CUBO, A Mathematical Journal Vol.22, N◦02, (155–175). August 2020 http://dx.doi.org/10.4067/S0719-06462020000200155 Received: 09 December, 2019 | Accepted: 28 May, 2020 A new iterative method based on the modified proximal-point algorithm for finding a common null point of an infinite family of accretive operators in Banach spaces T.M.M. Sow Amadou Mahtar Mbow University, Dakar Senegal sowthierno89@gmail.com ABSTRACT In this paper, we introduce and study a new iterative method for finding a common null point of an infinite family of accretive operators with a strongly accretive and Lip- schitzian operator, by using the proximal-point algorithm. And also we prove that the common null point is a unique solution of variational inequality without imposing any compactness-type condition on either the operators or the space considered. Finally, some applications of the main results to equilibrium problems and fixed point prob- lems with an infinite family of pseudocontractive mappings are given. The main result is a generalization and improvement of numerous well-known results in the available literature. RESUMEN En este art́ıculo, introducimos y estudiamos un nuevo método iterativo para encon- trar un cero común de una familia infinita de operadores acretivos con un operador Lischitziano fuertemente acretivo, usando el algoritmo punto-proximal. También de- mostramos que el cero común es la única solución de una desigualdad variacional sin imponer ninguna condición de tipo compacidad en ninguno de los operadores o los espacios considerados. Finalmente, se entregan algunas aplicaciones de los resultados principales a problemas de equilibrio y problemas de punto fijo con una familia in- finita de aplicaciones pseudo-contractivas. El resultado principal es una generalización y mejora de numerosos resultados bien conocidos en la literatura disponible. Keywords and Phrases: Proximal-point algorithm; Accretive operators; Variational inequality; Common zeros. 2020 AMS Mathematics Subject Classification: 46T05; 47H06; 47H09; 47H10. c©2020 by the author. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://dx.doi.org/10.4067/S0719-06462020000200155 156 T.M.M. Sow CUBO 22, 2 (2020) 1 Introduction Let H be a real Hilbert space and K be a nonempty subset of H. For a set-valued map A : H → 2H, the domain of A, D(A), the image of a subset S of H, A(S) the range of A, R(A) and the graph of A, G(A) are defined as follows: D(A) := {x ∈ H : Ax 6= ∅}, A(S) := ∪{Ax : x ∈ S}, R(A) := A(H), G(A) := {(x, u) : x ∈ D(A), u ∈ Ax}. A multi-valued map A : D(A) ⊂ H → 2H is called monotone if the inequality 〈u − v, x − y〉 ≥ 0 holds for each x, y ∈ D(A), u ∈ Ax, v ∈ Ay. A single-valued operator A : K → H is said to be strongly positive bounded linear if there exists a constant k > 0 such that 〈Ax, x〉 ≥ k‖x‖2, ∀ x, y ∈ K. Remark 1. It is immediate that if A is k-strongly positive bounded linear, then A is k-strongly monotone and ‖A‖-Lipschitz continuous. A monotone operator A is called maximal monotone if its graph G(A) is not properly contained in the graph of any other monotone operator. It is well known that A is maximal monotone if and only if A is monotone and R(I + rA) = H for all r > 0 and A is said to satisfy the range condition if D(A) ⊂ R(I + rA). Many problems arising in different areas of mathematics, such as optimization, variational analysis and differential equations, can be modeled by the equation 0 ∈ Ax, (1.1) where A is a monotone mapping. The solution set of this equation coincide to a null points set of A. Such operators have been studied extensively (see, e.g., Bruck Jr [5], Chidume [9], Rockafellar [29], Xu [30] and the references therein). Consider, for example, the following: let f : H → R ∪ {∞} be a proper lower semi continuous and convex function. The subdifferential, ∂f : H → 2H of f at x ∈ H is defined by ∂f(x) = { x∗ ∈ H : f(y) − f(x) ≥ 〈y − x, x∗〉 ∀ y ∈ H } . It is easy to check that ∂f : H → 2H is a monotone operator on H, and that 0 ∈ ∂f(x) if and only if x is a minimizer of f. Setting ∂f ≡ A, it follows that solving the inclusion 0 ∈ Au, in this case, is solving for a minimizer of f. CUBO 22, 2 (2020) A new iterative method based on the modified proximal-point . . . 