Inertial Halpern-type iterative algorithm for the generalized multiple-set split feasibility problem in Banach spaces

In this paper, we study the generalized multiple-set split feasibility problem including the common ﬁxed-point problem for a ﬁnite family of generalized demimetric mappings and the monotone inclusion problem in 2-uniformly convex and uniformly smooth Banach spaces. We propose an inertial Halpern-type iterative algorithm for obtaining a solution of the problem and derive a strong convergence theorem for the algorithm. Then, we apply our convergence results to the convex minimization problem, the variational inequality problem, the multiple-set split feasibility problem and the split common null-point problem in Banach spaces.


Introduction
Let E be a real Banach space with a dual space E * .We consider the monotone inclusion problem, which involves a single-valued monotone operator A : E → E * and a multivalued monotone operator B : E → 2 E * .The problem aims to find a solution x * ∈ E such that 0 ∈ (A + B)x * .The set of solutions to this problem is denoted by (A + B) -1 0. This problem has practical applications in various fields such as image recovery, signal processing, and machine learning.It also encompasses several mathematical problems as special cases, including variational inequalities, split feasibility problems, minimization problems, and Nash equilibrium problems in noncooperative games, see, for instance, [1][2][3][4] and the references therein.The forward-backward splitting algorithm, introduced in [1,2], is a wellknown method for approximating solutions to the monotone inclusion problem.In this method, defined in a Hilbert space H, we start with an initial iterate x 1 ∈ H and update subsequent iterates as follows: x n+1 = (I + λB) -1 (I -λA)x n , where n ≥ 1 and λ is a step-size parameter.Over the past few decades, the convergence properties and modified versions of the forward-backward splitting algorithm have been extensively studied in the literature.In [5] (see also [6]), Takahashi et al. introduced and analyzed a Halpern-type modification of the forward-backward splitting algorithm in real Hilbert spaces.They established the strong convergence of the sequence generated by their algorithm to a solution of the monotone inclusion problem.In 2019, Kimura and Nakajo [7] proposed a modified forward-backward splitting method and proved a strong convergence theorem for solutions of the monotone inclusion problem in a real 2-uniformly convex and uniformly smooth Banach space.Their convergence result is as follows: Theorem 1.1 Let C be a nonempty, closed, and convex subset of a real 2-uniformly convex and uniformly smooth Banach space E. Let A : C → E * be an α-inverse strongly monotone and B : E → 2 E * be a maximal monotone.Suppose that = (A + B) -1 0 = ∅.Let {x n } be a sequence generated by x 0 ∈ C, u ∈ E and by: where C is the generalized projection of E onto C, J B λ n is the resolvent of B, J E is the normalized duality mapping of E, {λ n } ⊂ (0, ∞) and {γ n } ⊂ (0, 1) such that lim n→∞ γ n = 0 and ∞ n=1 γ n = ∞.Then, if 0 < inf n∈N λ n ≤ sup n∈N λ n < 2αc, (where c is the constant in Lemma 2.2), the sequence {x n } converges strongly to u.
The inertial technique has gained significant interest among researchers due to its favorable convergence characteristics and the ability to improve algorithm performance.It was first considered in [8] for solving smooth, convex minimization problems.The key idea behind these methods is to utilize two previous iterates to update the next iterate, resulting in accelerated convergence.In [9], Lorenz and Pock proposed an inertial forward-backward splitting algorithm in real Hilbert spaces, formulated as follows: where θ n ∈ [0, 1) is an extrapolation factor, λ n is a step-size parameter, and H represents the Hilbert space.They proved that the sequence x n generated by this algorithm converges weakly to a zero of A + B.
In addition to the monotone inclusion problem, we consider the fixed-point problem for a nonlinear mapping U : E → E. A point x ∈ E such that Ux = x is called a fixed point of U. The set of fixed points of the nonlinear mapping U is denoted by Fix(U).
Let E and F be Banach spaces and T : E → F be a bounded linear operator.Let {C i } p i=1 be a family of nonempty, closed, and convex subsets in E and {Q j } r j=1 be a family of nonempty, closed, and convex subsets in F. The multiple-set split feasibility problem (MSSFP) is formulated as finding a point x with the property: The multiple-set split feasibility problem with p = r = 1 is known as the split feasibility problem (SFP).The MSSFP was first introduced by Censor et al. [10] in the framework of Hilbert spaces for modeling inverse problems that arise from phase retrievals and medical image reconstruction.Such models were successfully developed, for instance, in radiationtherapy treatment planning, sensor networks, resolution enhancement, and so on [11,12].Initiated by SFP, several split-type problems have been investigated and studied.For example, the split common fixed-point problem [13], the split monotone variational inclusions problem [14], and the split common null-point problem [15].Algorithms for solving these problems have received great attention, (see, e.g., [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] and some of the references therein).
In 2018, Kawasaki and Takahashi [32], introduced a new general class of mappings, called generalized demimetric mappings as follows: Definition 1.2 Let E be a smooth, strictly convex, and reflexive Banach space.Let ζ be a real number with ζ = 0.A mapping U : Such a class of mappings is fundamental because it includes many types of nonlinear mappings arising in applied mathematics and optimization, see [32][33][34] for details.
In this paper, we study the following generalized multiple-set split feasibility problem: let E be a 2-uniformly convex and uniformly smooth Banach space and let F be a smooth, strictly convex, and reflexive Banach space.Let for i = 1, 2, . . ., m, U i : F → F be a finite family of generalized demimetric mappings, A i : E → E * be an inverse strongly monotone and B i : E → 2 E * be maximal monotone mappings.Let for each i = 1, 2, . . ., m, T i : E → F be a bounded linear operator such that T i = 0. Consider the following problem: ( By combining the concepts and techniques from the inertial algorithm, forward-backward splitting algorithm, and the Halpern method, we introduce a new and efficient iterative method for solving the generalized multiple-set split feasibility problem.We establish strong convergence theorems for the proposed method under standard and mild conditions.The iterative scheme does not require prior knowledge of the operator norm.Finally, we apply our convergence results to the convex minimization problem, variational inequality problem, multiple-set split feasibility problem, and split common null-point problem in Banach spaces.The main results presented in this paper improve and generalize the previous results obtained by Kimura and Nakajo [7], Takahashi et al. [17] and others.

