# Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type

Authors: C. E. Chidume; N. Djitté
Journal: ISRN Mathematical Analysis
 (2012)

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## Abstract

Suppose thatH is a real Hilbert space and F,K:H→H are bounded monotone maps with D(K)=D(F)=H. Let u* denote a solution of the Hammerstein equation u+KFu=0. An explicit iteration process is shown to converge strongly to u*. No invertibility or continuity assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our result is a significant improvement on the Galerkin method of Brézis and Browder.


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## Body

## 1. Introduction

LetX be a real normed linear space with dual X*. For q>1, we denote by Jq the generalized duality mapping from X to 2X* defined by(1.1)Jq(x):={f*∈X*:〈x,f*〉=‖x‖⋅‖f*‖,‖f*‖=‖x‖q-1},
where 〈·,·〉 denotes the generalized duality pairing. J2 is denoted by J. If X* is strictly convex, then Jq is single-valued. A map G with domain D(G) in a normed linear space X is said to be strongly accretiveif there exists a constant k>0 such that for every x,y∈D(G), there exists jq(x-y)∈Jq(x-y) such that(1.2)〈Gx-Gy,jq(x-y)〉≥k‖x-y‖q.
If k=0, G is said to be accretive. If X 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(1.3)x′(t)+Ax(t)=0,x(0)=x0,
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 Schrödinger equations. If in (1.3), x(t) is independent of t, then (1.3) reduces to(1.4)Au=0,
whose solutions correspond to the equilibrium points of the system (1.3). Consequently, considerable research efforts have been devoted, especially within the past 30 years or so, to methods of finding approximate solutions (when they exist) of (1.4). An early fundamental result in the theory of accretive operators, due to Browder [1], states that the initial value problem (1.3) is solvable if A is locally Lipschitzian and accretive on X. Utilizing the existence result for (1.3), Browder [1] proved that if A is locally Lipschitzian and accretive on X, then A is m-accretive, that is, R(I+A)=X, where R(I+A) denotes the range of (I+A). Clearly, a consequence of this is that the equation(1.5)u+Au=0
has a solution. One important generalization of (1.5) 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:(1.6)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(1.7)Kv(x):=∫Ωκ(x,y)v(y)dy,x∈Ω,
and the so-called superpositionor Nemytskii operator by Fu(y):=f(y,u(y)) then, the integral equation (1.6) can be put in operator theoretic form as follows:(1.8)u+KFu=0,
where, without loss of generality, we have taken h≡0.Interest in (1.8) 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 (1.8) (see e.g., Pascali and Sburlan [5], Chapter IV). 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], Browder and Gupta [11], Cydotchepanovich [12], and De Figueiredo and Gupta [13]). For the iterative approximation of solutions of (1.4) and (1.5), 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 be monotone. In the special case in which the operators are defined on subsets D of X which are compact (or more generally, angle-bounded see e.g., Pascali and Sburlan [5] for definition), Brézis and Browder [7] have proved the strong convergence of a suitably defined Galerkin approximation to a solution of (1.8) (see also Brézis and Browder [9]).It is our purpose in this paper to prove that an explicit coupled iteration process recently introduced by Chidume and Zegeye [17] which does not involve K-1 which is also required to be monotone converges strongly to a solution of (1.8) when K and F are bounded and monotone. Our new method of proof is also of independent interest.


