# Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type with Lipschitz and Bounded Nonlinear Operators

Authors: N. Djitte; M. Sene
Journal: ISRN Applied Mathematics
 (2012)

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## Abstract

LetE be a reflexive real Banach space with uniformly Gâteaux differentiable norm and F, K : E→E be Lipschitz accretive maps with D(K)=R(F)=E. Suppose that the Hammerstein equation u+KFu=0 has a solution. An explicit iteration method is shown to converge strongly to a solution of the equation. No invertibility assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our theorems are significant improvements on important recent results (e.g., (Chiume and Djitte, 2012)).


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## Body

## 1. Introduction

LetE be a real normed space and let S:={x∈E:∥x∥=1}. E is said to have a Gâteaux differentiable norm if the limit
(1.1)lim⁡t→0+⁡‖x+ty‖-‖x‖t
exists for each x,y∈S. The normed space E is said to have a uniformly Gâteaux differentiable norm if for each y∈S the limit is attained uniformly for x∈S. Furthermore, E is said to be uniformly smooth if the limit exists uniformly for (x,y)∈S×S.LetE be a normed linear space with dimension greater than or equal to 2. The modulus of smoothness of E is the function ρE:[0,∞)→[0,∞) defined by
(1.2)ρE(t):=sup⁡{‖x+y‖+‖x-y‖2-1:‖x‖=1,‖y‖=t}.
In terms of the modulus of smoothness, the space E is called uniformly smooth if and only if limt→0+(ρE(t)/t)=0.E is called q-uniformly smooth if there exists a constant c>0 such that ρE(t)≤ctq,t>0. Lp (and ℓp) spaces, 1<p<+∞ are q-uniformly smooth. In particular, Lpis2-uniformly smooth if2≤p<+∞ and p-uniformly smooth if 1<p<2. It is easy to see that for 1<q<+∞, every q-uniformly smooth real Banach space is uniformly smooth and thus has a uniformly Gâteaux differentiable norm.LetE be a real normed linear space with dual E*. For q>1, we denote by Jq the generalized duality mapping from E to 2E* defined by
(1.3)Jq(x):={f*∈E*:〈x,f*〉=‖x‖⋅‖f*‖,‖f*‖=‖x‖q-1},
where 〈·,·〉 denotes the generalized duality pairing. J2 is denoted by J. If E* is strictly convex, then Jq is single-valued. We note that Jq(x)=∥x∥q-2J(x), for x≠0. It is known that if E is a real Banach space with uniform Gâteaux differentiable norm, then the duality map J is norm-to-weak* uniformly continuous on bounded subsets of E.A mapG with domain D(G) in a normed linear space E is said to be strongly accretive if there exists a constant k>0 such that for every x,y∈D(G), there exists jq(x-y)∈Jq(x-y) such that
(1.4)〈Gx-Gy,jq(x-y)〉≥k‖x-y‖q.
If k=0, G is said to be accretive. If E is a Hilbert space, accretive operators are called monotone.The accretive mappings were introduced independently in 1967 by Browder [1] and Kato [2]. Interest in such mappings stems mainly from their firm connection with equations of evolution. It is known (see, e.g., Zeidler [3]) that many physically significant problems can be modelled by initial-value problems of the form
(1.5)u′(t)+Au(t)=0,u(0)=u0,
where A is an accretive operator in an appropriate Banach space. Typical examples where such evolution equations occur can be found in the heat, wave, or the Schrödinger equations. If in (1.5), u(t) is independent of t, then (1.5) reduces to
(1.6)Au=0,
whose solutions correspond to the equilibrium points of the system (1.5). Consequently, considerable research efforts have been devoted, especially within the past 40 years or so, to methods of finding approximate solutions (when they exist) of (1.6). An early fundamental result in the theory of accretive operators, due to Browder [1], states that the initial-value problem (1.5) is solvable if A is locally Lipschitzian and accretive on E. Utilizing the existing result for (1.5), Browder [1] proved that if A is locally Lipschitzian and accretive on E, then A is m-accretive that is, R(I+A)=E. Clearly, a consequence of this is that the following equation
(1.7)u+Au=0
has a solution.One important generalization of (1.7) is the so-called equation of Hammerstein type (see e.g., Hammerstein [4]), where a nonlinear integral equation of Hammerstein type is one of the form:
(1.8)u(x)+∫Ωκ(x,y)f(y,u(y))dy=h(x),
where dy is a σ-finite measure on the measure space Ω; the real kernel κ is defined on 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
(1.9)Kv(x):=∫Ωκ(x,y)v(y)dy;x∈Ω,
and the so-called superposition or  Nemytskii operator by Fu(y):=f(y,u(y)) then, the integral equation (1.8) can be put in operator theoretic form as follows:
(1.10)u+KFu=0,
where without loss of generality, we have taken h≡0.Interest in (1.10) stems mainly from the fact that several problems that arise in differential equations, for instance, elliptic boundary value problems whose linear parts possess Greens functions can, as a rule, be transformed into the form (1.10). Among these, we mention the problem of the forced oscillations of finite amplitude of a pendulum (see, e.g., Pascali and Sburlan [5], Chapter IV).Example 1.1.
