Some notations and definitions on uniformly convex Banach spaces

Published online December 1, 2016 c Springer International Publishing 2016

Journal of Fixed Point Theory and Applications

An iterative method for split inclusion problems without prior knowledge of operator norms

J. Y. Bello Cruz and Y. Shehu

Abstract. In this paper, we study the approximation of solution (as- suming existence) for the split inclusion problem in uniformly convex Banach spaces which are also uniformly smooth. We introduce an it- erative algorithm in which the stepsizes are selected without the need for any prior information about the bounded linear operator norm and strong convergence obtained. The novelty of our algorithm is that the bounded linear operator norm is not given a priori and stepsizes are constructed step by step in a natural way. Our results extend and im- prove many recent and important results obtained in the literature on the split inclusion problem and its variations.

Mathematics Subject Classification. 49J53, 65K10, 49M37, 90C25.

Keywords. Strong convergence, Split inclusion problem,p-uniformly convex Banach space, Uniformly smooth Banach space.

1. Introduction

LetH1 andH2 be two real Hilbert spaces,T₁:H1⇒H1 andT₂: H2⇒H2

be two set-valued maximal monotone mappings,A:H1→ H2be a bounded linear operator, andA^∗be the adjoint ofA. Many authors [5,10,15,17,37,38]

have studied the following split inclusion problem (SIP) in Hilbert spaces:

Findx^∗∈ H₁such that 0∈T₁(x^∗), and

0∈T₂(Ax^∗). (SIP)

The split inclusion problem has studied extensively by many authors and applied to solving many real life problems, such as in modelling intensity- modulated radiation therapy treatment planning [7,8], modelling of inverse problems arising from phase retrieval, and in sensor networks in computer- ized tomography and data compression [4,12].

Next, we give some examples which serve as motivations for studying the split inclusion problem and to understand its relationship with other problems in the literature. Byrne et al. [6] studied a split inverse problem (SInvP) formulated in Sect. 2 of [9] which concerns a model in which there

are two given vector spacesX₁andX₂and a linear operatorA:X₁→ X₂. In this model, two Inverse Problems (IP) are involved. The first one, denoted by (IP1), is formulated in the spaceX1, and the second one, denoted by (IP2), is inX2. Given these data, the Split Inverse Problem (SInvP) is formulated

as follows:

Find x^∗∈ X₁ solving (IP1), and

Ax^∗∈ X₂ solves (IP2). (SInvP)

Now, recall the split variational inequality problem (SVIP) introduced in [9], which is an SIP with a variational inequality problem (VIP) in each one of the two spaces. LetH₁andH₂be two real Hilbert spaces, and assume that there are given two point-to-point operators F₁:H₁ → H₁ and F₂: H₂ → H₂, a bounded linear operator A: H₁ → H₂, and nonempty, closed, and convex subsets Ω₁⊆ H₁ and Ω₂⊆ H₂. The SVIP is then formulated as follows:

Findx^∗∈Ω₁ such that F1(x^∗), x−x^∗ ≥0, ∀x∈Ω₁, and

y^∗=Ax^∗∈Ω₂ solves F₂(y^∗), y−y^∗ ≥0, ∀y∈Ω₂. (SVIP) The above problem (SVIP) can be structurally considered a special case of the split monotonic variational inclusion problem (SMVIP) when the operators F₁ andF₂are monotones. It is formulated as follows:

Findx^∗∈ SOL(F₁,Ω₁) such that

Ax^∗∈ SOL(F₂,Ω₂), (SMVIP)

where we denote by SOL(F₁,Ω₁) and SOL(F₂,Ω₂) the solution sets of the VIPs in (SVIP). Taking in (SMVIP), Ω₁ =H1,Ω₂=H2 and choosing x:=

x^∗−F₁(x^∗)∈ H1 and y=Ax^∗−F₂(Ax^∗)∈ H2, we obtain the Split Zeros Problem (SZP) for two operators F₁:H1 → H1 and F₂:H2 → H2, which were introduced in Sect. 7.3 of [9]. It is formulated as follows:

