Computing the Fixed Points of Strictly Pseudocontractive Mappings by the Implicit and Explicit Iterations

Abstract

It is known that strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems. In this paper, we devote to study computing the fixed points of strictly pseudocontractive mappings by the iterations. Two iterative methods (one implicit and another explicit) for finding the fixed point of strictly pseudocontractive mappings have been constructed in Hilbert spaces. As special cases, we can use these two methods to find the minimum norm fixed point of strictly pseudocontractive mappings.

1. Introduction

In this paper, we devote to study computing the fixed points of strictly pseudocontractive mappings by the iterations. Our motivations are mainly in two respects.

Motivation 1
Iterative methods for finding fixed points of nonexpansive mappings have received vast investigations due to its extensive applications in a variety of applied areas of inverse problem, partial differential equations, image recovery, and signal processing; see [1–35] and the references therein. It is known [36] that strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems. Therefore it is interesting to develop the algorithms for strictly pseudocontractive mappings.

Motivation 2
In many problems, it is needed to find a solution with minimum norm. In an abstract way, we may formulate such problems as finding a point 𝑥^† with the property 𝑥^†∈𝐶,𝑥^†=min_𝑥∈𝐶‖𝑥‖, (1.1) where 𝐶 is a nonempty closed convex subset of a real Hilbert space 𝐻. A typical example is the least-squares solution to the constrained linear inverse problem [37]. Some related works for finding the minimum-norm solution (or fixed point of nonexpansive mappings) have been considered by some authors. The reader can refer to [38–41].
In the present paper, we present two iterative methods (one implicit and another explicit) for finding the fixed point of strictly pseudocontractive mappings in Hilbert spaces. As special cases, we can use these two methods to find the minimum norm fixed point of strictly pseudocontractive mappings.

2. Preliminaries

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively. Let 𝐶 be a nonempty closed convex subset of 𝐻.

2.1. Some Concepts

Recall that a mapping 𝑇∶𝐶→𝐶 is called nonexpansive, if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦, (2.1) for all 𝑥,𝑦∈𝐶. And a mapping 𝑇∶𝐶→𝐶 is said to be strictly pseudocontractive if there exists a constant 0≤𝜆<1 such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝜆(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦², (2.2) for all 𝑥,𝑦∈𝐶. For such a case, we also say that 𝑇 is a 𝜆-strictly pseudocontractive mapping. It is clear that, in a real Hilbert space 𝐻, (2.2) is equivalent to 𝑇𝑥−𝑇𝑦,𝑥−𝑦≤𝑥−𝑦²−1−𝜆2(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦², (2.3) for all 𝑥,𝑦∈𝐶. It is clear that the class of strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings.

Recall that the nearest point (or metric) projection from 𝐻 onto 𝐶 is defined as follows: for each point 𝑥∈𝐻, 𝑃_𝐶[𝑥] is the unique point in 𝐶 with the property: 𝑥−𝑃_𝐶𝑥≤𝑥−𝑦,𝑦∈𝐶. (2.4) Note that 𝑃_𝐶 is characterized by the inequality: 𝑃_𝐶𝑥∈𝐶,𝑥−𝑃_𝐶𝑥,𝑦−𝑃_𝐶𝑥≤0,𝑦∈𝐶. (2.5) Consequently, 𝑃_𝐶 is nonexpansive.

2.2. Several Useful Lemmas

Lemma 2.1 (see [42]). Let 𝐻 be a real Hilbert space. There holds the following identity: 𝑥+𝑦²=‖𝑥‖²+2𝑥,𝑦+𝑦², (2.6) for all 𝑥,𝑦∈𝐻.

Lemma 2.2 (see [43]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a 𝜆-strict pseudocontraction. Then, (i)𝐹(𝑇) is closed convex so that the projection 𝑃_𝐹(𝑇) is well defined; (ii)𝜅𝐼+(1−𝜅)𝑇 for 𝜅∈[𝜆,1), is nonexpansive.

Lemma 2.3 2.3 (see [42]). Let 𝐶 be a nonempty closed convex of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a 𝜆-strictly pseudocontractive mapping. Then 𝐼−𝑇 is demiclosed at 0 that is if 𝑥_𝑛⇀𝑥∈𝐶 and 𝑥_𝑛−𝑇𝑥_𝑛→0, then 𝑥=𝑇𝑥.

Lemma 2.4 (see [44]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose that 𝑥_𝑛+1=1−𝛽_𝑛𝑦_𝑛+𝛽_𝑛𝑥_𝑛 (2.7) for all 𝑛≥0 and limsup_𝑛→∞𝑦_𝑛+1−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (2.8) Then lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 2.5 (see [45]). Let {𝑎_𝑛}^∞_𝑛=0 be a sequence of nonnegative real numbers satisfying 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛾_𝑛𝜎_𝑛,𝑛≥0, (2.9) where {𝛾_𝑛}^∞_𝑛=0⊂(0,1) and {𝜎_𝑛}^∞_𝑛=0 satisfy (i)∑^∞_𝑛=0𝛾_𝑛=∞,(ii)either limsup_𝑛→∞𝜎_𝑛≤0 or ∑^∞_𝑛=0|𝛾_𝑛𝜎_𝑛|<∞. Then {𝑎_𝑛}^∞_𝑛=0 converges to 0.

We use the following notation: (i)Fix(𝑇) stands for the set of fixed points of 𝑇; (ii)𝑥_𝑛⇀𝑥 stands for the weak convergence of (𝑥_𝑛) to 𝑥; (iii)𝑥_𝑛→𝑥 stands for the strong convergence of (𝑥_𝑛) to 𝑥.

3. Iterations and Convergence Analysis

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 a 𝜆-strictly pseudocontractive mapping with Fix(𝑇)≠∅. Let 𝜅∈(0,1) be a constant. For 𝑢∈𝐻 and any 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence defined by the following implicit manner: 𝑥_𝑛=𝜅𝑇𝑥_𝑛+(1−𝜅)𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛,𝑛≥0. (3.1) Then the sequence {𝑥_𝑛} converges strongly to 𝑃_Fix(𝑇)(𝑢).