We consider the problem of the pendulum.(1.11)d2v(t)dt2+a2sin⁡v(t)=z(t),t∈[0,1],v(0)=v(1)=0,
where the driving force z is odd. The constant a, (a≠0) depends on the length of the pendulum and gravity. Since Green's function of the problem:
(1.12)v′′(t)=0,v(0)=v(1)=0
is given by:
(1.13)k(t,s):={t(1-s),0≤t≤s≤1,s(1-t),0≤s≤t≤1,
it follows that problem (1.11) is equivalent to the nonlinear integral equation
(1.14)v(t)=-∫01‍k(t,s)[z(s)-a2sin⁡v(s)]ds,t∈[0,1].
If we set
(1.15)g(t):=∫01‍k(t,s)z(s)ds,u(t):=v(t)+g(t),t∈[0,1],
then
(1.16)v=u-g
and (1.14) can be written as
(1.17)u(t)+∫01‍k(t,s)a2sin⁡(g(s)-u(s))ds=0,
which is in the Hammerstein equation form:
(1.18)u(t)+∫01‍k(t,s)f(s,u(s))ds=0,
where f(t,s)=a2sin(g(t)-s),t,s∈[0,1].
Equations of Hammerstein type play a crucial role in the theory of optimal control systems and in automation and network theory (see, e.g., Dolezal [6]).
Several existence and uniqueness theorems have been proved for equations of the Hammerstein type (see, e.g., Brézis and Browder [7–9], Browder [1],  Browder et al. [10], Bowder and Gupta [11], Chepanovich [12], and De Figueiredo and Gupta [13]).
For the iterative approximation of solutions of (1.6) and (1.7), the monotonicity/accretivity of A is crucial. The Mann iteration scheme (see, e.g., Mann [14]) has successfully been employed (see, e.g., the recent monographs of Berinde [15] and Chidume [16]). The recurrence formulas used involved K-1 which is also assumed to be strongly monotone, and this, apart from limiting the class of mappings to which such iterative schemes are applicable, is also not convenient in applications. Part of the difficulty is the fact that the composition of two monotone operators need not to be monotone. In the special case in which the operators are defined on subsets D of E which are compact, Brézis and Browder [7] proved the strong convergence of a suitably defined Galerkin approximation to a solution of (1.10) (see also Brézis and Browder [9]). In fact, they proved the following Theorem.Theorem BB (Brézis and Browder).
LetH be a separable real Hilbert space and C be a closed subspace of H. Let K:H→C be a bounded continuous monotone operator and F:C→H be an angle-bounded and weakly compact mapping. For a given f∈C, consider the Hammerstein equation
(1.19)(I+KF)u=f,
and its nth Galerkin approximation given by
(1.20)(I+KnFn)un=P*f,
where Kn=Pn*KPn:H→Cn and Fn=PnFPn*:Cn→H, where the symbols have their usual meanings (see, [5]). Then, for each n∈ℕ, the Galerkin approximation (1.20) admits a unique solution un in Cn and {un} converges strongly in H to the unique solution u∈C of (1.19).Theorem BB is a special case of the actual theorem of Brézis and Browder in which the Banach space is a separable real Hilbert space. The main theorem of Brézis and Browder is proved in an arbitrary separable real Banach space.We observe that the Galerkin method of Brézis and Browder is notiterative. The first attempt to construct an iterative method for the approximation of a solution of a Hammerstein equation, as far as we know, was made by Chidume and Zegeye [17] who constructed a sequence in the cartesian product E×E and proved the convergence of the sequence to a solution of the Hammerstein equation. In subsequent papers, [18, 19], these authors were able to construct explicit coupled algorithms in the original spaceE which converge strongly to a solution of the equation. Following this, Chidume and Djitte studied this explicit coupled algorithms and proved several strong convergence theorems (see, [20–22]).Recently, Chidume and Djitte [20] introduced and studied a coupled explicit iterative process (see Theorem CD below) to approximate solutions of nonlinear equations of Hammerstein-type in real Hilbert spaces when the operators K and F are bounded and maximal monotone. They proved the following theorem.Theorem CD (Chidume and Djitte, [20]).
