Approximation of Fixed Points of Weak Bregman Relatively Nonexpansive Mappings in Banach Spaces

Definition 2.1 (see [1], [13].)Let f : E → (−∞, +∞] be a Gâteaux differentiable and convex function. The function D_f : dom  f × int (dom  f) → [0, +∞), defined by

Remark 2.2 (see [24].)The Bregman distance has the following properties:

• (i)

  the three point identity, for any x ∈ dom  f and y, z ∈ int (dom  f),

• (ii)

  the four point identity, for any y, ω ∈ dom  f and x, z ∈ int (dom  f),

Definition 2.3 (see [1].)Let f : E → (−∞, +∞] be a Gâteaux differentiable and convex function. The Bregman projection of x ∈ int (dom  f) onto the nonempty closed and convex set C ⊂ dom  f is the necessarily unique vector $$ satisfying

Remark 2.4. (i) If E is a Hilbert space and f(y) = (1/2)∥x∥² for all x ∈ E, then the Bregman projection $$ is reduced to the metric projection of x onto C.

(ii) If E is a smooth Banach space and f(y) = (1/2)∥x∥² for all x ∈ E, then the Bregman projection $$ is reduced to the generalized projection Π_C(x) (see, e.g. [3]) which defined by

Definition 2.5 (see[12, 21]). Let C be a nonempty closed and convex set of dom  f. The operator T : C → int (dom  f) with F(T) ≠ ∅ is called:

• (i)

  quasi-Bregman nonexpansive if

• (ii)

  Bregman relatively nonexpansive if

  $$ (2.9)
  and $$,

• (iii)

  Bregman firmly nonexpansive if

  $$ (2.10)
  or equivalently
  $$ (2.11)

Definition 2.6. Let C be a nonempty closed and convex set of dom f. The operator T : C → int (dom f) with F(T) ≠ ∅ is called weak Bregman relatively nonexpansive if $$ and

Remark 2.7. It is easy to see that each nonexpansive mapping T is quasi-Bregman nonexpansive mapping with respect to f(x) = (1/2)∥x∥² for all x ∈ E. Moreover, every relatively nonexpansive mapping T also is Bregman relatively nonexpansive mapping, where T is called relatively nonexpansive mapping (see, e.g., [32]) if the following conditions are satisfied:

Now, we give an example which is weak Bregman relatively nonexpansive mapping but not Bregman relatively nonexpansive mapping.

Example 2.8. Let E = l², f(x) = (1/2)∥x∥² for all x ∈ E, where

Define a mapping T : E → E by

For any strong convergent sequence {y_n} ⊂ l² such that y_n → y₀ and ∥Ty_n − y_n∥ → 0 as n → ∞. Then, there exists a sufficiently large nature number M such that y_n ≠ x_m for any n, m > M. Thus, Ty_n = −y_n for n > M, which implies that 2y_n → 0 and y_n → y₀ = 0 as n → ∞. That is, y₀ = 0 is a strong asymptotic fixed point of T, and so, $$. Since

Definition 2.9 (see [12].)Let f : E → (−∞, +∞] be a convex and Gâteaux differentiable function. f is called:

• (i)

  totally convex at x ∈ int (dom  f) if its modulus of total convexity at x; that is, the function ν_f : int (dom  f) × [0, +∞) → [0, +∞) defined by

• (ii)

  totally convex if, it is totally convex at every point x ∈ int (dom   f),

• (iii)

  totally convex on bounded sets if ν_f(B, t) is positive for any nonempty bounded subset B of E and t > 0, where the modulus of total convexity of the function f on the set B is the function ν_f : int (dom  f) × [0, +∞) → [0, +∞) defined by

Definition 2.10 (see [12], [21].)The function f : E → (−∞, +∞] is called:

• (i)

  cofinite if dom f^* = E^*,

• (ii)

  sequentially consistent if, for any two sequences {x_n} and {y_n} in E such that the first is bounded, and

Lemma 2.11 (see [21], Proposition 2.3.)If f : E → (−∞, +∞] is Fréchet differentiable and totally convex, then f is cofinite.

