# Strong Convergence Theorems for Solutions of Equations of Hammerstein Type

Authors: Chih-Sheng Chuang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We consider an auxiliary operator, defined in a real Hilbert space in terms ofK and F, that is, monotone and Lipschitz mappings (resp., monotone and bounded mappings). We use an explicit iterative process that converges strongly to a solution of equation of Hammerstein type. Furthermore, our results improve related results in the literature.


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## Body

## 1. Introduction

LetH be a real Hilbert space. A mapping A:D(A)⊆H→H is said to be monotone if 〈Ax-Ay,x-y〉≥0 for every x,y∈D(A). A is called maximal monotone if it is monotone  and the R(I+rA)=H, the range of (I+rA), for each r>0, where I is the identity mapping on H.  A is said to satisfy the range condition if  cl(D(A))⊆R(I+rA) for each r>0. For monotone mappings, there are many related equations of evolution. Several problems that arise in differential equations, for instance, elliptic boundary value problems whose linear parts possess Green's function, can be put in operator form as
(1)u+KFu=0,
where K and F are monotone mappings. In fact, (1) comes from the following integral equation of Hammerstein type [1]:
(2)u(x)+∫Ω‍k(x,y)f(y,u(y))dy=h(x),
where dy is a σ-finite measure on the measure space Ω; the real kernel is defined by Ω×Ω, f is a real-valued function defined on Ω×ℝ and is, in general, nonlinear, and h is a given function on Ω. If we now define an operator K by
(3)Kv(x)=∫Ω‍k(x,y)v(y)dy,x∈Ω,
and the so-called superposition or Nemytskii operator by Fu(y):=f(y,u(y)), then (2) can be put in (1) (without loss of generality, we may assume that h≡0).Note that equations of Hammerstein type play a crucial role in the theory of optimal control systems and in automation and network theory, and several existence and uniqueness theorems have been proved for equations of the Hammerstein type. For details, one can refer to [2–7].In 2005, Chidume and Zegeye [8] constructed an iterative process as follows:
(4)un+1=un-λn(Fun-vn)-λnθn(un-w),vn+1=vn-λn(Kvn+un)-λnθn(vn-w),n∈ℕ,
where H is a real Hilbert space, F and K: H→H are bounded monotone mappings satisfying the range condition, w∈H, and {λn}n∈ℕ and {θn}n∈ℕ are sequences in (0,1). Chidume and Zegeye [8] show that this sequence converges strongly to the solution of (1) under suitable conditions.In 2011, Chidume and Ofoedu [9] introduced a coupled explicit iterative process as follows:
(5)un+1=un-λnαn(Fun-vn)-λnθn(un-u1),vn+1=vn-λnαn(Kvn+un)-λnθn(vn-v1),n∈ℕ,
where E is a uniformly smooth real Banach space, F and K: E→E are bounded and monotone mappings, and {λn}n∈ℕ, {θn}n∈ℕ, and {αn}n∈ℕ are sequences in (0,1). Chidume and Ofoedu [9] gave a strong convergence theorem for approximation of the solution of (1) under suitable conditions.In 2012, Chidume and Djitté [10] consider the following iterative process:
(6)un+1=un-λn(Fun-vn)-λnθn(un-u1),vn+1=vn-λn(Kvn+un)-λnθn(vn-v1),n∈ℕ,
where H is a real Hilbert space, K and F are bounded and maximal monotone mappings, {λn}n∈ℕ and {θn}n∈ℕ are sequences in (0,1) and Chidume and Djitté [10] show that this iterative process converges to an approximate solution of nonlinear equations of Hammerstein type under suitable conditions.Motivated by the previous works, in this paper, we consider an auxiliary operator, defined in a real Hilbert space in terms ofK and F, that is monotone and Lipschitz mapping, or monotone and bounded mappings. We use an explicit iterative process that converges strongly to a solution of equation of Hammerstein type. Furthermore, our results improve related results in the literature.


## 2. Preliminaries

Throughout this paper, letℕ be the set of positive integers and let ℝ be the set of real numbers. Let H be a (real) Hilbert space with inner product 〈·,·〉 and norm ||·||, respectively.Lemma 1.