LetH be a real Hilbert space and F,K:H→H be bounded and maximal monotone operators. Let {un} and {vn} be sequences in H defined iteratively from arbitrary points u1,v1∈H as follows:
(1.21)un+1=un-λn(Fun-vn)-λnθn(un-u1),n≥1,vn+1=vn-λn(Kvn+un)-λnθn(vn-v1),n≥1,
where {λn} and {θn} are sequences in (0,1) satisfying the following conditions:  (1)
limθn=0,(2)
∑n=1∞‍λnθn=∞,λn=o(θn),(3)
limn→∞(((θn-1)/θn)-1)/λnθn=0. Suppose that u+KFu=0 has a solution in H. Then, there exists a constant d0>0 such that if λn≤d0θn for all n≥n0 for some n0≥1, then the sequence {un} converges to u*, a solution of u+KFu=0.It is Kown thatLp spaces, p=2, are Hilbert and Lp spaces, 1<p<∞,p≠2, are not. So, the results of Chidume and Djitte [20] do not cover Lp spaces, 1<p<∞,p≠2.It is our purpose in this paper to prove convergence theorems in reflexive real Banach spaces that include allLp spaces, 1<p<∞. In fact, it is proved in this paper, that an iteration process converges strongly in reflexive real Banach spaces with uniformly Gâteaux differentiable norm, to a solution of the Hammerstein equation (assuming existence) when the operators K and F are Lipschitz and accretive. These spaces include Lp spaces, 1<p<∞. This complements the results of Chidume and Djitte [20] to provide iterative methods for the approximation of solutions of the Hammerstein equation u+KFu=0 in all Lp spaces, 1<p<∞. Our method of proof is different and is also of independent interest.


## 2. Preliminaries

In the sequel, we will need the followings results.Lemma 2.1 (see, e.g., [23]).
Let{λn} be a sequence of nonnegative numbers and {αn}⊆(0,1) a sequence such that ∑n=1∞αn=∞. Let the recursive inequality
(2.1)λn+1≤λn-2αnψ(λn+1)+σn,n=1,2,…,
be given where ψ:[0,∞)→[0,∞) is a strictly increasing function such that it is positive on (0,∞), ψ(0)=0. If σn=o(αn). Then λn→0, as n→∞.Lemma 2.2.
There exists a mapμn:ℓ∞(ℕ)→ℝ such that  (1)
μn is linear; that is, μn(x+y)=μn(x)+μn(y) and μn(c·x)=c·μn(x),(2)
μn is positive: μn(x)≥0 for every x∈ℓ∞(ℕ) with xn≥0,  for  all  n,(3)
μn is normalized: μn(e)=1, where e=(1,1,…),(4)
μn is shift invariant: μn(Sx)=μn(x), for all x∈ℓ∞(ℕ), where  S:ℓ∞(ℕ)→ℓ∞(ℕ) is the shift operator defined by
(2.2)S:(x1,x2,x3,…)⟼(x2,x3,…),
the above properties on the functional μn imply the following:(5)
μn has norm one; thus, |μn(x)|≤∥x∥,  for  all  x∈ℓ∞(ℕ),(6)
μn extends lim on the subspace of the convergent sequences:
(2.3)lim⁡n→∞(xn)=c⟹μn(x)=c,wherex=(xn),(7)
for anyx=(xn)∈ℓ∞(ℕ):
(2.4)liminf⁡n→∞xn≤μn(x)≤limsup⁡n→∞xn.Remark 2.3.
Functionsμn as above are called Banach Limits.Lemma 2.4 (see [24, 25]).
Letx=(xn)∈ℓ∞ be such that μn(x)≤0 for all Banach limits μn. If limsupn(xn+1-xn)≤0, then limsupnxn≤0.Lemma 2.5.
LetE be a real normed linear space. Then, the following inequality holds:
(2.5)‖x+y‖2≤‖x‖2+2〈y,j(x+y)〉∀j(x+y)∈J(x+y),∀x,y∈E.


## 3. Main Results

We now prove the following theorems.Theorem 3.1.
LetE be a real Banach space and F,K:E→E be Lipschitz and accretive maps with D(K)=R(F)=E and Lipschitz constants L1 and L2, respectively. Let {un} and {vn} be sequences in E defined iteratively from arbitrary points u1,v1∈E as follows:
(3.1)un+1=un-λnαn(Fun-vn)-λnθn(un-u1),n≥1,vn+1=vn-λnαn(Kvn+un)-λnθn(vn-v1),n≥1,
where {λn}, {αn}, and {θn} are sequences in (0,1) satisfying the following conditions:  (1)
limλn=0, limαn=0,(2)
λn=o(θn),αn=o(θn). Suppose that u+KFu=0 has a solution in E. Then the sequences {un} and {vn} are bounded.Proof.
LetX:=E×E with the norm ∥z∥X=(∥u∥E2+∥v∥E2)1/2, where z=(u,v). Define the sequence {wn} in X by: wn:=(un,vn). Let u*∈E be a solution of u+KFu=0, set v*:=Fu* and w*:=(u*,v*). We observe that u*=-Kv*. We show that the sequence {wn} is bounded in X.
For this, define(3.2)L:=max⁡{L1,L2},γ0:=2-18(L+1)(L+2).
Since λn=o(θn) and αn=o(θn), there exists N∈ℕ such that αn/θn<γ0 and λn/θn<γ0 for all n≥N. Let r>0 sufficiently large such that w1∈B(w*,r/2) and wN∈B(w*,r). Define B:=B(w*,r)¯.
Claim 1. wn is in B for all n≥N. The proof is by induction. By construction wN∈B. Suppose that wn∈B for some n≥N. We prove that wn+1∈B. Assume for contradiction that wn+1∉B. Then we have ∥wn+1-w*∥E>r. We compute as follows: (3.3)‖wn+1-w*‖X2=‖un+1-u*‖E2+‖vn+1-v*‖E2. Using Lemma 2.5, We have (3.4)‖un+1-u*‖E2=‖un-u*-λn(αn(Fun-vn)+θn(un-u1))‖E2≤‖un-u*‖E2-2λn〈αn(Fun-vn)+θn(un-u1),j(un+1-u*)〉.
Observing that
(3.5)〈αn(Fun-vn),j(un+1-u*)〉=〈αn(Fun+1-Fu*),j(un+1-u*)〉+〈αn(Fun-Fun+1)+αn(v*-vn),j(un+1-u*)〉,〈θn(un-u1),j(un+1-u*)〉=〈θn(un+1-u*),j(un+1-u*)〉+〈θn(un-un+1)+θn(u*-u1),j(un+1-u*)〉,
and using the fact that F is accretive, we obtain the following estimate:
(3.6)‖un+1-u*‖E2≤‖un-u*‖E2-2λnθn‖un+1-u*‖E2+2λn〈αn(Fun+1-Fun)+αn(vn-v*),j(un+1-u*)〉+2λn〈θn(un+1-un)+θn(u1-u*),j(un+1-u*)〉.
Using Schwartz's inequality and the fact that F is Lipschitz, it follows that:
(3.7)‖un+1-u*‖E2≤‖un-u*‖E2-2λnθn‖un+1-u*‖E2+2λn[αnL1‖un+1-un‖E+αn‖vn-v*‖E]‖un+1-u*‖E+2λn[θn‖un+1-un‖E+θn‖u1-u*‖E]‖un+1-u*‖E.
Thus, we obtain
(3.8)‖un+1-u*‖E2≤‖un-u*‖E2-2λnθn‖un+1-u*‖E2+2λn[(αnL1+θn)‖un+1-un‖E]‖un+1-u*‖E+2λn[αn‖vn-v*‖E+θn‖u1-u*‖E]‖un+1-u*‖E.