Preliminaries
Let E be a real Banach space with norm .and let E * be the dual space of E. We denote the value of y * ∈ E * at x ∈ E by x, y * .Let S(E) = {x ∈ E : x = 1} denote the unit sphere of E. A Banach space E is said to be strictly convex if x+y 2 < 1 for all x, y ∈ S(E) with x = y.The Banach space E is said to be smooth provided exists for each x, y ∈ S(E).The modulus of convexity of E is defined by The space E is called uniformly convex if δ E ( ) > 0 for any ∈ (0, 2], and 2-uniformly convex if there is a c > 0 so that δ E ( ) ≥ c 2 for any ∈ (0, 2].We know that a uniformly convex Banach space is strictly convex and reflexive.The modulus of smoothness of E is defined by The space E is called uniformly smooth if lim τ →0 ρ E (τ ) τ = 0, and 2-uniformly smooth if there exists a C > 0 so that ρ E (τ ) ≤ Cτ 2 for any τ > 0. It is observed that every 2-uniformly convex (2-uniformly smooth) space is a uniformly convex (uniformly smooth) space.It is known that E is 2-uniformly convex (2-uniformly smooth) if and only if its dual E * is 2uniformly smooth (2-uniformly convex).It is known that all Hilbert spaces and Lebesgue spaces L p for 1 < p ≤ 2 are uniformly smooth and 2-uniformly convex.See [35,36] for more details.
The normalized duality mapping of E into 2 E * is defined by for all x ∈ E. The normalized duality mapping J E has the following properties (see, e.g., [36]): • if E is a smooth, strictly convex, and reflexive Banach space, then J E is a single-valued bijection and in this case, the inverse mapping J -1 E coincides with the duality mapping J E * ; • if E is uniformly smooth, then J E is uniformly norm-to-norm continuous on each bounded subset of E.
The following lemma was proved by Xu [37].
Lemma 2.1 Let E be a 2-uniformly smooth Banach space.Then, there exists a constant γ > 0 such that for every x, y ∈ E there holds the following inequality: x + y 2 ≤ x 2 + 2 y, J E (x) + γ y 2 .

Lemma 2.2 ([37]
) Let E be a smooth Banach space.Then, E is 2-uniformly convex if and only if there exists a constant c > 0 such that for each x, y ∈ E, x + y 2 ≥ x 2 + 2 y, J E (x) + c y 2 holds.

Lemma 2.3 ([38]
) If E is a 2-uniformly convex Banach space, then there exists a constant d > 0 such that for all x, y ∈ E, Let E be a smooth Banach space.The function φ : Observe that, in Hilbert spaces, φ(x, y) reduces to xy 2 .It is obvious from the definition of function φ that We also know that if E is strictly convex and smooth, then, for x, y ∈ E, φ(x, y) = 0 if and only if x = y; see [39,40].We will use the following mapping V : Lemma 2.4 [39] Let E be a reflexive, strictly convex, and smooth Banach space with E * as its dual.Then, for all x ∈ E and x * , y * ∈ E * .
Lemma 2.5 ([41]) Let E be a uniformly convex and smooth real Banach space, and let {x n } and {y n } be two sequences of E. If either {x n } or {y n } is bounded and lim n→∞ φ(x n , y n ) = 0 then lim n→∞ x ny n = 0.
Let C be a nonempty, closed, and convex subset of a strictly convex, reflexive, and smooth Banach space E and let x ∈ E.Then, there exists a unique element x ∈ C such that We denote x by C x and call C the generalized projection of E onto C; see [39,41].We have the following well-known result [39,41] for the generalized projection.

Lemma 2.6
Let C be a nonempty, convex subset of a smooth Banach space E, x ∈ E and x ∈ C.Then, φ(x, x) = inf y∈C φ(y, x) if and only if xy, J E x -J E x ≥ 0 for every y ∈ C, or equivalently φ(y, x) + φ(x, x) ≤ φ(y, x) for all y ∈ C. Definition 2.7 Let E be a Banach space and T : E → E be a nonlinear mapping with Fix(T) = ∅.Then, I -T is said to be demiclosed at zero if {x n } is a sequence in E converges weakly to x and (I -T)x n converges strongly to zero, then (I -T)x = 0. Lemma 2.8 ([32]) Let E be a smooth, strictly convex, and reflexive Banach space and let ζ be a real number with ζ = 0. Let T : E → E be a ζ -generalized demimetric mapping.Then, the fixed point set Fix(T) of T is closed and convex.Definition 2.9 An operator A : E → E * is called an inverse, strongly monotone operator, if there exist α > 0 such that In this case, we say that A is an α-inverse strongly monotone.
Let B be a mapping of A monotone mapping B on E is said to be maximal if its graph is not properly contained in the graph of any other monotone mapping on E. The set of null points of B is denoted by B -1 0 = {x ∈ E : 0 ∈ Bx}.Let E be a reflexive, strictly convex, and smooth real Banach space and let B : E → 2 E * be a maximal monotone operator.Then, for any λ > 0 and u ∈ E, there exists a unique element u λ ∈ D(B) such that J E (u) ∈ J E (u λ ) + λB(u λ ).We define J B λ by J B λ u = u λ for every u ∈ E and λ > 0 and such J B λ is called the resolvent of B. Alternatively, J B λ = (J E + λB) -1 J E .Let E be a uniformly convex and smooth Banach space and B be a maximal monotone operator.Then, B -1 0 = Fix(J B λ ) for all λ > 0. See [36,42] for more details.
The lemma that follows is stated and proven in [ [7], Lemma 3.1].
Lemma 2.10 Let E be a 2-uniformly convex and uniformly smooth real Banach space.Let A : E → E * be an α-inverse strongly monotone and B : for all λ > 0 and x ∈ E.Then, the following hold: The following lemmas are very helpful for the convergence analysis of the algorithm.