## 2. Preliminaries

In the sequel, we will need the followings results.Lemma 2.1 (see Xu [18]).
Let{an} be a sequence of nonnegative real numbers satisfying the following relations:(2.1)an+1≤(1-αn)an+αnσn+γn,n≥0,
where (i){αn}⊂(0,1), ∑αn=∞; (ii)limsupσn≤0; (iii)γn≥0, (n≥0), ∑γn<∞. Then, an→0 as n→∞.Lemma 2.2 (see Chidume and Djitte, [19, Lemma  2.5]).
LetH be a real Hilbert space and A:H→H be a map with D(A)=H. Suppose that A is m-accretive, that is, (i) for allu,v∈H, 〈Au-Av,u-v〉≥0; (ii)R(I+s0A)=H for some s0>0. Then A satisfies the range condition, that is, R(I+sA)=H for all s>0.We now prove the following result.Lemma 2.3.
LetH be a real Hilbert space and F,K:H→H be maps with D(F)=D(K)=H. Let E=H×H and T:E→E be the map defined by:
(2.2)Tw=(Fu-v,Kv+u),∀w=(u,v)∈E.
Assume that F and K are monotones and satisfy the range condition. Then, T is monotone and also satisfies the range condition.Proof.
OnE we have the natural norm ∥·∥E and natural inner product 〈·,·〉E given by:
(2.3)‖w‖E=(‖u‖H2+‖v‖H2)1/2,forw=(u,v)∈E,〈w1,w2〉E=〈u1,u2〉H+〈v1,v2〉H,forw1=(u1,v1),w2=(u2,v2)∈E.Step 1.
We prove thatT is monotone. Let w1=(u1,v1), w2=(u2,v2)∈E. We have Tw1=(Fu1-v1,Kv1+u1) and Tw2=(Fu2-v2,Kv2+u2). So, Tw1-Tw2=(Fu1-Fu2+v2-v1,Kv1-Kv2+u1-u2). Therefore, using the fact that F and K are monotone, we obtain,(2.4)〈Tw1-Tw2,w1-w2〉E=〈Fu1-Fu2+v2-v1,u1-u2〉H+〈Kv1-Kv2+u1-u2,v1-v2〉H=〈Fu1-Fu2,u1-u2〉H+〈Kv1-Kv2,v1-v2〉H≥0.
So, T is monotone.Step 2.
We show thatℛ(IE+rT)=E for all r, 0<r<1. In fact let r0 such that 0<r0<1. Since F and K are monotone and satisfy the range condition, then it is known that (I+r0F) and (I+r0K) are bijective and moreover, the resolvent Jr0F:=(I+r0F)-1 of F and the resolvent Jr0K:=(I+r0K)-1 of K are nonexpansive.
Leth=(h1,h2)∈E. Define G:E→E by(2.5)Gw=(Jr0F(h1+r0v),Jr0K(h2-r0u)),∀w=(u,v)∈E.
Using the fact that Jr0F and Jr0K are nonexpansive, we have,
(2.6)‖Gw1-Gw2‖E≤r0‖w1-w2‖E,∀w1,w2∈E.
Therefore G is a contraction. So, by the Banach fixed point theorem, G has a unique fixed point w*=(u*,v*)∈E, that is Gw*=w* or equivalently,
(2.7)u*=Jr0F(h1+r0v*),v*=Jr0K(h2-r0u*).
These imply (IE+r0T)w*=h. Therefore, ℛ(IE+r0T)=E.
By Lemma2.2, it follows that T satisfies the range condition. This completes the proof.Theorem 2.4 (see Reich [20]).
LetH be a real Hilbert space. Let A:H→H be monotone with D(A)=H and suppose that A satisfies the range condition: ℛ(I+rA)=Hfor all r>0. Let Jtx:=(I+tA)-1x, t>0 be the resolvent of A, and assume that A-1(0) is nonempty. Then for each x∈H, limt→∞Jtx∈A-1(0).