Using the fact that
(3.9)‖un+1-un‖E=‖λnαn(Fun-vn)+λnθn(un-u1)‖E≤λnαnL1‖un-u*‖E+λnαn‖vn-v*‖E+λnθn‖un-u1‖E=λn(αnL1‖un-u*‖E+αn‖vn-v*‖E+θn‖un-u1‖E),
we have
(3.10)‖un+1-u*‖E2≤‖un-u*‖E2-2λnθn‖un+1-u*‖E2+2λn[(αnL1+θn)(λnαnL1‖un-u*‖E)]‖un+1-u*‖E+2λn[(αnL1+θn)(λnαn‖vn-v*‖E)]‖un+1-u*‖E+2λn[(αnL1+θn)(λnθn‖un-u1‖E)]‖un+1-u*‖E+2λn[αn‖vn-v*‖E+θn‖u1-u*‖E]‖un+1-u*‖E.
Following the same argument, we also obtain
(3.11)‖vn+1-v*‖E2≤‖vn-v*‖E2-2λnθn‖vn+1-v*‖E2+2λn[(αnL2+θn)(λnαnL2‖vn-v*‖E)]‖vn+1-v*‖E+2λn[(αnL2+θn)(λnαn‖un-u*‖E)]‖vn+1-v*‖E+2λn[(αnL2+θn)(λnθn‖vn-v1‖E)]‖vn+1-v*‖E+2λn[αn‖un-u*‖E+θn‖v1-v*‖E]‖vn+1-v*‖E.
Using the fact that
(3.12)‖u‖E+‖v‖E≤2‖w‖X,∀w=(u,v)∈X,
we have
(3.13)‖wn+1-w*‖X2≤‖wn-w*‖X2-2λnθn‖wn+1-w*‖X2+2λn[2(αnL+θn)(λnαnL‖wn-w*‖X)]‖wn+1-w*‖X+2λn[2(αnL+θn)(λnαn‖wn-w*‖X)]‖wn+1-w*‖X+2λn[2(αnL+θn)(λnθn‖wn-w1‖X)]‖wn+1-w*‖X+2λn[2αn‖wn-w*‖X+2θn‖w1-w*‖X]‖wn+1-w*‖X,
which implies that
(3.14)‖wn+1-w*‖X2≤‖wn-w*‖X2-2λnθn‖wn+1-w*‖X2+2λn[22(αnL+θn)(λnαnL‖wn-w*‖X)]‖wn+1-w*‖X+2λn[2(αnL+θn)(λnθn‖wn-w1‖X)]‖wn+1-w*‖X+2λn[2αn‖wn-w*‖X+2θn‖w1-w*‖X]‖wn+1-w*‖X.
Since wn∈B and w1∈B(w*,r/2), we have
(3.15)‖wn+1-w*‖X2≤‖wn-w*‖X2-2λnθn‖wn+1-w*‖X2+2λn[22(αnL+θn)λn(αnLr+2θnr)]‖wn+1-w*‖X+2λn[22αnr+22θnr]‖wn+1-w*‖X.
By assumption, ∥wn+1-w*∥X2>|wn-w*∥X2. So we have
(3.16)θn‖wn+1-w*‖X≤22(αnL+θn)λn(αnL+2θn)r+22αnr+22θnr.
Hence
(3.17)‖wn+1-w*‖X≤22λnθn(L+1)(L+2)r+22αnθnr+22r.
Therefore, using the fact that λn/θn<γ0 and αn/θn<γ0, it follows that ∥wn+1-w*∥X<r. A contradiction. So wn+1∈B. This prove the boundedness of the sequences {un} and {vn}.Theorem 3.2.
LetE be reflexive real Banach space with uniformly Gâteaux differentiable norm and F,K:E→E be Lipschitz accretive mappings. Let {un}n≥1 and {vn}n≥1 be sequences in E defined iteratively from arbitrary u1,v1∈E by
(3.18)un+1=un-λnαn(Fun-vn)-λnθn(un-u1),n≥1,vn+1=vn-λnαn(Kvn+un)-λnθn(vn-v1),n≥1,
where {λn}n≥1, {αn}n≥1, and {θn}n≥1 are real sequences in (0,1) such that λn=o(θn),αn=o(θn), and ∑n=1∞‍λnθn=+∞. Suppose that the equation u+KFu=0 has a solution u*. Then there exists a subset Kmin of E×E such that if (u*,v*)∈Kmin with v*=Fu*, then the sequence {un}n≥1 converges strongly to u*.Proof.
