
In this paper, three G-nonexpansive mappings are implemented and analyzed using an efficient modified three-step iteration algorithm. Assuming coordinate-convexity in a uniformly convex Banach space endowed with a directed graph, conditions for the weak and strong convergence of the scheme are determined. We give numerical comparisons to back up our main theorem, and compare our algorithm's convergence behavior to that of the three-step Noor iteration and SP-iteration. We use our proposed algorithm to solve image deblurring problems as an application. In addition, we discuss a novel approach to signal recovery in situations where the type of noise is unknown.
Citation: Damrongsak Yambangwai, Tanakit Thianwan. An efficient iterative algorithm for common solutions of three G-nonexpansive mappings in Banach spaces involving a graph with applications to signal and image restoration problems[J]. AIMS Mathematics, 2022, 7(1): 1366-1398. doi: 10.3934/math.2022081
[1] | Damrongsak Yambangwai, Chonjaroen Chairatsiripong, Tanakit Thianwan . Iterative manner involving sunny nonexpansive retractions for nonlinear operators from the perspective of convex programming as applicable to differential problems, image restoration and signal recovery. AIMS Mathematics, 2023, 8(3): 7163-7195. doi: 10.3934/math.2023361 |
[2] | Maryam Iqbal, Afshan Batool, Aftab Hussain, Hamed Al-Sulami . A faster iterative scheme for common fixed points of G-nonexpansive mappings via directed graphs: application in split feasibility problems. AIMS Mathematics, 2024, 9(5): 11941-11957. doi: 10.3934/math.2024583 |
[3] | Lu-Chuan Ceng, Yeong-Cheng Liou, Tzu-Chien Yin . On Mann-type accelerated projection methods for pseudomonotone variational inequalities and common fixed points in Banach spaces. AIMS Mathematics, 2023, 8(9): 21138-21160. doi: 10.3934/math.20231077 |
[4] | Nipa Jun-on, Raweerote Suparatulatorn, Mohamed Gamal, Watcharaporn Cholamjiak . An inertial parallel algorithm for a finite family of G-nonexpansive mappings applied to signal recovery. AIMS Mathematics, 2022, 7(2): 1775-1790. doi: 10.3934/math.2022102 |
[5] | Shahram Rezapour, Maryam Iqbal, Afshan Batool, Sina Etemad, Thongchai Botmart . A new modified iterative scheme for finding common fixed points in Banach spaces: application in variational inequality problems. AIMS Mathematics, 2023, 8(3): 5980-5997. doi: 10.3934/math.2023301 |
[6] | Aftab Hussain, Danish Ali, Amer Hassan Albargi . On assessing convergence and stability of a novel iterative method for fixed-point problems. AIMS Mathematics, 2025, 10(7): 15333-15357. doi: 10.3934/math.2025687 |
[7] | Hamza Bashir, Junaid Ahmad, Walid Emam, Zhenhua Ma, Muhammad Arshad . A faster fixed point iterative algorithm and its application to optimization problems. AIMS Mathematics, 2024, 9(9): 23724-23751. doi: 10.3934/math.20241153 |
[8] | Junaid Ahmad, Kifayat Ullah, Reny George . Numerical algorithms for solutions of nonlinear problems in some distance spaces. AIMS Mathematics, 2023, 8(4): 8460-8477. doi: 10.3934/math.2023426 |
[9] | Liliana Guran, Khushdil Ahmad, Khurram Shabbir, Monica-Felicia Bota . Computational comparative analysis of fixed point approximations of generalized α-nonexpansive mappings in hyperbolic spaces. AIMS Mathematics, 2023, 8(2): 2489-2507. doi: 10.3934/math.2023129 |
[10] | Buthinah A. Bin Dehaish, Rawan K. Alharbi . On fixed point results for some generalized nonexpansive mappings. AIMS Mathematics, 2023, 8(3): 5763-5778. doi: 10.3934/math.2023290 |
In this paper, three G-nonexpansive mappings are implemented and analyzed using an efficient modified three-step iteration algorithm. Assuming coordinate-convexity in a uniformly convex Banach space endowed with a directed graph, conditions for the weak and strong convergence of the scheme are determined. We give numerical comparisons to back up our main theorem, and compare our algorithm's convergence behavior to that of the three-step Noor iteration and SP-iteration. We use our proposed algorithm to solve image deblurring problems as an application. In addition, we discuss a novel approach to signal recovery in situations where the type of noise is unknown.
Fixed point theory is an extremely active research area because of its multifaceted applications. The typical situation is that a self-map on a set admits a fixed point under certain conditions. In a complete metric space, Banach [1] proved the existence of unique fixed point for contractions. The Banach contraction principle has been generalized and extended in several ways due to its applications in mathematics and other related disciplines. The most recent version of the theorem was proved in Banach spaces endowed with a graph G, where G=(V(G),E(G)) is a directed graph such that the set V(G) of its vertices of a graph and the set E(G) of its edges contains all loops.
In order to extend the Banach contraction principle, by combination of the concepts in fixed point theory and graph theory, Banach G-contractions were introduced by Jachymaski [2] in complete metric spaces accompanied with a graph G which has a convex subset of the underlying metric space as its set of vertices, see e.g., [3,4,5,6,7,8,9,10].
In the last few decades finding of fixed points by some iterative schemes for G-contraction, G-nonexpansive and G-monotone nonexpansive mappings have been studied extensively by various authors In 2012, Aleomraninejad et al. [11] presented some iterative scheme for G-contraction and G-nonexpansive mappings in a metric space with a directed graph and stated the convergence for such mappings. In 2015, Alfuraidan and Khamsi [12] defined the concept of G-monotone nonexpansive multivalued mappings defined on a hyperbolic metric space with a graph. Alfuraidan [13] studied the existence of fixed points of monotone nonexpansive mappings on a Banach space endowed with a directed graph. Tiammee et al. [14] proved the Browder convergence theorem for G-nonexpansive mappings and studied the convergence of the Halpern iteration process to the projection of initial point where the projection is onto the set of fixed points of the G-nonexpansive mapping in Hilbert space with the directed graph. Tripak [15] defined the Ishikawa type iteration process for two G-nonexpansive mappings and gave some weak and strong convergence theorems of such iterations in a Banach space endowed with a graph.
The three-step iterative scheme, usually called the Noor iteration method, [16] is used to approximate common fixed point of three G-nonexpansive mappings via
zn=(1−γn)xn+γnT3xn,yn=(1−βn)xn+βnT2zn,xn+1=(1−αn)xn+αnT1yn,n≥0, | (1.1) |
where {αn}, {βn} and {γn} are sequences in [0,1].
Glowinski and Le Tallec [17] employed three-step iterative approach in elastoviscoplasticity, eigenvalue computation and theory of liquid crystals. In [17], it was shown that the three-step iterative process yields better numerical results than the estimated iterations in two and one steps. In 1998, Haubruge, Nguyen and Strodiot [18] investigated the convergence analysis of three-step methods of Glowinski and Le Tallec [17] and applied these methods to obtain new splitting-type algorithms for solving variation inequalities, separable convex programming and minimization of a sum of convex functions. They also proved that three-step iterations lead to highly parallelized algorithms under certain conditions.
Recently, Sridarat et al. [19] modified the SP-iteration process for three G-nonexpansive mappings T1,T2 and T3 as follows:
zn=(1−γn)xn+γnT3xn,yn=(1−βn)zn+βnT2zn,xn+1=(1−αn)yn+αnT1yn,n≥0, | (1.2) |
where {αn}, {βn} and {γn} are appropriate real sequences in [0,1]. They studied the weak and strong convergence of the iterative scheme (1.2) under proper conditions in a uniformly convex Banach space endowed with a graph.
The main purpose of this paper is to construct an efficient modified three-step iteration algorithm for approximating common fixed points of three G-nonexpansive mappings in a uniformly convex Banach space endowed with a graph. This paper is organized as follows.
The notations, basic definitions, and some lemmas for proving our main result are given in Section 2. Our main result is presented in Section 3. In this section, we also prove weak and strong convergence results of the proposed method under mild conditions in a uniformly convex Banach space endowed with a graph. In Section 4, we study the convergence behavior of our method, and compare its efficiency with competing methods. In Section 5, we apply our proposed method to solve certain image deblurring and signal recovering problems.