157 In order to find a solution of problem (1.1), Rockafellar [29] introduced a powerful and suc- cessful algorithm which is recognized as Rockafellar proximal- point algorithm: for any initial point x0 ∈ H, a sequence {xn} is generated by: xn+1 = Jrn(xn + en), ∀ n ≥ 0, where Jr = (I + rA) −1 for all r > 0, is the resolvent of A and {en} is an error sequence in a Hilbert space. In the recent years, the problem of finding a common element of the set of solutions of convex minimization, variational inequality and the set of fixed point problems in real Hilbert spaces, Banach spaces and complete CAT(0) (Hadamard) spaces have been intensively studied by many authors; see, for example, [20, 21, 19, 29, 30] and the references therein. Very recently, Eslamian and Vahidi [10] introduced a new iterative method base on proximal point algorithm with strongly positive bounded linear operator for solving a system of inclusion problem. They established a strong convergence theorem which extends the corresponding results in [30, 2, 32, 28, 13, 14, 15, 16, 16, 17, 18]. Theorem 2 (Eslamian and Vahidi [10]). Let H be a real Hilbert space and K be a nonempty, closed and convex subset of H. Let {Bi}, i ∈ N ∗ := {1, 2, 3, ...} be an infinite family of operators of H such that ∞ ⋂ i=1 Bi −1(0) 6= ∅ and ∞ ⋂ i=1 D(Bi) ⊂ K ⊂ ∞ ⋂ i=1 R(I + rBi), for all r > 0. Let A : H → H be a k-strongly bounded linear operator with a coefficient γ̄ and f be a b− contraction mapping of K into itself with a constant b ≥ 0. Let {xn} be a sequence defined iteratively from arbitrary x0 ∈ K by:      yn = βn,0xn + ∞ ∑ i=1 βn,iJ Bi rn xn xn+1 = αnγf(xn) + (I − αnA)yn. (1.2) Let {rn} ⊂]0, ∞[, {βn,i} and {αn} be real sequences in (0, 1) satisfying: (i) lim n→∞ αn = 0; (ii) ∞ ∑ n=0 αn = ∞, ∞ ∑ i=0 βn,i = 1, (iii) lim n→∞ inf rn > 0, and lim n→∞ inf βn,0βn,i > 0, for all i ∈ N. Assume that 0 < γ < γ̄ b . Then, the sequence {xn} generated by (1.2) converges strongly to x ∗ ∈ ∞ ⋂ i=1 Bi −1(0). Above discussion yields the following questions. Question 1:Can results of Eslamian and Vahidi [10], and so on be extended from Hilbert spaces to Banach spaces? 158 T.M.M. Sow CUBO 22, 2 (2020) Question 2: We know that Lipschitzian mapping is more general than contraction. What hap- pens if the contraction is replaced by Lipschitzian mapping ? Question 3: We know that k-strongly accretive operators and L-Lipchizian operators is more general than the strong positive bounded linear operators. What happens if the strongly positive bounded linear operators is replaced by k- strongly accretive operators and L-Lipchizian operators ? The purpose of this paper is to give affirmative answers to these questions mentioned above. Applications are also included to valide our new findings. 2 Preliminairies Let E be a real Banach space and C be a nonempty, closed and convex subset of E. We denote by J the normalized duality map from E to 2E ∗ (E∗ is the dual space of E) defined by: J(x) := {x∗ ∈ E∗ : 〈x, x∗〉 = ||x||2 = ||x∗||2}, ∀ x ∈ E. Let S := {x ∈ E : ‖x‖ = 1}. E is said to be smooth if lim t→0+ ‖x + ty‖ − ‖x‖ t exists for each x, y ∈ S. E is said to be uniformly smooth if it is smooth and the limit is attained uniformly for each x, y ∈ S. Let E be a normed space with dimE ≥ 2. The modulus of smoothness of E is the function ρE : [0, ∞) → [0, ∞) defined by ρE(τ) := sup { ‖x + y‖ + ‖x − y‖ 2 − 1 : ‖x‖ = 1, ‖y‖ = τ } ; τ > 0. It is known that a normed linear space E is uniformly smooth if lim τ→0 ρE(τ) τ = 0. If there exists a constant c > 0 and a real number q > 1 such that ρE(τ) ≤ cτ q, then E is said to be q-uniformly smooth. Typical examples of such spaces are the Lp, ℓp and W m p spaces for 1 < p < ∞ where, Lp (or lp) or W m p is { 2 − uniformly smooth and p − uniformly convex if 2 ≤ p < ∞; 2 − uniformly convex and p − uniformly smooth if 1 < p < 2. (2.1) It is known that a normed linear space E is uniformly smooth if lim τ→0 ρE(τ) τ = 0. CUBO 22, 2 (2020) A new iterative method based on the modified proximal-point . . . 