Since, by Theorem3.1, the sequences {un}n≥1 and {vn}n≥1 are bounded and F is bounded, there exists an R>0 such that un∈B1:=B(u*,R)¯ and vn∈B2:=B(v*,R)¯ for all n≥1. Furthermore, the sets B1 and B2 are nonempty closed convex and bounded subsets of E. Let μn be a Banach limit. Define the maps φ1:E→ℝ and φ2:E→ℝ by
(3.19)φ1(u):=μn∥un-u∥2,φ2(v):=μn∥vn-v∥2.
Then for i=1,2,φi is weakly lower semicontinuous. Since E reflexive, then
(3.20)Argmin⁡u∈B1φ1(u)≠∅,Argmin⁡v∈B2φ2(v)≠∅.
Set
(3.21)Kmin⁡:=Argmin⁡u∈B1φ1(u)×Argmin⁡v∈B2φ2(v).
Now assume that w*=(u*,v*)∈Kmin and let t∈(0,1). Then, by convexity of B1, we have that (1-t)u*+tu1∈B1. Thus, φ1(u*)≤φ1((1-t)u*+tu1). It follows from Lemma 2.5 that
(3.22)‖un-u*-t(u1-u*)‖2≤‖un-u*‖2-2t〈u1-u*,j(un-u*-t(u1-u*))〉∀n∈N.
Thus, taking Banach limits gives
(3.23)μn(‖un-u*-t(u1-u*)‖2)≤μn(‖un-u*‖2)-2tμn(〈u1-u*,j(un-u*-t(u1-u*))〉).
That is,
(3.24)φ1((1-t)u*+tu1)≤φ1(u*)-2tμn(〈u1-u*,j(un-u*-t(u1-u*))〉).
This implies that μn(〈u1-u*,j(un-u*-t(u1-u*))〉)≤0foralln∈ℕ. Furthermore, the fact that E has a uniformly Gâteaux differentiable norm gives, as t→0, that
(3.25)〈u1-u*,j(un-u*)〉-〈u1-u*,j(un-u*-t(u1-u*))〉⟶0.
Thus, for all ϵ>0, there exists δϵ>0 such that for all t∈]0,δϵ[ and for all n∈ℕ,
(3.26)〈u1-u*,j(un-u*)〉-〈u1-u*,j(un-u*-t(u1-u*))〉<ϵ.
Thus,
(3.27)μn(〈u1-u*,j(un-u*)〉)-μn(〈u1-u*,j(un-u*-t(u1-u*))〉)≤ϵ,
which implies that
(3.28)μn(〈u1-u*,j(un-u*)〉)≤0.
Using the same argument, we have
(3.29)μn(〈v1-v*,j(vn-v*)〉)≤0.
Moreover, since {un}n≥1,{vn}n≥1, {Fun}n≥1, and {Kvn}n≥1 are all bounded, then, from (3.18) and that there exist positive constants M1 and M2 such that
(3.30)‖un+1-un‖≤λn[αn(‖Fun‖+‖vn‖)+(θn‖un-u1‖)]≤λnM1,‖vn+1-vn‖≤λn[αn(‖Kvn‖+‖un‖)+(θn‖vn-v1‖)]≤λnM2.
Thus, limn→∞∥un+1-un∥=0 and limn→∞∥vn+1-vn∥=0.
Again, using the fact thatE has a uniformly Gâteaux differentiable norm, we obtain that
(3.31)lim⁡n→∞(〈u1-u*,j(un+1-u*)〉-〈u1-u*,j(un-u*)〉)=0,lim⁡n→∞(〈v1-v*,j(vn+1-v*)〉-〈v1-v*,j(vn-v*)〉)=0.
Therefore, the sequences {〈u1-u*,j(un-u*)〉}n≥1 and {〈v1-v*,j(vn-v*)〉}n≥1 satisfy the conditions of Lemma 2.4.
Hence,(3.32)limsup⁡n→∞〈u1-u*,j(un-u*)〉≤0,limsup⁡n→∞〈v1-v*,j(vn-v*)〉≤0.
Define
(3.33)σn:=max⁡{〈u1-u*,j(un+1-u*)〉,0},ξn:=max⁡{〈v1-v*,j(vn+1-v*)〉,0},
then limn→∞σn=0=limn→∞ξn. Moreover,
(3.34)〈u1-u*,j(un-u*)〉≤σn,〈v1-v*,j(vn-v*)〉≤ξn,∀n∈N.