LetH be a real Hilbert space. One has ||x+y||2≤||x||2+2〈y,x+y〉 for all x,y∈H.Lemma 2 (see [11]).
Let{an}n∈ℕ be a sequence of nonnegative real numbers, {αn}n∈ℕ a sequence of real numbers in [0,1] with ∑n=1∞‍αn=∞, and  {un}n∈ℕ a sequence of nonnegative real numbers with ∑n=1∞‍un<∞, {tn}n∈ℕ a sequence of real numbers with limsuptn≤0. Suppose that an+1≤(1-αn)an+αntn+un for each n∈ℕ. Then, limn→∞an=0.Letℓ∞ be the Banach space of bounded sequences with the supremum norm. A linear functional μ on ℓ∞ is called a mean if μ(e)=||μ||=1, where e=(1,1,1,…). For x=(x1,x2,x3,…), the value μ(x) is also denoted by μn(xn). A mean μ on ℓ∞ is called a Banach limit if it satisfies μn(xn)=μn(xn+1). If μ is a Banach limit on ℓ∞, then for x=(x1,x2,x3,…)∈ℓ∞,
(7)liminfn→∞xn≤μn(xn)≤limsupn→∞xn.In particular, ifx=(x1,x2,x3,…)∈ℓ∞ and limn→∞xn=a∈ℝ, then we have μ(x)=μn(xn)=a. For details, we can refer to [12].Lemma 3 (see [13]).
Letα be a real number and (x0,x1,…)∈ℓ∞ such that μnxn≤α for all Banach limit μ on ℓ∞. If limsupn→∞(xn+1-xn)≤0, then, limsupn→∞xn≤α.Lemma 4 (see [14]).
LetC be a nonempty closed convex subset of a Hilbert space H, let {xn}n∈ℕ be a bounded sequence in H, and let μ be a Banach limit on ℓ∞. Let g:C→ℝ be defined by g(z)=μn||xn-z||2 for each z∈C. Then there exists a unique z0∈C such that g(z0)=min{g(z):z∈C}.Lemma 5 (see [15]).
LetH be a Hilbert space, let {xn}n∈ℕ be a bounded sequence in H, and let μ be a mean on ℓ∞. Then, there exists a unique point z0∈H such that μn〈xn,y〉=〈z0,y〉 for each y∈H. Indeed, z0∈co¯{xn:n∈ℕ}.LetH be a real Hilbert space. Let W:=H×H with norm
(8)∥z∥:=(∥u∥2+∥v∥2)1/2,wherez=(u,v)∈W.
Hence, W is a real Hilbert space with inner product 〈w1,w2〉=〈u1,u2〉+〈v1,v2〉 for all w1=(ul,v1), w2=(u2,v2)∈W [8].Lemma 6.
LetH be a real Hilbert space, and let W:=H×H. Let F, K:H→H be two mappings, and let A:W→W be defined by
(9)Aw:=(Fu-v,Kv+u)foreachw=(u,v)∈W.(i)
IfF and K are monotone mappings, then A is a monotone mapping [8, Lemma 3.1].(ii)
IfF and K are bounded mappings, then A is a bounded mapping [8, Lemma 3.1].(iii)
IfF and K are Lipschitz mappings with Lipschitz constants L1 and L2, respectively, then A is a Lipschitz mapping. Indeed, the Lipschitz constant of A is 2(L+1), where L:=max{L1,L2} [16, Remark 13.6].


## 3. Main Results (I)

LetH be a real Hilbert space. Let F, K:H→H be two mappings, and let A:H×H→H×H be defined by Aw=(Fu-v,Kv+u) for each w=(u,v)∈H×H. Then, we observe that u-∈H is a solution of u+KFu=0 if and only if   w-=(u-,v-) is a solution of Aw=0 in H×H for v-=Fu-.Theorem 7.
LetH be a real Hilbert space. Let F, K:H→H be Lipschitz and monotone mappings. Suppose that u+KFu=0 has a solution in H. Let {λn}n∈ℕ be a sequence in (0,1). Let {αn}n∈ℕ and {θn}n∈ℕ be sequences in (0,1]. Let {un}n∈ℕ and {vn}n∈ℕ be sequences in H defined iteratively from arbitrary u1, v1∈H by
(10)un+1=un-λnαn(Fun-vn)-λnθn(un-u1),vn+1=vn-λnαn(Kvn+un)-λnθn(vn-v1),n∈ℕ.