159 If there exists a constant c > 0 and a real number q > 1 such that ρE(τ) ≤ cτ q, then E is said to be q-uniformly smooth. Typical examples of such spaces are the Lp, ℓp and W m p spaces for 1 < p < ∞ where, Lp (or lp) or W m p is { 2 − uniformly smooth and p − uniformly convex if 2 ≤ p < ∞; 2 − uniformly convex and p − uniformly smooth if 1 < p < 2. Let Jq denote the generalized duality mapping from E to 2 E ∗ defined by Jq(x) := { f ∈ E∗ : 〈x, f〉 = ‖x‖q and ‖f‖ = ‖x‖q−1 } where 〈., .〉 denotes the generalized duality pairing. Notice that for x 6= 0, Jq(x) = ‖x‖ q−2J2(x), q > 1. Following Browder [3], we say that a Banach space has a weakly continuous normalized duality map if J is a single-valued and is weak-to-weak∗ sequentially continous, i.e., if {xn} ⊂ E, xn ⇀ x, then J(xn) ⇀ J(x) in E ∗. Weak continuity of duality map J plays an important role in the fixed point theory for nonlinear operators. Finally recall that a Banach space E satisfies Opial property (see, e.g., [24]) if lim sup n→+∞ ‖xn − x‖ < lim sup n→+∞ ‖xn − y‖ whenever xn ⇀ x, x 6= y. A Banach space E that has a weakly continuous normalized duality map satisfies Opial’s property. Remark 3. Note also that a duality mapping exists in each Banach space. We recall from [1] some of the examples of this mapping in lp, Lp, W m,p-spaces, 1 < p < ∞. (i) lp : Jx = ‖x‖ 2−p lp y ∈ lq, x = (x1, x2, · · · , xn, · · · ), y = (x1|x1| p−2, x2|x2| p−2, · · · , xn|xn| p−2, · · · ), (ii) Lp : Ju = ‖u‖ 2−p Lp |u|p−2u ∈ Lq, (iii) W m,p : Ju = ‖u‖ 2−p W m,p ∑ |α≤m|(−1) |α|Dα ( |Dαu|p−2Dαu ) ∈ W −m,q, where 1 < q < ∞ is such that 1/p + 1/q = 1. Finally recall that a Banach space E satisfies Opial’s property (see, e.g., [24]) if lim sup n→+∞ ‖xn − x‖ < lim sup n→+∞ ‖xn − y‖ whenever xn w −→ x, x 6= y. Recall that an operator A : K → E is said to be accretive if there exists j ∈ Jq(x − y) such that 〈Ax − Ay, j〉 ≥ 0, ∀x, y ∈ K. 160 T.M.M. Sow CUBO 22, 2 (2020) It is said to be strongly accretive if there exists a positive constant k ∈ (0, 1) and such that for all x, y ∈ K, such that 〈Ax − Ay, j〉 ≥ k‖x − y‖q, ∀x, y ∈ K. In a Hilbert space, the normalized duality map is the identity map. Hence, in Hilbert spaces, monotonicity and accretivity coincide. A multi-valued map A defined on a real Banach space E is called m-accretive if it is accretive and R(I + rA) = E for some r > 0 and it is said to satisfy the range condition R(I + rA) = E for all r > 0. The operator A in the following example satisfies range condition. Example 4. Let A : R → 2R defined by Ax = { sgn(x), x 6= 0, [−1, 1] , x = 0, (2.2) where A is the subdifferential of the absolute value function, ∂|.|, then A is m-accretive. It can be shown that if R(I + rA) = E for some r > 0, then this holds for all r > 0. Hence, m-accretive condition implies range condition. The demiclosedness of a nonlinear operator T usually plays an important role in dealing with the convergence of fixed point iterative algorithms. Definition 1. Let E be a real Banach space and T : D(T ) ⊂ E → E be a mapping. I − T is said to be demiclosed at 0 if for any sequence {xn} ⊂ D(T ) such that {xn} converges weakly to p and ‖xn − T xn‖ converges to zero, then p ∈ F(T ), where F(T ) denote the set of fixed points of the mapping T. Lemma 5 (Demiclosedness principle, [3]). Let E be a real Banach space satisfying Opial’s property, K be a closed convex subset of E, and T : K → K be a nonexpansive mapping such that F(T ) 6= ∅. Then I − T is demiclosed; that is, {xn} ⊂ K, xn ⇀ x ∈ K and (I − T )xn → y implies that (I − T )x = y. Lemma 6 ([22]). Let E be a smooth real Banach space. Then, we have ‖x + y‖2 ≤ ‖x‖2 + 2〈y, J(x + y)〉 ∀x, y ∈ E. Lemma 7 ([31]). Assume that {an} is a sequence of nonnegative real numbers such that an+1 ≤ (1 − αn)an + σn for all n ≥ 0, where {αn} is a sequence in (0, 1) and {σn} is a sequence in R such that (a) ∞ ∑ n=0 αn = ∞, (b) lim sup n→∞ σn αn ≤ 0 or ∞ ∑ n=0 |σn| < ∞. Then lim n→∞ an = 0. CUBO 22, 2 (2020) A new iterative method based on the modified proximal-point . . . 