In this section, we recall a few basic notions concerning the connectivity of graphs. All of these notions can be found, e.g., in [20].
Let C be a nonempty subset of a real Banach space X. We identify the graph G with the pair (V(G),E(G)), where the set V(G) of its vertices coincide with set C and the set of edges E(G) contains {(x,x):x∈C}. Also, G is such that no two edges are parallel. A mapping T:C→C is said to be G-contraction if T preserves edges of G (or T is edge-preserving), i.e.,
(x,y)∈E(G)⇒(Tx,Ty)∈E(G), |
and T decreases weights of edges of G in the following way: there exists α∈(0,1) such that
(x,y)∈E(G)⇒‖Tx−Ty‖≤α‖x−y‖. |
A mapping T:C→C is said to be G-nonexpansive (see [12], Definition 2.3 (iii)) if T preserves edges of G, i.e.,
(x,y)∈E(G)⇒(Tx,Ty)∈E(G), |
and T non-increases weights of edges of G in the following way:
(x,y)∈E(G)⇒‖Tx−Ty‖≤‖x−y‖. |
If x and y are vertices in a graph G, then a path in G from x to y of length N (N∈N∪{0}) is a sequence {xi}Ni=0 of N+1 vertices such that x0=x, xN=y and (xi,xi+1)∈E(G) for i=0,1,…N−1. A graph G is connected if there is a path between any two vertices. A directed graph G=(V(G),E(G)) is said to be transitive if, for any x,y,z∈V(G) such that (x,y) and (y,z) are in E(G), we have (x,z)∈E(G). We denote G−1 the conversion of a graph G and
E(G−1)={(x,y)∈X×X:(y,x)∈E(G)}. |
Let x0∈V(G) and A a subset of V(G). We say that A is dominated by x0 if (x0,x)∈E(G) for all x∈A. A dominates x0 if for each x∈A, (x,x0)∈E(G).
In this paper, we use → and ⇀ to denote the strong convergence and weak convergence, respectively.
A mapping T:C→C is said to be G-demiclosed at 0 if, for any sequence {xn} in C such that (xn,xn+1)∈E(G), xn⇀x and Txn→0 imply Tx=0.
Let X be a Banach space with dimension X≥2. The modulus of X is the function δX:(0,2]→[0,1] defined by
δX=inf{1−‖12(x+y)‖:‖x‖=1,ϵ=‖x−y‖}. |
Banach space X is uniformly convex if and only if δX(ϵ)>0 for all ϵ∈(0,2].
Recall that a Banach space X is said to satisfy Opial's condition [21] if xn⇀x and x≠y implying that
lim supn→∞‖xn−x‖<lim supn→∞‖xn−y‖. |
Let C be a nonempty closed convex subset of a real uniformly convex Banach space X. Recall that the mappings Ti(i=1,2,3) on C are said to satisfy condition (C) [19] if there exists a nondecreasing function f:[0,∞)→[0,∞) with f(0)=0 and f(r)>0 for all r>0 such that for all x∈C,
max{‖x−T1x‖,‖x−T2x‖,‖x−T3x‖}≥f(d(x,F)), |
where F=F(T1)∩F(T2)∩F(T3), F(Ti)(i=1,2,3) are the sets of fixed points of Ti and d(x,F)=inf{‖x−q‖:q∈F}.
Let C be a subset of a metric space (X,d). A mapping T:C→C is semi-compact [22] if for a sequence {xn} in C with limn→∞d(xn,Txn)=0, there exists a subsequence {xnj} of {xn} such that xnj→p∈C.
Let C be a nonempty subset of a normed space X and let G=(V(G),E(G)) be a directed graph such that V(G)=C. Then, C is said to have Property G (see [19]) if for each sequence in C converging weakly to x∈C and (xn,xn+1)∈E(G), there is a subsequence {xnj} of {xn} such that (xnj,x)∈E(G) for all j∈N.
Remark 1. If G is transitive, then Property G is equivalent to the property: if {xn} is a sequence in C with (xn,xn+1)∈E(G) such that for any subsequence {xnj} of the sequence {xn} converging weakly to x in X, then (xn,x)∈E(G) for all n∈N.
Definition 1. ([23], Definition 3.1) Let X be a vector space and D be a nonempty subset of X×X. Then D is said to be coordinate-convex if for all (p,u),(p,v),(u,p),(v,p)∈D and for all t∈[0,1], we have
t(p,u)+(1−t)(p,v)∈Dandt(u,p)+(1−t)(v,p)∈D. |
Remark 2. If D is convex in X×X, then D is coordinate-convex in X×X.
In the sequel, the following lemmas are needed to prove our main results.
Lemma 1. ([24]) Let X be a uniformly convex Banach space, and {αn} a sequence in [δ,1−δ] for some δ∈(0,1). Suppose that sequences {xn} and {yn} in X are such that lim supn→∞||xn||≤c,lim supn→∞||yn||≤c and lim supn→∞||αnxn+(1−αn)yn||=c for some c≥0. Then limn→∞||xn−yn||=0.
Lemma 2. ([25]) Let X be a Banach space that satisfies Opial's condition and let {xn} be a sequence in X. Let u,v∈X be such that limn→∞||xn−u|| and limn→∞||xn−v|| exist. If {xnj} and {xnk} are subsequences of {xn} that converge weakly to u and v, respectively, then u=v.
The following lemmas show some useful Property G of C on a uniformly convex Banach space.
Lemma 3. ([19]) Suppose that X is a Banach space having Opial's condition, C has Property G and let T:C→C be a G-nonexpansive mapping. Then, I−T is G-demiclosed at 0, i.e., if xn⇀x and xn−Txn→0, then x∈F(T), where F(T) is the set of fixed points of T.
Lemma 4. ([26]) Let C be a nonempty closed convex subset of a uniformly convex Banach space X and suppose that C has Property G. Let T be a G-nonexpansive mapping on C. Then I−T is G-demiclosed at 0.
The following useful result is due to [27].
Lemma 5. ([27]) Let {xn} be a bounded sequence in a reflexive Banach space X. If for any weakly convergent subsequence {xnj} of {xn}, both {xnj} and {xnj+1} converge weakly to the same point in X, then the sequence {xn} is weakly convergent.
Throughout the section, we let C be a nonempty closed convex subset of a Banach space X endowed with a directed graph G such that V(G)=C and E(G) is coordinate-convex. We also suppose that the graph G is transitive. The mappings Ti (i=1,2,3) are G-nonexpansive from C to C with F=F(T1)∩F(T2∩F(T3) nonempty. For an arbitrary x0∈C, defined the sequence {xn} by
zn=(1−γn)xn+γnT3xn,yn=(1−βn)T3xn+βnT2zn,xn+1=(1−αn)T1yn+αnT2zn,n≥0, | (3.1) |
where {αn}, {βn} and {γn} are appropriate real sequences in [0,1].
Proposition 1. Let q0∈F be such that (x0,q0),(q0,x0) are in E(G). Then (xn,q0),(zn,q0),(yn,q0), (q0,xn),(q0,zn),(q0,yn),(xn,yn),(xn,zn) and (xn,xn+1) are in E(G).
Proof. We proceed by induction. Since T3 is edge-preserving and (x0,q0)∈E(G), we have (T3x0,q0)∈E(G). Note that
(z0,q0)=((1−γ0)x0+γ0T3x0,q0)=(1−γ0)(x0,q0)+γ0(T3x0,q0). |
Using (x0,q0),(T3x0,q0)∈E(G) and the coordinate-convexity of E(G), we have (z0,q0)∈E(G). Again, by edge-preserving of T2 and (z0,q0)∈E(G), we have (T2z0,q0)∈E(G). We also have
(y0,q0)=((1−β0)T3x0+β0T2z0,q0)=(1−β0)(T3x0,q0)+β0(T2z0,q0). |
Since (T3x0,q0),(T2z0,q0)∈E(G) by the coordinate-convexity of E(G), we have (y0,q0)∈E(G). Then, since T1 is edge-preserving, we get (T1y0,q0)∈E(G). Note that
(x1,q0)=((1−α0)T1y0+α0T2z0,q0)=(1−α0)(T1y0,q0)+α0(T2z0,q0). |
By the coordinate-convexity of E(G) and (T2z0,q0),(T1y0,q0)∈E(G), we get (x1,q0)∈E(G). Thus, by edge-preserving of T3, (T3x1,q0)∈E(G). Again, by the coordinate-convexity of E(G) and (T3x1,q0),(x1,q0)∈E(G), we have
(z1,q0)=((1−γ1)x1+γ1T3x1,q0)=(1−γ1)(x1,q0)+γ1(T3x1,q0)∈E(G). |
Then, since T2 is edge-preserving, (T2z1,q0)∈E(G). By the coordinate-convexity of E(G) and (T2z1,q0),(T3x1,q0)∈E(G), we get
(y1,q0)=((1−β1)T3x1+β1T2z1,q0)=(1−β1)(T3x1,q0)+β1(T2z1,q0)∈E(G), |
and hence, (T1y1,q0)∈E(G).