Lemma 2.11 ([43]
) Let {γ n } be a sequence in (0,1) and {δ n } be a sequence in R satisfying (i) for each n ≥ 0, then lim n→∞ a n = 0. Lemma 2.12 ([44]) Let {s n } be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {s n i } of {s n } such that s n i ≤ s n i +1 for all i ≥ 0. For every n ∈ N, (sufficiently large) define an integer sequence {τ (n)} as

Main results
We first prove the following lemma.
Lemma 3.1 Let E be a Banach space and F be a smooth, strictly convex, and reflexive Banach space.Let J F be the duality mappings on F. Let A : E → F be a bounded linear operator such that A = 0 and let A * be the adjoint operator of A. Let ζ = 0 and U : which implies that Ax -UAx = 0. Now, in this position, we give our algorithm and its convergence analysis for the generalized multiple-set split feasibility problem in Banach spaces.
Theorem 3.2 Let E be a 2-uniformly convex and uniformly smooth Banach space and let F be a smooth, strictly convex, and reflexive Banach space.Let J E and J F be the duality mappings on E and F, respectively.Let for i = 1, 2, . . ., m, ζ i = 0 and U i : F → F be a finite family of ζ i -generalized demimetric mappings such that U i -I is demiclosed at 0. Let for each i = 1, 2, . . ., m, A i : E → E * be an η i -inverse strongly monotone and B i : E → 2 E * be a maximal monotone.Let for each i = 1, 2, . . ., m, T i : E → F be a bounded linear operator such that T i = 0 and let T i * : Suppose the stepsizes are chosen in such a way that for small enough > 0, otherwise r n,i = r i is any nonnegative real number (where γ is the constant in Lemma 2.1).Suppose that the following conditions are satisfied: Then, {x n } converges strongly to v, where is the generalized projection of E onto .
Proof From Lemma 3.1, we have that the stepsizes r n,i are well defined.We show that {x n } is bounded.Indeed, let x ∈ , since for each i = 1, 2, . . ., m, U i is a ζ i -generalized demimetric mapping and by Lemma 2.1 we have For n ∈ , from the condition of r n,i we have This implies that Utilizing Lemma 2.10 we have for each β > 0 and n ∈ N. Since 0 < inf n∈N λ n,i ≤ sup n∈N λ n,i < 2cη i , for each i ∈ {1, 2, . . ., m} there exists From inequalities ( 5), (6), and (7), we obtain From the definition of w n , we obtain From inequalities ( 8) and ( 9) we obtain Therefore, φ(x * , x n ) is bounded and by inequality (3), the sequence {x n } is also bounded.Consequently, {w n }, {y n,i }, {z n,i }, and {T i w n } are all bounded.We have θ n J E (x n ) -J E (x n-1 ) ≤ ε n for all n, which together with lim n→∞ Utilizing Lemma 2.3 we obtain that Since the sequence {x n } is bounded, there exists a constant M > 0 such that Hence, from (11) and (10) we obtain From inequalities (5), ( 6), (7), and (8) we obtain This implies that where Therefore, where From Lemma 2.10 and Lemma 2.8, we have that is closed and convex.Therefore, the generalized projection from E onto is well defined.Suppose x * = v.The next task is to prove that the sequence {x n } converges to the point x * .In order to prove this, we consider two possible cases.
Case 1. Suppose that there exists n 0 ∈ N such that {φ(x * , x n )} ∞ n=n 0 is nonincreasing.By the boundedness of {φ(x * , x n )}, we have {φ(x * , x n )} is convergent.Furthermore, we have φ(x * , x n )φ(x * , x n+1 ) → 0 as n → ∞.Therefore, from (13), for each i ∈ {1, 2, . . ., m} we obtain This implies that From Lemma 3.1 we obtain In a similar way we obtain that By our assumption we obtain Since J E is uniformly continuous on bounded subsets of E, we obtain Also, we have Furthermore, for i = 1, 2, . . ., m, we obtain that Therefore, we obtain This implies that By uniform continuity of J -1 E on bounded subset of E * , we conclude that Since {x n } is bounded and E is a reflexive Banach space, there exits a subsequence {x n k } of {x n } such that x n k z.Since y n,ix n → 0, we have y n k ,i z, (i = 1, 2, . . ., m).Thus, we obtain z ∈ m i=1 (A i + B i ) -1 0 (see, e.g., [7] for this proof).Since w nx n → 0, we have w n k z.From the continuity of T i , we have that T i w n k T i z.Now, from ( 16) and the demiclosedness of I -U i , we obtain T i z ∈ Fix(U i ).This implies that z ∈ .Next, we show that lim sup n→∞ x n+1x * , J E v -J E x * ≤ 0. We can choose a subsequence {x n k } of {x n } such that This, together with lim n→∞ x n+1x n = 0, implies that Applying inequality ( 14) and Lemma 2.11, we deduce that lim n→∞ φ(x * , x n ) = 0. Now, from Lemma 2.5 we obtain, lim n→∞ x nx * = 0. Case 2: Put n = φ(x * , x n ) for all n ∈ N. Suppose there exists a subsequence { n j } of { n } such that n j ≤ n j +1 for all j ∈ N. Define a mapping τ : N → N by τ (n) = max{k ≤ n : k < k+1 } for all n ≥ n 0 (for some n 0 large enough).Thus, by Lemma 2.12 we have τ (n) → ∞ and 0 ≤ τ (n) ≤ τ (n)+1 for all n ≥ n 0 .Following the proof line in Case 1, we can show that and Further, we can show that It follows from ( 14) that Since α τ (n) > 0 and lim sup n→∞ χ τ (n) ≤ 0, we deduce that Using this and inequality (24), we conclude that lim n→∞ φ(x * , x τ (n)+1 ) = 0. Now, from Lemma 2.12, we have that lim n→∞ φ(x * , x n ) = 0. Hence, lim n→∞ x *x n = 0.This completes the proof.
We know that every Hilbert space H, is a 2-uniformly convex and uniformly smooth Banach space and the normalized duality mapping is J H = I.Also, we have the following relation in Hilbert space H: Therefore, in Lemma 2.2 and Lemma 2.1 the constant c = γ = 1.Theorem 3.2 now yields the following result regarding an algorithm for solving the generalized multiple-set split feasibility problem in Hilbert spaces.
Theorem 3.3 Let E and F be Hilbert spaces.Let for i = 1, 2, . . ., m, ζ i = 0 and U i : F → F be a finite family of ζ i -generalized demimetric mappings such that U i -I is demiclosed at 0. Let for i = 1, 2, . . ., m, A i : E → E be an η i -inverse strongly monotone and B i : E → 2 E be a maximal monotone.Let for each i = 1, 2, . . ., m, T i : E → F be a bounded linear operator such that T i = 0 and let T i * : F → E be the adjoint operator of T i .Suppose that = {x * ∈ m i=1 (A i + B i ) -1 0 : T i x * ∈ Fix(U i ), i = 1, 2, . . ., m} = ∅.Let {x n } be a sequence generated by v, x 0 , x 1 ∈ E and by: Suppose the stepsizes are chosen in such a way that for small enough > 0, otherwise r n,i = r i is any nonnegative real number.Suppose that the following conditions are satisfied: Then, {x n } converges strongly to P v.

Applications
In this section we present some applications of our main result.

Common solutions to variational inequality problems
Let C be a nonempty, closed, and convex subset of a Banach space E and A : E → E * be an operator.The variational inequality problem (VIP) is formulated as follows: The set of solutions of this problem is denoted by VI(C, A).It is well known that the VIP is a fundamental problem in optimization theory and nonlinear analysis (see [45]).Let f : E → (-∞, +∞] be a proper, lower semicontinuous, and convex function.Then, it is known that the subdifferential ∂f of f defined by for x ∈ E is a maximal monotone operator; see [46].Let i C : E → (-∞, +∞] be the indicator function of C. We know that i C is proper lower semicontinuous and convex, and hence its subdifferential ∂i C is a maximal monotone operator.Let B = ∂i C .Then, it is easy to see that J B λ x = C x for every λ > 0 and x ∈ E. Further, we also obtain (A + B) -1 (0) = VI(C, A).Now, as an application of our main result, we obtain the following theorem for finding a common element of the set of common solutions of a system of a variational inequality problem and the set of common fixed points of a finite family of generalized demimetric mappings in 2-uniformly convex and uniformly smooth Banach spaces.
Theorem 4.1 Let E be a 2-uniformly convex and uniformly smooth Banach space.Let {C i } m i=1 be a finite family of nonempty, closed, and convex subsets of E. Let for i = 1, 2, . . ., m, Let {x n } be a sequence generated by v, x 0 , x 1 ∈ E and by: Suppose that the following conditions are satisfied: Proof Setting F = E, T i = I, and B i = ∂i C i , we obtain the desired result from Theorem 3.2.
Remark 4.2 Setting U i = I in Theorem 4.1, our result generalizes the result of [47] from the problem of finding common solutions to unrelated variational inequalities in a Hilbert space to a 2-uniformly smooth and uniformly convex Banach space.
Remark 4.3 We extend the main results of Kimura and Nakajo [7] from the problem of finding a solution of the variational inequality problem to the problem of finding a common element of the set of common solutions of a system of a variational inequality problem and a common fixed-point problem.