161 Theorem 8. [9] Let q > 1 be a fixed real number and E be a smooth Banach space. Then the following statements are equivalent: (i) E is q-uniformly smooth. (ii) There is a constant dq > 0 such that for all x, y ∈ E ‖x + y‖q ≤ ‖x‖q + q〈y , Jq(x)〉 + dq‖y‖ q. (iii) There is a constant c1 > 0 such that 〈x − y , Jq(x) − Jq(y)〉 ≤ c1‖x − y‖ q ∀ x, y ∈ E. Lemma 9 ( [8]). Let E be a uniformly convex real Banach space. For arbitrary r > 0, let B(0)r := {x ∈ E : ||x|| ≤ r}, a closed ball with center 0 and radius r > 0. For any given sequence {u1, u2, ....., un, .....} ⊂ B(0)r and any positive real numbers {λ1, λ2, ...., λn, ....} with ∞ ∑ k=1 λk = 1, there exists a continuous, strictly increasing and convex function g : [0, 2r] → R+, g(0) = 0, such that for any integer i, j with i < j, ‖ ∞ ∑ k=1 λkuk‖ 2 ≤ ∞ ∑ k=1 λk‖uk‖ 2 − λiλjg(‖ui − uj‖). Lemma 10. [33] Let H be a real Hilbert space and K a nonempty, closed convex subset of H. Let A : K → H be a k-strongly monotone and L-Lipschitzian operator with k > 0, L > 0. Assume that 0 < η < 2k L2 and τ = η ( k − L2η 2 ) . Then for each t ∈ ( 0, min{1, 1 τ } ) , we have ‖(I − tηA)x − (I − tηA)y‖ ≤ (1 − tτ)‖x − y‖ ∀x, y ∈ K. Let C be a nonempty subsets of a real Banach space E. A mapping QC : E → C is said to be sunny if QC(QCx + t(x − QCx)) = QCx for each x ∈ E and t ≥ 0. A mapping QC : E → C is said to be a retraction if QCx = x for each x ∈ C. Lemma 11. [26] Let C and D be nonempty subsets of a smooth real Banach space E with D ⊂ C and QD : C → D a retraction from C into D. Then QD is sunny and nonexpansive if and only if 〈z − QDz, J(y − QDz)〉 ≤ 0 (2.3) for all z ∈ C and y ∈ D. 162 T.M.M. Sow CUBO 22, 2 (2020) Remark 12. If K is a nonempty closed convex subset of a Hilbert space H, then the nearest point projection PK from H to K is the sunny nonexpansive retraction. The resolvent operator has the following properties: Lemma 13. [12] For any r > 0. (i) A is accretive if and only if the resolvent JAr of A is single-valued and nonexpansive; (ii) A is m-accretive if and only if JAr of A is single-valued and nonexpansive and its domain is the entire E; (iii) 0 ∈ A(x∗) if and only if x∗ ∈ F(JAr ), where F(J A r ) denotes the fixed-point set of J A r . Lemma 14. ( [23]) For any r > 0 and µ > 0, the following holds: µ r x + (1 − µ r )JAr x ∈ D(J A r ) and JAr x = J A µ ( µ r x + (1 − µ r )JAr x). Lemma 15. [7] Let A be a continuous accretive operator defined on a real Banach space E with D(A) = E. Then A is m-accretive. 3 Main results For our main theorem, we shall need the following lemma. Lemma 16. Let q > 1 be a fixed real number and E be a q-uniformly smooth real Banach space with constant dq. Let A : E → E be a k-strongly accretive and L-Lipschitzian operator with k > 0, L > 0. Assume that η ∈ ( 0, min { 1, ( kq dqLq ) 1 q−1 }) and τ = η ( k − dqL qηq−1 q ) . Then for each t ∈ ( 0, min{1, 1 τ } ) , we have ‖(I − tηA)x − (I − tηA)y‖ ≤ (1 − tτ)‖x − y‖, ∀ x, y ∈ E. (3.1) Proof. Without loss of generality, assume k < 1 q . Then, as η < ( kq dqLq ) 1 q−1 , we have 0 < qk − dqL qηq−1. Furthermore, from k < 1 q , we have qk − dqL qηq−1 < 1 so that 0 < qk − dqL qηq−1 < 1. By using (ii) of Theorem 8 and properties of A, it follows that ‖(I − tηA)x − (I − tηA)y‖q ≤ ‖x − y‖q + q〈tηAy − tηAx , Jq(x − y)〉 + dq‖tηAx − tηAy‖ q ≤ ‖x − y‖q − qtη〈Ax − Ay , Jq(x − y)〉 + dq(tη) q‖Ax − Ay‖q ≤ ‖x − y‖q − qtkη‖x − y‖q + dq(Ltη) q‖x − y‖q ≤ ( 1 − qtkη + dqL qtqηq ) ‖x − y‖q. CUBO 22, 2 (2020) A new iterative method based on the modified proximal-point . . . 163 Therefore ‖(I − tηA)x − (I − tηA)y‖ ≤ ( 1 − qtkη + dqL qtηq ) 1 q ‖x − y‖. (3.2) Using definition of τ, inequality (3.2) and inequality (1 + x)s ≤ 1 + sx, for x > −1 and 0 < s < 1, we have ‖(I − tηA)x − (I − tηA)y‖ ≤ ( 1 − tkη + dqL qtηq q ) ‖x − y‖ ≤ ( 1 − tη(k − dqL qηq−1 q ) ) ‖x − y‖ ≤ (1 − tτ)‖x − y‖, which gives us the required result (3.1). This completes the proof. Remark 17. Lemma 16 is one generalization of Lemma 10 for a Banach space. We are now in a position to state and prove our main result. Theorem 18. Let q > 1 be a fixed real number and E be a q-uniformly smooth and uniformly convex real Banach space having a weakly continuous duality map. Let K be a nonempty, closed and convex subset of E which is a nonexpansive retract of E with QK as the nonexpansive retraction. Let {Bi}, i ∈ N ∗ be an infinite family of accretive operators of E such that F := ∞ ⋂ i=1 Bi −1(0) 6= ∅ and ∞ ⋂ i=1 D(Bi) ⊂ K ⊂ ∞ ⋂ i=1 R(I + rBi), for all r > 0. Let A : K → E be a k-strongly accretive and L-Lipschitzian operator and f : K → E be a b-Lipschitzian mapping with a constant b ≥ 0. Let {xn} and {yn} be sequences defined iteratively from arbitrary x0 ∈ K by:        yn = βn,0xn + ∞ ∑ i=1 βn,iJ Bi rn xn, xn+1 = QK ( αnγf(xn) + (I − ηαnA)yn ) . (3.3) Let {rn} ⊂]0, ∞[, {βn,i} and {αn} be real sequences in (0, 1) satisfying: (i) lim n→∞ αn = 0; (ii) ∞ ∑ n=0 αn = ∞, ∞ ∑ i=0 βn,i = 1, (iii) lim n→∞ inf rn > 0, and lim n→∞ inf βn,0βn,i > 0, for all i ∈ N. Assume that 0 < η < ( kq dqLq ) 1 q−1 and 0 < bγ < τ, where τ = η ( k − dqL qηq−1 q ) . Then the sequence {xn} generated by (3.3) converges strongly to x ∗ ∈ F, which is a unique solution of variational inequality 〈ηAx∗ − γf(x∗), J(x∗ − p)〉 ≤ 0, ∀p ∈ F. (3.4) 164 T.M.M. Sow CUBO 22, 2 (2020) Proof. First of all, we show that the uniqueness of a solution of the variational inequality (3.4). Suppose both x∗ ∈ F and x∗∗ ∈ F are solutions to (3.4). Then 〈ηAx∗ − γf(x∗), J(x∗ − x∗∗)〉 ≤ 0 (3.5) and 〈ηAx∗∗ − γf(x∗∗), J(x∗∗ − x∗)〉 ≤ 0. (3.6) Adding up (3.5) and (4.3) yields 〈ηAx∗∗ − ηAx∗ + γf(x∗) − γf(x∗∗), J(x∗∗ − x∗)〉 ≤ 0. (3.7) dqL qηq−1 q > 0 ⇐⇒ k − dqL qηq−1 q < k ⇐⇒ η ( k − dqL qηq−1 q ) < kη ⇐⇒ τ < kη. It follows that 0 < bγ < τ < kη. Noticing that 〈ηAx∗∗ − ηAx∗ + γf(x∗) − γf(x∗∗), Jϕ(x ∗∗ − x∗)〉 ≥ (kη − bγ)‖x∗ − x∗∗‖2, which implies that x∗ = x∗∗ and the uniqueness is proved. Below we use x∗ to denote the unique solution of (3.4). Without loss of generality, we can assume αn ∈ ( 0, min{1 , 1 τ } ) . Now, we prove that the sequences {xn} and {yn} are bounded. Let p ∈ F. Using (3.3) and the fact that JBirn are nonexpansive, we have ‖yn − p‖ = ‖βn,0xn + ∞ ∑ i=1 βn,iJ Bi rn xn − p‖ ≤ βn,0‖xn − p‖ + ∞ ∑ i=1 βn,i‖J Bi rn xn − p‖ ≤ βn,0‖xn − p‖ + ∞ ∑ i=1 βn,i‖xn − p‖ ≤ ‖xn − p‖. CUBO 22, 2 (2020) A new iterative method based on the modified proximal-point . . . 165 Using Lemma 16, we have ‖xn+1 − p‖ = ‖QK ( αnγf(xn) + (I − ηαnA)yn ) − p‖ ≤ ‖αnγf(xn) + (I − ηαnA)yn − p‖ ≤ αnγ‖f(xn) − f(p)‖ + (1 − ταn)‖yn − p‖ + αn‖γf(p) − ηAp‖ ≤ (1 − αn(τ − bγ))‖xn − p‖ + αn‖γf(p) − ηAp‖ ≤ max {‖xn − p‖, ‖γf(p) − ηAp‖ τ − bγ }. By induction, it is easy to see that ‖xn − p‖ ≤ max {‖x0 − p‖, ‖γf(p) − ηAp‖ τ − bγ }, n ≥ 1. Hence {xn} is bounded also are {f(xn)}, and {Axn}. Let k ∈ N∗, from Lemma 9 and (3.3), we have ‖yn − p‖ 2 = ‖βn,0xn + ∞ ∑ i=1 βn,iJ Bi rn xn − p‖ 2 ≤ βn,0‖xn − p‖ 2 + ∞ ∑ i=1 βn,i‖J Bi rn xn − p‖ 2 − βn,0βn,kg(‖J Bk rn xn − xn‖) ≤ ‖xn − p‖ 2 − βn,0βn,kg(‖J Bk rn xn − xn‖). Consequently, we obtain ‖xn+1 − p‖ 2 = ‖QK ( αnγf(xn) + (I − ηαnA)yn ) − p‖2 ≤ ‖αn(γf(xn) − ηAp) + (I − ηαnA)(yn − p)‖ 2 ≤ α2n‖γf(xn) − ηAp‖ 2 + (1 − ταn) 2‖yn − p‖ 2 + 2αn(1 − ταn)‖γf(xn) −ηAp‖‖yn − p‖ ≤ α2n‖γf(xn) − ηAp‖ 2 + (1 − ταn) 2‖xn − p‖ 2 − (1 − ταn) 2βn,0βn,kg(‖J Bk rn xn − xn‖) +2αn(1 − ταn)‖γf(xn) − ηAp‖‖xn − p‖ Thus, for every k ∈ N∗, we get (1 − ταn) 2βn,0βn,kg(‖J Bk rn xn − xn‖) ≤ ‖xn − p‖ 2 − ‖xn+1 − p‖ 2 + α2n‖γf(xn) − ηAp‖ 2 +2αn(1 − ταn)‖γf(xn) − ηAp‖‖xn − p‖. (3.8) Since {xn} and {f(xn)} are bounded, there exists a constant C > 0 such that (1 − ταn) 2βn,0βn,kg(‖J Bk rn xn − xn‖) ≤ ‖xn − p‖ 2 − ‖xn+1 − p‖ 2 + αnC. (3.9) Let V I(A, F) the solutions set of variational inequality (3.4). Now, we prove V I(A, F) is nonempty. Let t0 be a fixed real number such that t0 ∈ ( 0, min{1 , 1 τ } ) . We observe that QF (I+(t0γf−t0ηA)) 166 T.M.M. Sow CUBO 22, 2 (2020) is a contraction, where QF is the sunny nonexpansive retraction from E to F. Indeed, for all x, y ∈ K, by Lemma 16, we have ‖QF (I + (t0γf − t0ηA))x − QF (I + (t0γf − t0ηA))x‖ ≤ ‖(I + (t0γf − t0ηA))x −(I + (t0γf − t0ηA))x‖ ≤ t0γ‖f(x) − f(y)‖ +‖(I − t0ηA)x − (I − t0ηA)y‖ ≤ (1 − t0(τ − γ))‖x − y‖. Banach’s Contraction Mapping Principle guarantees that QF (I +(t0γf −t0ηA)) has a unique fixed point, say x1 ∈ E. That is, x1 = QF (I + (t0γf − t0ηA))x1. Thus, in view of Lemma 11, it is equivalent to the following variational inequality problem 〈ηAx1 − γf(x1), J(x1 − p)〉 ≤ 0, ∀ p ∈ F. Hence, x1 ∈ V I(A, F). By the uniqueness of the solution of (3.4), we have x1 = x ∗. Next, we prove that {xn} converges strongly to x ∗. We divide the proof into two cases. Case 1. Assume that the sequence {‖xn − p‖} is monotonically decreasing. Then {‖xn − p‖} is convergent. Clearly, we have ‖xn − p‖ 2 − ‖xn+1 − p‖ 2 → 0. It then implies from (3.9) that lim n→∞ βn,0βn,kg(‖J Bk rn xn − xn‖) = 0. (3.10) Since limn→∞ inf βn,0βn,k > 0 and property of g, we have lim n→∞ ‖xn − J Bk rn xn‖ = 0. (3.11) By using the resolvent identity (Lemma 14), for any r > 0, we conclude that ‖xn − J Bk r xn‖ ≤ ‖xn − J Bk rn xn‖ + ‖J Bk rn xn − J Bk r xn‖ ≤ ‖xn − J Bk rn xn‖ + ‖J Bk r xn ( r rn xn + (1 − r rn )JBkrn xn ) − JBkr xn‖ ≤ ‖xn − J Bk rn xn‖ + ‖ r rn xn + ( 1 − r rn ) JBkrn xn − xn‖ ≤ ‖xn − J Bk rn xn‖+|1 − r rn | ‖JBkrn xn − xn‖ → 0, n → ∞, ∀k ∈ N ∗. Hence, lim n→∞ ‖xn − J Bk r xn‖ = 0. (3.12) We show that lim sup n→+∞ 〈ηAx∗ − γf(x∗), J(x∗ − xn)〉 ≤ 0. Since E is reflexive and {xn} is bounded, there exists a subsequence {xnj } of {xn} such that {xnj } converges weakly to a in K and lim sup n→+∞ 〈ηAx∗ − γf(x∗), J(x∗ − xn)〉 = lim j→+∞ 〈ηAx∗ − γf(x∗), J(x∗ − xnj )〉. CUBO 22, 2 (2020) A new iterative method based on the modified proximal-point . . . 167 From (3.12), the fact that JBkr , k ∈ N ∗ are nonexpansive and Lemma 5, we obtain a ∈ F. On the other hand, the assumption that the duality mapping is weakly continuous and the fact that x∗ ∈ V I(A, F), we then have lim sup n→+∞ 〈ηAx∗ − γf(x∗), J(x∗ − xn)〉 = lim j→+∞ 〈ηAx∗ − γf(x∗), J(x∗ − xnj )〉 = 〈ηAx∗ − γf(x∗), J(x∗ − a)〉 ≤ 0. Finally, we show that xn → x ∗. Applying Lemma 6, we get that ‖xn+1 − x ∗‖2 = ‖QK(αnγf(xn) + (I − ηαnA)yn) − x ∗‖2 ≤ 〈αnγf(xn) + (I − ηαnA)yn − x ∗, J(xn+1 − x ∗)〉 = 〈αnγf(xn) + (I − ηαnA)yn − x ∗ − αnγf(x ∗) + αnγf(x ∗) − αnηAx ∗ +αnηAx ∗, J(xn+1 − x ∗)〉 ≤ ( αnγ‖f(xn) − f(x ∗)‖ + ‖(I − αnηA)(yn − x ∗)‖ ) ‖xn+1 − x ∗‖ +αn〈ηAx ∗ − γf(x∗), J(x∗ − xn+1)〉 ≤ (1 − αn(τ − bγ))‖xn − x ∗‖‖xn+1 − x ∗‖ + αn〈ηAx ∗ − γf(x∗), J(x∗ − xn+1)〉 ≤ (1 − αn(τ − bγ))‖xn − x ∗‖2 + 2αn〈ηAx ∗ − γf(x∗), J(x∗ − xn+1)〉. From Lemma 7, its follows that xn → x ∗. Case 2. Assume that the sequence {‖xn−x ∗‖} is not monotonically decreasing. Set Bn = ‖xn−x ∗‖ and τ : N → N be a mapping for all n ≥ n0 (for some n0 large enough) by τ(n) = max{k ∈ N : k ≤ n, Bk ≤ Bk+1}. We have τ is a non-decreasing sequence such that τ(n) → ∞ as n → ∞ and Bτ(n) ≤ Bτ(n)+1 for n ≥ n0. Let i ∈ N ∗, from (3.9), we have (1 − τατ(n)) 2βτ(n),0βτ(n),ig(‖J Bi rτ(n) xτ(n) − xτ(n)‖) ≤ ατ(n)C → 0 as n → ∞. Furthermore, we have βτ(n),0βτ(n),ig(‖J Bi rτ(n) xτ(n) − xτ(n)‖) → 0 as n → ∞. Hence, lim n→∞ ‖JBirτ(n)xτ(n) − xτ(n)‖ = 0. (3.13) By same argument as in Case 1, we can show that xτ(n) and yτ(n) are bounded in K and lim sup τ(n)→+∞ 〈ηAx∗ − γf(x∗), J(x∗ − xτ(n))〉 ≤ 0. We have for all n ≥ n0, 0 ≤ ‖xτ(n)+1−x ∗‖2−‖xτ(n)−x ∗‖2 ≤ ατ(n)[−(τ−bγ)‖xτ(n)−x ∗‖2+2〈ηAx∗−γf(x∗), J(x∗−xτ(n)+1)〉], which implies that ‖xτ(n) − x ∗‖2 ≤ 2 τ − bγ 〈ηAx∗ − γf(x∗), J(x∗ − xτ(n)+1)〉. 168 T.M.M. Sow CUBO 22, 2 (2020) Then, we have lim n→∞ ‖xτ(n) − x ∗‖2 = 0. Therefore, lim n→∞ Bτ(n) = lim n→∞ Bτ(n)+1 = 0. Furthermore, for all n ≥ n0, we have Bτ(n) ≤ Bτ(n)+1 if n 6= τ(n) (that is, n > τ(n)); because Bj > Bj+1 for τ(n) + 1 ≤ j ≤ n. As a consequence, we have for all n ≥ n0, 0 ≤ Bn ≤ max{Bτ(n), Bτ(n)+1} = Bτ(n)+1. Hence, lim n→∞ Bn = 0, that is {xn} converges strongly to x ∗. This completes the proof. As a consequence of Theorem 18, we have the following theorem. Theorem 19. Let q > 1 be a fixed real number and E be a q-uniformly smooth and uniformly convex real Banach space having a weakly continuous duality map. Let {Bi}, i ∈ N ∗ be an infinite family of m-accretive operators of E such that F := ∞ ∩ i=1 Bi −1(0) 6= ∅. Let A : E → E be a k- strongly accretive and L-Lipschitzian operator and and f : K → E be a b-Lipschitzian mapping with a constant b ≥ 0. Let {xn} and {yn} be sequences defined iteratively from arbitrary x0 ∈ E by:      yn = βn,0xn + ∞ ∑ i=1 βn,iJ Bi rn xn, xn+1 = αnγf(xn) + (I − ηαnA)yn. (3.14) Let {rn} ⊂]0, ∞[, {βn,i} and {αn} be real sequences in (0, 1) satisfying: (i) lim n→∞ αn = 0; (ii) ∞ ∑ n=0 αn = ∞, ∞ ∑ i=0 βn,i = 1, (iii) lim n→∞ inf rn > 0, and lim n→∞ inf βn,0βn,i > 0, for all i ∈ N. Assume that 0 < η < ( kq dqLq ) 1 q−1 and 0 < bγ < τ, where τ = η ( k − dqL qηq−1 q ) . Then the sequence {xn} generated by (3.14) converges strongly to x ∗ ∈ F, which is a unique solution of variational inequality (3.4) . Proof. Since Bi are m-accretive operators, we conclude that Bi are accretive and satisfy the con- dition R(I + rBi) = E for all r > 0. Setting K = E in Theorem 18, we obtain the desired result. Corollary 1. Let H be a real Hilbert space. Let K be a nonempty, closed and convex subset of H. Let {Bi}, i ∈ N ∗ be an infinite family of monotone operators of H such that F := ∞ ⋂ i=1 Bi −1(0) 6= ∅ and ∞ ⋂ i=1 D(Bi) ⊂ K ⊂ ∞ ⋂ i=1 R(I + rBi), for all r > 0. Let A : K → H be a strongly bounded linear CUBO 22, 2 (2020) A new iterative method based on the modified proximal-point . . . 169 operator and and f : K → E be a b-Lipschitzian mapping with a constant b ≥ 0. Let {xn} and {yn} be sequences defined iteratively from arbitrary x0 ∈ K by:        yn = βn,0xn + ∞ ∑ i=1 βn,iJ Bi rn xn, xn+1 = PK ( αnγf(xn) + (I − ηαnA)yn ) . (3.15) Let {rn} ⊂]0, ∞[, {βn,i} and {αn} be real sequences in (0, 1) satisfying: (i) lim n→∞ αn = 0; (ii) ∞ ∑ n=0 αn = ∞, ∞ ∑ i=0 βn,i = 1, (iii) lim n→∞ inf rn > 0, and lim n→∞ inf βn,0βn,i > 0, for all i ∈ N. Assume that 0 < η < 2k ‖A‖2 and 0 < bγ < τ, where τ = η ( k − ‖A‖2η 2 ) . Then the sequence {xn} generated by (3.15) converges strongly to x ∗ ∈ F, which is the optimality condition for the minimization problem min x∈F η 2 〈Ax, x〉 − h(x), (3.16) where h is a potential function for γf (i.e. h ′ (x) = γf(x) on K ). Proof. From Remark 1, we have A is strongly monotone and ‖A‖-Lipschitz. the proof follows Theorem 18. 4 Applications In this section, as applications, we will utilize Theorem 18 to deduced several results. As a direct consequence of Theorem 18, we have the following results: 4.1 Application to equilibrium problems Let H be a real Hilbert space and let C be a nonempty, closed and convex subset of H. Let F be a bifunction of C × C into R, where R is the real numbers. The equilibrium problem for F is to find x ∈ C such that F(x, y) ≥ 0, ∀y ∈ C. (4.1) The set of solutions is denoted by EP(F). Equilibrium problems which were introduced by Fan [11] and Blum and Oettli [4] have had a great impact and influence on the development of sev- eral branches of pure and applied sciences. For solving the equilibrium problem for a bifunction F : C × C → R, let us assume that f satisfies the following conditions: (A1) F(x, x) = 0 for all x ∈ C; (A2) F is monotone, i.e., F(x, y) + F(y, x) ≤ 0 for all x, y ∈ C; 170 T.M.M. Sow CUBO 22, 2 (2020) (A3) for each x, y, z ∈ C, lim t→0 F(tz + (1 − t)x, y) ≤ F(x, y) (A4) for each x ∈ C, y → F(x, y) is convex and lower semicontinuous. Lemma 20. [6] Assume that F : C × C → R satisfying (A1)-(A4). For r > 0 and x ∈ H, define a mapping Tr : H → C as follows Tr(x) = {z ∈ C, F(z, y) + 1 r 〈y − z, z − x〉 ≥ 0, ∀y ∈ C}, for all x ∈ H. Then, the following hold: 1.Tr is single-valued; 2.Tr is firmly nonexpansive, i.e., ‖Tr(x) − Tr(y)‖ 2 ≤ 〈Trx − Try, x − y〉 for any x, y ∈ H; 3.F(Tr) = EP(F); 4.EP(F) is closed and convex. The following lemma appears implicitly in [29]. Lemma 21. [29] Let H be a Hilbert space and let C be a nonempty closed convex subset of H. Let F : C × C → R satisfy (A1) − (A4). Let AF be a set-valued mapping of H into itself defined by: AF x = { {z ∈ H, F(x, y) ≥ 〈y − x, z〉, ∀y ∈ C, } ∀x ∈ C ∅, x /∈ C. (4.2) Then EP(F) = AF −1(0) and AF is a maximal monotone operator with D(AF ) ⊂ C. Furthermore, for any x ∈ H and r > 0, the map Tr defined as Lemma 20 coincides with the resolvent of AF , i.e, Trx = ( I + rAF )−1 x. Using Theorem 18 , we prove a strong convergence theorem for an equilibrium problem in a Hilbert space. Theorem 22. Let H be a real Hilbert space and F : H × H → R satisfying (A1)-(A4) such that EP(F) 6= ∅. Let A : H → H be a k-strongly monotone and L-Lipschitzian operator and f : K → E be a b-Lipschitzian mapping with a constant b ≥ 0. Let {xn}, {un} and {yn} be a sequences defined iteratively from arbitrary x0 ∈ H by:      F(un, y) + 1 rn 〈y − un, un − xn〉 ≥ 0, ∀y ∈ H yn = βnxn + (1 − βn)un, xn+1 = αnγf(xn) + (I − ηαnA)yn. (4.3) Let {rn} ⊂]0, ∞[, {βn} and {αn} be real sequences in (0, 1) satisfying: (i) lim n→∞ αn = 0; (ii) ∞ ∑ n=0 αn = ∞, βn ∈ [a, b] ⊂ (0, 1). CUBO 22, 2 (2020) A new iterative method based on the modified proximal-point . . . 171 (iii) lim n→∞ inf rn > 0. Assume that 0 < η < 2k L2 and 0 < bγ < τ, where τ = η ( k − L2η 2 ) . Then the sequence {xn} generated by (4.3) converge strongly to x∗ ∈ EP(f), which is a unique solution of variational inequality 〈ηAx∗ − γf(x∗), x∗ − p〉 ≤ 0, ∀p ∈ EP(F). (4.4) Proof. Since F : H × H → R satisfying (A1)-(A4), we have that the mapping AF defined by Lemma 21 is a maximal and monotone operator. Put B = AF in Theorem 19 (with i=1). Then, we obtain that un = Trnxn = J B rn xn. Therefore, we arrive at the desired results. 4.2 Application to an infinite family of continuous pseudocontractive mappings. Let K be a nonempty, closed convex subset of a real Banach spaceE. A mapping T : K → K is said to be pseudocontractive if there exists j(x − y) ∈ J(x − y) such that 〈T x − T y, j(x − y)〉 ≤ ‖x − y‖2, ∀x, y ∈ K. It is well known that the class of pseudocontractive mapping is more general than the class of non- expansive mapping. Moreover, there exists a relationship between the class of accretive mappings and the class of pseudocontractive mappings. A mapping A : K → E is said to be pseudocontrac- tive if T := I − A is accretive. We can observe that x∗ is a zero of the accretive mapping A if and only if it is a fixed point of the pseudocontractive mapping T := I − A. Hence, one has the following result. Theorem 23. Let q > 1 be a fixed real number and E be a q-uniformly smooth and uniformly convex real Banach space having a weakly continuous duality map. Let K be a nonempty, closed and convex subset of E which is a nonexpansive retract of E with QK as the nonexpansive retraction. Let Ti : K → E, i ∈ N ∗ be an infinite family of continuous pseudo-contractive mappings of such that ∞ ⋂ i=1 F(Ti) 6= ∅. For each r > 0, let J i r := (I + r(I − Ti)) −1, i ∈ N∗. Let A : K → E be a k-strongly accretive and L-Lipschitzian operator and f : K → E be an b-Lipschitzian mapping with a constant b ≥ 0. Let {xn} and {yn} be sequences defined iteratively from arbitrary x0 ∈ K by:        yn = βn,0xn + ∞ ∑ i=1 βn,iJ i rn xn xn+1 = QK ( αnγf(xn) + (I − ηαnA)yn ) . (4.5) Let {rn} ⊂]0, ∞[, {βn,i} and {αn} be real sequences in (0, 1) satisfying: (i) lim n→∞ αn = 0; (ii) ∞ ∑ n=0 αn = ∞, ∞ ∑ i=0 βn,i = 1, 172 T.M.M. Sow CUBO 22, 2 (2020) (iii) lim n→∞ inf rn > 0, and lim n→∞ inf βn,0βn,i > 0, for all i ∈ N ∗. Assume that 0 < η < ( kq dqLq ) 1 q−1 and 0 < bγ < τ, where τ = η ( k − dqL qηq−1 q ) . 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