Proof. Step 1. The sequence {𝑥_𝑛} is well defined.
Set 𝛽=𝜅/(1−(1−𝜅)𝜆). It is easily to check that 𝛽∈(0,1). Then, we can rewrite (3.1) as 𝑥_𝑛=𝛽(1−𝜆)1−𝛽𝜆𝑇𝑥_𝑛+1−𝛽𝑃1−𝛽𝜆^𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛,𝑛≥0, (3.2) which is equivalent to the following: 𝑥_𝑛=𝛽𝜆𝑥_𝑛+(1−𝜆)𝑇𝑥_𝑛+1−𝛽𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛,𝑛≥0. (3.3) Note that 𝜆𝐼+(1−𝜆)𝑇 is nonexpansive (see Lemma 2.2). For fix 𝑛, we define a mapping 𝑆_𝑛∶𝐶→𝐶 by 𝑆_𝑛𝑥=𝛽(𝜆𝑥+(1−𝜆)𝑇𝑥)+1−𝛽𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥,𝑥∈𝐶. (3.4) For 𝑥,𝑦∈𝐶, we have 𝑆_𝑛𝑥−𝑆_𝑛𝑦=𝛽(𝜆𝐼+(1−𝜆)𝑇)𝑥+1−𝛽𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥−𝛽(𝜆𝐼+(1−𝜆)𝑇)𝑦−1−𝛽𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑦≤𝛽𝑥−𝑦+1−𝛽1−𝛼_𝑛=𝛼𝑥−𝑦1−1−𝛽_𝑛𝑥−𝑦, (3.5) which implies that 𝑆_𝑛 is a self-contraction of 𝐶 for every 𝑛. Hence 𝑆_𝑛 has a unique fixed point 𝑥_𝑛∈𝐶 which is the unique solution of the fixed point equation (3.3).
Step 2. The sequence {𝑥_𝑛} is bounded.
Pick up any 𝑥^∗∈Fix(𝑇). From (3.3), we have 𝑥_𝑛−𝑥^∗=𝛽𝜆𝑥_𝑛+(1−𝜆)𝑇𝑥_𝑛+1−𝛽𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛−𝑥^∗≤𝛽𝜆𝑥_𝑛+(1−𝜆)𝑇𝑥_𝑛−𝑥^∗+1−𝛽𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛−𝑥^∗≤𝛽𝑥_𝑛−𝑥^∗+1−𝛽𝛼_𝑛𝑢−𝑥^∗+1−𝛼_𝑛𝑥_𝑛−𝑥^∗≤𝛽𝑥_𝑛−𝑥^∗+1−𝛽1−𝛼_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛𝑢−𝑥^∗. (3.6) It follows that 𝑥_𝑛−𝑥^∗≤𝑢−𝑥^∗. (3.7)
Hence, {𝑥_𝑛} is bounded and so is {𝑇𝑥_𝑛}.
Step 3. lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0.
From (3.3), we have 𝑥_𝑛−𝑇𝑥_𝑛≤𝛽𝜆𝑥_𝑛−𝑇𝑥_𝑛+1−𝛽𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛−𝑃_𝐶𝑇𝑥_𝑛≤𝛽𝜆𝑥_𝑛−𝑇𝑥_𝑛+1−𝛽𝛼_𝑛𝑢−𝑇𝑥_𝑛+1−𝛼_𝑛𝑥_𝑛−𝑇𝑥_𝑛=1−1−𝛼_𝑛−𝜆𝛽𝑥_𝑛−𝑇𝑥_𝑛+1−𝛽𝛼_𝑛𝑢−𝑇𝑥_𝑛. (3.8)
It follows that 𝑥_𝑛−𝑇𝑥_𝑛≤1−𝛽1−𝛼_𝑛𝛽𝛼−𝜆^𝑛𝑢−𝑇𝑥_𝑛⟶0. (3.9)
Step  4.  𝑥_𝑛→𝑥∈𝑃_Fix(𝑇)(𝑢).
Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛}, which converges weakly to a point 𝑥∈𝐶. Noticing (3.9) we can use Lemma 2.3 to get 𝑥∈Fix(𝑇).
By using the convexity of the norm and Lemma 2.1, for any 𝑥∈Fix(𝑇), we have 𝑥_𝑛−𝑥²=𝛽𝜆𝑥_𝑛+(1−𝜆)𝑇𝑥_𝑛−𝑥+1−𝛽𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛−𝑥²≤𝛽𝜆𝑥_𝑛+(1−𝜆)𝑇𝑥_𝑛−𝑥²+1−𝛽𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛−𝑥²≤𝛽𝑥_𝑛−𝑥²+1−𝛽𝛼_𝑛(𝑢−𝑥)+1−𝛼_𝑛𝑥_𝑛−𝑥²=𝛽𝑥_𝑛−𝑥²+1−𝛽1−𝛼_𝑛²𝑥_𝑛−𝑥²+2𝛼_𝑛1−𝛼_𝑛𝑢−𝑥,𝑥_𝑛−𝑥+𝛼²_𝑛‖𝑢−𝑥‖²=𝛽𝑥_𝑛−𝑥²+1−𝛽𝑥_𝑛−𝑥²−2𝛼_𝑛𝑥_𝑛−𝑥²+2𝛼_𝑛𝑢−𝑥,𝑥_𝑛−𝑥+𝛼²_𝑛‖𝑢−𝑥‖²+𝑥_𝑛−𝑥²−2𝑢−𝑥,𝑥_𝑛−𝑥. (3.10) It turns out that 𝑥_𝑛−𝑥²≤𝑢−𝑥,𝑥_𝑛+𝛼−𝑥^𝑛2‖𝑢−𝑥‖²+𝑥_𝑛−𝑥²+2‖𝑢−𝑥‖𝑥_𝑛≤−𝑥𝑢−𝑥,𝑥_𝑛−𝑥+𝛼_𝑛𝑀, (3.11) where 𝑀>0 is some constant such that 1sup2‖𝑢−𝑥‖²+𝑥_𝑛−𝑥²+2‖𝑢−𝑥‖𝑥_𝑛−𝑥≤𝑀. (3.12) Therefore we can substitute 𝑥 for 𝑥 in (3.11) to get 𝑥_𝑛−𝑥²≤𝑢−𝑥,𝑥_𝑛−𝑥+𝛼_𝑛𝑀. (3.13) However, 𝑥_𝑛⇀𝑥. This together with (3.13) guarantees that 𝑥_𝑛→𝑥. It is clear that 𝑥=𝑃_{𝐹𝑖𝑥(𝑇)}(𝑢). As a matter of fact, in (3.11), if we let 𝑛→∞, then we get ⟨𝑢−𝑥,𝑥−𝑥⟩≥0,∀𝑥∈Fix(𝑇). (3.14) This is equivalent to ⟨𝑢−𝑥,𝑥−𝑥⟩≥0,∀𝑥∈Fix(𝑇). (3.15) Hence, 𝑥=𝑃_Fix(𝑇)(𝑢). Therefore, 𝑥_𝑛→𝑥=𝑃_Fix(𝑇)(𝑢). This completes the proof.