## 3. Main Results

LetH be a real Hilbert space and F,K:H→H be maps with D(K)=D(F)=H such that the following conditions hold:(i)
F is bounded and monotone, that is,
(3.1)〈Fu1-Fu2,u1-u2〉≥0,∀u1,u2∈H,(ii)
K is bounded and monotone, that is,
(3.2)〈Ku1-Ku2,u1-u2〉≥0,∀u1,u2∈H,(iii)
F and K satisfy the range condition.With these assumptions, we prove the following theorem.Theorem 3.1.
LetH be a real Hilbert space. Let {un} and {vn} be sequences in H defined iteratively from arbitrary points u1,v1∈H as follows:
(3.3)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.Proof.
LetE:=H×H with the norm ∥z∥E=(∥u∥H2+∥v∥H2)1/2, where z=(u,v). Define the sequence {wn} in E by: wn:=(un,vn). Let u*∈H be a solution of u+KFu=0, v*:=Fu* and w*:=(u*,v*). We observe that u*=-Kv*. It suffices to show that {wn} converges to w* in E.
For this, letn0∈ℕ, there exists r>0 sufficiently large such that w1∈B(w*,r/2), wn0∈B(w*,r), where B(w*,r) denotes the ball of center w* and radius r. Define B:=B(w*,r)¯. Since F and K are bounded, we set M1:=sup{∥Fx-y∥H2+r2:(x,y)∈B}<∞ and M2:=sup{∥Ky+x∥H+r2:(x,y)∈B}<∞. Let M:=M1+M2. We split the proof in three steps.
Step 1.
We first prove that the sequence{wn} is bounded in E. Indeed, it suffices to show that wn is in B for all n≥n0. The proof is by induction. By construction, wn0∈B. Suppose that wn∈B for n≥n0. 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:
(3.4)‖wn+1-w*‖2=‖un+1-u*‖H2+‖vn+1-v*‖H2.
We have
(3.5)‖un+1-u*‖H2=‖un-u*-λn(Fun-vn)-λnθn(un-u1)‖2=‖un-u*‖H2-2λn〈Fun-vn+θn(un-u1),un-u*〉+λn2‖Fun-vn+θn(un-u1)‖H2≤‖un-u*‖H2-2λn〈Fun-vn+θn(un-u1),un-u*〉+λn2M1.
Observing that
(3.6)〈Fun-vn+θn(un-u1),un-u*〉=〈Fun-Fu*,un-u*〉-〈vn-v*,un-u*〉+θn‖un-u*‖H2+θn〈u*-u1,un-u*〉,
and using (3.1), we obtain the following estimate:
(3.7)‖un+1-u*‖H2≤[1-2λnθn]‖un-u*‖H2+λn2M1+2λn〈vn-v*,un-u*〉-2λnθn〈u*-u1,un-u*〉.
Following the same argument, we also obtain
(3.8)‖vn+1-v*‖H2≤[1-2λnθn]‖vn-v*‖H2+λn2M2-2λn〈un-u*,vn-v*〉-2λnθn〈v*-v1,vn-v*〉.
Thus, we obtain
(3.9)‖wn+1-w*‖E2≤[1-2λnθn]‖wn-w*‖E2+Mλn2-2λnθn〈u*-u1,un-u*〉-2λnθn〈v*-v1,vn-v*〉.
Using
(3.10)0≤‖u*-u1+(un-u*)‖H2=‖u*-u1‖H2+2〈u*-u1,un-u*〉+‖un-u*‖H2,0≤‖v*-v1+(vn-v*)‖H2=‖v*-v1‖H2+2〈v*-v1,vn-v*〉+‖vn-v*‖H2,
we have
(3.11)-2〈u*-u1,un-u*〉≤‖u*-u1‖H2+‖un-u*‖H2,-2〈v*-v1,vn-v*〉≤‖v*-v1‖H2+‖vn-v*‖H2.
Therefore
(3.12)‖wn+1-w*‖E2≤[1-2λnθn]‖wn-w*‖E2+Mλn2+λnθn‖wn-w*‖E2+λnθn‖w*-w1‖E2.
So we obtain the following estimate:
(3.13)‖wn+1-w*‖E2≤[1-λnθn]‖wn-w*‖E2+λnθn‖w*-w1‖E2+Mλn2.
Let d0=r2/4M. Then using the induction assumptions, the fact that w1∈B(w*,r/2) and λn≤d0θn, we obtain
(3.14)‖wn+1-w*‖H2≤[1-λnθn4]r2<r2,
a contradiction. Therefore, wn+1∈B. Thus by induction, {wn} is bounded and so are {un} and {vn}.Step 2.
We show that there exists a unique sequencezn=(xn,yn)∈E such that