Next, we assume that (xk,q0)∈E(G). Since T3 is edge-preserving, we get (T3xk,q0)∈E(G). Thus
(zk,q0)=((1−γk)xk+γkT3xk,q0)=(1−γk)(xk,q0)+γk(T3xk,q0)∈E(G), |
since E(G) is coordinate-convex. Hence, by edge-preserving of T2 and (zk,q0)∈E(G), we have (T2zk,q0)∈E(G). By the coordinate-convexity of E(G) and (T3x1,q0),(T2zk,q0)∈E(G), we get
(yk,q0)=((1−βk)T3xk+βkT2zk,q0)=(1−βk)(T3xk,q0)+βk(T2zk,q0)∈E(G). |
Also, since T1 is edge-preserving and (yk,q0)∈E(G), we get (T1yk,q0)∈E(G). By the coordinate-convexity of E(G) and (T2zk,q0),(T1yk,q0)∈E(G), we obtain
(xk+1,q0)=(1−αk)(T1yk+αkT2zk,q0)=(1−αk)(T1yk,q0)+αk(T2zk,q0)∈E(G). |
Hence, by edge-preserving of T3, we get (T3xk+1,q0)∈E(G), and so (zk+1,q0)=((1−γk+1)xk+1+γk+1T3xk+1,q0)=(1−γk+1)(xk+1,q0) +γk+1(T3xk+1,q0)∈E(G), since E(G) is coordinate-convex. Again, by edge-preserving of T2, we obtain (T2zk+1,q0)∈E(G). By the coordinate-convexity of E(G) and (T3xk+1,q0), (T2zk+1,q0) ∈E(G), we obtain (yk+1,q0)=((1−βk+1)T3xk+1+βk+1T2zk+1,q0)=(1−βk+1)(T3xk+1,q0)+βk+1(T2zk+1,q0)∈E(G). Therefore, (xn,q0),(zn,q0), (yn,q0)∈E(G) for all n≥1.
Since T3 is edge-preserving and (q0,x0)∈E(G), we have (q0,T3x0)∈E(G), and so (q0,z0)=(q0,(1−γ0)x0+γ0T3x0)=(1−γ0)(q0,x0)+γ0(q0,T3x0)∈E(G). Again, since T2 is edge-preserving and (q0,z0)∈E(G), we have (q0,T2z0) ∈E(G). By the coordinate-convexity of E(G) and (q0,T3x0),(q0,T2z0) ∈E(G), so (q0,y0)=(q0,(1−β0)T3x0+β0T2z0)=(1−β0)(q0,T3x0)+β0(q0,T2z0) ∈E(G).
Using a similar argument, we can show that (q0,xn),(q0,zn),(q0,yn)∈E(G) under the assumption that (q0,x0),(q0,z0), (q0,y0)∈E(G). By transitivity of G, we get (xn,yn),(xn,zn),(xn,xn+1)∈E(G). This completes the proof.
Lemma 6. Let X be a uniformly convex Banach space. Suppose that {αn}, {βn} and {γn} are real sequences in [δ,1−δ] for some δ∈(0,1) and (x0,q0), (q0,x0)∈E(G) for arbitrary x0∈C and q0∈F. Then
(i) limn→∞‖xn−q0‖ exists;
(ii) limn→∞‖xn−T1xn‖=limn→∞‖xn−T2xn‖=limn→∞‖xn−T3xn‖=0.
Proof. (i) Let q0∈F. By Proposition 1, we have (xn,q0),(yn,q0) and (zn,q0)∈E(G). Then, by G-nonexpansiveness of Ti(i=1,2,3) and using (3.1), we have
‖zn−q0‖=‖(1−γn)xn+γnT3xn−q0‖=‖(1−γn)(xn−q0)+γn(T3xn−q0)‖≤(1−γn)‖xn−q0‖+γn‖T3xn−q0‖≤(1−γn)‖xn−q0‖+γn‖xn−q0‖=‖xn−q0‖, | (3.2) |
‖yn−q0‖=‖(1−βn)T3xn+βnT2zn−q0‖=‖(1−βn)(T3xn−q0)+βn(T2zn−q0)‖≤(1−βn)‖T3xn−q0‖+βn‖T2zn−q0‖≤(1−βn)‖xn−q0‖+βn‖zn−q0‖≤(1−βn)‖xn−q0‖+βn‖xn−q0‖=‖xn−q0‖, | (3.3) |
and so
‖xn+1−q0‖=‖(1−αn)T1yn+αnT2zn−q0‖=‖(1−αn)(T1yn−q0)+αn(T2zn−q0)‖≤(1−αn)‖T1yn−q0‖+αn‖T2yn−q0‖≤(1−αn)‖yn−q0‖+αn‖zn−q0‖≤(1−αn)‖xn−q0‖+αn‖xn−q0‖≤‖xn−q0‖. | (3.4) |
As {‖xn−q0‖:n∈N} is decreasing sequence which is obviously bounded below. Therefore limn→∞‖xn−q0‖ exists. In particular, the sequence {xn} is bounded.