Findx^∗∈ H1 such thatF₁(x^∗) = 0 and

F₂(Ax^∗) = 0. (SZP)

Furthermore, we can define the set-valued mappingT₁: Ω₁⊂ H₁⇒H₁ by T₁(x) :=

F₁(x) +NΩ1(x), x∈Ω₁

∅, Otherwise,

andT₂: Ω₂⊂ H₂⇒H₂ by T₂(y) :=

F₂(y) +N_Ω2(y), y∈Ω₂

∅, Otherwise,

whereNΩ: H⇒His the normal cone of some nonempty, closed, and convex set Ω⊂ H, which is defined at a point u∈Ω as

N_Ω(u) :={d∈ H:d, u−v ≤0, ∀u∈Ω},

and the empty set otherwise. Then, under maximal monotonicity assumption onF₁andF₂, Theorem 3 of [19] implies thatT₁andT₂are maximal monotone mappings and so x^∗ ∈ SOL(F₁,Ω₁) and Ax^∗ ∈ SOL(F₂,Ω₂) if and only if 0 ∈T₁(x^∗) and 0 ∈T₂(Ax^∗), respectively. Hence, we observe that problem (SMVIP) is a special case of Split Inclusion Problem defined in (SIP). Thus, SIP is a remalkable problem, which contains a large class of applications;

for further discussions of this problem, the reader can see [10,17] and the references therein.

Some notations and definitions on uniformly convex Banach spaces

We start this part by recalling some definitions and notation used in this paper, which are standard and follows from [3,39]. Throughout this paper, we write S := R to indicate that S is defined to be equal to R. Let B be a real Banach space. The modulus of smoothness of B is the function ρ_B: [0,∞)→[0,∞) defined by

ρ_B(t) := sup 1

2(x+y+x−y)−1 :x ≤1,y ≤t

.

We assume that 1< p, q <∞such that ¹_p+¹_q = 1. Bisuniformly smoothif and only if

t→0lim ρ_B(t)

t = 0,

andq-uniformly smooth if there exists aC_q >0, such that ρ_B(t)≤C_qt^q for anyt >0.

Let dim B ≥ 2. The modulus of convexity of B is the function δ_B : (0,2]→[0,1] defined by

δ_B() := inf

1−x+y 2

:x=y= 1;=x−y

.

Bisuniformly convexif and only ifδ_B()>0 for all∈(0,2] andp-uniformly convexif there is aC_p>0, so thatδ_B()≥C_p^pfor any∈(0,2]. The uniform convexity is a geometric property of the unit ball. In particular, the unit sphere must be round and cannot include any line segment. Surprisingly, this geometric property ensures a topological one. That is, it follows from Milman–

Pettis’ Theorem that every uniformly convex Banach space is reflexive; see Theorem 3.31 of [3].

It is well known thatB isp-uniformly convex and uniformly smooth if and only if its dualB^∗ is q-uniformly smooth and uniformly convex. Then, from now on, mainly in Sect.3, we assume thatBisp-uniformly convex and uniformly smooth with constantC_pand we denoteC_q with 1/p+ 1/q= 1 the associated constant toB^∗. It is also a common knowledge that the duality mappingJ_p^B is one-to-one, single valued and satisfiesJ_p^B= (J_q^B∗)⁻¹, where J_q^B∗ is the duality mapping ofB^∗ (see [1,11]).

Definition 2.1. The duality mappingJ_p^B:B⇒B^∗ is defined by J_p^B(x) :=

x^∗∈ B^∗:x, x^∗=x^p,x^∗=x^p−1 .

We denote it asJ whenp= 1 and the domain spaceBis clear. The duality mappingJ_p^B is said to be weak-to-weak star continuous if

x_n x⇒ J_p^B(x_n), y → J_p^B(x), y

holds true for any y ∈ B. We note here that l_p(p > 1) spaces have such a property, butL_p(p >2) does not share this property.