Suppose that one of the following conditions holds:
(11)(i)limn→∞(αn2θn)=limn→∞λnαn=0;(ii)limn→∞(λnθn)=0;(iii)limn→∞(αnθn)=0.
Then, the sequences {un}n∈ℕ, {vn}n∈ℕ, {Fun}n∈ℕ, and {Kvn}n∈ℕ are bounded.Proof.
SinceF and K are Lipschitz mappings, we may assume that the Lipschitz constants of F and K are L1 and L2, respectively. Let
(12)L=max{L1,L2},r0:=132(L+1)2.
LetW:=H×H with the norm ||w||:=(||u||2+||v||2)1/2 for each w=(u,v)∈H×H. Take any u-∈H such that u- is solution of u+KFu=0, and let u- be fixed. Let v-=Fu- and w-=(u-,v-). We observe that u-=-Kv-. For each n∈ℕ, let wn:=(un,vn).
For eachn∈ℕ, it follows from Lemma 1 that
(13)∥un+1-u-∥2=∥(1-λnθn)(un-u-)+λn(θnu1-αnFun+αnvn-θnu-)∥2≤(1-λnθn)2∥un-u-∥2+2λn〈θnu1-αnFun+αnvn-θnu-,un+1-u-〉≤(1-λnθn)∥un-u-∥2+2λn〈θnu1-αnFun+αnvn-θnu-,un+1-u-〉.
Similarly, we have
(14)∥vn+1-v-∥2≤(1-λnθn)∥vn-v-∥2+2λn〈θnv1-αnKvn-αnun-θnu-,vn+1-v-〉.
For each n∈ℕ, by (13) and (14), we know that
(15)∥wn+1-w-∥2≤(1-λnθn)∥wn-w-∥2+2λn〈θnw1-αnAwn-θnw-,wn+1-w-〉=(1-λnθn)∥wn-w-∥2+2λnθn〈w1-w-,wn+1-w-〉-2λnαn〈Awn,wn+1-w-〉.
For each n∈ℕ, since A is monotone and Aw-=0, we know that
(16)〈Awn,wn+1-w-〉=〈Awn,wn-λnαnAwn-λnθn(wn-w1)-w-〉=〈Awn,wn-w-〉+〈Awn,-λnαnAwn-λnθn(wn-w1)〉≥〈Awn,-λnαnAwn-λnθn(wn-w1)〉.
Hence, for each n∈ℕ, it follows from (15) and (16) that
(17)∥wn+1-w-∥2≤(1-λnθn)∥wn-w-∥2+2λnθn〈w1-w-,wn+1-w-〉-2λnαn〈Awn,wn+1-w-〉≤(1-λnθn)∥wn-w-∥2+2λnθn〈w1-w-,wn+1-w-〉+2λnαn〈Awn,λnαnAwn+λnθnwn-λnθnw1〉≤(1-λnθn)∥wn-w-∥2+2λnθn〈w1-w-,wn+1-w-〉+2λnαn∥Awn∥·(λnαn∥Awn∥+λnθn∥wn-w1∥)≤(1-λnθn)∥wn-w-∥2+2λnθn〈w1-w-,wn+1-w-〉+4λnαn(L+1)∥wn-w-∥·(2λnαn(L+1)∥wn-w-∥+λnθn∥wn-w1∥)=(1-λnθn)∥wn-w-∥2+2λnθn〈w1-w-,wn+1-w-〉+8λn2αn2(L+1)2∥wn-w-∥2+4λn2αnθn(L+1)∥wn-w-∥·∥wn-w1∥.
For conditions (i)–(iii), we only need to consider one case since the proof is similar. Now, we assume thatlimn→∞(λn/θn)=0. Then, there exists n0∈ℕ such that λn/θn<r0 for each n≥n0. Choose r>0 such that w1∈B(w-,r/4) and wn0∈B(w-,r/4). Let B:=B(w-,r)¯.