Lemma 2.12 (see [14], Theorem 2.10.)Let f : E → (−∞, +∞] be a convex function whose domain contains at least two points. Then, the following statements hold:

• (i)

  f is sequentially consistent if and only if it is totally convex on bounded sets,

• (ii)

  if f is lower semicontinuous, then f is sequentially consistent if and only if it is uniformly convex on bounded sets,

• (iii)

  if f is uniformly strictly convex on bounded sets, then it is sequentially consistent and the converse implication holds when f is lower semicontinuous, Fréchet differentiable on its domain, and the Fréchet derivative f′ is uniformly continuous on bounded sets.

Lemma 2.13 (see [20], Proposition 2.1.)Let f : E → R be a uniformly Fréchet differentiable and bounded on bounded subsets of E. Then, ∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E^*.

Lemma 2.14 (see [21], Lemma 3.1.)Let f : E → R be a Gâteaux differentiable and totally convex function. If x₀ ∈ E and the sequence $$ is bounded, then the sequence $$ is also bounded.

Lemma 2.15 (see [21], Proposition 2.2.)Let f : E → R be a Gâteaux differentiable and totally convex function, x₀ ∈ E, and let C be a nonempty closed convex subset of E. Suppose that the sequence $$ is bounded and any weak subsequential limit of $$ belongs to C. If $$ for any n ∈ N, then $$ converges strongly to $$.

Lemma 2.16 (see  [23, Lemma 15.5]). Let f : E → (−∞, +∞] be a Legendre function. Let C be a nonempty closed convex subset of int (dom  f) and T : C → C a Bregman firmly nonexpansive mapping with respect to f. Then, F(T) is closed and convex.

Proposition 2.17. Let f : E → (−∞, +∞] be a Legendre function. Let C be a nonempty closed convex subset of int (dom  f) and T : C → C a quasi-Bregman nonexpansive mapping with respect to f. Then, F(T) is closed and convex.

Proof. Without loss of generality, set F(T) is nonempty. Firstly, we show that F(T) is closed. Let $$ be a sequence in F(T) such that $$. By the definition of quasi-Bregman nonexpansive mapping, we have

We now show that F(T) is convex. For any x, y ∈ F(T) and t ∈ (0,1), it yields that z = tx + (1 − t)y ∈ C. From the definition of quasi-Bregman nonexpansive mapping, it follows that

Lemma 2.18. Let f : E → (−∞, +∞] be a Gâteaux differentiable and proper convex lower semicontinuous. Then, for all z ∈ E,

Lemma 2.19 (see [14], Corollary 4.4.)Let f : E → (−∞, +∞] be a Gâteaux differentiable and totally convex on int (dom  f). Let x ∈ int (dom  f) and C ⊂ int (dom  f) a nonempty closed convex set. If $$, then the following statements are equivalent:

• (i)

  the vector $$ is the Bregman projection of x onto C with respect to f,

• (ii)

  the vector $$ is the unique solution of the variational inequality

• (iii)

  the vector $$ is the unique solution of the inequality

Theorem 3.1. Let C be a nonempty closed convex subset of a real reflexive Banach space E and f : E → R a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of E, and let T : C → C be a weak Bregman relatively nonexpansive mapping such that F(T) ≠ ∅. Define a sequence {x_n} in C by the following algorithm:

Proof. By Proposition 2.17, it follows that F(T) is a nonempty closed and convex subset of E. It is easy to verify that C₀, C₁, Q₀, and Q₁ are closed and convex. Suppose that C_k and Q_k  (k ≥ 1) are closed and convex. Then, C_k∩Q_k is closed and convex. For any z ∈ C_k+1, y ∈ Q_k+1,

Secondly, we show that {x_n} is a Cauchy sequence and bounded. Since

Thirdly, we show that {x_n} converges strongly to a point of F(T). Since f is uniformly Fréchet differentiable on bounded subsets of E, from Lemma 2.12, ∇f is norm-to-norm uniformly continuous on bounded subsets of E. So, by (3.15),

Finally, we show $$. Since $$, it follows from $$ that $$. By Lemma 2.15, $$ as n → ∞. Therefore, {x_n} and {y_n} converge strongly to $$. This completes the proof.