(3.15)θn(xn-u1)+Fxn-yn=0,(3.16)θn(yn-v1)+Kyn+xn=0,
and xn→x*, yn→y*, with x*+KFx*=0 and y*=Fx*.
In fact, letT:E→E be defined by T(u,v)=(Fu-v,Kv+u),for all (u,v)∈E. Using the fact that F and K are monotone and satisfy the range condition, it follows from Lemma 2.3 that T is monotone and also satisfies the range condition.
Applying Theorem2.4, with t=1/θn and x=(u1,v1), we obtain that limt→+∞Jtx∈T-1(0) implies that
(3.17)lim⁡n→+∞(I+1θnT)-1(u1,v1)∈T-1(0).
Set zn=(xn,yn):=(I+(1/θn)T)-1(u1,v1). Then (I+(1/θn)T)(xn,yn)=(u1,v1), for all n≥1. So we have,
(3.18)xn+1θn(Fxn-yn)=u1,yn+1θn(Kyn+xn)=v1.
Therefore,
(3.19)θn(xn-u1)+Fxn-yn=0,θn(yn-v1)+Kxn+xn=0.
Since T is monotone and satisfies the range condition, then it is known that (I+rT) is bijective for every r>0. So, the sequence {zn} is unique. Using (3.17) and Theorem 2.4, we have, limzn∈T-1(0). Let xn→x* and yn→y*. Then (x*,y*)∈T-1(0). So, T(x*,y*)=0, that is,
(3.20)Fx*-y*=0,Ky*+x*=0.
Therefore, y*=Fx* and x*+KFx*=0.Step 3.
We show that{wn}→(u*,v*), where u*+KFu*=0 and v*=Fu*.Claim 1.
wn+1-zn→0 as n→∞. We compute as follows:
(3.21)‖wn+1-zn‖E2=‖un+1-xn‖H2+‖vn+1-yn‖H2.
We have
(3.22)‖un+1-xn‖H2=‖un-xn-λn(Fun-vn+θn(un-u1))‖H2=‖un-xn‖H2-2λn〈Fun-vn+θn(un-u1),un-xn〉+λn2‖Fun-vn+θn(un-u1)‖H2.
From the boundness of {un}, {vn}, and F, there exists M3>0 such that ∥Fun-vn+θn(un-u1)∥H2≤M3. Using (3.15) and the fact that F is monotone, we obtain
(3.23)‖un+1-xn‖H2≤(1-λnθn)‖un-xn‖H2-2λn〈Fxn-vn,un-xn〉-2λn〈yn-Fxn,un-xn〉+M3λn2=(1-λnθn)‖un-xn‖H2-2λn〈yn-vn,un-xn〉+M3λn2,
for some constant M3>0. Using (3.16) and similar arguments, we obtain:
(3.24)‖vn+1-vn‖H2≤(1-λnθn)‖vn-yn‖H2-2λn〈Kyn+un,vn-yn〉+2λn〈xn+Kyn,vn-yn〉+M4λn2=(1-λnθn)‖vn-yn‖H2+2λn〈xn-un,vn-yn〉+M4λn2,
for some constant M4>0. Therefore, we have the following estimate:
(3.25)‖wn+1-zn‖E2≤(1-λnθn)‖wn-zn‖E2+M′λn2,whereM′=M3+M4.
On the other hand, using the monotonicity of F and K we have
(3.26)‖zn-1-zn‖E2≤‖xn-1-xn+θn-1(Fxn-1-yn-1-Fxn+yn)‖H2+‖yn-1-yn+θn-1(Kyn-1+xn-1-Kyn-xn)‖H2.
Using (3.15) and (3.16), we observe that
(3.27)xn-1-xn+1θn(Fxn-1-yn-1-Fxn+yn)=θn-θn-1θn(xn-1-u1),yn-1-yn+1θn(Kyn-1+xn-1-Kyn-xn)=θn-θn-1θn(yn-1-v1).
Therefore,
(3.28)‖zn-1-zn‖E≤θn-1-θnθn‖zn-1-w1‖E.
Using (3.25) and the boundness of {xn} and {yn}, we obtain that there exists C>0 such that:
(3.29)‖wn+1-zn‖E2≤(1-λnθn)‖wn-zn-1‖H2+C(θn-1-θnθn)+M′λn2.
Thus, by Lemma 2.1, wn+1-zn→0. Since zn→(x*,y*), we obtain that wn→(x*,y*). But since wn=(un,vn), this implies that un→u* and vn→v*. This completes the proof.Corollary 3.2.
LetH be a real Hilbert space and F,K:H→H be maps with D(K)=D(F)=H such that the following conditions hold:(i)
F and K are Lipschitz and monotone,(ii)
F and K satisfy the range condition. Let {un} and {vn} be sequences in H defined iteratively from arbitrary points u1,v1∈H as follows:
