The Split Common Fixed Point Problem for Total Asymptotically Strictly Pseudocontractive Mappings

Remark 2.1. It is well known that each Hilbert space possesses the Opial property.

Definition 2.2. Let H be a real Hilbert space, and let K be nonempty and closed convex subset of H.

• (1)

  A mapping G : K → K is said to be (γ, {μ_n}, {ξ_n}, ϕ)-totally asymptotically strictly pseudocontractive, if there exist a constant γ ∈ [0,1) and sequences {μ_n}⊂[0, ∞) and {ξ_n}⊂[0, ∞) with μ_n → 0 and ξ_n → 0 such that for all x, y ∈ K 

  $$ (2.2)
  where ϕ : [0, ∞)→[0, ∞) is a continuous and strictly increasing function with ϕ(0) = 0.

• (2)

  A mapping G : K → K is said to be (γ, {k_n})-asymptotically strictly pseudocontractive, if there exist a constant γ ∈ [0,1) and a sequence {k_n}⊂[1, ∞) with k_n → 1 such that

• (3)

  Especially, if there exists γ ∈ [0,1) such that

  $$ (2.4)
  then G : K → K is called a γ-strictly pseudocontractive mapping.

• (4)

  A mapping G : K → K is said to be uniformly L-Lipschitzian, if there exists a constant L > 0, such that

  $$ (2.5)

• (5)

  A mapping G : K → K is said to be semicompact, if for any bounded sequence {x_n} ⊂ K with lim _n→∞∥x_n − Gx_n∥ = 0, there exists a subsequence $$ such that $$ converges strongly to some point x^* ∈ K.

Remark 2.3. If ϕ(λ) = λ²,   λ ≥ 0, and ξ_n = 0, then a (γ, {μ_n}, {ξ_n}, ϕ)-total asymptotically strictly pseudocontractive mapping is an (γ, {k_n})-asymptotically strict pseudocontractive mapping, where {k_n = 1 + μ_n}.

Proposition 2.4. Let G : K → K be a (γ, {μ_n}, {ξ_n}, ϕ)-total asymptotically strictly pseudocontractive mapping. If F(G) ≠ ∅, then for each q ∈ F(G) and for each x ∈ K, the following inequalities hold and they are equivalent:

Proof. (I) Inequality (2.6) can be obtained from (2.2) immediately.

(II) (2.6) ⇔ (2.7) In fact, since

from (2.6) we have that Simplifying it, the inequality (2.7) is obtained.

Conversely, from (2.7) the inequality (2.6) can be obtained immediately.

(III) (2.7) ⇔ (2.8) In fact, since

it follows from (2.7) that Simplifying it, the inequality (2.8) is obtained.

Conversely, the inequality (2.7) can be obtained from (2.8) immediately.

This completes the proof of Proposition 2.4.

Lemma 2.5 (see [11].)Let {a_n}, {b_n}, and {δ_n} be sequences of nonnegative real numbers satisfying

If $$ and $$, then the limit lim _n→∞a_n exists.

Lemma 2.6 (see [12].)Let H be a real Hilbert space. If {x_n} is a sequence in H weakly convergent to z, then

Proposition 2.7. Let H be a real Hilbert space and let T : H → H be a uniformly L-Lipschitzian and γ, {μ_n}, {ξ_n}, ϕ-total asymptotically strictly pseudocontractive mapping. Then the demiclosedness principle holds for T in the sense that if {x_n} is a sequence in H such that x_n⇀x^*, and limsup _m→∞limsup _n→∞∥x_n − T^mx_n∥ = 0, then (I − T)x^* = 0. In particular, if x_n⇀x^*, and ∥(I − T)x_n∥→0, then (I − T)x^* = 0, that is, T is demiclosed at origin.

Proof. Since {x_n} is bounded, we can define a function f on H by

Since x_n⇀x^*, it follows from Lemma 2.6 that In particular, for each m ≥ 1, On the other hand, since T is a (γ, {μ_n}, {ξ_n}, ϕ)-total asymptotically strictly pseudocontraction mapping, we get Taking limsup _m→∞ on both sides and observing the facts that lim _m→∞μ_m = 0, lim _m→∞ξ_m = 0, and limsup _m→∞limsup _n→∞∥x_n − T^mx_n∥ = 0, we derive that On the other hand, it follows from (2.17) that Since κ < 1, this together with (2.19) shows that limsup _m→∞∥x^* − T^mx^*∥² = 0. That is, lim _m→∞T^mx^* = x^*; hence Tx^* = x^*.

Theorem 3.1. Let $$, and ϕ be the same as mentiond before. Let {x_n} be the sequence generated by

where {α_n} is a sequence in [0,1] and γ > 0 is a constant satisfying the following conditions:

• (iv)

  α_n ∈ (δ, 1 − β),   for all  n ≥ 1 and γ ∈ (0, (1 − κ)/∥A∥²), where δ ∈ (0,1 − β) is a positive constant.

  • (I)

    If Γ ≠ ∅ (where Γ is the set of solutions to (SCFP)-(1.1)), then {x_n} converges weakly to a point x^* ∈ Γ.

  • (II)

    In addition, if S is also semicompact, then {x_n} and {u_n} both converge strongly to x^* ∈ Γ.