Now, we want to show thatwn∈B for each n≥n0. Clearly, wn0∈B(w-,r). Suppose that wn∈B for some n≥n0. Then, wn+1∈B. Indeed, if not, then we have
(18)∥wn-w-∥≤r<∥wn+1-w-∥.
Hence, by (17) and (18), we get
(19)∥wn+1-w-∥2≤(1-λnθn)∥wn+1-w-∥2+2λnθn∥w1-w-∥·∥wn+1-w-∥+8λn2αn2(L+1)2∥wn-w-∥·∥wn+1-w-∥+4λn2αnθn(L+1)∥wn+1-w-∥·∥wn-w1∥.
By (19), we have
(20)λnθn∥wn+1-w-∥≤2λnθn∥w1-w-∥+8λn2αn2(L+1)2∥wn-w-∥+4λn2αnθn(L+1)∥wn-w1∥≤2λnθn·r4+8λn2αn2(L+1)2r+4λn2αnθn(L+1)·(∥wn-w-∥+∥w1-w-∥)≤2λnθn·r4+8λn2αn2(L+1)2r+4λn2αnθn(L+1)·5r4.
This implies that
(21)r<∥wn+1-w-∥≤12r+8λnαn2θn(L+1)2r+5λnαnθnθn(L+1)r≤12r+8λnθn(L+1)2r+8λnθn(L+1)r≤r.
This leads to a contradiction. So, wn+1∈B. Hence, by mathematical induction, we know that {wn}n≥n0⊆B. Therefore, {un}n∈ℕ and {vn}n∈ℕ are bounded sequences. Furthermore, {Fun}n∈ℕ and {Kvn}n∈ℕ are bounded sequences since F and K are Lipschitz mappings. For conditions (ii) and (iii), the proof is similar. Therefore, the proof is completed.Remark 8.
(i) Theorem7 improves the conditions of [17, Theorem 3.1] if the space E in [17] is reduced to a real Hilbert space. Indeed, [17, Theorem 3.1] assumes that limn→∞λn=limn→∞αn=limn→∞(λn/θn)=limn→∞(αn/θn)=0.
(ii) Furthermore, we know that it is impossible to assume thatαn=θn=1 in [17, Theorem 3.1]. However, we can choose αn=θn=1 in our result. Indeed, if αn=θn=1 and λn=βn, then we have the following result as a special case of Theorem 7.Corollary 9.
LetH be a real Hilbert space. Let F, K:H→H be Lipschitz and monotone mappings. Suppose that u+KFu=0 has a solution in H. Let {βn}n∈ℕ be a sequence in (0,1). Let {un}n∈ℕ and {vn}n∈ℕ be sequences in H defined iteratively from arbitrary u1, v1∈H by
(22)un+1=un-βn(Fun-vn)-βn(un-u1),vn+1=vn-βn(Kvn+un)-βn(vn-v1),n∈ℕ.
If limn→∞βn=0, then the sequences {un}n∈ℕ, {vn}n∈ℕ, {Fun}n∈ℕ, and {Kvn}n∈ℕ are bounded.In fact, following the same argument as the proof of Theorem7, we can get the following result.Theorem 10.
LetH be a real Hilbert space. Let F, K:H→H be Lipschitz and monotone mappings. Suppose that u+KFu=0 has a solution in H. Let {αn}n∈ℕ and {θn}n∈ℕ be sequences in (0,1]. Let {un}n∈ℕ and {vn}n∈ℕ be sequences in H defined iteratively from arbitrary u1, v1∈H by
(23)un+1=un-αn(Fun-vn)-θn(un-u1),vn+1=vn-αn(Kvn+un)-θn(vn-v1),n∈ℕ.
Suppose that one of the following conditions holds:
(24)(i)limn→∞(αn2θn)=limn→∞αn=0;(ii)limn→∞(αnθn)=0.
Then, the sequences {un}n∈ℕ, {vn}n∈ℕ, {Fun}n∈ℕ, and {Kvn}n∈ℕ are bounded.Remark 11.
Corollary9 is also a special case of Theorem 10.Theorem 12.
LetH be a real Hilbert space. Let F, K:H→H be Lipschitz and monotone mappings. Suppose that u+KFu=0 has a solution in H. Let {λn}n∈ℕ be a sequence in (0,1). Let {αn}n∈ℕ and {θn}n∈ℕ be sequences in (0,1]. Let {un}n∈ℕ and {vn}n∈ℕ be sequences in H defined iteratively from arbitrary u1, v1∈H by
(25)un+1=un-λnαn(Fun-vn)-λnθn(un-u1),vn+1=vn-λnαn(Kvn+un)-λnθn(vn-v1),n∈ℕ.
Assume that (i)
∑n=1∞‍λnθn=∞; limn→∞λnαn=limn→∞λnθn=0;(ii)
one of the following conditions holds:(26)(a)limn→∞(αn2θn)=0;(b)limn→∞(λnθn)=0;(c)limn→∞(αnθn)=0;(iii)
one of the following conditions holds:(27)(d)∑n=1∞λn2αn2<∞;(e)limn→∞(λnαn2θn)=0.Then, there exists a subset Kmin of H×H such that if (u-,v-)∈Kmin with v-=Fu-, then the sequence {un} converges strongly to u-.Proof.
LetB and n0 be the same as the proof of Theorem 7. Let μ be a Banach limit on ℓ∞. Let {xn}n∈ℕ be a sequence with x1=w1 and xn=wn0+n-2 for each n≥2. Clearly, {xn}n∈ℕ⊆B. By Lemma 4, there is a unique x∈B such that
(28)μn∥xn-x∥2=miny∈Bμn∥xn-y∥2.
Let Kmin={x}, and we assume that w-=(u-,v-)∈Kmin. Let t∈[0,1]. Then, tw1+(1-t)w-∈B. Hence, for each t∈(0,1), it follows from Lemma 1 that
(29)μn∥xn-w-∥2≤μn∥xn-tw1-(1-t)w-∥2≤μn∥xn-w-∥2-2tμn〈w1-w-,xn-tw1-(1-t)w-〉.
By Lemma 5, there exists z0∈H such that μn〈xn,y〉=〈z0,y〉 for each y∈H. By (29), for each t∈(0,1),
(30)〈w1-w-,z0-tw1-(1-t)w-〉=μn〈w1-w-,xn-tw1-(1-t)w-〉≤0.
In (30), letting t→0, we get
(31)μn〈w1-w-,xn-w-〉=〈w1-w-,z0-w-〉≤0.
Clearly,
(32)limsupn→∞(〈w1-w-,xn+1-w-〉-〈w1-w-,xn-w-〉)≤0.
By (31), (32), and Lemma 3, we know that limsupn→∞〈w1-w-,xn-w-〉≤0.
Hence,(33)limsupn→∞〈w1-w-,wn-w-〉≤0.
By (17), (33), and Lemma 2, we know that limn→∞||wn-w-||=0. Therefore, {un} converges strongly to u-.Remark 13.
(i) The conclusion of Theorem12 is still true if ∑n=1∞‍λnθn=∞, limn→∞(αn2/θn)=0, and limn→∞λnαn=limn→∞λnθn=0.
(ii) The conclusion of Theorem12 is still true if ∑n=1∞‍λnθn=∞, limn→∞(αn/θn)=0, and limn→∞λn=0.Remark 14.
Following the same argument as in Remark8, we know that Theorem 12 improves the conditions of [17, Theorem 3.2] if the space E is reduced to a real Hilbert space.In Theorem12, if αn=θn=1 and λn=βn for each n∈ℕ, then we have the following result.Corollary 15.
LetH be a real Hilbert space. Let F, K:H→H be Lipschitz and monotone mappings. Suppose that u+KFu=0 has a solution u- in H. Let {βn}n∈ℕ be a sequence in (0,1). Let {un}n∈ℕ and {vn}n∈ℕ be sequences in H defined iteratively from arbitrary u1, v1∈H by
(34)un+1=un-βn(Fun-vn)-βn(un-u1),vn+1=vn-βn(Kvn+un)-βn(vn-v1),n∈ℕ.
Assume that limn→∞βn=0 and ∑n=1∞‍βn=∞. Then there exists a subset Kmin of H×H such that if (u-,v-)∈Kmin with v-=Fu-, then the sequence {un} converges strongly to u-.In Theorem12, if αn=1 for each n∈ℕ, then we have the following result.Corollary 16.
LetH be a real Hilbert space. Let F, K:H→H be Lipschitz and monotone mappings. Suppose that u+KFu=0 has a solution in H. Let {λn}n∈ℕ be a sequence in (0,1). Let {θn}n∈ℕ be a sequence in (0,1]. Let {un}n∈ℕ and {vn}n∈ℕ be sequences in H defined iteratively from arbitrary u1, v1∈H by
(35)un+1=un-λn(Fun-vn)-λnθn(un-u1),vn+1=vn-λn(Kvn+un)-λnθn(vn-v1),n∈ℕ.
Assume that ∑n=1∞‍λnθn=∞, limn→∞λnθn=0, and limn→∞(λn/θn)=0. Then, there exists a subset Kmin of H×H such that if (u-,v-)∈Kmin with v-=Fu-, then the sequence {un} converges strongly to u-.In fact, following the same argument as the proof of Theorem12, we get the following result.Theorem 17.
LetH be a real Hilbert space. Let F, K:H→H be Lipschitz and monotone mappings. Suppose that u+KFu=0 has a solution in H. Let {αn}n∈ℕ and {θn}n∈ℕ be sequences in (0,1]. Let {un}n∈ℕ and {vn}n∈ℕ be sequences in H defined iteratively from arbitrary u1, v1∈H by
(36)un+1=un-αn(Fun-vn)-θn(un-u1),vn+1=vn-αn(Kvn+un)-θn(vn-v1),n∈ℕ.
Assume that ∑n=1∞‍θn=∞, limn→∞θn=0, and limn→∞(αn2/θn)=0. Then, there exists a subset Kmin of H×H such that if (u-,v-)∈Kmin with v-=Fu-, then the sequence {un} converges strongly to u-.Remark 18.
Corollary15 is also a special case of Theorem 17.


## 4. Main Results (II)

In this section, we consider thatF and K are bounded mappings.Theorem 19.
LetH be a real Hilbert space. Let F, K:H→H be bounded and monotone mappings. Suppose that u+KFu=0 has a solution in H. Let {λn}n∈ℕ be a sequence in (0,1). Let {αn}n∈ℕ and {θn}n∈ℕ be sequences in (0,1]. Let {un}n∈ℕ and {vn}n∈ℕ be sequences in H defined iteratively from arbitrary u1, v1∈H by
(37)un+1=un-λnαn(Fun-vn)-λnθn(un-u1),vn+1=vn-λnαn(Kvn+un)-λnθn(vn-v1),n∈ℕ.
Suppose that one of the following conditions holds:
(38)(i)limn→∞(αn2θn)=limn→∞λnαn=0;(ii)limn→∞(λnθn)=0;(iii)limn→∞(λnθn)=limn→∞λnαn=0;(iv)limn→∞(αnθn)=0.
Then, the sequences {un}n∈ℕ, {vn}n∈ℕ, {Fun}n∈ℕ, and {Kvn}n∈ℕ are bounded.Proof.
SinceF and K are bounded, we know that A is bounded. Hence, ||A||<∞. From the proof of Theorem 7, we know that
(39)∥wn+1-w-∥2≤(1-λnθn)∥wn-w-∥2+2λnθn〈w1-w-,wn+1-w-〉+2λnαn∥Awn∥·(λnαn∥Awn∥+λnθn∥wn-w1∥)≤(1-λnθn)∥wn-w-∥2+2λnθn〈w1-w-,wn+1-w-〉+2λn2αn2∥A∥2·∥wn∥2+2λn2αnθn∥A∥·∥wn∥·∥wn-w1∥
for each n∈ℕ.
Chooser0>0 such that max{||w1||,||w-||}≤r0/8. For conditions (i)–(iv), we only need to consider one case since the proof is similar. Now, we assume that limn→∞(λn/θn)=0. Then, there exists n0∈ℕ such that
(40)λnθn<max{881∥A∥2,110∥A∥}
for each n≥n0. Choose r>1 such that r0<r and ||wn0||≤r/8. Let B:=B(w-,r). Clearly, ||wn0-w-||≤r/4.
Now, we want to show thatwn∈B for each n≥n0. Clearly, wn0∈B(w-,r). Suppose that wn∈B for some n≥n0. Then, wn+1∈B. Indeed, if not, then we have
(41)∥wn-w-∥≤r<∥wn+1-w-∥.
Clearly, ||wn||≤||wn-w-||+||w-||≤9r/8. Hence, by (39) and (41), we get
(42)∥wn+1-w-∥2≤(1-λnθn)∥wn-w-∥2+2λnθn∥w1-w-∥·∥wn+1-w-∥+2λn2αn2∥A∥2·∥wn∥2+2λn2αnθn∥A∥·∥wn∥·∥wn-w1∥≤(1-λnθn)r2+2λnθnr4·∥wn+1-w-∥+2λn2αn2∥A∥2·8164r2+2λn2αnθn∥A∥·108r2≤(1-λnθn)∥wn+1-w-∥2+λnθnr2·∥wn+1-w-∥+λn2αn2∥A∥2·8132r2+λn2αnθn∥A∥·52r2.
This implies that
(43)∥wn+1-w-∥2≤r2·∥wn+1-w-∥+8132·λnαn2θn∥A∥2·r2+52·λnαnθnθn∥A∥·r2.
Furthermore, we get
(44)r<∥wn+1-w-∥≤r2+8132λnαn2θn∥A∥2·r+52·λnαnθnθn∥A∥·r≤r2+8132λnθn∥A∥2·r+52·λnθn∥A∥·r≤r.
This leads to a contradiction. So, wn+1∈B. Hence, by mathematical induction, we know that {wn}n≥n0⊆B. Therefore, {un}n∈ℕ and {vn}n∈ℕ are bounded sequences. Furthermore, {Fun}n∈ℕ and {Kvn}n∈ℕ are bounded sequences since F and K are bounded mappings.Remark 20.
(i) Theorem19 improves the conditions of [9, Theorem 3.1] if the space E is reduced to a real Hilbert space. Indeed, [9, Theorem 3.1] assumes that limn→∞(λn/θn)=limn→∞(αn/θn)=0.
(ii) Furthermore, we know that it is impossible to assume thatαn=θn=1 in [9, Theorem 3.1]. However, we can choose αn=θn=1 in our result. Indeed, if αn=θn=1 and λn=βn, then we have the following result as special case of Theorem 19.Corollary 21.
LetH be a real Hilbert space. Let F, K:H→H be bounded and monotone mappings. Suppose that u+KFu=0 has a solution in H. Let {βn}n∈ℕ be a sequence in (0,1). Let {un}n∈ℕ and {vn}n∈ℕ be sequences in H defined iteratively from arbitrary u1, v1∈H by
(45)un+1=un-βn(Fun-vn)-βn(un-u1),vn+1=vn-βn(Kvn+un)-βn(vn-v1),n∈ℕ.
If limn→∞βn=0, then the sequences {un}n∈ℕ, {vn}n∈ℕ, {Fun}n∈ℕ, and {Kvn}n∈ℕ are bounded.Following the same argument as the proof of Theorem19, we get the following result. Note that Corollary 21 is also a special case of the following result.Theorem 22.
LetH be a real Hilbert space. Let F, K:H→H be bounded and monotone mappings. Suppose that u+KFu=0 has a solution in H. Let {αn}n∈ℕ and {θn}n∈ℕ be sequences in (0,1]. Let {un}n∈ℕ and {vn}n∈ℕ be sequences in H defined iteratively from arbitrary u1, v1∈H by
(46)un+1=un-αn(Fun-vn)-θn(un-u1),vn+1=vn-αn(Kvn+un)-θn(vn-v1),n∈ℕ.
Suppose that one of the following conditions holds:
(47)(i)limn→∞(αn2θn)=limn→∞αn=0;(ii)limn→∞(αnθn)=0.
Then, the sequences {un}n∈ℕ, {vn}n∈ℕ, {Fun}n∈ℕ, and {Kvn}n∈ℕ are bounded.Following the similar argument as the proof of Theorem12, we get the following result.Theorem 23.
LetH be a real Hilbert space. Let F, K:H→H be bounded and monotone mappings. Suppose that u+KFu=0 has a solution in H. Let {λn}n∈ℕ be a sequence in (0,1). Let {αn}n∈ℕ and {θn}n∈ℕ be sequences in (0,1]. Let {un}n∈ℕ and {vn}n∈ℕ be sequences in H defined iteratively from arbitrary u1, v1∈H by
(48)un+1=un-λnαn(Fun-vn)-λnθn(un-u1),vn+1=vn-λnαn(Kvn+un)-λnθn(vn-v1),n∈ℕ.
