A Perturbed Projection Algorithm with Inertial Technique for Split Feasibility Problem

Definition 2.1. Let N be an operator on a Hilbert space H; let N_k, k = 0,1, 2, … be a family of operators on a Hilbert space. {N_k} is said to be convergent to N if ∥N_k(x) − N(x)∥ → 0 as k → +∞ for all x ∈ H.

Definition 2.2. The ρ-distance (ρ ≥ 0) for operators N₁ and N₂ on H is given by

Definition 2.3 (see [13].)Let X be a reflexive Banach space and C and {C_k} _k∈N (N is a set of natural numbers) a sequence of subsets of X. The sequence {C_k} _k∈N is Mosco-convergent to C, denoted by $$, if

where X_s and X_w denote the strong and weak topologies, respectively. In particular, if {C_k} and C are in Rⁿ, then $$ is equivalent to

Definition 2.4. Let C₁ and C₂ be elements in NCCS(Rⁿ). The ρ-distance is defined by

Let C and C_k be sets in NCCS(Rⁿ), then $$ if and only if d_ρ(C_k, C) → 0 for all ρ ≥ 0.

Lemma 2.5 (see [14].)Let {δ_k} and {γ_k} be nonnegative sequences satisfying ∑_kδ_k < +∞ and γ_k+1 ≤ γ_k + δ_k, k = 0,1, …. Then, {γ_k} is a convergent sequence.

Lemma 2.6 (see [10].)Let ϕ_k ∈ [0, ∞), k = 1,2, …, and δ_k ∈ [0, ∞), k = 1,2, … satisfying

•  

  (1) ϕ_k+1 − ϕ_k ≤ θ_k(ϕ_k − ϕ_k−1) + δ_k,

•  

  (2) ∑_kδ_k < ∞,

•  

  (3) θ_k ∈ [0, θ], k = 1,2, …, where θ ∈ [0,1).

Then, {ϕ_k} is a converging sequence and $$, where (t) ₊∶ = max  {t, 0} for any t ∈ R.

Lemma 2.7 (Opial [15]). Let H be a Hilbert space, and let {x^k} be a sequence in H such that there exists a nonempty set S ⊂ H satisfying:

•  

  (1) for every x^* ∈ S,   lim _k ∥x^k − x^*∥ exists,

•  

  (2) any weak cluster point of {x^k} belongs to S.

Then, there exists $$ such that {x^k} weakly converges to $$.

Lemma 2.8 (see [15].)Let H be a Hilbert space, T : H → H a nonexpansive operator, and y a weak cluster point of a sequence {x^k}, and let ∥T(x^k) − x^k∥ → 0. Then y ∈ Fix (T).

Lemma 3.1 (see [7].)Let C and C_k be sets in NCCS(Rⁿ), and let Q and Q_k be sets in NCCS(R^m), for k = 0,1, …, with $$ and $$. Then, the operators N and N_k defined in (3.1) are nonexpansive operators for γ ∈ (0, 2/λ). Moreover, the operator sequence {N_k} converges to N.

Algorithm 3.2. Given arbitrary elements in Rⁿ for k = 0,1, …, let

where $$, α_k ∈ (0,1), for any k and γ ∈ (0, 2/λ) with λ = ρ(A^TA), {θ_k}⊂[0, θ], θ ∈ [0.1).

Theorem 3.3. Let N and N_k for k = 0,1, 2, … be nonexpansive (ne) operators in finite-dimensional Hilbert space, with N_k → N, and let {α_k} be a sequence in (0,1) satisfying

for all ρ > 0. Then, the sequence {x^k} defined by the iterative step converges to a fixed point of N provided that we choose parameter θ_k satisfying whenever such fixed points exist.

Proof. We first prove that the sequence {x^k} is bounded and {∥x^k − z∥} is convergent for all z ∈ Fix (N), where Fix (N) denotes the set of the fixed points of the operator N, that is, N(z) = z. Since N and N_k are ne operators, we have

where $$. From the selection of parameter θ_k, we have It is easy to get By (3.4), (3.8)-(3.10), we obtain from Lemma 2.5 that the sequence {∥x^k − z∥} is convergent and hence the sequence {x^k} is bounded.

We next prove that lim  inf _k→∞ ∥y^k − N(y^k)∥ = 0. Let e^k = N_k(y^k) − N(y^k) and notice the fact that

Then, we have From (3.6), we obtain By (3.12) and observing that $$ (since θ_k ∈ [0,1]), we have Combining 〈a, b〉 = −(1/2)∥a−b∥² + (1/2)∥a∥² + (1/2)∥b∥² with (3.14), we get We have known that the sequence {x^k} is bounded and {∥x^k − z∥} is convergent; hence, there exist $$ and G > 0 such that ∥x^k∥ ≤ ρ and ∥x^k − z∥ ≤ G for all k.

Thus

Moreover, one has that Denoting we get Similarly, from the selection of parameter θ_k, we have It is easy to get Both (3.10) and (3.21) manifest Then, from (3.4), (3.21), and (3.22), we get According to Lemma 2.6, we obtain $$.

From (3.17), it follows that

Because of (3.3) $$, we conclude that lim  inf _k→∞ ∥y^k − N(y^k)∥ = 0.

Finally, we prove that {x^k} converges to a fixed point of N. From the above computation, we know that the sequence {y^k} is also bounded; hence there exist x^* and a subsequence of {y^k} (denoted $$) such that

From Lemmas 2.7 and 2.8, we have x^* ∈ Fix (N). It is easy to obtain that $$, because ∥y^k − x^k∥ = θ_k∥x^k − x^k−1∥ → 0 by (3.10). Since {∥x^k − x^*∥} is convergent, it follows that $$. The proof is completed.

Remark 1. Since the current value of ∥x^k − x^k−1∥ is known when choosing the parameter θ_k, then θ_k is well defined in Theorem 3.3. In fact, from the process of proof for the Theorem 3.3, we can get the following assert: the convergence result of Theorem 3.3 always holds provided that we select θ_k ∈ [0, θ], θ ∈ [0,1), for  all  k ≥ 0 with

Theorem 3.4. Let the hypotheses in Lemma 3.1 be satisfied. Then the sequence {x^k} generated by (3.2) converges to a fixed point of N if

where parameter θ_k is satisfying (3.10) and (3.21).

Proof. For any y ∈ Rⁿ, ∥y∥ ≤ ρ, by reason of the nonexpansive properties of projection, we have

Obviously, where $$.

Since $$ for any given $$, the result of this theorem can be obtained using Theorem 3.3.