Corollary 3.2. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 a nonexpansive mapping with Fix(𝑇)≠∅. Let 𝜅∈(0,1) be a constant. For 𝑢∈𝐻 and any 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence defined by the following implicit manner: 𝑥_𝑛=𝜅𝑇𝑥_𝑛+(1−𝜅)𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛,𝑛≥0. (3.16) Then the sequence {𝑥_𝑛} converges strongly to 𝑃_Fix(𝑇)(𝑢).

Corollary 3.3. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 a 𝜆-strictly pseudocontractive mapping with Fix(𝑇)≠∅. Let 𝜅∈(0,1) be a constant. For any 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence defined by the following implicit manner: 𝑥_𝑛=𝜅𝑇𝑥_𝑛+(1−𝜅)𝑃_𝐶1−𝛼_𝑛𝑥_𝑛,𝑛≥0. (3.17) Then the sequence {𝑥_𝑛} converges strongly to 𝑃_Fix(𝑇)(0) which is the minimum norm fixed point of 𝑇.

Corollary 3.4. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 a nonexpansive mapping with Fix(𝑇)≠∅. Let 𝜅∈(0,1) be a constant. For any 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence defined by the following implicit manner: 𝑥_𝑛=𝜅𝑇𝑥_𝑛+(1−𝜅)𝑃_𝐶1−𝛼_𝑛𝑥_𝑛,𝑛≥0. (3.18) Then the sequence {𝑥_𝑛} converges strongly to 𝑃_Fix(𝑇)(0) which is the minimum norm fixed point of 𝑇.

Next, we introduce an explicit algorithm for finding the fixed point of 𝑇.

Theorem 3.5. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 a 𝜆-strictly pseudocontractive mapping with Fix(𝑇)≠∅. Let 𝛽 and 𝛿 be two constants in (0,1) satisfying 𝛽+𝛿<1. For 𝑢∈𝐻 and any 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence defined by the following explicit manner: 𝑥_𝑛+1=𝛽𝜆+𝛿𝑥_𝑛+𝛽(1−𝜆)𝑇𝑥_𝑛+1−𝛽−𝛿𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛,𝑛≥0, (3.19) where 𝛼_𝑛∈(0,1) satisfies the following conditions: (𝐶1)lim_𝑛→∞𝛼_𝑛=0, (𝐶2)∑^∞_𝑛=0𝛼_𝑛=∞.
Then the sequence {𝑥_𝑛} converges strongly to 𝑃_Fix(𝑇)(𝑢).