Assume that (i)
∑n=1∞‍λnθn=∞; limn→∞λnαn=limn→∞λnθn=0;(ii)
one of the following conditions holds:(49)(a)limn→∞(αn2θn)=0;(b)limn→∞(λnθn)=0;(c)limn→∞(αnθn)=0;(iii)
one of the following conditions holds:(50)(d)∑n=1∞λn2αn2<∞;∑n=1∞λn2αnθn<∞;(e)∑n=1∞λn2αn<∞;(f)limn→∞(λnαn2θn)=0.Then, there exists a subset Kmin of H×H such that if (u-,v-)∈Kmin with v-=Fu-, then the sequence {un} converges strongly to u-.Remark 24.
(i) The conclusion of Theorem23 is still true if ∑n=1∞‍λnθn=∞, limn→∞(αn2/θn)=0, and limn→∞λnαn=limn→∞λnθn=0. Furthermore, the conclusion of Theorem 23 is still true if ∑n=1∞‍λnθn=∞, limn→∞(αn/θn)=0, and limn→∞λn=0.
(ii) Theorem23 improves the conditions of [9, Theorem 3.2] if the space E is reduced to a real Hilbert space. Indeed, [9, Theorem 3.2] assumes that limn→∞(λn/θn)=limn→∞(αn/θn)=0.The following is a special case of Theorem23.Corollary 25.
LetH be a real Hilbert space. Let F, K:H→H be bounded and monotone mappings. Suppose that u+KFu=0 has a solution in H. Let {βn}n∈ℕ be a sequence in (0,1). Let {un}n∈ℕ and {vn}n∈ℕ be sequences in H defined iteratively from arbitrary u1, v1∈H by
(51)un+1=un-βn(Fun-vn)-βn(un-u1),vn+1=vn-βn(Kvn+un)-βn(vn-v1),n∈ℕ.
Assume that ∑n=1∞‍βn=∞ and limn→∞βn=0. Then, there exists a subset Kmin of H×H such that if (u-,v-)∈Kmin with v-=Fu-, then the sequence {un} converges strongly to u-.The following is also a special case of Theorem23.Corollary 26.
LetH be a real Hilbert space. Let F, K:H→H be bounded and monotone mappings. Suppose that u+KFu=0 has a solution in H. Let {λn}n∈ℕ be a sequence in (0,1). Let {θn}n∈ℕ be a sequence in (0,1]. Let {un}n∈ℕ and {vn}n∈ℕ be sequences in H defined iteratively from arbitrary u1, v1∈H by
(52)un+1=un-λnαn(Fun-vn)-λnθn(un-u1),vn+1=vn-λnαn(Kvn+un)-λnθn(vn-v1),n∈ℕ.
Assume that ∑n=1∞‍λnθn=∞, limn→∞λnθn=0, and limn→∞(λn/θn)=0. Then, there exists a subset Kmin of H×H such that if (u-,v-)∈Kmin with v-=Fu-, then the sequence {un} converges strongly to u-.Furthermore, we get the following result. Note that Corollary25 is also a special case of the following result.Theorem 27.
LetH be a real Hilbert space. Let F, K:H→H be bounded and monotone mappings. Suppose that u+KFu=0 has a solution in H. Let {αn}n∈ℕ and {θn}n∈ℕ be sequences in (0,1]. Let {un}n∈ℕ and {vn}n∈ℕ be sequences in H defined iteratively from arbitrary u1, v1∈H by
(53)un+1=un-αn(Fun-vn)-θn(un-u1),vn+1=vn-αn(Kvn+un)-θn(vn-v1),n∈ℕ.
Assume that ∑n=1∞‍θn=∞, limn→∞θn=0, and limn→∞(αn2/θn)=0. Then, there exists a subset Kmin of H×H such that if (u-,v-)∈Kmin with v-=Fu-, then the sequence {un} converges strongly to u-.



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*Source: 541079-2013-05-09.xml*