nonempty, closed, and convex set and x ∈ int ( dom f ) , then z = P C f ( x ) if and only if ⟨ ∇ f ( x ) − ∇ f ( z ) , y − z ⟩ ≤ 0 , for all y ∈ C

If C is a smooth Banach space and f ( x ) = ∣ ∣ x ∣ ∣ 2 , then the Bregman projection P C f x reduces to the generalized projection Π C ( x ) , which is defined by

ϕ ( Π C x , x ) = min y ∈ C ϕ ( y , x ) ,

where ϕ ( y , x ) = ∣ ∣ y ∣ ∣ 2 − 2 ⟨ y , J x ⟩ + ∣ ∣ x ∣ ∣ 2 and J is the normalized duality mapping.

Lemma 6

[32] Let f : E → ( − ∞ , + ∞ ] be a Gâteaux differentiable and totally convex function. For x 0 ∈ E , if the sequence { D f ( x , x n ) } is bounded, then the sequence { x n } is bounded.

The Legendre function f is associated with bifunction V f : E × E ∗ → [ 0 , + ∞ ) , which is defined by

V f ( x , x ∗ ) = f ( x ) − ⟨ x , x ∗ ⟩ + f ∗ ( x ∗ ) , for all x ∈ E , x ∗ ∈ E ∗ .

We remark that V f is a nonnegative function satisfying the following conditions: for all x ∈ E and x ∗ ∈ E ∗ (see [34])

(19) V f ( x , x ∗ ) + ⟨ y ∗ , ∇ f ∗ ( x ∗ ) − x ⟩ ≤ V f ( x , x ∗ + y ∗ )

and

(20) V f ( x , x ∗ ) = D f ( x , ∇ f ∗ ( x ∗ ) ) .

Lemma 7

[40] Let { a n } be a sequence of nonnegative real numbers and { δ n } be a sequence of real numbers. Let { α n } ⊂ ( 0 , 1 ) such that ∑ n = 1 + ∞ α n = + ∞ satisfying the following relation:

a n + 1 ≤ ( 1 − α n ) a n + α n δ n , f o r a l l n ≥ n 0 , f o r s o m e n 0 > 0 .

If limsup n → + ∞ δ n ≤ 0 , then a n → 0 as n → + ∞ .

Lemma 8

[22] Suppose { Γ n } is a sequence of real numbers such that there exists a subsequence { n i } of { n } such that Γ n i < Γ n i + 1 for all i ∈ N . Let the sequence of { m k } be defined by m k = max { j ≤ k : Γ j < Γ j + 1 } . Then, { m k } is a non decreasing sequence satisfying m k → + ∞ as k → + ∞ and the following properties hold:

Γ m k ≤ Γ m k + 1 a n d Γ k ≤ Γ m k + 1 ,

for all k ≥ N 0 , for some N 0 > 0 .

Lemma 9

[4,31] Let E be a real reflexive Banach space. Let f : E → ( − ∞ , + ∞ ] be a Legendre function and A : C → E ∗ be a continuous monotone mapping, where C is a nonempty, closed, and convex subset of E . For r > 0 and x ∈ E , the following hold:

There exists z ∈ C , such that
⟨ A z , y − z ⟩ + 1 r ⟨ ∇ f z − ∇ f x , y − z ⟩ ≥ 0 , f o r a l l y ∈ C .
Define a map F r A : E → C by
F r A x = { z ∈ C : ⟨ A z , y − z ⟩ + 1 r ⟨ ∇ f z − ∇ f x , y − z ⟩ ≥ 0 , f o r a l l y ∈ C } ,
for all x ∈ E . Then, the following hold:
1. F r A is single valued;
2. F ( F r A ) = V I ( C , A ) ;
3. V I ( C , A ) is closed and convex;
4. D f ( q , F r A x ) + D f ( F r A x , x ) ≤ D f ( q , x ) , f o r a l l q ∈ F ( F r A ) , x ∈ E , provided that F ( F r A ) ≠ ∅ .

3 Main result

We shall need the following lemma in the sequel.

Lemma 10

Let E be a real reflexive Banach space and f : E → ( − ∞ , + ∞ ] be a Legendre function and S : E → E ∗ be a continuous f-pseudocontractive mapping. Then, the following hold:

There exists z ∈ E , such that
⟨ S z , y − z ⟩ − 1 r ⟨ ( 1 + r ) ∇ f z − ∇ f x , y − z ⟩ ≤ 0 , f o r a l l y ∈ E .
Define a map T r S : E → E by
T r S x = z ∈ E : ⟨ S z , y − z ⟩ − 1 r ⟨ ( 1 + r ) ∇ f z − ∇ f x , y − z ⟩ ≤ 0 , f o r a l l y ∈ E ,
for all x ∈ E . Then, the following conditions hold:
1. T r S is single valued;
2. F ( T r S ) = F f ( S ) ;
3. F f ( S ) is closed and convex;
4. D f ( q , T r S x ) + D f ( T r S x , x ) ≤ D f ( q , x ) , f o r a l l q ∈ F f ( S ) , x ∈ E .