Convex minimization problem
For a convex differentiable function : E → R and a proper convex lower semicontinuous function : E → (-∞, +∞], the convex minimization problem is to find a point x ∈ E such that If ∇ and ∂ represent the gradient of and subdifferential of , respectively, then Fermat's rule ensures the equivalence of problem (26) to the problem of finding a point Many optimization problems from image processing, statistical regression, and machine learning (see, e.g., [48,49]) can be adapted into the form of (26).In this setting, we assume that is Gâteaux-differentiable with derivative ∇ that is an inverse strongly monotone.Now, as an application of our main result we obtain the following theorem.Let {x n } be a sequence generated by v, x 0 , x 1 ∈ E and by: Suppose that the following conditions are satisfied: (i) α n ∈ (0, 1), lim n→∞ α n = 0 and ∞ n=1 α n = ∞; (ii) 0 < inf n∈N λ n ≤ sup n∈N λ n < 2ηc, (c is the constant in Lemma 2.2); (iii) γ n ∈ (0, 1), α n + γ n = 1 and lim inf n→∞ γ n > 0; (iv) r n ∈ ( , 2l γ ζ -) for some > 0, (γ is the constant in Lemma 2.1); (v) ε n > 0 and lim n→∞ ε n α n = 0.Then, {x n } converges strongly to v.
Proof We know that the subdifferential mapping ∂ of a proper, convex, and lower semicontinuous function is a maximal monotone.Also, we have (see [50]): Now, setting F = E, m = 1, T i = I and A = ∇ , we obtain the desired result from Theorem 3.2.

The multiple-set split feasibility problem
Let C be a nonempty, closed, and convex subset of a strictly convex and reflexive Banach space E.Then, we know that for any x ∈ E, there exists a unique element z ∈ C such that xz ≤ xy for all y ∈ C. Putting z = P C x, we call P C the metric projection of E onto C. Let E be a uniformly convex and smooth Banach space and let C be a nonempty, closed, and convex subset of E.Then, P C is 1-generalized demimetric and I -P C is demiclosed at zero (see [33] for details).
As another application of our main result, we obtain the following strong convergence theorem for the multiple-set split feasibility problem.
Theorem 4.5 Let E be a 2-uniformly convex and uniformly smooth Banach space and let F be a uniformly convex and smooth Banach space.Let {C i } m i=1 and {Q i } m i=1 be two finite families of nonempty, closed, and convex subsets of E and F, respectively.Let for each i = 1, 2, . . ., m, T i : E → F be a bounded linear operator such that T i = 0. Suppose that = {x * ∈ m i=1 C i : T i x * ∈ Q i , i = 1, 2, . . ., m} = ∅.Let {x n } be a sequence generated by v, x 0 , x 1 ∈ E and by: where 0 ≤ θ n ≤ θ n and θ * ∈ (0, 1) such that Suppose the stepsizes are chosen in such a way that for small enough > 0, otherwise r n,i = r i is any nonnegative real number (where γ is the constant in Lemma 2.1).Suppose that the following conditions are satisfied: (i) α n ∈ (0, 1), lim n→∞ α n = 0 and ∞ n=1 α n = ∞; (ii) γ n,i ∈ (0, 1), α n + m i=1 γ n,i = 1 and lim inf n→∞ γ n,i > 0; (iii) ε n > 0 and lim n→∞ ε n α n = 0.Then, {x n } converges strongly to v.