Using Lemma 2.5, and (3.18), we have:
(3.35)‖un+1-u*‖E2≤‖un-u*‖E2-2λnθn‖un+1-u*‖E2+2λn〈αn(Fun+1-Fun)+αn(vn-v*),j(un+1-u*)〉+2λn〈θn(un+1-un)+θn(u1-u*),j(un+1-u*)〉,‖vn+1-v*‖E2≤‖vn-v*‖E2-2λnθn‖vn+1-v*‖E2+2λn〈αn(Kvn+1-Kvn)+αn(v*-vn),j(vn+1-v*)〉+2λn〈θn(vn+1-vn)+θn(v1-v*),j(vn+1-v*)〉.
Using the fact that F and K are Lipschitz and the sequences {un} and {vn} are bounded, we obtain the following estimate:
(3.36)‖wn+1-w*‖E2≤‖wn-w*‖E2-2λnθn‖wn+1-w*‖E2+2(λnαn+λn2)c+2λnθn(σn+ξn)=‖wn-w*‖E2-2λnθn‖wn+1-w*‖E2+γn,
where γn=2(λnαn+λn2)c+2λnθn(σn+ξn)=o(λnθn) for some c>0. Hence, by Lemma 2.1, wn→w* as n→∞. But wn=(un,vn) and w*=(u*,v*). This implies that un→u* as n→∞. This completes the proof.Corollary 3.3.
LetE be a q-uniformly real Banach spaces, q>1 and F,K:E→E be Lipschitz accretive mappings. Let {un}n≥1 and {vn}n≥1 be sequences in E defined iteratively from arbitrary u1,v1∈E by
(3.37)un+1=un-λnαn(Fun-vn)-λnθn(un-u1),n≥1,vn+1=vn-λnαn(Kvn+un)-λnθn(vn-v1),n≥1,
where {λn}n≥1, {αn}n≥1, and {θn}n≥1 are real sequences in (0,1) such that λn=o(θn),αn=o(θn), and ∑n=1∞‍λnθn=+∞. Suppose that the equation u+KFu=0 has a solution u*. Then there exists a set Kmin in E×E such that if (u*,v*)∈Kmin with v*=Fu*, then the sequence {un}n≥1 converges strongly to u*.Corollary 3.4.
LetE=Lp (or ℓp) space (1<p<+∞) and F,K:E→E be Lipschitz accretive mappings. Let {un}n≥1 and {vn}n≥1 be sequences in E defined iteratively from arbitrary u1,v1∈E by
(3.38)un+1=un-λnαn(Fun-vn)-λnθn(un-u1),n≥1,vn+1=vn-λnαn(Kvn+un)-λnθn(vn-v1),n≥1,
where {λn}n≥1, {αn}n≥1, and {θn}n≥1 are real sequences in (0,1) such that λn=o(θn),αn=o(θn), and ∑n=1∞‍λnθn=+∞. Suppose that the equation u+KFu=0 has a solution u*. Then there exists a subset Kmin of E×E such that if (u*,v*)∈Kmin with v*=Fu*, then the sequence {un}n≥1 converges strongly to u*.Corollary 3.5.
Let H be a real Hilbert space andF,K:H→H be Lipschitz monotone mappings. Let {un}n≥1 and {vn}n≥1 be sequences in H defined iteratively from arbitrary u1,v1∈H by
(3.39)un+1=un-λnαn(Fun-vn)-λnθn(un-u1),n≥1,vn+1=vn-λnαn(Kvn+un)-λnθn(vn-v1),n≥1,
where {λn}n≥1, {αn}n≥1, and {θn}n≥1 are real sequences in (0,1) such that λn=o(θn),αn=o(θn), and ∑n=1∞‍λnθn=+∞. Suppose that the equation u+KFu=0 has a solution u*. Then there exists a subset Kmin of E×E such that if (u*,v*)∈Kmin with v*=Fu*, then the sequence {un}n≥1 converges strongly to u*.Remark 3.6.
Real sequences that satisfy the hypotheses of Theorems3.1 are λn=αn=n-a and θn=n-b, n≥1 with 0<b<a and a+b<1.



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*Source: 963802-2012-07-12.xml*