Proof

For (1), set A z = ∇ f ( z ) − S ( z ) , then we have that A is continuous and monotone from E into E ∗ . Thus, by Lemma 9 (1), there exists z ∈ E such that

⟨ ∇ f z − S z , y − z ⟩ + 1 r ⟨ ∇ f z − ∇ f x , y − z ⟩ ≥ 0 , for all y ∈ E .

Equivalently, we obtain

⟨ S z , y − z ⟩ − 1 r ⟨ ( 1 + r ) ∇ f z − ∇ f x , y − z ⟩ ≤ 0 , for all y ∈ E ,

and this completes the proof of (1).

For (2), since A z = ∇ f z − S z is continuous and monotone, for each x ∈ E , the mapping F r A : E → E defined by

F r A x = z ∈ E : ⟨ A z , y − z ⟩ + 1 r ⟨ ∇ f z − ∇ f x , y − z ⟩ ≥ 0 , for all y ∈ E ,

satisfies conditions (i)–(iv) of Lemma 9 (2). We note that one can re-write F r A x in terms of the mapping S as follows:

T r S x = z ∈ E : ⟨ S z , y − z ⟩ − 1 r ⟨ ( 1 + r ) ∇ f z − ∇ f x , y − z ⟩ ≤ 0 , for all y ∈ E .

Thus, conditions (a)–(d) of Lemma 10 hold. This completes the proof.□

Let C be a nonempty, closed, and convex subset of a reflexive real Banach space with its dual E ∗ . Let S : E → E ∗ be a continuous f-pseudocontractive mapping and A : C → E ∗ be a continuous monotone mapping. In what follows T r S and F r A will be defined as follows: For x ∈ E and r ∈ ( 0 , + ∞ ) ,

T r S x = z ∈ E : ⟨ S z , y − z ⟩ − 1 r ⟨ ( 1 + r ) ∇ f z − ∇ f x , y − z ⟩ ≤ 0 , for all y ∈ E ,

and

F r A x = z ∈ C : ⟨ A z , y − z ⟩ + 1 r ⟨ ∇ f z − ∇ f x , y − z ⟩ ≥ 0 , for all y ∈ C .

Now, we prove our main theorem.

Theorem 1

Let E be a real reflexive Banach space with its dual E ∗ and C be a nonempty, closed, and convex subset of E. Let f : E → ( − ∞ , + ∞ ] be a strongly coercive, bounded, and uniformly Frêchet differentiable Legendre function, which is strongly convex with constant α > 0 on bounded subsets of E. Let A i : C → E ∗ , i = 1 , 2 , be continuous monotone mappings and S i : E → E ∗ , i = 1 , 2 , be continuous f-pseudocontractive mappings satisfying ℱ ≔ ∩ i = 1 2 [ F f ( S i ) ∩ V I ( C , A i ) ] ≠ ∅ . Let { u n } be a sequence generated from an arbitrary u 0 , u ∈ C by

(21) w n = ∇ f ∗ [ a n ∇ f u n + b n ∇ f T r n S 1 T r n S 2 u n + c n ∇ f F r n A 1 F r n A 2 u n ] , u n + 1 = ∇ f ∗ [ α n ∇ f u + ( 1 − α n ) ∇ f w n ] ,

where { a n } ⊂ ( 0 , 1 ) and { b n } , { c n } ⊂ [ e , 1 ) ⊂ ( 0 , 1 ) such that a n + b n + c n = 1 , { α n } ⊂ ( 0 , 1 ) satisfying lim n → + ∞ α n = 0 and ∑ n = 1 + ∞ α n = + ∞ and { r n } ⊂ ( 0 , + ∞ ) satisfying liminf n → + ∞ r n > 0 . Then, { u n } converges strongly to a point u ∗ in ℱ , which is nearest to u with respect to the Bregman distance.

Proof

Let p ∈ ℱ . Set h n = T r n S 2 u n and v n = F r n A 2 u n . From Lemma 10(d) and Lemma 9(iv), we obtain

(22) D f ( p , T r n S 1 T r n S 2 u n ) = D f ( p , T r n S 1 h n ) ≤ D f ( p , h n ) − D f ( T r n S 1 h n , h n ) = D f ( p , T r n S 2 u n ) − D f ( T r n S 1 h n , h n ) ≤ D f ( p , u n ) − D f ( T r n S 2 u n , u n ) − D f ( T r n S 1 h n , h n ) = D f ( p , u n ) − D f ( h n , u n ) − D f ( T r n S 1 h n , h n ) ,

and

(23) D f ( p , F r n A 1 F r n A 2 u n ) = D f ( p , F r n A 1 v n ) ≤ D f ( p , v n ) − D f ( F r n A 1 v n , v n ) = D f ( p , F r n A 2 u n ) − D f ( F r n A 1 v n , v n ) ≤ D f ( p , u n ) − D f ( F r n A 2 u n , u n ) − D f ( F r n A 1 v n , v n ) = D f ( p , u n ) − D f ( v n , u n ) − D f ( F r n A 1 v n , v n ) .

