Bregman Asymptotic Pointwise Nonexpansive Mappings in Banach Spaces

Definition 4 (see [10].)Let E be a Banach space. The function g : E → ℝ is said to be a Bregman function if the following conditions are satisfied:

• (1)

  g  is continuous, strictly convex, and Gâteaux differentiable;

• (2)

  the set {y ∈ E : D_g(x, y) ≤ r} is bounded for all x ∈ E and r > 0.

Lemma 5 (see [9], [16].)Let E be a reflexive Banach space and g : E → ℝ a strongly coercive Bregman function. Then

• (1)

   ∇g : E → E^* is one-to-one, onto, and norm-to-weak* continuous;

• (2)

  〈x − y, ∇g(x)−∇g(y)〉 = 0 if and only if x = y;

• (3)

  {x ∈ E : D_g(x, y) ≤ r} is bounded for all y ∈ E and r > 0;

• (4)

  dom g^* = E^*, g^* is Gâteaux differentiable and ∇g^* = (∇g) ⁻¹.

Theorem 6. Let E be a reflexive Banach space and g : E → ℝ a convex function which is bounded on bounded subsets of E. Then the following assertions are equivalent:

• (1)

  g is strongly coercive and uniformly convex on bounded subsets of E;

• (2)

  dom g^* = E^*, g^* is bounded on bounded subsets and uniformly smooth on bounded subsets of E^*;

• (3)

  dom g^* = E^*, g^* is Fr$$chet differentiable and ∇g^* is uniformly norm-to-norm continuous on bounded subsets of E^*.

Theorem 7. Let E be a reflexive Banach space and g : E → ℝ a continuous convex function which is strongly coercive. Then the following assertions are equivalent:

• (1)

  g is bounded on bounded subsets and uniformly smooth on bounded subsets of E;

• (2)

  g^* is Fr$$chet differentiable and ∇g^* is uniformly norm-to-norm continuous on bounded subsets of E^*;

• (3)

  dom g^* = E^*, g^* is strongly coercive and uniformly convex on bounded subsets of E^*.

Lemma 8 (see [23].)Let E be a Banach space and g : E → ℝ a Gâteaux differentiable function which is uniformly convex on bounded subsets of E. Let {x_n} _n∈ℕ and {y_n} _n∈ℕ be bounded sequences in E. Then the following assertions are equivalent:

• (1)

   lim _n→∞D_g(x_n, y_n) = 0;

• (2)

  lim _n→∞∥x_n − y_n∥ = 0.

Lemma 9 (see [10], [24].)Let E be a reflexive Banach space, g : E → ℝ a strongly coercive Bregman function, and V the function defined by

Then the following assertions hold:

• (1)

  D_g(x, ∇g^*(x^*)) = V(x, x^*) for all x ∈ E and x^* ∈ E^*;

• (2)

  V(x, x^*)+〈∇g^*(x^*) − x, y^*〉≤V(x, x^* + y^*) for all x ∈ E and x^*, y^* ∈ E^*.

Lemma 10 (see [25].)Suppose {r_k} _k∈ℕ is a bounded sequence of real numbers and {d_k,n} _k,n∈ℕ is a doubly index sequence of real numbers which satisfy

for each k, n ≥ 1. Then {r_k} _k∈ℕ converges to an r ∈ ℝ.

Lemma 11 (see [26].)Let E be a Banach space and let g : E → (−∞, +∞] be a proper strictly convex function so that it is Gâteaux differentiable on int  dom g. Suppose {x_n} _n∈ℕ is a sequence in dom g such that x_n⇀v for some v ∈ int  dom g. Then

Theorem 12 (see [16].)Let g : E → (−∞, +∞] be a function. Then the following assertions are equivalent:

• (1)

  g is convex and lower semicontinuous;

• (2)

  g is convex and weakly lower semicontinuous;

• (3)

  epi (g) is convex and closed;

• (4)

  epi (g) is convex and weakly closed,

where epi (g) = {(x, t) ∈ E × ℝ : g(x) ≤ t} denotes the epigraph of g.