Theorem 3.2. Let C be a nonempty closed convex subset of a real reflexive Banach space E and f : E → R a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of E, and let T : C → C be a Bregman relatively nonexpansive mapping such that F(T) ≠ ∅. Assume that {α_n}, {β_n} ⊂ [0,1] such that liminf _n→∞(1 − α_n)β_n > 0, and {e_n} is an error sequence in E with e_n → 0 as n → ∞. Then, the sequences {x_n} and {y_n} generated by (3.1) converge strongly to the point $$, where $$ is the Bregman projection of C onto F(T).

Proof. As in the proof of Theorem 3.1, we know that the sequences {x_n} and {y_n} converge strongly to $$, and so,

Corollary 3.3. Let C be a nonempty closed convex subset of a real reflexive, smooth, and strictly convex Banach space E, and let T : C → C be a relatively nonexpansive mapping such that F(T) ≠ ∅. Define a sequence {x_n} in C by the following algorithm:

In [27], Matsushita and Takahashi proved the following result.

Theorem MT (see [27], Theorem 3.1.)Let C be a nonempty closed convex subset of a real uniformly convex and uniformly smooth Banach space E, and let T : C → C be a relatively nonexpansive mapping such that F(T) ≠ ∅. Assume that {α_n} is a sequence of real numbers such that 0 ≤ α_n < 1 and limsup _n→∞α_n < 1. Then, the sequence {x_n} generated by (1.3) converges strongly to the point Π_F(T)(x₀), where Π_F(T)(x₀) is the generalized projection (see, e.g., [2, 3, 28]) of C onto F(T).

Remark 3.4. Corollary 3.3 extends Theorem MT [27] from uniformly convex and uniformly smooth Banach spaces to reflexive, smooth, and strictly convex Banach space.

Now, we investigate convergence theorems for Halpern-type iterative algorithms with errors.

Theorem 3.5. Let C be a nonempty closed convex subset of a real reflexive Banach space E and f : E → R a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of E, and let T : C → C be a weak Bregman relatively nonexpansive mapping such that F(T) ≠ ∅. Define a sequence {x_n} in C by the following algorithm:

Proof. By Proposition 2.17, it follows that F(T) is a nonempty closed and convex subset of E. It is easy to see that C_n is closed and Q_n is closed and convex for all n ≥ 0. For any z ∈ C_n, n ≥ 1,

Secondly, we show that {x_n} converges strongly to a point of F(T). Since f is uniformly Fréchet differentiable on bounded subsets of E, from Lemma 2.12, ∇f is norm-to-norm uniformly continuous on bounded subsets of E. So, by (3.33),

Corollary 3.6. Let C be a nonempty closed convex subset of a real reflexive Banach space E and f : E → R a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of E, and let T be a weak Bregman relatively nonexpansive mapping from C into itself such that F(T) ≠ ∅. Define a sequence {x_n} in C by the following algorithm:

Theorem 3.7. Let C be a nonempty closed convex subset of a real reflexive Banach space E and f : E → R a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of E, and let T : C → C be a Bregman relatively nonexpansive mapping such that F(T) ≠ ∅. Define a sequence {x_n} in C by the following algorithm:

Proof. The proof is similar to Theorem 3.5 and so is omitted. This completes the proof.

Corollary 3.8. Let E be a real reflexive Banach space and f : E → R a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of E, and let T : E → E be a Bregman relatively nonexpansive mapping such that F(T) ≠ ∅. Assume that {β_n} is a real sequence in [0,1] such that lim _n→∞β_n = 0. Define a sequence {x_n} by the following algorithm:

Theorem QS (see [30], Theorem 2.2.)Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, and let T : C → C be a relatively nonexpansive mapping such that F(T) ≠ ∅. Assume that {β_n} is a real sequence in [0,1) such that lim _n→∞β_n = 0. Then, the sequence {x_n} generated by (1.5) converges strongly to Π_F(T)x₀, where Π_F(T) is the generalized projection (see, e.g., [2, 3]) from E onto F(T).

Remark 3.9. Corollary 3.8 extends Theorems QS [30] from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces.