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

: 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 map - pings and the set of f - ﬁ xed points of continuous f - pseudocontractive mappings in re ﬂ exive 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 - ﬁ xed 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.


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: The solution set of the VIP is denoted by VI 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

Ax Ay x y
x y C , 0, for every , , and it is said to be γ-inverse strongly monotone, if there exists γ 0 > such that Ax Ay x y γ Ax Ay x y C , , f o r a l l, . 2 An example of a monotone mapping is the subdifferential mapping f E : 2 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: x E x A find such that 0 , 1 ( ) where A 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 Tx Ty x y fx fy x y x y C , , ,f o r a l l , , ⟨ − − ⟩≤⟨∇ −∇ − ⟩ ∈ where f ∇ represents the gradient of f, and it is called γ-strictly f-pseudocontractive, if there exists γ 0 > such that Tx Ty x y fx fy x y γ fx fy Tx Ty x y C , , , f o r a l l , . 2 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

=
, 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 Tx Ty x y Jx Jy x y x y C , , ,f o r a l l , , ⟨ − − ⟩≤⟨ − − ⟩ ∈ (5) and it is called a γ-strictly semi-pseudocontractive if there exists γ 0 > such that 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][27][28]38,44] and the references therein).Recall that a mapping 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 C 0 ∈ , let the sequence x n { } be generated by 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 α 0, 2 ( ).They proved that the sequence generated by (7) converges weakly to some . 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 x C , 0 ∈ , let the sequence x n { } be generated by 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 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 x C , 0 ∈ , let the sequence x n { } be generated by where A C H : → is an α-inverse strongly monotone mapping, S C H : → is a Lipschitz continuous map- ping, and where P C is the metric projection from H onto C, K r S x z E Sz y z r z x : , 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.
A common solution of nonlinear problems  3 { } 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.

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 f dom , is the set and strongly exists for any y E ∈ .In this case, f x y , 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 : → 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: (1) The interior of the domain of f denoted by 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 − * (see [7]).An example of a Legendre function is the function given by f when E is a smooth and strictly convex Banach space, whose conjugate is the function , where + = , (see [16]).In this case, f ∇ coincides with the generalized duality map- ping, We remark that J x x Jx +∞ is defined by 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 is a proper, weak * lower semi-continuous, and convex function.In addition, for all z E ∈ , we have where Let f be a Gâteaux differentiable function.The function f is said to be strongly convex with constant or equivalently (see, e.g., [24]) if for all x y f , dom ∈ , we have We note that the function f x x A function f is said to be a uniformly convex with modulus g, if for all x x y y Gph f , , , , there exists modulus g such that 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 t is positive, whenever t 0 > .The function f is called totally convex when it is totally convex at every point of f int dom( ).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.(i) f is strongly coercive and uniformly convex; .
Concerning the Bregman projection, the followings are well known.
, which is defined by We remark that V f is a nonnegative function satisfying the following conditions: for all x E ∈ and x E ∈ * * (see [34]) and { } is a sequence of real numbers such that there exists a subsequence Then, m k { } is a non decreasing sequence satisfying m k → +∞ as k → +∞ and the following properties hold: for all k N 0 ≥ , for some N 0 0 > .
Then, the following hold: ) is closed and convex; (iv) D q F x D F x x D q x for all q F F , , , ,

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: (1) There exists z E, ∈ such that

Sz y z r r fz fx y z for all y E
Then, the following conditions hold: (a) T r S is single valued; − , then we have that A is continuous and monotone from E into E * .Thus, by Lemma 9 (1), there exists z E ∈ such that and this completes the proof of (1).For (2), since Az fz Sz = ∇ − is continuous and monotone, for each x E ∈ , the mapping F E E : 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 map- ping.In what follows T r S and F r A will be defined as follows: For x E ∈ and r 0, where a 0, 1

D p T T u D p T h D p h D T h h D p T u D T h h D p u D T u u D T h h D p u D h u D T h h
, , and Now, from (21)-( 23), we have the following:

D p w D p f a fu b fT T u c fF F u a D p u b D p T T u c D p F F u D p u b D h u b D T h h
Again from (21) and ( 24) , .
Now, we consider two cases.Case 1. Suppose that there exists n 0 ∈ such that D u u , is decreasing for all n n 0 ≥ .Then, we obtain that D u u , 28) and the fact that b c e , 0 n n ≥ > , for all n 0 ≥ and α 0 n → as n → +∞, we have and Moreover, from (30), (31), and Lemma 3, we obtain and A common solution of nonlinear problems  9 The limits in (32) and (33) imply that In addition, by the property of D f and Lemma 2, we have .
This together with (34) . But, from (32) and the uniform continuity of f ∇ , we obtain fh fu n 0, as .
Thus, Lemma 10 yields + − for any h E ∈ .By the inequality in (38) and the definition of f- pseudocontractivity of S 2 , we obtain that

S h h h S h h h S h h h r r fh fu h h S h S h h h r r fh fu h h fh fh h h r r fh fu h h fh
for some constant D 0 0 > .Taking limits on both sides of the inequality (39) as j → +∞ and using the fact that r a 0 n ≥ > , for some a 0 > and for all n 1 ≥ , and (37), we have Thus, from inequality (40), we obtain 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 . Since E * is strictly convex and f ∇ * is monotone, we obtain that , and hence, z and v z nj ⇀ , as j → +∞.Now, from Lemma 9, we obtain Set v tv t z Thus, by using the fact that r a 0 n ≥ > , for all n 1 ≥ , it follows that . Similarly, by using the definition of F v for all k ∈ .Now, from (28) and the fact that b c e , 0 n n ≥ > , for all n 0 ≥ and α 0 n → as n → +∞, we obtain that and Thus, by following the method in Case . .
Thus, by using (47) and (48), we obtain that D u u , 0 as k → +∞.This together with (49) imply that D u u , 0 for all k ∈ gives that u u k → * as k → +∞.Therefore, from the aforementioned two cases, we can conclude that u n { } converges strongly to a point . 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.
Then, u n { } converges strongly to a point u * in 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.
where a 0, 1 where c e, 1 0,  ( ) = from C. In this case, we observe that the rate of convergence becomes the same with different initial points after few number of iterations.
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.

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.

p 2 =
, then we write J J 2 = , which is the normalized duality mapping.Let f E : , ( ] → −∞ +∞ be a convex and Gâteaux differentiable function.The function D , strongly coercive, and uniformly Fréchet differentiable Legendre function, which is strongly convex with a constant α * in a Banach space E, which is smooth and 2-uniformly convex.

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: (1) There exists z C, ⟨∇ − ∇ − ⟩ ≥ ∈satisfies conditions (i)-(iv) of Lemma 9(2).We note that one can re-write F x r A in terms of the mapping S as follows:A common solution of nonlinear problems  7 (a)-(d) of Lemma 10 hold.This completes the proof.□

. 2 =
Then, u n { } converges strongly to a point u * in , which is nearest to u with respect to the Bregman distance.Proof.Let p ∈ .Set h .From Lemma 10(d) and Lemma 9(iv), we obtain

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 func- tion, which is strongly convex with constant α 0 > on bounded subsets of E. Let A C E :

.Corollary 3 .Figure 1
Figure 1 indicates the behavior of the convergence of the sequence u n { } generated by Algorithm 54 to a point u t 0 ( ) = * 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 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 * and f ∇ * is uniformly continuous on bounded subsets of E.
we prove our main theorem. .
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 i If, in Theorem 1, we assume that A 0 i = , 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 E E : →+∞.Then, u n { } converges strongly to a minimum norm point u * of with respect to the Bregman distance.
Then, u n { } converges strongly to a point u * in , which is nearest to u with respect to the Bregman distance.+∞ be a strongly coercive, bounded and uniformly Frêchet differentiable Legendre function, which is strongly convex constant α 0 > on bounded subsets of E. Let A C E : →+∞.