Proof. Step 1. The sequence {𝑥_𝑛} is bounded.
First, we can rewrite (3.19) as 𝑥_𝑛+1=𝛽𝜆𝑥_𝑛+(1−𝜆)𝑇𝑥_𝑛+𝛿𝑥_𝑛+1−𝛽−𝛿𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛,𝑛≥0. (3.20) Take 𝑥^∗∈Fix(𝑇). From (3.20), we have 𝑥_𝑛+1−𝑥^∗=𝛽𝜆𝑥_𝑛+(1−𝜆)𝑇𝑥_𝑛−𝑥^∗+𝛿𝑥_𝑛−𝑥^∗+1−𝛽−𝛿𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛−𝑥^∗≤𝛽𝜆𝑥_𝑛+(1−𝜆)𝑇𝑥_𝑛−𝑥^∗+𝛿𝑥_𝑛−𝑥^∗+1−𝛽−𝛿𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛−𝑥^∗≤𝛽𝑥_𝑛−𝑝+𝛿𝑥_𝑛−𝑥^∗+1−𝛽−𝛿𝛼_𝑛𝑢−𝑥^∗+1−𝛼_𝑛𝑥_𝑛−𝑥^∗=1−1−𝛽−𝛿𝛼_𝑛𝑥_𝑛−𝑥^∗+1−𝛽−𝛿𝛼_𝑛𝑢−𝑥^∗≤max𝑥_𝑛−𝑥^∗,𝑢−𝑥^∗. (3.21) By induction, 𝑥_𝑛+1−𝑥^∗≤max𝑥_𝑛−𝑥^∗,𝑢−𝑥^∗. (3.22) Hence, the sequence {𝑥_𝑛} is bounded and {𝑇𝑥_𝑛} is also bounded.
Step 2. lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0.
We can rewrite (3.20) as 𝑥_𝑛+1=𝛿𝑥_𝑛+(1−𝛿)𝑦_𝑛, (3.23) where 𝑦_𝑛=𝛽1−𝛿𝜆𝑥_𝑛+(1−𝜆)𝑇𝑥_𝑛+1−𝛽−𝛿𝑃1−𝛿^𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛,𝑛≥0. (3.24) It follows that 𝑦_𝑛+1−𝑦_𝑛≤𝛽1−𝛿𝜆𝑥_𝑛+1+(1−𝜆)𝑇𝑥_𝑛+1−𝜆𝑥_𝑛+(1−𝜆)𝑇𝑥_𝑛+1−𝛽−𝛿𝑃1−𝛿^𝐶𝛼_𝑛+1𝑢+1−𝛼_𝑛+1𝑥_𝑛+1−𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛≤𝛽𝑥1−𝛿^𝑛+1−𝑥_𝑛+1−𝛽−𝛿𝛼1−𝛿^𝑛+1−𝛼_𝑛‖𝑢‖+𝑥_𝑛+1−𝛽−𝛿(1−𝛿)1−𝛼_𝑛+1𝑥_𝑛+1−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+1−𝛽−𝛿𝛼1−𝛿^𝑛+1−𝛼_𝑛‖𝑢‖+𝑥_𝑛. (3.25) Thus, limsup_𝑛→∞𝑦_𝑛+1−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤limsup_𝑛→∞1−𝛽−𝛿𝛼1−𝛿^𝑛+1−𝛼_𝑛‖𝑢‖+𝑥_𝑛=0. (3.26) This together with Lemma 2.4 implies that lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (3.27) Note that 𝑥_𝑛−𝑇𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑇𝑥_𝑛=𝑥_𝑛−𝑦_𝑛+𝛽1−𝛿𝜆𝑥_𝑛+(1−𝜆)𝑇𝑥_𝑛+1−𝛽−𝛿𝑃1−𝛿^𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛−𝑇𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝛽𝜆𝑥1−𝛿^𝑛−𝑇𝑥_𝑛+1−𝛽−𝛿𝑃1−𝛿^𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛−𝑇𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝛽𝜆𝑥1−𝛿^𝑛−𝑇𝑥_𝑛+1−𝛽−𝛿𝛼1−𝛿^𝑛𝑢−𝑇𝑥_𝑛+1−𝛽−𝛿1−𝛿1−𝛼_𝑛𝑥_𝑛−𝑇𝑥_𝑛. (3.28) It follows that 𝑥_𝑛−𝑇𝑥_𝑛≤1−𝛿𝛽()𝑥1−𝜆^𝑛−𝑦_𝑛+1−𝛽−𝛿𝛽()𝛼1−𝜆^𝑛𝑢−𝑇𝑥_𝑛. (3.29) Thus, lim_𝑛→∞𝑥_𝑛−𝑇𝑥_𝑛=0. (3.30)
Step 3.  limsup_𝑛→∞⟨𝑢−𝑥,𝑥_𝑛−𝑥⟩≤0, where 𝑥=𝑃_Fix(𝑇)(𝑢).
To see this, we can take a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} satisfying the properties limsup_𝑛→∞𝑢−𝑥,𝑥_𝑛−𝑥=lim_𝑘→∞𝑢−𝑥,𝑥_𝑛𝑘−𝑥, (3.31)𝑥_𝑛𝑘⇀𝑥^∗as𝑘⟶∞. (3.32) By the demiclosed principle (see Lemma 2.3) and (3.30), we have that 𝑥^∗∈Fix(𝑇). So, limsup_𝑛→∞𝑢−𝑥,𝑥_𝑛−𝑥=𝑢−𝑥,𝑥^∗−𝑥≤0. (3.33)Step 4. 𝑥_𝑛→𝑥.
From (3.20), we get 𝑥_𝑛+1−𝑥²≤𝛽𝜆𝑥_𝑛+(1−𝜆)𝑇𝑥_𝑛−𝑥²+𝛿𝑥_𝑛−𝑥²+1−𝛽−𝛿𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛−𝑥²≤𝛽+𝛿𝑥_𝑛−𝑥²+1−𝛽−𝛿𝛼_𝑛(𝑢−𝑥)+1−𝛼_𝑛𝑥_𝑛−𝑥²=𝛽+𝛿𝑥_𝑛−𝑥²+1−𝛽−𝛿1−2𝛼_𝑛𝑥_𝑛−𝑥²+2𝛼_𝑛𝑢−𝑥,𝑥_𝑛−𝑥+𝛼²_𝑛‖𝑢−𝑥‖²+𝑥_𝑛−𝑥²−2𝑢−𝑥,𝑥_𝑛=𝛼−𝑥1−21−𝛽−𝛿_𝑛𝑥_𝑛−𝑥²+21−𝛽−𝛿𝛼_𝑛𝑢−𝑥,𝑥_𝑛+𝛼−𝑥^𝑛2‖𝑢−𝑥‖²+𝑥_𝑛−𝑥²−2𝑢−𝑥,𝑥_𝑛=−𝑥1−𝛿_𝑛𝑥_𝑛−𝑥²+𝛿_𝑛𝜃_𝑛, (3.34) where 𝛿_𝑛=2(1−𝛽−𝛿)𝛼_𝑛 and 𝜃_𝑛=𝑢−𝑥,𝑥_𝑛+𝛼−𝑥^𝑛2‖𝑢−𝑥‖²+𝑥_𝑛−𝑥²−2𝑢−𝑥,𝑥_𝑛−𝑥. (3.35) It is easy to see that lim_𝑛→∞𝛿_𝑛=0 and limsup_𝑛→∞𝜃_𝑛≤0. We can therefore apply Lemma 2.5 to (3.34) and conclude that 𝑥_𝑛+1→𝑥 as 𝑛→∞. This completes the proof.