LetB be a reflexive, strictly convex, and smooth Banach space and let T:B ⇒ B^∗ be a maximal monotone operator. Then, for λ >0 and x∈ B, consider the metric resolvent ofT, forλ >0, as

J_λT(x) :={z∈ B: 0∈ J(z−x) +λT(z)}

or equivalentlyJ_λT = (I+λJ⁻¹T)⁻¹:B → B^∗, which is point-to-point, full domain, and nonexpansive operator. Then

0≤ J_λT(x)−J_λT(y),J(x−J_λT(x))− J(y−J_λT(y)) holds, for allx, y∈ B; see, for instance, Proposition 57.5(b) of [40].

In addition, we consider the relative resolvent ofT, forλ >0, as Q_λT(x) :={z∈ B:J x∈J z+λT(z)},

or equivalently Q_λT = (J +λT)⁻¹J: B → B^∗, which is point-to-point and relatively nonexpansive mapping. That is, forλ >0,

0≤ Q_λT(x)−Q_λT(y),J(x)− J(Q_λT(x))−(Jy−Q_λT(y)) for allx, y∈ B; see, Theorem 5.2. of [27].

Previous related schemes and our proposal

In this part, we describe some previous schemes, which use the norm of the operatorA to find the stepsizes. In 2011, Byrne et al. [5] gave the following convergence theorem for the split inclusion problem (SIP) working in Hilbert spaces.

Theorem 1.1 (Theorem 3.2 of [5]). Let H1 and H2 be real Hilbert spaces, A: H1 → H2 be a linear and bounded operator, and A^∗ denote the adjoint ofA. LetT₁:H1 ⇒H1 and T₂:H2⇒H2 be set-valued maximal monotone mappings,λ >0andγ∈

0,_A²_∗A

. Suppose thatΓ=∅, the solution set of (SIP). The sequence(x_n)_n∈N defined as follows:

x_n+1=J_λT1 x_n−γA^∗

I−J_λT2

Ax_n

converges weakly to an elementx^∗∈Γ.

Furthermore, in 2013, Chuang [10] gave the following strong convergence theorem for problem (SIP).

Theorem 1.2 (Theorem 4.1 of [10]). Let H1 and H2 be real Hilbert spaces, A: H1 → H2 be a linear and bounded operator, and A^∗ denote the ad- joint of A. Let T₁: H₁ ⇒ H₁ and T₂: H₂ ⇒ H₂ be set-valued maximal monotone mappings. Let(α_n)_n∈N⊂(0,1), (λ_n)_n∈N be a sequence in(0,∞) and (γ_n)_n∈N ⊂

0,2/(A²+ 2)

. Let Γ be the solution set of (SIP) and suppose thatΓ=∅. Let (x_n)_n∈N be defined by

x_n+1=J_λnT1 (1−α_nγ_n)x_n−γ_nA^∗

I−J_λnT2

Ax_n

, n≥1.

Assume that

n→∞lim α_n= 0, ∞ n=1

α_nγ_n=∞, lim inf

n→∞γ_n>0, lim inf

n→∞λ_n>0.

Then, lim_n→∞x_n = ¯x:= P_Γ0, that is, ¯x is the minimal norm solution of (SIP).

Recently, Takahashi and Takahashi [29] considered the problem (SIP) in uniformly convex and smooth Banach spaces, which are higher spaces other than Hilbert spaces. In other words, let B1 and B2 be two real uniformly convex and smooth Banach spaces,T₁: B₁ ⇒B₁^∗ and T₂: B₂ ⇒ B₂^∗ be two set-valued maximal monotone mappings,A: B₁ → B₂ be a bounded linear operator, and A^∗:B^∗₂ → B^∗₁ be the adjoint ofA. Takahashi and Takahashi [29] studied the Split Inclusion Problem given in (SIP) in Banach spaces, that

is

Findx^∗∈ B₁ such that 0∈T₁(x^∗), and

0∈T₂(Ax^∗). (SIP_B)

Let the set of solutions of the split inclusion problem (SIP_B) be denoted by Γ.

Using the shrinking projection method, Takahashi and Takahashi [29] proved the following strong convergence theorem for finding a solution of (SIP_B) in uniformly convex and smooth Banach spaces.

Theorem 1.3 (Theorem 6 of [29]). Let B1 and B2 be uniformly convex and smooth Banach spaces and let J^B1 and J^B2 be the duality mappings on B1 and B2, respectively. Let T₁: B1 ⇒ B₁^∗ and T₂: B2 ⇒ B₂^∗ be maximal monotone operators, such thatT₁⁻¹(0)=∅ andT₂⁻¹(0)=∅, respectively. Let Q_λT2 be the relative resolvent ofT₂ forλ >0. Let A:B1→ B2 be a bounded linear operator, such that A = 0 and let A^∗ be the adjoint operator of A.

Suppose that Γ = ∅. Let x₁ ∈ B1 and let C₁ = T₁⁻¹(0). Let (x_n)_n∈N be a sequence generated by

⎧⎨

⎩

z_n =x_n−t_n(J^B1)⁻¹A^∗J^B2(Ax_n−Q_λnT2(Ax_n)), C_n+1=

z∈C_n :zn−z,J^B1(x_n−z_n) ≥0 x_n+1 =P_Cn+1x₁, ∀n≥1,

(1.1) where {t_n},(λ_n)_n∈N ⊂ (0,∞) satisfy the conditions, such that for some a, b, c∈R

0< a≤t_nA²≤b <1 and 0< c≤λ_n, ∀n≥1.

Then, the sequence(x_n)_n∈Nconverges strongly to a pointz₀∈Γ, where z₀= P_Γx₁.

Closely studying the result of Takahashi and Takahashi [29], we see that the choice of step-sizet_n is operator norm dependent. This is a drawback in terms of efficiency and implementation of the iteration process (1.1). Further- more, the setC_n has to be computed in each iteration step in the iteration process (1.1). This makes the iteration process (1.1), above, computationally expensive, because the complexity of the proposed iteration grows dramat- ically whenn is large. Note that the set where is doing the projection step can be written as

C_n+1=C_n∩ {z∈ B₁:z_n−z,J^B1(x_n−z_n) ≥0}

=∩ⁿ₌₂{z∈C₁:z−z,J^B1(x−z) ≥0}.

Moreover, it is worth mentioning that in most of the results on the split inclusion problem studied so far, the choice of the stepsize depends on the operator normA. This involves knowing a priori the norm (or at least an estimate of the norm) of the bounded linear operator, which is in general not an easy work in practice. Hence, it makes the implementation of the iteration process inefficient when the computation of the operator normA is not explicit (see [13,16]). Even in finite dimensions, computing the norm of bounded linear operator is a difficult task as shown by the following Theorem of Hendrickx and Olshevsky [14]:

Theorem 1.4 (Theorem 2.3 of [14]). For any rational p∈ [1,∞)except p= 1,2, unless P =N P, there is no algorithm which computes the p-norm of a matrix with entries in{−1,0,1} to relative error with running time poly- nomial in the dimensions.

These observations lead to the following natural question.

Question: Can we obtain strong convergence result for the split inclusion problem in higher spaces other than Hilbert spaces in which the iteration process is operator norm independent for solving the Split Inclusion Problem (SIP)? Moreover, could the proposed iteration process be more efficient and implementable than the iteration processes already obtained for the split inclusion problem in the literature?

Our main purpose, in this paper, is to give an affirmative answer to the above question. Thus, we propose an iterative scheme, which generated a sequence strongly convergent to some solution of the split inclusion problem inp-uniformly convex real Banach spaces which are also uniformly smooth.

The proposed iteration process is constructed in such a way that we do not need to know a priori the norm or an estimate of the norm of the bounded linear operator for approximating solution of the split inclusion problems and so it can be more efficiently implemented. Our results complement and extend many recent and important results on the split inclusion problems both in Hilbert spaces and Banach spaces.

Organization of the paper: The next subsection provides some preliminar- ies results that will be used in the remainder of this paper. In Sect. 3, the proposed algorithm and its strong convergence are analyzed by choosing step- sizes without norm dependence, and moreover, we remark some consequences.

Finally, Sect.4gives some concluding remarks.

2. Preliminaries

This section discusses some preliminary results which will be used throughout the paper.

Lemma 2.1 (Corollary 1 of [34]). Let x, y∈ B. If B isq-uniformly smooth Banach space, then there existsC_q >0, such that

x−y^q≤ x^q−qJ_q(x), y+C_qy^q. (2.1) Definition 2.2. Let B be a p-uniformly convex real Banach spaces which is also uniformly smooth. Given a Gˆateaux differentiable convex function f:B →R, theBregman distancewith respect to f is defined as

Δ_f(x, y) =f(y)−f(x)− f(x), y−x, x, y∈ B.

We note here also that the duality mappingJ_p^B is in fact the derivative of the functionf_p(x) := ¹_px^p, for 2≤p <∞. Hence, the Bregman distance with respect tof_p (see [24]) is given by

Δ_p(x, y) =1

qx^p− J_p^B(x), y+1

py^p. (2.2)

In addition, the Bregman distance possesses the following important properties:

Δ_p(x, y) = Δ_p(x, z) + Δ_p(z, y) +

z−y,J_p^B(x)− J_p^B(y)

, ∀x, y, z∈ B. (2.3) Δ_p(x, y) + Δ_p(y, x) =

x−y,J_p^B(x)− J_p^B(y)

, ∀x, y∈ B. (2.4) When considering thep-uniformly convex space, the Bregman distance and the metric distance have the following relation (see (7) of [21]):

τx−y^p≤Δ_p(x, y)≤

x−y,J_p^B(x)− J_p^B(y)

, (2.5)

whereτ >0 is some fixed number.

Let Ω be a nonempty, closed, and convex subset ofB. The metric pro- jection

P_Ωx:= arg min

y∈Ωx−y, x∈ B,

is the unique minimizer of the norm distance, which can be characterized by a variational inequality:

J_p^B(x−P_Ωx), z−P_Ωx ≤0, ∀z∈Ω. (2.6) Similar to the metric projection, the Bregman projection is defined as

Π_Ωx= arg min

y∈ΩΔ_p(x, y), x∈ B,

the unique minimizer of the Bregman distance (see Proposition 1.25 of [20]).

The Bregman projection can also be characterized by a variational inequality:

J_p^B(x)− J_p^B(Π_Ωx), z−Π_Ωx ≤0, ∀z∈Ω, (2.7) from which one has

Δ_p(Π_Ωx, z)≤Δ_p(x, z)−Δ_p(x,Π_Ωx), ∀z∈Ω. (2.8) Following [1] [see equation (6.3), page 29], we make use of the functionV_p : B^∗× B →[0,+∞), which is defined by

V_p(x, y) := 1

qx^q− x, y+1

py^p, ∀x∈ B^∗, y∈ B.

Then, V_p is nonnegative and V_p(x, y) = Δ_p(J_q^B∗(x), y) for all x ∈ B^∗ and y∈ B. Moreover, by the subdifferential inequality

f(x), y−x ≤f(y)−f(x),

withf(x) =¹_qx^q, x∈ B^∗, then f(x) =J_q^B∗. Then, we have J_q^B∗(x), y ≤1

qx+y^q−1

qx^q (2.9)

and from (2.9), we obtain (see [24])

V_p(x^∗+y^∗, x)≥V_p(x^∗, x) +y^∗,J_q^B∗(x^∗)−x, (2.10) for all x∈ B and x^∗, y^∗ ∈ B^∗. In addition, since f = f_p is a proper lower semi-continuous and convex function, we have f^∗ = f_p^∗ is a proper weak^∗ lower semi-continuous and convex function; see [18]. Hence, V_p is convex in the second variable. Thus, for allz∈ B, we can show that (see [24])

Δ_p

J_q^B∗N

i=1

t_iJ_p^B(x_i) , z

≤V_p ^N

i=1

t_iJ_p^B(x_i), z

= N i=1

t_iΔ_p(x_i, z) (2.11) where (x_i)_i=1,...,N ⊂ Band (t_i)_i=1,...,N ⊂(0,1) with _N

i=1t_i = 1.For more details, see [22,23,25,26,32].

We need the following useful lemma in the sequel.

Lemma 2.2 (Lemma 2.5 of [35]). Let (a_n)_n∈N be a sequence of nonnegative real numbers satisfying the following relation:

a_n+1≤(1−α_n)a_n+α_nσ_n+γ_n, n≥0,

where(α_n)_n∈N, (γ_n)_n∈Nand(σ_n)_n∈Nare sequences of real numbers satisfying (i) (α_n)_n∈N⊂[0,1],

α_n =∞; (ii) lim sup_n→∞σ_n ≤0;

(iii) γ_n ≥0,

γ_n<∞.

Then,a_n→0 asn→ ∞.

3. Main results

Using the ideas of [41], we introduce an iterative algorithm in which the step- size does not depend on the normAand then prove the strong convergence of the sequence generated by the algorithm inp-uniformly convex real Banach spaces which are also uniformly smooth.

AlgorithmLetB1andB2be twop-uniformly convex real Banach spaces which are also uniformly smooth. LetT₁:B1 ⇒B1 and T₂:B2 ⇒B2 be maximal monotone operators, such thatT₁⁻¹(0)=∅andT₂⁻¹(0)=∅, respectively. Let J_λT1 andQ_λT2, forλ >0, be the metric and relative resolvents ofT₁ andT₂, respectively. LetA:B₁→ B₂ be a bounded linear operator, such thatA= 0 andA^∗:B^∗₂ → B^∗₁ be the adjoint of A. For a fixedu∈ B₁, choose an initial guessx₁∈ B₁ arbitrarily. Let (α_n)_n∈N ⊂[0,1]. Assume that the nth iterate x_n ∈ B₁has been constructed; then, we calculate the (n+ 1)th iteratex_n+1 via the formula:

u_n=Jq^B∗1

J_p^B1(x_n)−t_nA^∗J_p^B2(I−Q_λT2)Ax_n , x_n+1=Jq^B∗1 α_nJ_p^B1(u) + (1−α_n)J_p^B1(J_λT1(u_n))

. (3.1)

Let the stepsizet_n be chosen in such a way that t^q−1_n ∈

0,

q(I−Q_λT2)Ax_n^p C_qA^∗Jp^B2(I−Q_λT2)Ax_n^q

, n∈Ω, (3.2) where the index set Ω :={n∈N: (I−Q_λT2)Ax_n = 0} otherwiset_n =t (t being any nonnegative value).

Before the formal analysis of the convergence properties of the algorithm (3.1)–(3.2), we note that the step-sizet_n does need compute the norm of any operator, and also, it is possible to use a backtracking procedure to find it without multiple evaluations of the vector norm.

In the following, we present two lemmas establishing that the proposed algorithm is well-defined and the generated sequences are bounded.

Lemma 3.1. Suppose that the split inclusion problem (SIP_B) has a nonempty solution setΓ. Then,t_n defined by (3.2)is well defined.

Proof. First, we observe that in algorithm (3.1)–(3.2) the choice of the step- sizet_n is independent of the normA. In addition, the value of tdoes not influence the considered algorithm, but it was introduced just for the sake of clarity. Furthermore, we show thatt_n is well defined. Now, letx∈Γ. Then, Ax=Q_λT2Ax. Hence

(I−Q_λT2)Ax_n^p

=J_p^B2(I−Q_λT2)Ax_n,(I−Q_λT2)Ax_n

=J_p^B2(I−Q_λT2)Ax_n, Ax_n−Ax+Q_λT2Ax−Q_λT2Ax_n

=J_p^B2(I−Q_λT2)Ax_n, Ax_n−Ax

+J_p^B2(I−Q_λT2)Ax_n, Q_λT2Ax−Q_λT2Ax_n

=A^∗J_p^B2(I−Q_λT2)Ax_n, x_n−x

+J_p^B2(I−Q_λT2)Ax_n, Q_λT2Ax−Q_λT2Ax_n

≤ A^∗J_p^B2(I−Q_λT2)Ax_nxn−x

+J_p^B2(I−Q_λT2)Ax_nQ_λT2Ax−Q_λT2Ax_n

=A^∗J_p^B2(I−Q_λT2)Ax_nx_n−x

+(I−Q_λT2)Ax_n^p−1Q_λT2Ax−Q_λT2Ax_n.

Consequently, forn∈Ω, that is,(I−Q_λT2)Ax_n>0, we getA^∗J_p^B2(I− Q_λT2)Ax_nx_n−x>0 and(I−Q_λT2)Ax_n^p−1Q_λT2Ax−Q_λT2Ax_n>0.

SinceA^∗J_p^B2(I−Q_λT2)Ax_nx_n−x>0, then we obtain thatA^∗J_p^B2(I− Q_λT2)Ax_n = 0. This implies thatt_n is well defined.

Lemma 3.2. Let B₁ and B₂ be two p-uniformly convex real Banach spaces which are also uniformly smooth. Let T₁: B₁ ⇒ B₁ and T₂: B₂ ⇒ B₂ be

maximal monotone operators, such thatT₁⁻¹(0)=∅andT₂⁻¹(0)=∅, respec- tively. LetQ_λT2 be the metric resolvent of T₂ forλ >0. Let A:B₁→ B₂ be a bounded linear operator, such thatA= 0andA^∗:B^∗₂ → B₁^∗ be the adjoint ofA. Suppose that Γ=∅ and(α_n)_n∈N⊂(0,1). Let the sequence (x_n)_n∈N be generated by (3.1)–(3.2). Assume that for small enough >0,

t_n ∈

,

q(I−Q_λT2)Ax_n^p

C_qA^∗Jp^B2(I−Q_λT2)Ax_n^q − _q−1¹

, n∈Ω.

Then, the sequences(x_n)_n∈N and(u_n)_n∈N are bounded.

Proof. It follows from (2.6) that for anyz∈Γ, where Γ is the solution set of problem (SIP_B), we have

J_p^B2((I−Q_λT2)Ax_n), Ax_n−Az

=Ax_n−Q_λT2(Ax_n)^p+J_p^B2((I−Q_λT2)Ax_n), Q_λT2(Ax_n)−Az

≥ Ax_n−Q_λT2(Ax_n)^p. (3.3)

In addition, from (3.3) and Lemma2.1, we obtain that

Δ_p(u_n, z)≤Δ_p(Jq^B∗1[J_p^B1(x_n)−t_nA^∗J_p^B2(I−Q_λT2)Ax_n], z)

=1

qJ_p^B1(x_n)−t_nA^∗J_p^B2((I−Q_λT2)Ax_n)^q− J_p^B1(x_n), z +t_nJ_p^B2((I−Q_λT2)Ax_n), Az+1

pz^p

≤1

qJ_p^B1(x_n)^q−t_nAx_n,J_p^B2((I−Q_λT2)Ax_n) +C_qt^q_n

q A^∗J_p^B2(I−Q_λT2)Ax_n^q

− J_p^B1(x_n), z+t_nAz,J_p^B2((I−Q_λT2)Ax_n)+1 pz^p

=1

qxn^p− J_p^B1(x_n), z+1 pz^p +t_nJ_p^B2((I−Q_λT2)Ax_n), Az−Ax_n +C_qt^q_n

q A^∗J_p^B2(I−Q_λT2)Ax_n^q

= Δ_p(x_n, z) +t_nJ_p^B2(I−Q_λT2)Ax_n, Az−Ax_n +C_qt^q_n

q A^∗J_p^B2(I−Q_λT2)Ax_n^q. (3.4) Now, using (3.3) in (3.4), we obtain

Δ_p(un, z)≤Δ(xn, z)−tn(I−QλT2)Axn^p+ Cqt^q_n

q A^∗J_p^B2(I−QλT2)Axn^q

= Δ(xn, z)−tn (I−QλT2)Axn^p−Cqt^q−1_n

q A^∗J_p^B2(I−QλT2)Axn^q . (3.5)

By the condition ont_n, it follows from (3.5) that Δ_p(u_n, z)≤Δ_p(x_n, z).

Furthermore, using (2.8) in (3.1), we get that Δ_p(x_n+1, z)≤Δ_p

Jq^B∗1 α_nJ_p^B1(u) + (1−α_n)J_p^B1(u_n)

, z

=α_nΔ_p(u, z) + (1−α_n)Δ_p(u_n, z)

≤α_nΔ_p(u, z) + (1−α_n)Δ_p(x_n, z)

≤max{Δ_p(u, z),Δ_p(x_n, z)} ...

≤max{Δ_p(u, z),Δ_p(x₁, z)}. (3.6) Therefore, (Δ_p(x_n, z))_n∈N is bounded and consequently, the sequence (Δ_p(u_n, z))_n∈N is bounded. Thus, the sequences (x_n)_n∈N and (u_n)_n∈N are

bounded.

Now, we proceed by proving strong convergence of the sequence gener- ated by Algorithm (3.1)–(3.2).

Theorem 3.3. Let B₁ and B₂ be two p-uniformly convex real Banach spaces which are also uniformly smooth. Let T₁: B₁ ⇒ B₁ and T₂: B₂ ⇒ B₂ be maximal monotone operators, such thatT₁⁻¹(0)=∅andT₂⁻¹(0)=∅, respec- tively. LetQ_λT2 be the metric resolvent of T₂ forλ >0. Let A:B₁→ B₂ be a bounded linear operator such that A= 0 andA^∗:B₂^∗ → B₁^∗ be the adjoint ofA. Suppose that Γ=∅ and(α_n)_n∈N⊂(0,1). Let the sequence (x_n)_n∈N be generated by (3.1)–(3.2). Assume that for small enough >0:

t_n ∈

,

q(I−Q_λT2)Ax_n^p

C_qA^∗Jp^B2(I−Q_λT2)Ax_n^q − _q−1¹

, n∈Ω.

Suppose the following conditions are satisfied:

(i) lim_n→∞α_n = 0;

(ii) _∞

n=1α_n=∞.

Then,(u_n)_n∈N and(x_n)_n∈N both converge strongly tox, where¯ x¯= Π_Γu.

Proof. Let ¯x= Π_Γuand let us setv_n :=J_q^B1∗ α_nJ_p^B1(u) + (1−α_n)J_p^B1(u_n)

, n≥1. Then, using (2.10) and (2.11), we obtain

Δ_p(xn+1,x¯)≤Δ_p(Jq^B1∗ αnJ_p^B1(u) + (1−αn)J_p^B1(un) ,x¯)

=Vp(αnJ_p^B1(u) + (1−αn)J_p^B1(un),x)¯

≤Vp(αnJp^B1(u) + (1−αn)Jp^B1(un)−αn(Jp^B1(u)− Jp^B1(¯x)),x¯)

− −αn(Jp^B1(u)− Jp^B1(¯x)),Jq^B∗1 αnJp^B1(u) + (1−αn)Jp^B1(un) −x¯

=Vp(αnJ_p^B1(¯x) + (1−αn)J_p^B1(un),¯x) +αnJ_p^B1(u)− J_p^B1(¯x), vn−¯x

= Δ_p(Jq^B∗1 αnJ_p^B1(¯x) + (1−αn)J_p^B1(un)

,x¯) +αnJ_p^B1(u)− J_p^B1(¯x), vn−x¯

≤αnΔ_p(¯x,x) + (1¯ −αn)Δ_p(un,x) +¯ αnJ_p^B1(u)− J_p^B1(¯x), vn−x¯