(3.30)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.LetX be a real Banach space with dual space X* and let A:X→X* be a monotone linear operator. The mapping A is said to be angle-bounded with constant α≥0 if(3.31)|(Ax,y)-(Ay,x)|≤2α(Ax,x)1/2(Ay,y)1/2,∀x,y∈D(A),
where (·,·) denotes the duality pairing between elements of X* and those of X. The class of angle-bounded operators is a subclass of the class of monotone operators. The angle-boundness of A with α=0 corresponds to the symmetry of A, that is,(3.32)(Ax,y)=(Ay,x),∀x,y∈D(A).
(See Pascali and Sburlan [5, Chapter IV, page 189]).LetH be a separable real Hilbert space and C be a closed subspace of H. For a given f∈C, consider the Hammerstein equation:(3.33)(I+KF)u=f,
and its nth Galerkin approximation given by(3.34)(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] for the meaning of the symbols). Under this setting, Brézis and Browder (see [9]) proved the following approximation theorem.Theorem BB.
LetH be a separable real Hilbert space. Let K:H→C be a bounded continuous monotone operator and F:C→H be an angle-bounded and weakly compact mapping. Then, for each n∈ℕ, the Galerkin approximation (3.34) admits a unique solution un in Cn and {un} converges strongly in H to the unique solution u∈C of the (3.33).Remark 3.3.
Theorem BB is the 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 Banach space.Remark 3.4.
The class of mappings considered in our theorem (Theorem3.1) is larger than that considered in Theorem BB. In particular, in Theorem BB, in addition to assuming that the operator K is bounded and monotone, the authors also required K to be continuous. Furthermore, the operator F is restricted to the class of angle-bounded operators (a subclass of the monotone operators) and is also assumed to be weakly compact. In Theorem 3.1, the operators K and F are only assumed to be bounded and monotone and satisfy the range condition. We remark that continuity of the monotone map K implies that K is m-accretive (see Martin [21]) and it is known that m-accretive implies range condition.Remark 3.5.
Theorem BB guarantees theexistence of a sequence {un} which converges strongly to a solution of the Hammerstein equation (3.33). Our theorem provides an iterative sequence which converges strongly to a solution of (3.33).Remark 3.6.
Real sequences that satisfy the hypotheses of Theorem3.1 are λn=(n+1)-a and θn=(n+1)-b with 0<b<a and a+b<1.We verify that these choices satisfy, in particular, condition (3) of Theorem3.1. In fact, using the fact that (1+x)p≤1+px, for x>-1 and 0<p<1, we have(3.35)0≤((θn-1/θn)-1)λnθn=[(1+1n)b-1]⋅(n+1)a+b≤b⋅(n+1)a+bn=b⋅n+1n⋅1(n+1)1-(a+b)⟶0,
as n→∞.



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*Source: 169751-2012-03-25.xml*