Proof. The following is the proof of Theorem 3.1.

In fact, since ϕ is a continuous and increasing function, it results that ϕ(λ) ≤ ϕ(M), if λ ≤ M, and ϕ(λ) ≤ M^*λ², if λ ≥ M. In either case, we can obtain that

For any given p ∈ Γ, hence p ∈ C : = F(S), and Ap ∈ Q : = F(T), from (3.1) and (2.7) we have

On the other hand, since In (2.8) taking x = Ax_n,   Gⁿ = Tⁿ,    and   q = Ap and noting Ap ∈ F(T), from (2.8) we have Substituting (3.8) into (3.7), after simplifying it and then substituting the resultant result into (3.5), we have Substituting (3.9) into (3.4) and simplifying it we have where By condition (iii) we have By condition (ii), $$ and $$. Hence it follows from Lemma 2.5 that the following limit exists: Consequently, from (3.10) and (3.13) we have that This together with the condition (iii) implies that It follows from (3.5), (3.13), and (3.16) that the limit lim _n→∞∥u_n − p∥ exists and The conclusion (1) is proved.

(2) Next we prove that

In fact, it follows from (3.1) that

In view of (3.15) and (3.16) we have that Similarly, it follows from (3.1), (3.16), and (3.20) that The conclusion (3.18) is proved.

(3) Next we prove that

In fact, from (3.15) we have

Since S is uniformly L-Lipschitzian continuous, it follows from (3.18) and (3.23) that

Similarly, from (3.16) we have

Since T is uniformly $$-Lipschitzian continuous, by the same way as above, from (3.18) and (3.25), we can also prove that

(4) Finally we prove that x_n⇀x^* and u_n⇀x^* which is a solution of (SCFP)-(1.1).

Since {u_n} is bounded, there exists a subsequence $$ such that $$ (some point in H₁). From (3.22) we have

By Proposition 2.7, S is demiclosed at zero; hence we know that x^* ∈ F(S).

Moreover, from (3.1) and (3.16) we have

Since A is a linear bounded operator, it gets $$. In view of (3.22) we have Again by Proposition 2.7, T is demiclosed at zero, and we have Ax^* ∈ F(T). Summing up the above argument, it shows that x^* ∈ Γ; that is, x^* is a solution to the (SCFP)-(1.1).

Now we prove that x_n⇀x^* and u_n⇀x^*.

Suppose to the contrary, if there exists another subsequence $$ such that $$ with y^* ≠ x^*, then by virtue of (3.2) and the Opial property of Hilbert space, we have

This is a contradiction. Therefore, u_n⇀x^*. By using (3.1) and (3.16), we have

This completes the proof of Theorem 3.1.

Theorem 3.2. Let H₁ and H₂ be two real Hilbert spaces, let A : H₁ → H₂ be a bounded linear operator, let S : H₁ → H₁ be a uniformly L-Lipschitzian and $$-asymptotically strictly pseudocontractive mapping, and let T : H₂ → H₂ be a uniformly $$-Lipschitzian and $$- asymptotically strictly pseudocontractive mapping satisfying the following conditions:

• (i)

  C : = F(S) ≠ ∅, Q : = F(T) ≠ ∅;

• (ii)

  $$, and $$.

Let {x_n} be the sequence defined by (3.1), where {α_n} is a sequence in [0,1] and γ > 0 is a constant satisfying the following condition:

• (iii)

  α_n ∈ (δ, 1 − β), for all n ≥ 1 and γ ∈ (0, (1 − κ)/∥A∥²), where δ ∈ (0,1 − β) is a constant. If Γ ≠ ∅, then the conclusions of Theorem 3.1 still hold.

Theorem 3.3. Let H₁ and H₂ be two real Hilbert spaces, let A : H₁ → H₂ be a bounded linear operator, S : H₁ → H₁ be a uniformly L-Lipschitzian and β-strictly pseudocontractive mapping, and let T : H₂ → H₂ be a uniformly $$-Lipschitzian and κ-strictly pseudocontractive mapping satisfying the following conditions:

• (i)

  C : = F(S) ≠ ∅, Q : = F(T) ≠ ∅;

• (ii)

  T and S both are demiclosed at origin.

Let {x_n} be the sequence generated by where {α_n} is a sequence in [0,1] and γ > 0 is a constant satisfying the following condition:

• (iii)

  α_n ∈ (δ, 1 − β), for all n ≥ 1 and γ ∈ (0, (1 − κ)/∥A∥²), where δ ∈ (0,1 − β) is a constant. If Γ ≠ ∅, then the conclusions of Theorem 3.1 still hold.

Proof. By the same way as given in the proof of Theorems 3.1 and 3.2 and noting that in the case of strictly pseudocontractive mapping the sequence {k_n = 1} in Theorem 3.2. Therefore we can prove that for each p ∈ Γ, the limits lim _n→∞∥x_n − p∥ and lim _n→∞∥u_n − p∥ exist and

In addition, if S is also semicompact, we can also prove that {x_n} and {u_n} both converge strongly to x^*.

Remark 3.4. Theorems 3.1 and 3.2 improve and extend the corresponding results of Censor et al. [4, 5], Yang [7], Moudafi [9, 10], Xu [13], Censor and Segal [14], Masad and Reich [15], and others.