Proposition 13. Let E be a reflexive Banach space and let g : E → ℝ be a strongly admissible function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E and let T ∈ ℬ𝒯_r(C). Then T has a fixed point. Moreover, F(T) is closed and convex.

Proof. We first show that F(T) is nonempty. Let x in C be fixed. We define a function f : C → [0, ∞) by

In view of Remark 1, it is easy to see that f is convex. Since g is continuous, by Theorem 12 we conclude that the Bregman distance D_g is weakly lower-semicontinuous in the first argument. Since E is a uniformly convex Banach space and C is weakly compact, in view of [1] there exists a unique point z ∈ C such that We show that {Tⁿz} _n∈ℕ is convergent in norm. To this end, let s₁ = sup {∥Tⁿz∥, ∥z∥ : n ∈ ℕ} and $$ be the gauge of uniform convexity of g. For any k, m ∈ ℕ, put u_k,m = (1/2)T^kz + (1/2)T^mz. Then we have u_k,m ∈ C. In view of Remark 1, we obtain Applying to both sides of the above inequalities limsup _n→∞ we obtain This, together with f(z) ≤ f(u_k,m), implies that Letting k, m → ∞ in (39) we conclude that From the properties of $$, we deduce that lim _k,m→∞∥T^kz − T^mz∥ = 0. Thus, {T^kz} _k∈ℕ is a norm-Cauchy sequence and hence convergent. Let Since T is a Bregman asymptotic pointwise nonexpansive mapping, we have, for all n ∈ ℕ, Letting n → ∞ in (42), we conclude that Tv = v. This shows that F(T) ≠ ∅.

Now, we show that F(T) is closed. Let {p_n} _n∈ℕ be a sequence in F(T) such that p_n → p as n → ∞. Then we have that {p_n} _n∈ℕ is a bounded sequence in E. We claim that p ∈ F(T). Since g is continuous, we conclude that g(p_n) → g(p) as n → ∞. This implies that

In view of the definition of T, we obtain This implies that It follows from Lemma 8 that Tp = p. Thus we have p ∈ F(T).

Let us show that F(T) is convex. For any p, q ∈ F(T), t ∈ (0,1), and n ∈ ℕ, we set x = tp + (1 − t)q and e_n(p, q) = max {a_n(p), a_n(q)}. We prove that x ∈ F(T). By the definition of Bregman distance (see (12)), we get

This implies that lim _n→∞D_g(x, Tⁿx) = 0. Thus for each ϵ > 0, there exists n₀ ∈ ℕ such that This means that the sequence {D_g(x, Tⁿx)} _n∈ℕ is bounded. In view of Definition 4, we conclude that the sequence {Tⁿx} _n∈ℕ is bounded. Then, by Lemma 8, we obtain lim _n→∞∥x − Tⁿx∥ = 0. Thus we have Tⁿ⁺¹x → x; that is, T(Tⁿx) → x. On the other hand, in view of three-point identity (see (26)), we deduce that Letting n → ∞ in the above inequalities we deduce that D_g(x, Tx) = 0 and hence by Lemma 8 we conclude that Tx = x, which completes the proof.

Lemma 14. Let E be a reflexive Banach space and let g : E → ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E and let T ∈ ℬ𝒯_r(C). If {x_n} _n∈ℕ is a sequence of C such that lim _n→∞∥Tx_n − x_n∥ = 0, then, for any m ∈ ℕ, lim _n→∞∥T^mx_n − x_n∥ = 0.

Proof. In view of (25), there exists a finite constant M₁ > 0 such that

It follows from three-point identity (see (26)) that where M₂ : = sup {∥∇g(T^jx_n)∥ : j = 1,2, …, m  and  n ∈ ℕ}. This, together with Lemma 8, implies that lim _n→∞∥T^mx_n − x_n∥ = 0. This completes the proof.

Theorem 15. Let E be a reflexive Banach space and let g : E → ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E and let T ∈ ℬ𝒯_r(C). If {x_n} _n∈ℕ converges weakly to z and lim _n→∞∥Tx_n − x_n∥ = 0, then Tz = z. That is, I − T is demiclosed at zero, where I is the identity mapping on E.

Proof. Let the function ϕ : E → [0, ∞) be defined by

For any m, n ∈ ℕ with m > 2, in view of three-point identity (see (26)), we obtain where $$. This, together with Lemma 8, implies that In view of (25), there exists a finite constant M₃ > 0 such that where M₄ : = sup {∥∇g(T^m−1x_n)∥ : m = 1,2, …, m  and  n ∈ ℕ}. This, together with Lemma 8, implies that lim _n→∞∥T^mx_n − x_n∥ = 0. Employing Lemma 8, we conclude that This means that Since T is a Bregman asymptotic pointwise nonexpansive mapping, it follows that In view of (57), we deduce that By the Bregman Opial-like inequality ((34)) we obtain that for any x ≠ z This shows that ϕ(z) = inf {ϕ(x) : x ∈ C}. Thus we have Put s₂ = sup {∥∇g(z)∥, ∥∇g(T^mz)∥ : m ∈ ℕ}, z_m = ∇g^*((1/2)∇g(z) + (1/2)∇g(T^mz)), and $$ for all m ∈ ℕ. Then we have u_m ∈ C. In view of Remark 1, we obtain a continuous strictly increasing convex function $$ with $$ such that Applying to both sides limsup _n→∞ and remembering that ϕ(z) = inf {ϕ(x) : x ∈ C} we obtain This implies that Letting m → ∞ in (63) we conclude that From the properties of $$, we deduce that lim _m→∞∥∇g(z)−∇g(T^mz)∥. Since ∇g^* is uniformly norm-to-norm continuous on bounded subsets of E^*, we arrive at lim _m→∞∥z − T^mz∥ = 0. Thus we have T^m+1z → z; that is, T(T^mz) → z. On the other hand, in view of three-point identity (see (26)), we deduce that Letting m → ∞ in the above inequalities we deduce that D_g(z, Tz) = 0 and hence by Lemma 8 we conclude that Tz = z, which completes the proof.

Definition 16. Let E be a reflexive Banach space and let g : E → ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E and let T ∈ ℬ𝒯_r(C). Let {n_k} _k∈ℕ be an increasing sequence in ℕ and let {γ_k} _k∈ℕ ⊂ (0,1) be such that lim  inf _k→γ_k(1 − γ_k) > 0. The generalized Mann iteration process generated by the mapping T, the sequence {γ_k} _k∈ℕ, and the sequence {n_k} _k∈ℕ, denoted by gM(T, {γ_k} _k∈ℕ, {n_k} _k∈ℕ), is defined by the following iterative formula:

Definition 17. We say that a generalized Mann iteration process gM(T, {γ_k} _k∈ℕ, {n_k} _k∈ℕ) is well defined if

Remark 18. Observe that by the definition of Bregman asymptotic pointwise nonexpansive, $$ for every x ∈ C. Hence we can always select a subsequence $$ of {a_n} _n∈ℕ such that (67) holds. In other words, by a suitable choice of {n_k} _k∈ℕ we can always make gM(T, {γ_k} _k∈ℕ, {n_k} _k∈ℕ) well defined.

Lemma 19. Let E be a reflexive Banach space and let g : E → ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E. Let T ∈ ℬ𝒯_r(C) and let {n_k} _k∈ℕ ⊂ ℕ. Let {γ_k} _k∈ℕ ⊂ (0,1) such that lim  inf _k→∞γ_k(1 − γ_k) > 0. Let w ∈ F(T) and let gM(T, {γ_k} _k∈ℕ, {n_k} _k∈ℕ) be a generalized Mann process. Then there exists λ ∈ ℝ such that lim _k→∞D(x_k, w) = λ.

Proof. Let w ∈ F(T) be arbitrary chosen. In view of (66), we obtain

This implies that for every n ∈ ℕ, Put r_p = D_g(x_p, w) for every p ∈ ℕ and $$. Since T ∈ ℬ𝒯_r(C), we obtain that limsup _k→∞limsup _n→∞d_k,n = 0. In view of Lemma 11, there exists λ ∈ ℝ such that lim _k→∞D_g(x_k, w) = λ. This completes the proof.

Lemma 20. Let E be a reflexive Banach space and let g : E → ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E. Let T ∈ ℬ𝒯_r(C) and let {n_k} _k∈ℕ ⊂ ℕ. Let {γ_k} _k∈ℕ ⊂ (0,1) such that lim  inf _k→∞γ_k(1 − γ_k) > 0 and let gM(T, {γ_k} _k∈ℕ, {n_k} _k∈ℕ) be a generalized Mann process. Then

Proof. In view of Proposition 13, we conclude that F(T) ≠ ∅. Let w ∈ F(T) be fixed. It follows from Lemma 19 that there exists λ ∈ ℝ such that lim _k→∞D_g(x_k, w) = λ. Let $$ and let $$ be the gauge of uniform convexity of g. By the definition of T, we obtain

This implies that Letting k → ∞ in (72) we conclude that From the properties of $$, we deduce that $$. Employing Lemma 8, we conclude that This completes the proof.

Definition 21. A strictly increasing sequence {n_i} _i∈ℕ ⊂ ℕ is called quasiperiodic if the sequence {n_i+1 − n_i} _i∈ℕ is bounded or equivalently if there exists a number p ∈ ℕ such that any block of consecutive natural numbers must contain a term of the sequence {n_i} _i∈ℕ. The smallest of such numbers will be called a quasi period of {n_i} _i∈ℕ.

Lemma 22. Let E be a reflexive Banach space and let g : E → ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E. Let T ∈ ℬ𝒯_r(C) and let {n_k} _k∈ℕ ⊂ ℕ. Let {γ_k} _k∈ℕ ⊂ (0,1) such that liminf _k→∞γ_k(1 − γ_k) > 0. Let {n_k} _k∈ℕ ⊂ ℕ be such that the generalized Mann process gM(T, {γ_k} _k∈ℕ, {n_k} _k∈ℕ) is well defined. If, in addition, the set of indices 𝒥 = {j ∈ ℕ : n_j+1 = 1 + n_j} is quasi-periodic, then {x_k} _k∈ℕ is an approximate fixed point sequence; that is,

Proof. In view of (66), we have

This, together with Lemmas 8 and 20, implies that In view of Lemma 8, we conclude that Let p ∈ ℕ be a quasi-period of 𝒥. We first prove that ∥x_k − T(x_k)∥ → 0 as k → ∞ through 𝒥. Since n_k+1 = n_k + 1 for such k, we obtain where M₅ = sup {a₁(x) : x ∈ C} < ∞. This implies that It follows from Lemma 8 that On the other hand, we have Thus, we obtain ∥x_k − T(x_k)∥ → 0 as k → ∞ through 𝒥. In view of Lemma 8, we conclude that D_g(x_k, T(x_k)) → 0 as k → ∞ through 𝒥. Now, let ϵ > 0 be fixed. It follows from D_g(x_k, T(x_k)) → 0 as k → ∞ through 𝒥 that there exists N₀ ∈ ℕ such that Since 𝒥 is quasi-periodic, for any k ∈ ℕ there exists j_k ∈ 𝒥 such that |k − j_k| ≤ p. Assume that k − p ≤ j_k ≤ k (the proof of the other case is identical). Since T is a Bregman M₆-Lipschitzian mapping where M₆ = sup {a₁(x) : x ∈ C}, there exists 0 < δ < ϵ/3 such that In view of (66) and (83), there exists N₁ ∈ ℕ such that We also obtain This implies that If follows from (77)-(78) that This completes the proof.

Theorem 23. Let E be a reflexive Banach space and let g : E → ℝ be a convex, continuous, strongly coercive and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets. Let C be a nonempty, bounded, closed and convex subset of E. Let T ∈ ℬ𝒯_r(C) and let {n_k} _k∈ℕ ⊂ ℕ. Let {γ_k} _k∈ℕ ⊂ (0,1) such that lim  inf _k→∞γ_k(1 − γ_k) > 0. Let {n_k} _k∈ℕ ⊂ ℕ be such that the generalized Mann process gM(T, {γ_k} _k∈ℕ, {n_k} _k∈ℕ) is well defined. If, in addition, the set of indices 𝒥 = {j ∈ ℕ : k_j+1 = 1 + k_j} is quasi-periodic, then the sequence {x_k} _k∈ℕ generated by gM(T, {γ_k} _k∈ℕ, {n_k} _k∈ℕ) converges weakly to a fixed point of T.

Proof. In view of Lemma 22, we obtain

Let y, z ∈ C be weak cluster points of the sequence {x_k} _k∈ℕ. Then there exist subsequences {y_k} _k∈ℕ and {z_k} _k∈ℕ of {x_k} _k∈ℕ such that y_k⇀y and z_k⇀z as k → ∞. In view of (89) and Theorem 15, we conclude that Ty = y and Tz = z. It follows from Lemma 19 that there exist real numbers λ₁ and λ₂ such that We claim that y = z. Assume on the contrary that y ≠ z. By the Bregman Opial-like property we obtain This is a contradiction and hence there exists w ∈ C such that x_k⇀w as k → ∞. Since C is weakly sequentially compact, such a weak cluster point w is unique. In view of Theorem 15, we conclude that T(w) = w, which completes the proof.

Definition 24. Let E be a reflexive Banach space and let g : E → ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets. Let C be a nonempty, bounded, closed, and convex subset of E. Let T ∈ ℬ𝒯_r(C) and let {n_k} _k∈ℕ ⊂ ℕ be an increasing sequence. Let {β_k} _k∈ℕ ⊂ (0,1) and {γ_k} _k∈ℕ ⊂ (0,1) be sequences of real numbers such that lim  inf _k→∞β_k(1 − β_k) > 0 and lim  inf _k→∞γ_k(1 − γ_k) > 0. The generalized Ishikawa iteration process generated by the mapping T, the sequences {β_k} _k∈ℕ ⊂ (0,1), and {γ_k} _k∈ℕ ⊂ (0,1), and the sequence {n_k} _k∈ℕ denoted by gI(T, {β_k} _k∈ℕ, {γ_k} _k∈ℕ, {n_k} _k∈ℕ) is defined by the following iterative scheme:

Definition 25. We say that a generalized Ishikawa iteration process gI(T, {β_k} _k∈ℕ, {γ_k} _k∈ℕ, {n_k} _k∈ℕ) is well defined if

Lemma 26. Let E be a reflexive Banach space and let g : E → ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E. Let T ∈ ℬ𝒯_r(C) and let {n_k} _k∈ℕ ⊂ ℕ. Let {β_k} _k∈ℕ ⊂ (0,1) and {γ_k} _k∈ℕ ⊂ (0,1) be sequences of real numbers such that lim  inf _k→∞β_k(1 − β_k) > 0 and lim  inf _k→∞γ_k(1 − γ_k) > 0. Let w ∈ F(T) and gI(T, {β_k} _k∈ℕ, {γ_k} _k∈ℕ, {n_k} _k∈ℕ) be a generalized Ishikawa iteration process. Then there exists θ ∈ ℝ such that lim _k→∞D_g(x_k, w) = θ.

Proof. Let M₇ > 1 be fixed. Since $$, there exists k₀ ∈ ℕ such that for any k > k₀, $$. Let $$ and let $$ be the gauge of uniform convexity of g. By the definition of T and in view of (93), we obtain

This implies that for every n ∈ ℕ, Put r_p = D_g(x_p, w) for every p ∈ ℕ and $$. Since T ∈ ℬ𝒯_r(C), we obtain that lim  sup _k→∞lim  sup _n→∞d_k,n = 0. In view of Lemma 11, there exists θ ∈ ℝ such that lim _k→∞D_g(x_k, w) = θ. Let M₈ > 1 be fixed. Since $$, there exists k₀ ∈ ℕ such that for any k > k₀, $$. Therefore, by the same argument, as in the proof of Lemma 19, we conclude that for k > k₀ and n > 1 By the same manner as in the proof of Lemma 19, we deduce that there exists θ ∈ ℝ such that lim _k→∞D_g(x_k, w) = θ, which completes the proof.

Lemma 27. Let E be a reflexive Banach space and let g : E → ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets. Let C be a nonempty, bounded, closed, and convex subset of E. Let T ∈ ℬ𝒯_r(C) and let {n_k} _k∈ℕ ⊂ ℕ. Let {β_k} _k∈ℕ ⊂ (0,1) and {γ_k} _k∈ℕ ⊂ (0,1) be sequences of real numbers such that lim  inf _k→∞γ_k(1 − γ_k) > 0 and lim  inf _k→∞β_k(1 − β_k) > 0. Let w ∈ F(T) and gI(T, {γ_k} _k∈ℕ, {β_k} _k∈ℕ, {n_k} _k∈ℕ) be a generalized Ishikawa iteration process. Then

Proof. In view of Proposition 13, F(T) ≠ ∅. Take any w ∈ F(T) arbitrarily chosen. Then, by Lemma 26, there exists θ ∈ ℝ such that lim _k→∞D_g(x_k, w) = θ. By the same arguments, as in the proof of Lemma 26, we conclude that

This implies that This implies that Therefore, from the property of $$ we deduce that In a similar way, as in the proof of Lemma 20, we can prove that This completes the proof.

Lemma 28. Let E be a reflexive Banach space and let g : E → ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets. Let C be a nonempty, bounded, closed, and convex subset of E. Let T ∈ ℬ𝒯_r(C) and let {n_k} _k∈ℕ ⊂ ℕ. Let {γ_k} _k∈ℕ ⊂ (0,1) such that lim  inf _k→∞γ_k(1 − γ_k) > 0. Let {n_k} _k∈ℕ ⊂ ℕ be such that the generalized Ishikawa process gI(T, {γ_k} _k∈ℕ, {β_k} _k∈ℕ, {n_k} _k∈ℕ) is well defined. If, in addition, the set of indices 𝒥 = {j ∈ ℕ : n_j+1 = 1 + n_j} is quasi-periodic, then {x_k} _k∈ℕ is an approximate fixed point sequence; that is,

Theorem 29. Let E be a reflexive Banach space and let g : E → ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets. Let C be a nonempty, bounded, closed, and convex subset of E. Let T ∈ ℬ𝒯_r(C) and let {n_k} _k∈ℕ ⊂ ℕ. Let {γ_k} _k∈ℕ ⊂ (0,1) such that lim  inf _k→∞γ_k(1 − γ_k) > 0. Let {n_k} _k∈ℕ ⊂ ℕ be such that the generalized Ishikawa process gI(T, {γ_k} _k∈ℕ, {β_k} _k∈ℕ, {n_k} _k∈ℕ) is well defined. If, in addition, the set of indices 𝒥 = {j ∈ ℕ : k_j+1 = 1 + k_j} is quasi-periodic, then the sequence {n_k} _k∈ℕ generated by gI(T, {γ_k} _k∈ℕ, {β_k} _k∈ℕ, {n_k} _k∈ℕ) converges weakly to a fixed point of T.

Remark 30. Theorem 29 improves Theorems 3.1 and 4.1 of [18] in the following aspects.

• (1)

  For the structure of Banach spaces, we extend the duality mapping to more general case, that is, a convex, continuous, strongly coercive Bregman function which is bounded on bounded sets and uniformly convex and uniformly smooth on bounded sets.

• (2)

  For the mappings, we extend the mapping from an asymptotic pointwise nonexpansive mapping to a Bregman asymptotic pointwise nonexpansive mapping.

• (3)

  Since we do not need the weak sequential continuity of the duality mapping in Theorems 23 and 29 as was the case in [18], we can apply Theorem 29 in the Lebesgue space L^p where 1 < p < ∞ and p ≠ 2 while this space is not applicable for Theorems 3.1, 4.1, and 5.1 of [18].