Now, from (21)–(23), we have the following:

(24) D f ( p , w n ) = D f ( p , ∇ f ∗ [ a n ∇ f u n + b n ∇ f T r n S 1 T r n S 2 u n + c n ∇ f F r n A 1 F r n A 2 u n ] ) ≤ a n D f ( p , u n ) + b n D f ( p , T r n S 1 T r n S 2 u n ) + c n D f ( p , F r n A 1 F r n A 2 u n ) ≤ D f ( p , u n ) − ( b n D f ( h n , u n ) + b n D f ( T r n S 1 h n , h n ) + c n D f ( v n , u n ) + c n D f ( F r n A 1 v n , v n ) ) .

Again from (21) and (24)

(25) D f ( p , u n + 1 ) ≤ D f ( p , ∇ f ∗ ( α n ∇ f u + ( 1 − α n ) ∇ f w n ) ≤ α n D f ( p , u ) + ( 1 − α n ) D f ( p , w n ) ≤ α n D f ( p , u ) + ( 1 − α n ) D f ( p , u n ) − ( 1 − α n ) ( b n D f ( h n , u n ) + b n D f ( T r n S 1 h n , h n ) + c n D f ( v n , u n ) + c n D f ( F r n A 1 v n , v n ) ) .

Thus, from (25) we obtain,

(26) D f ( p , u n + 1 ) ≤ α n D f ( p , u ) + ( 1 − α n ) D f ( p , u n ) ≤ max { D f ( p , u ) , D f ( p , u n ) } ≤ ⋯ ≤ max { D f ( p , u ) , D f ( p , u 0 ) } .

Consequently, from (26) and Lemma 6, we obtain that { u n } , and hence, { w n } is bounded.

Let u ∗ = P ℱ f u . Then, from (21), Lemma 2, (20) and (19), we obtain

(27) D f ( u ∗ , u n + 1 ) = D f ( u ∗ , ∇ f ∗ ( α n ∇ f u + ( 1 − α n ) ∇ f w n ) ) = V f ( u ∗ , α n ∇ f u + ( 1 − α n ) ∇ f w n ) ≤ V f ( u ∗ , α n ∇ f u + ( 1 − α n ) ∇ f w n − α n ( ∇ f u − ∇ f u ∗ ) ) + α n ⟨ ∇ f u − ∇ f u ∗ , u n + 1 − u ∗ ⟩ = V f ( u ∗ , α n ∇ f u ∗ + ( 1 − α n ) ∇ f w n ) + α n ⟨ ∇ f u − ∇ f u ∗ , u n + 1 − u ∗ ⟩ = D f ( u ∗ , ∇ f ∗ ( α n ∇ f u ∗ + ( 1 − α n ) ∇ f w n ) ) + α n ⟨ ∇ f u − ∇ f u ∗ , u n + 1 − u ∗ ⟩ ≤ ( 1 − α n ) D f ( u ∗ , w n ) + α n ⟨ ∇ f u − ∇ f u ∗ , u n + 1 − u ∗ ⟩ ,

and hence, from (27) and (24) we obtain

(28) D f ( u ∗ , u n + 1 ) ≤ ( 1 − α n ) D f ( u ∗ , u n ) − ( 1 − α n ) ( b n D f ( h n , u n ) + b n D f ( T r n S 1 h n , h n ) + c n D f ( v n , u n ) + c n D f ( F r n A 1 v n , v n ) ) + α n ⟨ ∇ f u − ∇ f u ∗ , u n + 1 − u ∗ ⟩ .

Thus, we obtain that

(29) D f ( u ∗ , u n + 1 ) ≤ ( 1 − α n ) D f ( u ∗ , u n ) + α n ⟨ ∇ f u − ∇ f u ∗ , u n + 1 − u ∗ ⟩ ≤ ( 1 − α n ) D f ( u ∗ , u n ) + α n ∣ ∣ u n + 1 − u n ∣ ∣ . ∣ ∣ ∇ f u − ∇ f u ∗ ∣ ∣ + α n ⟨ ∇ f u − ∇ f u ∗ , u n − u ∗ ⟩ .

Now, we consider two cases.

Case 1. Suppose that there exists n 0 ∈ N such that { D f ( u ∗ , u n ) } is decreasing for all n ≥ n 0 . Then, we obtain that { D f ( u ∗ , u n ) } is convergent. Thus, from (28) and the fact that b n , c n ≥ e > 0 , for all n ≥ 0 and α n → 0 as n → + ∞ , we have

(30) D f ( h n , u n ) → 0 , D f ( T r n S 1 h n , h n ) → 0 , as n → + ∞ ,

and

(31) D f ( v n , u n ) → 0 , D f ( F r n A 1 v n , v n ) → 0 as n → + ∞ .

Moreover, from (30), (31), and Lemma 3, we obtain

(32) h n − u n → 0 , T r n S 1 h n − h n → 0 , as n → + ∞ ,

and

(33) v n − u n → 0 , F r n A 1 v n − v n → 0 as n → + ∞ .

The limits in (32) and (33) imply that

(34) T r n S 1 h n − u n → 0 and F r n A 1 v n − u n → 0 as n → + ∞ .