(ii) Assume that limn→∞‖xn−q0‖=c. If c=0, then by G-nonexpansiveness of Ti (i=1,2,3), we get
‖xn−Tixn‖≤‖xn−q0‖+‖q0−Tixn‖≤‖xn−q0‖+‖q0−xn‖. |
Therefore, the result follows. Suppose that c>0. Taking the lim sup on both sides in the inequality (3.3), we obtain
lim supn→∞‖yn−q0‖≤lim supn→∞‖xn−q0‖=c. | (3.5) |
In addition, by G-nonexpansiveness of T1, we have ‖T1yn−q0‖≤‖yn−q0‖, taking the lim sup on both sides in this inequality and using (3.5), we obtain
lim supn→∞‖T1yn−q0‖≤c. | (3.6) |
Taking the lim sup on both sides in the inequality (3.2), we obtain
lim supn→∞‖zn−q0‖≤lim supn→∞‖xn−q0‖=c. | (3.7) |
In addition, by G-nonexpansiveness of T2, we have ‖T2zn−q0‖≤‖zn−q0‖, taking the lim sup on both sides in this inequality and using (3.7), we obtain
lim supn→∞‖T2zn−q0‖≤c. | (3.8) |
Since limn→∞‖xn+1−q0‖=c. Letting n→∞ in the inequality (3.4), we have
limn→∞‖(1−αn)(T1yn−q0)+αn(T2zn−q0)‖=c. | (3.9) |
By using (3.6), (3.8) and (3.9) and Lemma 1, we have
limn→∞‖T2zn−T1yn‖=0. | (3.10) |
Moreover, we see that
‖xn+1−q0‖=‖(1−αn)T1yn+αnT2zn−q0‖≤(1−αn)‖T1yn−q0‖+αn‖T2zn−q0‖≤(1−αn)(‖T1yn−T2zn‖+‖T2zn−q0‖)+αn(‖T2zn−q0‖)=(1−αn)‖T1yn−T2zn‖+(1−αn)‖T2zn−q0‖+αn(‖T2zn−q0‖)=(1−αn)‖T1yn−T2zn‖+‖T2zn−q0‖. | (3.11) |
Using (3.10) and (3.11), we have
lim infn→∞‖T2zn−q0‖≥c, | (3.12) |
and so by (3.8) and (3.12), we have
limn→∞‖T2zn−q0‖=c. | (3.13) |
On the other hand,
‖T2zn−q0‖≤‖T2zn−T1yn‖+‖T1yn−q0‖≤‖T2zn−T1yn‖+‖yn−q0‖, |
and this yields to
lim infn→∞‖yn−q0‖≥c. | (3.14) |
From (3.5) and (3.14),
limn→∞‖yn−q0‖=c. | (3.15) |
In addition, by G-noneapansives of T3 we have ‖T3xn−q0‖≤‖xn−q0‖, taking the lim sup on both sides in this inequality, we obtain
lim supn→∞‖T3xn−q0‖≤c. | (3.16) |
By (3.3) and (3.15), we have
limn→∞‖(1−βn)(T3xn−q0)+βn(T2zn−q0)‖=c. | (3.17) |
Using (3.8), (3.16) and (3.17) and Lemma 1,
limn→∞‖T2zn−T3xn‖=0. | (3.18) |
Moreover,
‖xn+1−q0‖=‖(1−αn)T1yn+αnT2zn−q0‖≤(1−αn)‖T1yn−q0‖+αn‖T2zn−q0‖≤(1−αn)‖T1yn−q0‖+αn(‖T2zn−T1yn‖+‖T1yn−q0‖)=(1−αn)‖T1yn−q0‖+αn‖T2zn−T1yn‖+αn‖T1yn−q0‖=‖T1yn−q0‖+αn‖T2zn−T1yn‖. | (3.19) |
Using (3.10) and (3.19), we have
lim infn→∞‖T1yn−q0‖≥c. | (3.20) |
and so by (3.6) and (3.20), we get
limn→∞‖T1yn−q0‖=c. | (3.21) |
On the other hand,
‖T1yn−q0‖≤‖T1yn−T2zn‖+‖T2zn−q0‖≤‖T1yn−T2zn‖+‖zn−q0‖, |
and this yields to
lim infn→∞‖zn−q0‖≥c. | (3.22) |
From (3.7) and (3.22), we get
limn→∞‖zn−q0‖=c. | (3.23) |
From (3.2) and (3.23), we have
limn→∞‖(1−γn)(xn−q0)+γn(T3xn−q0)‖=c. | (3.24) |
By using (3.16), (3.24) and Lemma 1,
limn→∞‖T3xn−xn‖=0. | (3.25) |
Thus, it follows from (3.25) that
‖zn−xn‖=‖(1−γn)xn+γnT3xn−xn‖≤γn‖T3xn−xn‖→0 (as n→∞). | (3.26) |
Using (3.18), (3.25) and (3.26) together with G-nonexpansiveness of T2,
‖T2xn−xn‖=‖T2xn−T2zn+T2zn−xn‖≤‖xn−zn‖+‖T2zn−T3xn‖+‖T3xn−xn‖→0 (as n→∞). | (3.27) |
From (3.18) and (3.25), we have
‖yn−xn‖=‖(1−βn)(T3xn−xn)+βn(T2zn−xn)‖≤(1−βn)‖T3xn−xn‖+βn‖T2zn−xn‖≤(1−βn)‖T3xn−xn‖+βn(‖T2zn−T3xn‖+‖T3xn−xn‖)=(1−βn)‖T3xn−xn‖+βn‖T2zn−T3xn‖+βn‖T3xn−xn‖=‖T3xn−xn‖+βn‖T2zn−T3xn‖→0 (as n→∞). | (3.28) |
Using (3.10), (3.18), (3.25) and (3.28) together with G-nonexpansiveness of T1, we have
‖T1xn−xn‖≤‖T1xn−T1yn‖+‖T1yn−xn‖≤‖xn−yn‖+‖T1yn−T2zn‖+‖T2zn−T3xn‖+‖T3xn−xn‖→0 (as n→∞). |
Therefore, we conclude that limn→∞‖xn−T1xn‖=limn→∞‖xn−T2xn‖=limn→∞‖xn−T3xn‖=0. This completes the proof.
We now prove the weak convergence of the sequence generated by the new modified three-step iteration method (3.1) for three G-nonexpansive mappings in a uniformly convex Banach space satisfying Opial's condition.
Theorem 1. Let X be a uniformly convex Banach space which satisfies Opial's condition and C has Property G. Suppose that {αn}, {βn} and {γn} are real sequences in [δ,1−δ] for some δ∈(0,1). If (x0,q0),(q0,x0)∈E(G) for arbitrary x0∈C and q0∈F, then {xn} converges weakly to a common fixed point of T1,T2 and T3.
Proof. Let q0∈F be such that (x0,q0),(q0,x0)∈E(G). From Lemma 6 (i), limn→∞‖xn−q0‖ exists, thus {xn} is bounded. It follows from Lemma 6 (ii) that limn→∞‖xn−T1xn‖=limn→∞‖xn−T2xn‖=limn→∞‖xn−T3xn‖=0. Since X is uniformly convex and {xn} is bounded, we may assume that xn⇀u as n→∞, without loss of generality. By Lemma 3, we have u∈F. Suppose that subsequences {xnk} and {xnj} of {xn} converge weakly to u and v, respectively. By Lemma 6 (ii), we obtain that ‖xnk−Tixnk‖→0 and ‖xnj−Tixnj‖→0 as k,j→∞. Using Lemma 3, we have u,v∈F. By Lemma 6 (i), limn→∞‖xn−u‖ and limn→∞‖xn−v‖ exist. It follows from Lemma 3 that u=v. Therefore, {xn} converges weakly to a common fixed point of T1, T2 and T3.
It is worth noting that the Opial's condition has remained crucial in proving weak convergence theorems. However, each lp (1≤p<∞) satisfies the Opial's condition, while Lp do not have the property unless p=2.
Next, we deal with the weak convergence of the sequence {xn} generated by (3.1) for two G-nonexpansive mappings without assuming the Opial's condition in a uniformly convex Banach space with a directed graph.
Theorem 2. Let X be a uniformly convex Banach space. Suppose that C has Property G, {αn},{βn} and {γn} are real sequences in [δ,1−δ] for some δ∈(0,1), F is dominated by x0 and F dominates x0. If (x0,q0),(q0,x0)∈E(G) for arbitrary x0∈C and q0∈F, then {xn} converges weakly to a common fixed point of T1, T2 and T3.
Proof. Let q0∈F be such that (x0,q0),(q0,x0) are in E(G). From Lemma 6 (i), limn→∞‖xn−q0‖ exists, so {xn} is bounded in C. Since C is nonempty closed convex subset of a uniformly convex Banach space X, it is weakly compact and hence there exists a subsequence {xnj} of the sequence {xn} such that {xnj} converges weakly to some point p∈C. By Lemma 6 (ii) we obtain that
limj→∞‖xnj−T1xnj‖=limj→∞‖xnj−T2xnj‖=limj→∞‖xnj−T3xnj‖=0. | (3.29) |
In addition, ‖yn−zn‖≤‖yn−xn‖+‖xn−zn‖. Using (3.26) and (3.28), we have
limn→∞‖yn−zn‖=0. | (3.30) |
From (3.26) and G-nonexpansiveness of T2,
‖T2zn−zn‖≤‖T2zn−T2xn‖+‖T2xn−xn‖+‖xn−zn‖≤‖zn−xn‖+‖T2xn−xn‖+‖xn−zn‖→0 (as n→∞). | (3.31) |
Thus, it follows from (3.10), (3.30) and (3.31) that
‖T1yn−yn‖≤‖T1yn−T2zn‖+‖T2zn−zn‖+‖yn−zn‖→0 (as n→∞). | (3.32) |
Using Lemma 4, we have I−T1, I−T2 and I−T3 are G-demiclosed at 0 so that p∈F. To complete the proof it suffices to show that {xn} converges weakly to p. To this end we need to show that {xn} satisfies the hypothesis of Lemma 5. Let {xnj} be a subsequence of {xn} which converges weakly to some q∈C. By similar arguments as above q is in F. Now for each j≥1, using (3.1), we have
xnj+1=(1−αnj)T1ynj+αnjT2znj. | (3.33) |
It follows from (3.29) that
T3xnj=(T3xnj−xnj)+xnj⇀q. | (3.34) |
Now from (3.1) and (3.34),
znj=(1−γnj)xnj+γnjT3xnj⇀q. | (3.35) |
Using (3.31) and (3.35), we have
T2znj=(T2znj−znj)+znj⇀q. | (3.36) |
Now from (3.1), (3.34) and (3.36),
ynj=(1−βnj)T3xnj+βnjT2znj⇀q. | (3.37) |
Also from (3.32) and (3.37),
T1ynj=(T1ynj−ynj)+ynj⇀q. | (3.38) |
It follows from (3.33), (3.36) and (3.38) that
xnj+1⇀q. |
Therefore, the sequence {xn} satisfies the hypothesis of Lemma 5 which in turn implies that {xn} weakly converges to q so that p=q. This completes the proof.