Split common null-point problem
Let E be a uniformly convex and smooth Banach space and let G : E → 2 E * be a maximal monotone operator.For each x ∈ E and μ > 0, we define the metric resolvent of G for μ > 0 by It is observed that 0 ), (see [51]).It is known that Q G μ is 1-generalized demimetric.Also, we know that I -Q G μ is demiclosed at zero (see [33] for details).
We obtain the following strong convergence result for split common null-point problem.
where 0 ≤ θ n ≤ θ n and θ * ∈ (0, 1) such that Suppose the stepsizes are chosen in such a way that for small enough > 0, otherwise r n,i = r i is any nonnegative real number (where γ is the constant in Lemma 2.1).Suppose that the following conditions are satisfied: Then, {x n } converges strongly to v.
We obtain the following strong convergence result for the split common fixed-point problem.
Theorem 4.7 Let E and F be Hilbert spaces.Let for each i = 1, 2, . . ., m, U i : F → F be a ς i -strict pseudocontraction mapping and let S i : E → E be a κ i -strict pseudocontraction mapping.Let for each i = 1, 2, . . ., m, T i : E → F be a bounded linear operator such that T i = 0. Suppose that = {x * ∈ m i=1 Fix(S i ) : T i x * ∈ Fix(U i ), i = 1, 2, . . ., m} = ∅.Let {x n } be a sequence generated by v, x 0 , x 1 ∈ E and by: where 0 ≤ θ n ≤ θ n and θ * ∈ (0, 1) such that Suppose the stepsizes are chosen in such a way that for small enough > 0, otherwise r n,i = r i is any nonnegative real number.Suppose that the following conditions are satisfied: Then, {x n } converges strongly to P v.
Proof Put A i = I -S i and B i = 0, for each i = 1, 2, . . ., m.Then, A i is 1-κ i 2 -inverse strongly monotone and Fix(S i ) = A i -1 (0).Since for each i ∈ {1, 2, . . ., m}, S i is a ς i -strict pseudocontraction, we have S i is a 2 1-ς i -generalized demimetric mapping and I -S i is demiclosed at zero.Hence, by Theorem 3.2, we obtain the desired result.
The numerical results that we have obtained are presented in Table 1.

Conclusions
In this paper, a new iterative scheme with inertial effect is proposed for solving the generalized multiple-set split feasibility problem in a 2-uniformly convex and uniformly smooth Banach space E and a smooth, strictly convex, and reflexive Banach space F. The strong convergence of the iterative sequences generated by the presented algorithm is established without requiring the prior knowledge of operator norm.Finally, we applied our result to study and approximate the solutions of certain classes of optimization problems.The results obtained in this paper improved and extended many others known in the field.As part of our future research, we would like to extend the results in this paper to a more general space, such as the p-uniformly convex Banach space.Furthermore, we would consider the generalized multiple-set split feasibility problem involving a sum of maximal monotone and Lipschitz continuous monotone mappings.It would be interesting if the algorithm proposed in this paper could be applied to some practical optimization problems.

Theorem 4 . 4
Let E be a 2-uniformly convex and uniformly smooth Banach space.Let ζ = 0 and U : E → E be a ζ -generalized demimetric mapping such that U -I is demiclosed at 0. Let : E → R be a convex differentiable function such that its gradient ∇ is an ηinverse strongly monotone and let : E → (-∞, +∞] be a proper function with convexity and lower semicontinuity.Suppose that = {x ∈ Fix(U) : x = argmin z∈E (z) + (z)} = ∅.
6heorem 4.6Let E be a 2-uniformly convex and uniformly smooth Banach space and let F be a uniformly convex and smooth Banach space.Let for i = 1, 2, . . ., m, B , be metric resolvent operators of G i for μ i > 0. Let for each i = 1, 2, . . ., m, T i : E → F be a bounded linear operator such that T i : E → 2 E * and G i : F → 2 F * be maximal monotone operators.Let for i = 1, 2, . . ., m, J

Table 1
Table of numerical results for Theorem 4.5