In addition, by the property of D f and Lemma 2, we have

(35) D f ( u n , u n + 1 ) = D f ( u n , ∇ f ∗ ( α n ∇ f u + ( 1 − α n ) [ a n ∇ f u n + b n ∇ f T r n S 1 T r n S 2 u n + c n ∇ f F r n A 1 F r n A 2 u n ] ) ) ≤ α n D f ( u n , u ) + ( 1 − α n ) [ b n D f ( u n , T r n S 1 T r n S 2 u n ) + c n D f ( u n , F r n A 1 F r n A 2 u n ) ] = α n D f ( u n , u ) + ( 1 − α n ) [ b n D f ( u n , T r n S 1 h n ) + c n D f ( u n , F r n A 1 v n ) ] .

This together with (34), condition of { α n } and Lemma 3, imply that D f ( u n , u n + 1 ) → 0 , as n → + ∞ , which in turn implies that

(36) u n − u n + 1 → 0 as n → + ∞ .

Furthermore, since { u n } is a bounded subset of E which is reflexive, we can choose a subsequence { u n j } of { u n } such that u n j ⇀ z and limsup n → + ∞ ⟨ ∇ f u − ∇ f u ∗ , u n − u ∗ ⟩ = lim j → + ∞ ⟨ ∇ f u − ∇ f u ∗ , u n j − u ∗ ⟩ . Consequently, from (32) and (33), we have that h n j ⇀ z , v n j ⇀ z , T r n j S 1 h n j ⇀ z and F r n j A 1 v n j ⇀ z as j → + ∞ .

Now, first we show that z ∈ ∩ i = 1 2 F f ( S i ) . But, from (32) and the uniform continuity of ∇ f , we obtain

(37) ∇ f h n − ∇ f u n → 0 , as n → + ∞ .

Thus, Lemma 10 yields

(38) ⟨ S 2 h n , y − h n ⟩ − 1 r n ⟨ ( 1 + r n ) ∇ f h n − ∇ f u n , y − h n ⟩ ≤ 0 , for all y ∈ E .

Let α ∈ ( 0 , 1 ) and set h α = α h + ( 1 − α ) z for any h ∈ E . By the inequality in (38) and the definition of f-pseudocontractivity of S 2 , we obtain that

(39) ⟨ S 2 h α , h n j − h α ⟩ ≥ ⟨ S 2 h α , h n j − h α ⟩ + ⟨ S 2 h n j , h α − h n j ⟩ − 1 r n j ⟨ ( 1 + r n j ) ∇ f h n j − ∇ f u n j , h α − h n j ⟩ = − ⟨ S 2 h n j − S 2 h α , h n j − h α ⟩ − 1 r n j ⟨ ( 1 + r n j ) ∇ f h n j − ∇ f u n j , h α − h n j ⟩ ≥ − ⟨ ∇ f h n j − ∇ f h α , h n j − h α ⟩ − 1 r n j ⟨ ( 1 + r n j ) ∇ f h n j − ∇ f u n j , h α − h n j ⟩ ≥ ⟨ ∇ f h α , h n j − h α ⟩ − D 0 ∣ ∣ ∇ f h n j − ∇ f u n j ∣ ∣ r n j ,

for some constant D 0 > 0 . Taking limits on both sides of the inequality (39) as j → + ∞ and using the fact that r n ≥ a > 0 , for some a > 0 and for all n ≥ 1 , and (37), we have

(40) ⟨ S 2 h α , z − h α ⟩ ≥ ⟨ ∇ f h α , z − h α ⟩ .

Thus, from inequality (40), we obtain

(41) ⟨ S 2 ( z + α ( h − z ) ) , z − h ⟩ ≥ ⟨ ∇ f ( z + α ( h − z ) ) , z − h ⟩ .

By using the facts that S 2 is continuous and ∇ f is uniformly continuous on bounded subsets of E and letting α → 0 , we have from inequality (41) that

⟨ S 2 ( z ) , z − h ⟩ ≥ ⟨ ∇ f z , z − h ⟩ , for all h ∈ E ,

if and only if

⟨ ∇ f z − S 2 ( z ) , z − h ⟩ ≤ 0 , for all h ∈ E .

Now, set h = ∇ f ∗ ( S 2 z ) . Since E ∗ is strictly convex and ∇ f ∗ is monotone, we obtain that

⟨ ∇ f z − S 2 z , z − ∇ f ∗ S 2 z ⟩ = 0 ,

which implies that S 2 z = ∇ f z . Thus, z ∈ F f ( S 2 ) . Similarly, using the definition of T r n S 1 h n , we obtain that z ∈ F f ( S 1 ) , and hence, z ∈ ∩ i = 1 2 F f ( S i ) .

Next, we show that z ∈ V I ( C , A i ) , for each i ∈ { 1 , 2 } . From (33), we have that v n − u n → 0 as n → + ∞ , where v n = F r n A 2 u n and v n j ⇀ z , as j → + ∞ . Now, from Lemma 9, we obtain

⟨ A 2 v n , y − v n ⟩ + 1 r n ⟨ ∇ f v n − ∇ f u n , y − v n ⟩ ≥ 0 , for all y ∈ C

and

(42) ⟨ A 2 v n j , y − v n j ⟩ + 1 r n j ⟨ ∇ f v n j − ∇ f u n j , y − v n j ⟩ ≥ 0 , for all y ∈ C .

Set v t = t v + ( 1 − t ) z for all t ∈ ( 0 , 1 ) and v ∈ C . Consequently, we obtain v t ∈ C . From (42), it follows that

⟨ A 2 v t , v t − v n j ⟩ ≥ ⟨ A 2 v t , v t − v n j ⟩ − ⟨ A 2 v n j , v t − v n j ⟩ − 1 r n j ⟨ ∇ f v n j − ∇ f u n j , v t − v n j ⟩ = ⟨ A 2 v t − A 2 v n j , v t − v n j ⟩ − 1 r n j ⟨ ∇ f v n j − ∇ f u n j , v t − v n j ⟩ .

From the fact that v n j − u n j → 0 as j → + ∞ , we obtain that ∇ f v n j − ∇ f u n j → 0 as j → + ∞ . Since A 2 is monotone, we also have that ⟨ A 2 v t − A 2 v n j , v t − v n j ⟩ ≥ 0 . Thus, by using the fact that r n ≥ a > 0 , for all n ≥ 1 , it follows that

0 ≤ lim j → + ∞ ⟨ A 2 v t , v t − v n j ⟩ = ⟨ A 2 v t , v t − z ⟩ ,

and hence, ⟨ A 2 v t , v − z ⟩ ≥ 0 , for all v ∈ C . If t → 0 , the continuity of A 2 gives that

0 ≤ ⟨ A 2 z , v − z ⟩ , for all v ∈ C .

This implies that z ∈ V I ( C , A 2 ) . Similarly, by using the definition of F r n A 1 v n , we obtain that z ∈ V I ( C , A 1 ) and hence z ∈ ∩ i = 1 2 V I ( C , A i ) . Consequently, we obtain that z ∈ ∩ i = 1 2 [ F f ( S i ) ∩ V I ( C , A i ) ] . Therefore, by Lemma 5, we immediately obtain

(43) limsup n → + ∞ ⟨ ∇ f u − ∇ f u ∗ , u n − u ∗ ⟩ = lim j → + ∞ ⟨ ∇ f u − ∇ f u ∗ , u n j − u ∗ ⟩ = ⟨ ∇ f u − ∇ f u ∗ , z − u ∗ ⟩ ≤ 0 .

Hence, it follows from (29), (36), (43), and Lemma 7 that D f ( u ∗ , u n ) → 0 as n → + ∞ . Thus, Lemma 3 implies that u n → u ∗ = P ℱ f u .

Case 2. Suppose that there exists a subsequence { n i } of { n } such that

D f ( u ∗ , u n i ) < D f ( u ∗ , u n i + 1 ) ,

for all i ∈ N . Then, by Lemma 8, there exists a nondecreasing sequence { m k } ⊂ N such that m k → + ∞ , and

(44) D f ( u ∗ , u m k ) ≤ D f ( u ∗ , u m k + 1 ) and D f ( u ∗ , u k ) ≤ D f ( u ∗ , u m k + 1 ) ,

for all k ∈ N . Now, from (28) and the fact that b n , c n ≥ e > 0 , for all n ≥ 0 and α n → 0 as n → + ∞ , we obtain that

(45) h m k − u m k → 0 , T r m k S 1 h m k − h m k → 0 , as k → + ∞ ,

and

(46) v m k − u m k → 0 , F r m k A 1 v m k − v m k → 0 as k → + ∞ .

Thus, by following the method in Case 1, we obtain

(47) limsup k → + ∞ ⟨ ∇ f u − ∇ f u ∗ , u m k − u ∗ ⟩ ≤ 0

and

(48) u m k + 1 − u m k → 0 as k → + ∞ .

Now, from (29), we have that

(49) D f ( u ∗ , u m k + 1 ) ≤ ( 1 − α m k ) D f ( u ∗ , u m k ) + α m k ⟨ ∇ f u − ∇ f u ∗ , u m k − u ∗ ⟩ + α m k ∣ ∣ u m k + 1 − u m k ∣ ∣ . ∣ ∣ ∇ f u − ∇ f u ∗ ∣ ∣ ,

and hence, (44) and (49) imply that

(50) α m k D f ( u ∗ , u m k ) ≤ D f ( u ∗ , u m k ) − D f ( u ∗ , u m k + 1 ) + α m k ⟨ ∇ f u − ∇ f u ∗ , u m k − u ∗ ⟩ + α m k ∣ ∣ u m k + 1 − u m k ∣ ∣ . ∣ ∣ ∇ f u − ∇ f u ∗ ∣ ∣ ≤ α m k ⟨ ∇ f u − ∇ f u ∗ , u m k − u ∗ ⟩ + α m k ∣ ∣ u m k + 1 − u m k ∣ ∣ . ∣ ∣ ∇ f u − ∇ f u ∗ ∣ ∣ .

But the fact that α m k > 0 implies that

D f ( u ∗ , u m k ) ≤ ⟨ ∇ f u − ∇ f u ∗ , u m k − u ∗ ⟩ + ∣ ∣ u m k + 1 − u m k ∣ ∣ . ∣ ∣ ∇ f u − ∇ f u ∗ ∣ ∣ .

Thus, by using (47) and (48), we obtain that D f ( u ∗ , u m k ) → 0 as k → + ∞ . This together with (49) imply that D f ( u ∗ , u m k + 1 ) → 0 as k → + ∞ . But D f ( u ∗ , u k ) ≤ D f ( u ∗ , u m k + 1 ) for all k ∈ N gives that u k → u ∗ as k → + ∞ . Therefore, from the aforementioned two cases, we can conclude that { u n } converges strongly to a point u ∗ = P ℱ f u . The proof is complete.□

We note that the method of proof of Theorem 1 provides the following theorem for approximating a common solution of a finite family of VIP for continuous monotone mappings and f-fixed point problems for continuous f-pseudocontractive mappings in reflexive real Banach spaces.

Theorem 2

Let C be a nonempty, closed, and convex subset of a real reflexive Banach space E with its dual E ∗ . Let f : E → ( − ∞ , + ∞ ] be a strongly coercive, bounded, and uniformly Frêchet differentiable Legendre function, which is strongly convex with constant α > 0 on bounded subsets of E. Let A i : C → E ∗ , i = 1 , 2 , … N be continuous monotone mappings and S i : E → E ∗ , i = 1 , 2 , … , N , be continuous f-pseudocontractive mappings satisfying ℱ ≔ ∩ i = 1 N [ F f ( S i ) ∩ V I ( C , A i ) ] ≠ ∅ . Let { u n } be a sequence generated from an arbitrary u 0 , u ∈ C by

w n = ∇ f ∗ [ a n ∇ f u n + b n ∇ f T r n S 1 T r n S 2 … T r n S N u n + c n ∇ f F r n A 1 F r n A 2 … F r n A N u n ] , u n + 1 = ∇ f ∗ [ α n ∇ f u + ( 1 − α n ) ∇ f w n ] ,

where { a n } ⊂ ( 0 , 1 ) and { b n } , { c n } ⊂ [ e , 1 ) ⊂ ( 0 , 1 ) such that a n + b n + c n = 1 , { α n } ⊂ ( 0 , 1 ) satisfying lim n → + ∞ α n = 0 and ∑ n = 1 + ∞ α n = + ∞ and { r n } ⊂ ( 0 , + ∞ ) satisfying liminf n → + ∞ r n > 0 . Then, { u n } converges strongly to a point u ∗ in ℱ which is nearest to u with respect to the Bregman distance.

We note that the following theorem for approximating the minimum norm point of the common solution of f-fixed point and VIP follows from Theorem 1.

Theorem 3

Let E be a real reflexive Banach space with its dual E ∗ and C be a nonempty, closed, and convex subset of E. Let f : E → ( − ∞ , + ∞ ] be a strongly coercive, bounded, and uniformly Frêchet differentiable Legendre function, which is strongly convex with constant α > 0 on bounded subsets of E. Let A i : C → E ∗ , i = 1 , 2 , be continuous monotone mappings and S i : E → E ∗ , i = 1 , 2 , be continuous f-pseudocontractive mappings satisfying ℱ ≔ ∩ i = 1 2 [ F f ( S i ) ∩ V I ( C , A i ) ] ≠ ∅ . Let { u n } be a sequence generated from an arbitrary u 0 ∈ C by

(51) w n = ∇ f ∗ [ a n ∇ f u n + b n ∇ f T r n S 1 T r n S 2 u n + c n ∇ f F r n A 1 F r n A 2 u n ] , u n + 1 = ∇ f ∗ [ ( 1 − α n ) ∇ f w n ] ,

where { a n } ⊂ ( 0 , 1 ) and { b n } , { c n } ⊂ [ e , 1 ) ⊂ ( 0 , 1 ) such that a n + b n + c n = 1 , { α n } ⊂ ( 0 , 1 ) satisfying lim n → + ∞ α n = 0 and ∑ n = 1 + ∞ α n = + ∞ and { r n } ⊂ ( 0 , + ∞ ) satisfying liminf n → + ∞ r n > 0 . Then, { u n } converges strongly to a minimum norm point u ∗ of ℱ with respect to the Bregman distance.

If, in Theorem 1, we assume that A i = 0 , for i = 1 , 2 , then Theorem 1 provides the following corollary.

Corollary 1

Let E be a real reflexive Banach space with its dual E ∗ and C be a nonempty, closed, and convex subset of E. Let f : E → ( − ∞ , + ∞ ] be a strongly coercive, bounded, and uniformly Frêchet differentiable Legendre function, which is strongly convex with constant α > 0 on bounded subsets of E. Let S i : E → E ∗ , i = 1 , 2 , be continuous f-pseudocontractive mappings satisfying ℱ ≔ ∩ i = 1 2 F f ( S i ) ≠ ∅ . Let { u n } be a sequence generated from an arbitrary u 0 , u ∈ C by

w n = ∇ f ∗ [ ( 1 − c n ) ∇ f u n + c n ∇ f T r n S 1 T r n S 2 u n ] , u n + 1 = ∇ f ∗ [ α n ∇ f u + ( 1 − α n ) ∇ f w n ] ,

where { c n } ⊂ [ e , 1 ) ⊂ ( 0 , 1 ) , { α n } ⊂ ( 0 , 1 ) satisfying lim n → + ∞ α n = 0 and ∑ n = 1 + ∞ α n = + ∞ and { r n } ⊂ ( 0 , + ∞ ) satisfying liminf n → + ∞ r n > 0 . Then, { u n } converges strongly to a point u ∗ in ℱ , which is nearest to u with respect to the Bregman distance.

If, in Theorem 1, we assume that S i = ∇ f , for i = 1 , 2 , then Theorem 1 provides the following corollary.

Corollary 2

Let E be a real reflexive Banach space with its dual E ∗ and C be a nonempty, closed and convex subset of E. Let f : E → ( − ∞ , + ∞ ] be a strongly coercive, bounded and uniformly Frêchet differentiable Legendre function, which is strongly convex with constant α > 0 on bounded subsets of E . Let A i : C → E ∗ , i = 1 , 2 , be continuous monotone mappings satisfying ℱ ≔ ∩ i = 1 2 V I ( C , A i ) ≠ ∅ . Let { u n } be a sequence generated from an arbitrary u 0 , u ∈ C by

w n = ∇ f ∗ [ c n ∇ f u n + ( 1 − c n ) ∇ f F r n A 1 F r n A 2 u n ] , u n + 1 = ∇ f ∗ [ α n ∇ f u + ( 1 − α n ) ∇ f w n ] ,

where { c n } ⊂ ( 0 , e ] ⊂ ( 0 , 1 ) , { α n } ⊂ ( 0 , 1 ) satisfying lim n → + ∞ α n = 0 and ∑ n = 1 + ∞ α n = + ∞ and { r n } ⊂ ( 0 , + ∞ ) satisfying liminf n → + ∞ r n > 0 . Then, { u n } converges strongly to a point u ∗ in ℱ , which is nearest to u with respect to the Bregman distance.

If, in Theorem 1, E is smooth and 2-uniformly convex, then the function f ( x ) = 1 2 ∣ ∣ x ∣ ∣ 2 is a strongly coercive, bounded, and uniformly Fréchet differentiable Legendre function, which is strongly convex with constant α = 1 and conjugate f ∗ ( x ∗ ) = 1 2 ∣ ∣ x ∗ ∣ ∣ 2 . Moreover, we have ∇ f = J and ∇ f ∗ = J − 1 . In this case, for x ∈ E and r ∈ ( 0 , + ∞ ) , we have

T r S x = z ∈ E : ⟨ S z , y − z ⟩ − 1 r ⟨ ( 1 + r ) J z − J x , y − z ⟩ ≤ 0 , for all y ∈ E ,

and

F r A x = z ∈ C : ⟨ A z , y − z ⟩ − 1 r ⟨ J z − J x , y − z ⟩ ≥ 0 , for all y ∈ C .

Thus, we obtain the following corollary.

Corollary 3

Let E be a real 2-uniformly convex and smooth Banach space with its dual E ∗ and C be a nonempty, closed and convex subset of E. Let A i : C → E ∗ , i = 1 , 2 , be continuous monotone mappings and S i : E → E ∗ , i = 1 , 2 , be continuous semi-pseudocontractive mappings satisfying ℱ ≔ ∩ i = 1 2 [ F f ( S i ) ∩ V I ( C , A i ) ] ≠ ∅ . Let { u n } be a sequence generated from an arbitrary u 0 , u ∈ C by

w n = J − 1 [ a n J u n + b n J T r n S 1 T r n S 2 u n + c n J F r n A 1 F r n A 2 u n ] , u n + 1 = J − 1 [ α n J u + ( 1 − α n ) J w n ] ,

If, in Corollary 3, we assume that E = H is a real Hilbert space, and f ( x ) = 1 2 ∣ ∣ x ∣ ∣ 2 , then we have ∇ f = J = I and ∇ f ∗ = J − 1 = I , where I is identity mapping on H . Moreover, f-pseudocontractive mapping becomes pseudocontractive mapping. In this case, for x ∈ E and r ∈ ( 0 , + ∞ ) , we have

(52) T r S x = z ∈ E : ⟨ S z , y − z ⟩ − 1 r ⟨ ( 1 + r ) z − x , y − z ⟩ ≤ 0 , for all y ∈ E

and

(53) F r A x = z ∈ C : ⟨ A z , y − z ⟩ − 1 r ⟨ z − x , y − z ⟩ ≥ 0 , for all y ∈ C .

We use the notations in (52) and (53) to state the next corollary in Hilbert spaces.

Corollary 4

Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let A i : C → H , i = 1 , 2 , be continuous monotone mappings and S i : H → H , i = 1 , 2 , be continuous pseudocontractive mappings satisfying ℱ ≔ ∩ i = 1 2 [ F ( S i ) ∩ V I ( C , A i ) ] ≠ ∅ . Let { u n } be a sequence generated from an arbitrary u 0 , u ∈ C by

w n = a n u n + b n T r n S 1 T r n S 2 u n + c n F r n A 1 F r n A 2 u n , u n + 1 = α n u + ( 1 − α n ) w n ,

4 Numerical example

Now, we give an example of continuous f-pseudocontractive and continuous monotone mappings satisfying Theorem 1 with i = 1 and some numerical experiment result to explain the conclusion of the theorem as follows:

Example 1

Let E = L p R ( [ 0 , 1 ] ) , where 1 < p ≤ 2 , with norm ∣ ∣ x ∣ ∣ L p ≔ ( ∫ 0 1 ∣ x ( t ) ∣ p d t ) 1 p , for x ∈ E . Then, E ∗ = L q R ( [ 0 , 1 ] ) with 1 p + 1 q = 1 . Let C ≔ B p ( 0 , 1 ) ≔ { x ∈ E : ∣ ∣ x ∣ ∣ L p ≤ 1 } and f : E → ( − ∞ , + ∞ ] be given by f ( x ) = 1 2 ∣ ∣ x ∣ ∣ 2 . Let A : C → E ∗ and S : E → E ∗ be mappings defined by ( A x ) ( t ) = 2 t ∇ f x ( t ) = 2 t J x ( t ) , for all t ∈ [ 0 , 1 ] and S x ( t ) = − t 2 J x ( t ) , for all t ∈ [ 0 , 1 ] . Clearly, A is a continuous and monotone mapping with V I ( C , A ) = { 0 } and S is a continuous f-pseudocontractive mapping with F f ( S ) = { 0 } , and hence, ℱ = F f ( S ) ∩ V I ( C , A ) = { 0 } . Furthermore, from Lemma 9 and 10, for any x ∈ E , we obtain F r n A x ( t ) = J − 1 1 1 + 2 t r n J x ( t ) and T r n S x ( t ) = J − 1 1 ( 1 + r n ( 1 + t 2 ) ) J x ( t ) .

We note that the aforementioned set of mappings satisfy the conditions of Theorem 1 with i = 1 . Now, for implementation, we choose E = H , a real Hilbert space, a n = b n = c n = 1 3 , u ( t ) = t , r n = 1 , for n ≥ 0 and u ∈ C , we compute the ( n + 1 ) th iteration as follows:

(54) w n ( t ) = 1 3 u n ( t ) + 1 3 ( 1 + r n ( 1 + t 2 ) ) u n ( t ) + 1 3 ( 1 + 2 t r n ) u n ( t ) , u n + 1 ( t ) = α n u ( t ) + ( 1 − α n ) w n ( t ) ,

Figure 1 indicates the behavior of the convergence of the sequence { u n } generated by Algorithm 54 to a point u ∗ ( t ) = 0 in ℱ using MATLAB version 7.5.0.342(R2007b) for α n = 1 1,000 ( n + 100 ) and different initial points: u 0 ( t ) = 2 t 2 , u 0 ( t ) = 2 t 4 , and u 0 ( t ) = 2 t 6 from C . In this case, we observe that the rate of convergence becomes the same with different initial points after few number of iterations.

$Figure 1 Convergence of the sequence { u n } \left\{{u}_{n}\right\} generated by (54) with different initial points.$

Figure 1

Convergence of the sequence { u n } generated by (54) with different initial points.

Remark 1

Theorem 1 extends Theorem 3.1 of Iiduka and Takahashi [16] and Theorem 3.1 of Zegeye and Shahzad [44] to the more general class of f-pseudocontractive mappings in Banach spaces. Moreover, Theorem 1 also extends Theorem 1.1 of Alghamdi et al. [3] and Theorem 1.2 of Bello and Nnakwe [6] to the more general class of f-pseudocontractive mappings.

5 Conclusion

In this article, we constructed a new Halpern-type subgradient-extragradient iterative algorithm that converges strongly to a common point of f-fixed point set of continuous f-pseudocontractive mappings and solution set of VIP for continuous monotone mappings in reflexive real Banach spaces. As a result, we obtained strong convergence results for a common f-fixed point of continuous f-pseudocontractive mappings and for a common zero of continuous monotone mappings in Banach spaces. In addition, a numerical example is given to illustrate the implementability of our algorithm. Our results provide an affirmative answer to the question raised.

Acknowledgements

The authors would like to thank the anonymous reviewers for their comments on the manuscript which helped us very much in improving and presenting the original version of this article.

Author contributions: All authors contributed equally and have both reviewed the manuscript.
Conflict of interest: There are no conflicts of interest to this work.

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Received: 2021-10-09

Revised: 2023-01-20

Accepted: 2023-01-23

Published Online: 2023-03-13

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

Articles in the same Issue

https://doi.org/10.1515/taa-2022-0133

Keywords for this article

f-fixed point; f-pseudocontractive mapping; monotone mapping; semi-pseudocontractive mapping; variational inequality; strong convergence; zero points

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