The strong convergence of the sequence generated by the new modified three-step iteration method (3.1) for three G-nonexpansive mappings in a uniformly convex Banach space with a directed graph is discussed in the rest of this section.
Theorem 3. Let X be a uniformly convex Banach space. Suppose that {αn}, {βn} and {γn} are real sequences in [δ,1−δ] for some δ∈(0,1), Ti(i=1,2,3) satisfy condition (C), F is dominated by x0 and F dominates x0. Then {xn} converges strongly to a common fixed point of T1,T2 and T3.
Proof. By Lemma 6 (i), limn→∞‖xn−q‖ exists and so limn→∞d(xn,F) exists for any q∈F. Also by Lemma 6 (ii), limn→∞‖xn−T1xn‖=limn→∞‖xn−T2xn‖=limn→∞‖xn−T3xn‖=0. It follows from condition (C) that limn→∞f(d(xn, F))=0. Since f:[0,∞)→[0,∞) is a nondecreasing function satisfying f(0)=0, f(r)>0 for all r∈(0,∞), we obtain that limn→∞d(xn,F)=0. Hence, we can find a subsequence {xnj} of {xn} and a sequence {uj}⊂F such that ‖xnj−uj‖≤12j. Put nj+1=nj+k for some k≥1. Then
‖xnj+1−uj‖≤‖xnj+k−1−uj‖≤‖xnj−uj‖≤12j. |
We obtain that ‖uj+1−uj‖≤32j+1, thus {uj} is a Cauchy sequence. We assume that uj→q0∈C as j→∞. Since F is closed, we get q0∈F. Therefore, we have xnj→q0∈C as j→∞. Since limn→∞‖xn−q0‖ exists, we obtain xn→q0. This completes the proof.
Theorem 4. Let X be a uniformly convex Banach space. Suppose that C has Property G, {αn},{βn} and {γn} are real sequences in [δ,1−δ] for some δ∈(0,1), F is dominated by x0 and F dominates x0. If one of Ti (i=1,2,3) is semi-compact, then {xn} converges strongly to a common fixed point of T1, T2 and T3.
Proof. It follows from Lemma 6 that {xn} is bounded and limn→∞‖xn−T1xn‖=limn→∞‖xn−T2xn‖=limn→∞‖xn−T3xn‖=0. Since one of T1, T2 and T3 is semi-compact, then there exists a subsequence {xnj} of {xn} such that xnj→q∈C as j→∞. Since C has Property G and transitivity of graph G, we obtain (xnj,q)∈E(G). Notice that, for each i∈{1,2,3}, limj→∞‖xnj−Tixnj‖=0. Then
‖q−Tiq‖≤‖q−xnj‖+‖xnj−Tixnj‖+‖Tixnj−Tiq‖≤‖q−xnj‖+‖xnj−Tixnj‖+‖xnj−q‖→0(asj→∞). |
Hence q∈F. Thus limn→∞d(xn,F) exists by Theorem 3. We note that d(xnj,F) ≤d(xnj,q)→0 as j→∞, hence limn→∞d(xn,F)=0. It follows, as in the proof of Theorem 3, that {xn} converges strongly to a common fixed point of T1, T2 and T3. This completes the proof.
For supporting our main theorem, the approximate solutions of common fixed-point problems for a class of G-nonexpansive mapping on the closed interval are discussed. The following definition will be useful in this context.
Definition 2. [28] Suppose {ζn} is a sequence that converges to ζ, with ζn≠ζ for all n. If positive constants ν and ψ exist with
limn→∞|ζn+1−ζ||ζn−ζ|ψ=ν, |
then {ζn} converges to ζ of order ψ, with asymptotic error constant ν. If ψ=1 (and ν<1), the sequence is linearly convergent.
In 2011, Phuengrattana and Suantai [29] showed that the Ishikawa iteration converges faster than the Mann iteration for a class of continuous functions on the closed interval in a real line. In order to study the order of convergence of a real sequence {ζn} converging to ζ, we usually use the well-known terminology in numerical analysis, see [28], for example.
Example 1. Let X=R, C=[0,2] and G=(V(G),E(G)) be a directed graph defined by V(G)=C and (x,y)∈E(G) if and only if 0.50≤x≠y≤1.70 or x=y∈C. Then E(G) is coordinate-convex and {(x,x):x∈V(G)}⊂E(G). Define mappings T1,T2,T3:C→C where
T1x=23arcsin(x−1)+1,T2x=13tan(x−1)+1,T3x=√x, |
for any x∈C. It is easy to show that T1,T2,T3 are G-nonexpansive but T1,T2,T3 are not nonexpansive because
|T1x−T1y|>0.50=|x−y|, |
|T2u−T2v|>0.07=|u−v|, |
and
|T3p−T3q|>0.45=|p−q|, |
when x=1.95,y=1.45,u=0.08,v=0.01,p=0.5 and q=0.05.
Let
αn=n+15n+3,βn=n+2√8n+5andγn=n+410n+7. | (4.1) |
Let {xn} be a sequence generated by (3.1) and {yn}, {zn} be sequences generated by the three-step Noor iteration (1.1) and SP-iteration, respectively. Example 1 shows the convergence behavior of these three comparative methods and x=1 is a common fixed point of T1,T2 and T3. We choose z1=y1=w1=x1=1.65 and set the relative error |ζn−x|/|x|<1.00e−08 as stopping criterion where {ζn} be all of comparative sequences. The results of three comparative algorithms with the permutation of the operators T1,T2 and T3 are presented. However, the permutation of the group of these three operators gives 27 case studies that we need to consider. But we are interested in the absence of duplicate operator being applied to the proposed algorithm (3.1). That is the following 6 cases of three comparative algorithms consists of the proposed algorithm, three-step Noor iteration (1.1) and SP-iteration (1.2) are demonstrated and discussed.
Case I. Three comparative algorithms with the operator order T1−T2−T3.
Case II. Three comparative algorithms with the operator order T1−T3−T2.
Case III. Three comparative algorithms with the operator order T2−T1−T3.
Case IV. Three comparative algorithms with the operator order T2−T3−T1.
Case V. Three comparative algorithms with the operator order T3−T1−T2.
Case VI. Three comparative algorithms with the operator order T3−T2−T1.
All numerical experiments for common fixed point solution by using three comparative methods with the permutation of the operators T1,T2 and T3 are shown in Figures 1–6. Each figure contains three graphs showing the behavior of numerical solution sequences, relative error sequences and asymptotic error sequences for three comparative methods, respectively. The first two graphs of each figure shows that all sequences generated by these three comparative methods converge to a common fixed point solution x=1 and the relative errors of these three comparative methods are also decreased to zero when the number of iteration increased. The remainning graph of each figure shows the tendency of the asymptotic error constant σ for sequence {ζn} results from the formula |ζn+1−1|/|ζn−1| of three-step Noor, SP and proposed methods.
It can bee seen from the remainning graph of each figure that all methods are linearly convergent. This message is being made more confident by using Definition 2. Since, the smaller of asymptotic error constant gives us the faster convergence of the considering sequence then it can be concluded from the remainning graph that the proposed method converge faster than three-step Noor and SP methods.
Thus, it can be concluded from Figures 1–6 that the proposed method with the absence of duplicate operator being applied to the proposed algorithm give us faster convergence compare with three-step Noor and SP methods. And the fastest case occurs when the proposed method with the operator order T3−T1−T2 is used.
Now, we apply our proposed algorithms to solve certain image deblurring and signal recovering problems where all codes were written in Matlab and run on laptop Intel core i7, 16.00 GB RAM, windows 10 (64-bit).