Corollary 3.6. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 a nonexpansive mapping with Fix(𝑇)≠∅. Let 𝛽 and 𝛿 be two constants in (0,1) satisfying 𝛽+𝛿<1. For 𝑢∈𝐻 and any 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence defined by the following explicit manner: 𝑥_𝑛+1=𝛽𝜆+𝛿𝑥_𝑛+𝛽(1−𝜆)𝑇𝑥_𝑛+1−𝛽−𝛿𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛,𝑛≥0, (3.36) where 𝛼_𝑛∈(0,1) satisfies the following conditions: (𝐶1)lim_𝑛→∞𝛼_𝑛=0, (𝐶2)∑^∞_𝑛=0𝛼_𝑛=∞. Then the sequence {𝑥_𝑛} converges strongly to 𝑃_Fix(𝑇)(𝑢).

Corollary 3.7. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 a 𝜆-strictly pseudocontractive mapping with Fix(𝑇)≠∅. Let 𝛽 and 𝛿 be two constants in (0,1) satisfying 𝛽+𝛿<1. For any 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence defined by the following explicit manner: 𝑥_𝑛+1=𝛽𝜆+𝛿𝑥_𝑛+𝛽(1−𝜆)𝑇𝑥_𝑛+1−𝛽−𝛿𝑃_𝐶1−𝛼_𝑛𝑥_𝑛,𝑛≥0, (3.37) where 𝛼_𝑛∈(0,1) satisfies the following conditions: (𝐶1)lim_𝑛→∞𝛼_𝑛=0, (𝐶2)∑^∞_𝑛=0𝛼_𝑛=∞. Then the sequence {𝑥_𝑛} converges strongly to 𝑃_Fix(𝑇)(0) which is the minimum norm fixed point of 𝑇.

Corollary 3.8. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 a nonexpansive mapping with Fix(𝑇)≠∅. Let 𝛽 and 𝛿 be two constants in (0,1) satisfying 𝛽+𝛿<1. For any 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence defined by the following explicit manner: 𝑥_𝑛+1=𝛽𝜆+𝛿𝑥_𝑛+𝛽(1−𝜆)𝑇𝑥_𝑛+1−𝛽−𝛿𝑃_𝐶1−𝛼_𝑛𝑥_𝑛,𝑛≥0, (3.38) where 𝛼_𝑛∈(0,1) satisfies the following conditions: (𝐶1)lim_𝑛→∞𝛼_𝑛=0; (𝐶2)∑^∞_𝑛=0𝛼_𝑛=∞. Then the sequence {𝑥_𝑛} converges strongly to 𝑃_Fix(𝑇)(0) which is the minimum norm fixed point of  𝑇.

4. Conclusion

Finding fixed points of nonlinear mappings (especially, nonexpansive mappings) has received vast investigations due to its extensive applications in a variety of applied areas of inverse problem, partial differential equations, image recovery and signal processing. It is wellknown that strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems. In this paper, we devote to construct the methods for computing the fixed points of strictly pseudocontractive mappings. Two iterative methods have been presented. Especially, we can use these two methods to find the minimum norm fixed point of strictly pseudocontractive mappings. The ideas contained in the present paper can help us to solve the minimum norm problems in the applied science.

Acknowledgment

The author was supported in part by NSC 100-2221-E-230-012.