The minimization problem of the sum of two functions is to find a solution of
minx∈Rn{F(x):=f(x)+h(x)}, | (5.1) |
where h:Rn→R∪{∞} is proper convex and lower semicontinuous function, and f:Rn→R is convex differentiable function with gradient ∇f being L-Lipschitz constant for some L>0. The solution of (5.1) can be characterized by using Fermat's rule, Theorem 16.3 of Bauschke and Combettes [30] as follows:
x∗is a minimizer of(f+h)if and only if0∈∂h(x∗)+∇f(x∗), |
where ∂h is the subdifferential of h and ∇f is the gradient of f. The subdifferential of h at x∗, denoted by ∂h(x∗), is defined by
∂h(x∗):={u:h(x)−h(x∗)≥⟨u,x−x∗⟩for allx}. |
It is also well-known that the solution of (5.1) is characterized by the following fixed point problem:
x∗is a minimizer of(f+h)if and only ifx∗=proxμh(I−μ∇f)(x∗), |
where c>0, proxh is the proximity operator of h defined by proxh:=argmin{h(y)+12‖x−y‖22}, see [31] for more details. It is also known that proxμh(I−μ∇f) is a nonexpansive mapping when μ∈(0,2L).
Let B is the degraded image of the true image X in the matrix form of ˜m rows and ˜n columns (B,X∈R˜mטn). The key to obtaining the image restoration model is to rearrange the elements of the images B and X into the column vectors by stacking the columns of these images into two long vectors b and x where b=vec(B) and x=vec(X), both of length n=˜m˜n. The image restoration problem can be modelled in one dimensional vector by the following linear equation system:
b=Mx, | (5.2) |
where x∈Rn is an original image, b∈Rn is the observed image, M∈Rn×n is the blurring matrix and n=˜m˜n. In order to solve problem (5.2), we aim to approximate the original image, vector b, which is known as the following least squares (LS) problem:
minx12‖b−Mx‖22, | (5.3) |
where ‖.‖2 is defined by ‖x‖2=√∑ni=1|xi|2. We can apply the minimization problem of the sum of two functions (5.1) to the LS-problem (5.3) by setting h(x)=0 and f(x)=12‖b−Mx‖22 where ∇f(x)=MT(Mx−y). Thus, LS-problem can be solved with our method (3.1) by setting
Tx=x−μMT(Mx−b),μ⊂(0,2||MTM||2). |
And, the following proposed method is obtained in finding the solution of the image deblurring problem:
zn=(1−γn)xn+γn(xn−μMT(Mxn−b)),yn=(1−βn)(xn−μMT(Mxn−b))+βn(zn−μMT(Mzn−b)),xn+1=(1−αn)(yn−μMT(Myn−b))+αn(zn−μMT(Mzn−b)), | (5.4) |
where μ∈(0,2‖MTM‖2) and {αn}, {βn} and {γn} are sequences in [δ,1−δ] for all n∈N and for some δ in (0,1). The algorithm (5.4) is used in solving the image deblurring problem (5.2) with the default parameter (4.1) and μ=1/‖MTM‖2.
The goal on image deblurring problem is to find the original image from the observed image without knowing which one is the blurring matrix. However, the blurring matrix M must be known in applying algorithm (5.4). Now, we present the new idea in solving the image deblurring problem when the observed image b1,b2,...,bN can be restored by using the blurring matrices M1,M2,...,MN, repectively in which
bi=Mix,i=1,2,…,N. | (5.5) |
That is, the original image x is a common solution of the N-determinated linear Eq (5.5). Now, Let us consider the following LS-problems:
minx∈Rn12‖M1x−b1‖22,minx∈Rn12‖M2x−b2‖22,...,minx∈Rn12‖MNx−bN‖22, | (5.6) |
where x is the common solution of problem (5.6). In applying the proposed algorithm (5.4) in finding the original image x, Only a group of the chosen three blurring matrices Mj,Mk,Ml from known blurring matrices M1,M2,...,MN. And, we can apply the minimization problem of the sum of two functions with our method (3.1) for solving the LS-problems by setting
Tix=x−μi∇fi(x)c |
with hi(x)=0 and fi(x)=12‖bi−Mix‖22 where ∇fi(x)=MTi(Mix−bi). Now, we presented the proposed algorithm with Mj−Mk−Ml:
zn=(1−γn)xn+γn(xn−μjMTj(Mjxn−bj)),yn=(1−βn)(xn−μjMTj(Mjxn−bj))+βn(zn−μkMTk(Mkzn−bk)),xn+1=(1−αn)(yn−μlMTl(Mlyn−bl))+αn(zn−μkMTk(Mkzn−bk)), | (5.7) |
with the default parameter (4.1), μj=1/‖MTjMj‖2,μk=1/‖MTkMk‖2,μl=1/‖MTlMl‖2 and called it as the proposed algorithm with Mj−Mk−Ml. The implemented algorithm (5.7) is proposed in solving the image deblurring problem by using every three blurring matrices from N known blurring matrices with the default parameter (4.1). The original RGB format for color image shown in Figure 7 is used to demonstrate the practicability of the proposed algorithm. The relative errors with the stopping criterion of the proposed algorithms are defined as ‖xn−x‖∞/‖x‖∞<10−3. The performance of the comparing algorithms at xn on image deblurring process is measured quantitatively by the means of the peak signal-to-noise ratio (PSNR), which is defined by
PSNR(xn)=20log10(2552MSE), |
where MSE=‖xn−x‖22. Three different types of the original RGB image degraded by the blurring matrices M1,M2 and M3 are shown in Figure 8. These are used to test the implemented algorithm.
Next, we present restoration of images that have been corrupted by the following blur types:
Type I. Gaussian blur of filter size 9×9 with standard deviation σ=4 (The original image has been degraded by the blurring matrix M1).
Type II. Out of focus blur (Disk) with radius r=6 (The original image has been degraded by the blurring matrix M2).
Type III. Motion blur specifying with motion length of 21 pixels (len =21) and motion orientation 11∘ (θ=11) (The original image has been degraded by the blurring matrix M3).
Since, the using image X and three different types of the blurring image B (See on Figure 8) are represented in the red-green-blue component. Then, we denote Xr,Xg,Xb and Br,Bg,Bb as the gray-scale images that constitute the red-green-blue channels of the using image X and the blurring image B respectively. Thus, we define the column vector x and b from color image X and B and both of length n=3˜m˜n. After that, we apply the proposed algorithms in getting the common solution of the image deblurring problem with these three blurring matrices. And, we are also compare with three-step Noor iteration (1.1) and SP-iteration (1.2).
The permutation of the group of these three blurring matrices gives 27 case studies that we need to consider. First, we are interested in the simple three cases in which all three blurring matrices are the same. Second, we are also interested in the case of the absence of duplicate blurring matrices. That is the following 9 cases of three comparative algorithms consists of the proposed algorithm (5.7), three-step Noor iteration (1.1) and SP-iteration (1.2) are demonstrated and discussed.
Case I. Three comparative algorithms with M1−M1−M1.
Case II. Three comparative algorithms with M1−M2−M3.
Case III. Three comparative algorithms with M1−M3−M2.
Case IV. Three comparative algorithms with M2−M2−M2.
Case V. Three comparative algorithms with M2−M1−M3.
Case VI. Three comparative algorithms with M2−M3−M1.
Case VII. Three comparative algorithms with M3−M3−M3.
Case VIII. Three comparative algorithms with M3−M1−M2.
Case IX. Three comparative algorithms with M3−M2−M1.
Figures 9–17 show the comparative plots behavior of Cauchy error, relative error and PSNR quality of the reconstructed RGB image with 9 cases.
It is remarkable that, the Cauchy error and relative error plots on each case of three comparative algorithms decrease as the iteration number increases. Since, the Cauchy and relative errors plot show the validity and confirm the convergence of the presented methods then, it can guarantee that all presented methods on Figures 9–17 converge to common solution of deblurring problem (5.7). Based on the PSNR plots of each case, all restored image using these three comparative algorithms in solving the deblurring problem get the quality improvements when the iteration number increases. It can be conclude from the comparative plots behavior on Figures 9–17 that the proposed method is more efficiency than three-step Noor and SP methods. Moreover, the PSNR quality of the observed image is much better when the proposed method with Mj−Mk−Ml and j≠k≠l is used for solving deblurring problem compare with the proposed methods with M1−M1−M1, M2−M2−M2 and M3−M3−M3. And, the best case in recovering the observed image occurs when the proposed methods with M2−M1−M3 and M3−M1−M2 are used. Figure 11 demonstrates the crop of reconstructed RGB image presented in 10,000th iteration by using the proposed algorithms in getting the common solution of the deblurring problem with operators M1−M1−M1 and M2−M2−M2, M3−M3−M3 and M3−M1−M2 respectively.
It can be seen from these figures that the quality of restored image by using the proposed algorithms with operators M3−M2−M1 get the smooth quality of the using degraded image.
Next, we give some numerical examples to the signal recovery.
In signal processing, compressed sensing can be modeled as the following undetermined linear equation:
y = Ax + ν, |
where x \in \mathbb{R}^{n} is an original signal with n components to be recovered, ν, y \in \mathbb{R}^{m} are noise and the observed signal with noisy for m components respectively and A \in \mathbb{R}^{m \times n} is a degraded matrix. Finding the solutions of previous undetermined linear equation can be seen as solving the LASSO problem:
\begin{eqnarray} \min\limits_{x \in \mathbb{R}^{N}} \frac{1}{2} \| y - Ax \|^{2}_{2} +\lambda\|x\|_1, \end{eqnarray} | (5.8) |
where \lambda > 0 . The new idea in solving the signal recovering problem is presented when the observed signal y_1, y_2, ..., y_N can be recovered by using the known degraded matrices A_1, A_2, ..., A_N in which
\begin{equation} y_i = A_i \mathbf{x} + \nu_i, \quad i = 1, 2, \ldots, N. \end{equation} | (5.9) |
where a true signal x is common solution of problem (5.9).
Let us consider the following LASSO problems:
\begin{eqnarray} \min\limits_{x \in \mathbb{R}^{N}} \frac{1}{2} \| A_{1}x - y_{1} \|^{2}_{2} &+&\lambda_1\|x\|_1 , \\ \min\limits_{x \in \mathbb{R}^{N}} \frac{1}{2} \| A_{2}x - y_{2} \|^{2}_{2} &+&\lambda_2\|x\|_1 , \\ &\vdots& \\ \min\limits_{x \in \mathbb{R}^{N}} \frac{1}{2} \| A_{N}x - y_{N} \|^{2}_{2} &+&\lambda_N\|x\|_1 , \end{eqnarray} | (5.10) |
where a true signal x is common solution of problem (5.10). We can apply the minimization problem of the sum of two functions with our method (3.1) for solving the LASSO problems (5.10) by setting
\mathcal{T}_ix = prox_{\mu_i h_i}\big(x -\mu_i\nabla f_i(x)\big), \quad \mu_i \subset \left( 0, \frac{2}{||A_i^T A_i||_2} \right), \quad i = 1, 2, \ldots, N, |
with h_i(x) = \lambda_i\|x\|_1 and f_i(x) = \frac{1}{2}\|y_i-A_ix\|_2^2 where \nabla f_i(x) = A_i^T(A_ix-y_i) . Now, we obtain the following proposed method to find the common solution of LASSO problems (5.10) for a group of three blurring matrices A_j, A_k, A_l chosen from N blurring matrices A_1, A_2, ..., A_N :
\begin{align} z_{n} & = (1 - \gamma_{n})x_{n} + \gamma_{n} prox_{\lambda g}\left( x_{n} - \mu_j A_j^T(A_jx_n - y_j) \right), \\ y_{n} & = (1 - \beta_{n}) prox_{\lambda g}\left( x_{n} - \mu_j A_j^T(A_j x_n - y_j) \right) + \beta_{n} prox_{\lambda g}\left( z_{n} - \mu_k A_k^T(A_k z_{n} - y_k) \right), \\ x_{n+1} & = (1 - \alpha_{n})prox_{\lambda g} \left( y_{n} -\mu_l A_l^T(A_l y_{n} - y_l) \right) + \alpha_{n} prox_{\lambda g}\left( z_{n} -\mu_k A_k^T(A_k z_{n} - y_k) \right), \end{align} | (5.11) |
with the default parameter (4.1), \mu_j = 1/\|A_j^T A_j\|_2, \mu_k = 1/\|A_k^T A_k\|_2, \mu_l = 1/\|A_l^T A_l\|_2 and called it as the proposed algorithm with A_j-A_k-A_l . The implemented algorithm (5.11) is proposed in solving the signal recovering problem by using every three blurring matrices from N blurring matrices on Eq (5.9) with the default parameter (4.1). And, we are also compare the proposed algorithm (5.11) with three-step Noor iteration (1.1) and SP-iteration (1.2). The following 9 cases of three comparative algorithms consists of the proposed algorithm (5.11), three-step Noor iteration (1.1) and SP-iteration (1.2) are demonstrated and discussed.
Case I. Three comparative algorithms with A_1-A_1-A_1 .
Case II. Three comparative algorithms with A_1-A_2-A_3 .
Case III. Three comparative algorithms with A_1-A_3-A_2 .
Case IV. Three comparative algorithms with A_2-A_2-A_2 .
Case V. Three comparative algorithms with A_2-A_1-A_3 .
Case VI. Three comparative algorithms with A_2-A_3-A_1 .
Case VII. Three comparative algorithms with A_3-A_3-A_3 .
Case VIII. Three comparative algorithms with A_3-A_1-A_2 .
Case IX. Three comparative algorithms with A_3-A_2-A_1 .
Next, some experiments are provided to illustrate the convergence and the effectiveness of the proposed algorithm (5.11). The original signal x with n = 1024 generated by the uniform distribution in the interval [-2, 2] with 70 nonzero elements is used to create the observation signal y_i = A_i x + \nu_i, i = 1, 2, 3 with m = 512 (See on Figure 19).
The observation signal y_i = A_ix + n_i shows on Figure 20.
The process is started when the signal initial data x_0 with n = 1024 is picked randomly (See on Figure 21).
The matrices A_i that generated by the normal distribution with mean zero and variance one and the white Gaussian noise \nu_i for all i = 1, 2, 3 (See on Figure 22).
The Cauchy error and the relative signal error are measured by using max-norm \|x_{n} - x_{n-1} \|_\infty and \|x_{n} -x\|_\infty/\|x\|_\infty , respectively. The performance of the comparing method at n^{th} iteration is measured quantitatively by the means of the the signal-to-noise ratio (SNR), which is defined by
\text{SNR}(x_{n} ) = 20 \log_{10} \left( \dfrac{ \|x_{n} \|_2}{ \|x_{n} - x\|_2} \right), |
where x_{n} is the recovered signal at n^{th} iteration by using the proposed method. The Cauchy error plots and relative error plots on each case of three comparative methods on Figures 23–31 guarantee that all presented methods converge to common solution of signal recovering problem. Based on the SNR plots on each case of three comparative methods, all recovering signal using the proposed method, three-step Noor method and Sridarat method witness quality improvement when the iteration number increases. It can be conclude from the comparative plots behavior on Figures 23–31 that the proposed method is more efficiency than three-step Noor and SP methods.
Moreover, the SNR quality of the recovering signal is significantly improved when the proposed method with A_j-A_k-A_l and j \neq k \neq l is used compare with the proposed methods with A_1-A_1-A_1 , A_2-A_2-A_2 and A_3-A_3-A_3 . And, the best case in recovering the observed signal occurred when the proposed method with the operator order A_2-A_1 -A_3 and A_3-A_1 -A_2 are used. Figure 32 shows an excellent quality of the restored signal using the proposed algorithm with A_3-A_1-A_2 .
In this article, the efficient modified three-step iteration algorithm is proposed for approximating a common fixed point of three G-nonexpansive mappings on Banach spaces involving a graph. By assuming coordinate-convexity in a uniformly convex Banach space endowed with a directed graph, we have proved strong convergence theorem for above said algorithm and mappings by using condition (C) which is a generalization of condition (A) [32] and a weak convergence theorem by using Opial's condition [21]. Also we have proved weak convergence theorem without using Opial's condition. The conditions for convergence of the method are established by systematic proof. Numerical example illustrating the performance of the suggested algorithm was provided. All numerical experiments for common fixed point solution by using the three-step Noor iteration, SP-iteration and the proposed method with the permutation of three operators are shown in Figures 1–6. Our algorithm was found to be faster than Noor and SP iterations. As applications, we applied our proposed algorithm to solve the image restoration problems with the permutation of the three blurring operators (see Figures 9–17). We also applied our algorithms for solving signal recovery in situations where the type of noise is unknown (see Figures 23–31). We found that our proposed algorithm is flexible and has good quality for common types of blur and noise effects in image deblurring and signal recovering problems. Moreover, we found that when the proposed is used in solving the common solution of image and signal recovering problems, it enhanced the quality range of the recovered image and signal.
This research project was supported by the thailand science research and innovation fund and the University of Phayao (Grant No. FF64-UoE011). The authors acknowledge the partially support provided by Thailand Science Research and Innovation under the Project IRN62W0007. The opinions in the research report belong to the researchers; Thailand Science Research and Innovation do not always have to agree. Also, T.Thianwan was supported by Contract No. PBTSC64019.
The authors declare no conflict of interest.
[1] |
S. Banach, Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales, Fund. Math., 3 (1922), 133–181. doi: 10.4064/fm-3-1-133-181. doi: 10.4064/fm-3-1-133-181
![]() |
[2] |
J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136 (2008), 1359–1373. doi: 10.1090/S0002-9939-07-09110-1. doi: 10.1090/S0002-9939-07-09110-1
![]() |
[3] |
R. P. Kelisky, T. J. Rivlin, Iterates of Bernstein polynomials, Pac. J. Math., 21 (1967), 511–520. doi: 10.2140/pjm.1967.21.511. doi: 10.2140/pjm.1967.21.511
![]() |
[4] |
M. Samreen, T. Kamran, Fixed point theorems for integral G-contractions, Fixed Point Theory Appl., 2013 (2013), 149. doi: 10.1186/1687-1812-2013-149. doi: 10.1186/1687-1812-2013-149
![]() |
[5] |
J. Tiammee, S. Suantai, Coincidence point theorems for graph-preserving multivalued mappings, Fixed Point Theory Appl., 2014 (2014), 70. doi: 10.1186/1687-1812-2014-70. doi: 10.1186/1687-1812-2014-70
![]() |
[6] |
A. Nicolae, D. O. Regan, A. Petrusel, Fixed point theorems for single-valued and multivalued generalized contractions in metric spaces endowed with a graph, Georgian Math. J., 18 (2011), 307–327. doi: 10.1515/gmj.2011.0019. doi: 10.1515/gmj.2011.0019
![]() |
[7] | F. Bojor, Fixed point of \phi-contraction in metric spaces endowed with a graph, Ann. Univ. Craiova Mat., 37 (2010), 85–92. |
[8] |
F. Bojor, Fixed point theorems for Reich type contractions on metric spaces with a graph, Nonlinear Anal., 75 (2012), 3895–3901. doi: 10.1016/j.na.2012.02.009. doi: 10.1016/j.na.2012.02.009
![]() |
[9] | F. Bojor, Fixed points of Kannan mappings in metric spaces endowed with a graph, An. St. Univ. Ovidius Constanta, 20 (2012), 31–40. |
[10] |
J. H. Asl, B. Mohammadi, S. Rezapour, S. M. Vaezpour, Some fixed point results for generalized quasi-contractive multifunctions on graphs, Filomat, 27 (2013), 311–315. doi: 10.2298/FIL1302311A. doi: 10.2298/FIL1302311A
![]() |
[11] |
S. M. A. Aleomraninejad, S. Rezapour, N. Shahzad, Some fixed point result on a metric space with a graph, Topol. Appl., 159 (2012), 659–663. doi: 10.1016/j.topol.2011.10.013. doi: 10.1016/j.topol.2011.10.013
![]() |
[12] |
M. R. Alfuraidan, M. A. Khamsi, Fixed points of monotone nonexpansive mappings on a hyperbolic metric space with a graph, Fixed Point Theory Appl., 2015 (2015), 44. doi: 10.1186/s13663-015-0294-5. doi: 10.1186/s13663-015-0294-5
![]() |
[13] |
M. R. Alfuraidan, Fixed points of monotone nonexpansive mappings with a graph, Fixed Point Theory Appl., 2015 (2015), 49. doi: 10.1186/s13663-015-0299-0. doi: 10.1186/s13663-015-0299-0
![]() |
[14] |
J. Tiammee, A. Kaewkhao, S. Suantai, On Browder's convergence theorem and Halpern iteration process for G-nonexpansive mappings in Hilbert spaces endowed with graphs, Fixed Point Theory Appl., 2015 (2015), 187. doi: 10.1186/s13663-015-0436-9. doi: 10.1186/s13663-015-0436-9
![]() |
[15] |
O. Tripak, Common fixed points of G-nonexpansive mappings on Banach spaces with a graph, Fixed Point Theory Appl., 2016 (2016), 87. doi: 10.1186/s13663-016-0578-4. doi: 10.1186/s13663-016-0578-4
![]() |
[16] | M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217–229. doi: 10.1006/jmaa.2000.7042. |
[17] | R. Glowinski, P. L. Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanic, Society for Industrial and Applied Mathematics, 1989. |
[18] |
S. Haubruge, V. H. Nguyen, J. J. Strodiot, Convergence analysis and applications of the Glowinski Le Tallec splitting method for finding a zero of the sum of two maximal monotone operators, J. Optimiz. Theory App., 97 (1998), 645–673. doi: 10.1023/a:1022646327085. doi: 10.1023/a:1022646327085
![]() |
[19] |
P. Sridarat, R. Suparaturatorn, S. Suantai, Y. J. Cho, Covergence analysis of SP-iteration for G-nonexpansive mappings with directed graphs, Bull. Malays. Math. Sci. Soc., 42 (2019), 2361–2380. doi: 10.1007/s40840-018-0606-0. doi: 10.1007/s40840-018-0606-0
![]() |
[20] | R. Johnsonbaugh, Discrete mathematics, Prentice Hall, 1997. |
[21] |
Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591–597. doi: 10.1090/S0002-9904-1967-11761-0. doi: 10.1090/S0002-9904-1967-11761-0
![]() |
[22] |
S. Shahzad, R. Al-Dubiban, Approximating common fixed points of nonexpansive mappings in Banach spaces, Georgian Math. J., 13 (2006), 529–537. doi: 10.1515/GMJ.2006.529
![]() |
[23] |
N. V. Dung, N. T. Hieu, Convergence of a new three-step iteration process to common fixed points of three G-nonexpansivemappings in Banach spaces with directed graphs, RACSAM, 114 (2020), 140. doi: 10.1007/s13398-020-00872-w. doi: 10.1007/s13398-020-00872-w
![]() |
[24] |
J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, B. Aust. Math. Soc., 43 (1991), 153–159. doi: 10.1017/S0004972700028884. doi: 10.1017/S0004972700028884
![]() |
[25] |
S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 331 (2005), 506–517. doi: 10.1016/j.jmaa.2005.03.002. doi: 10.1016/j.jmaa.2005.03.002
![]() |
[26] |
D. Yambangwai, S. Aunruean, T. Thianwan, A new modified three-step iteration method for Gnonexpansive mappings in Banach spaces with a graph, Numer. Algor., 20 (2019), 1–29. doi: 10.1007/s11075-019-00768-w. doi: 10.1007/s11075-019-00768-w
![]() |
[27] |
M. G. Sangago, Convergence of iterative schemes for nonexpansive mappings, Asian-Eur. J. Math., 4 (2011), 671–682. doi: 10.1142/S1793557111000551. doi: 10.1142/S1793557111000551
![]() |
[28] | R. L. Burden, J. D. Faires, Numerical analysis: Cengage learning, Brooks/Cole, 2010. |
[29] |
W. Phuengrattana, S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math., 235 (2011), 3006–3014. doi: 10.1016/j.cam.2010.12.022. doi: 10.1016/j.cam.2010.12.022
![]() |
[30] | H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, New York: Springer, 2011. |
[31] | J. J. Moreau, Proximit\acute{e} et dualit\acute{e} dans un espace hilbertien, B. Soc. Math. Fr., 93 (1965), 273–299. |
[32] |
H. F. Senter, W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44 (1974), 375–380. doi: 10.1090/S0002-9939-1974-0346608-8. doi: 10.1090/S0002-9939-1974-0346608-8
![]() |