OF HAMMERSTEIN TYPE

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ITERATIVE APPROXIMATION OF SOLUTIONS OF NONLINEAR EQUATIONS

OF HAMMERSTEIN TYPE

C. E. CHIDUME AND H. ZEGEYE Received 27 October 2001

SupposeX is a realq-uniformly smooth Banach space andF,K:X→X with D(K)=F(X)=Xare accretive maps. Under various continuity assumptions on FandKsuch that 0=u+KFuhas a solution, iterative methods which converge strongly to such a solution are constructed. No invertibility assumption is im- posed onKand the operatorsKandFneed not be defined on compact subsets ofX. Our method of proof is of independent interest.

1. Introduction

LetXbe a real normed linear space with dualX^∗. For 1< q <∞, we denote by J_q, the generalized duality mapping fromXto 2^X^∗ defined by

Jq(x) :=

f^∗∈X^∗:x, f^∗= xf^∗,f^∗= x^q⁻¹

, (1.1)

where·,·denotes the generalized duality pairing. Ifq=2,Jq=J2and is de- noted byJ. IfX^∗is strictly convex, thenJq is single-valued (see, e.g., [32]). A multivalued mapAwith domainD(A) in a normed linear spaceXis said to be accretiveif for everyx, y∈D(A), there existsjq(x−y)∈Jq(x−y) such that

ξ−η, jq(x−y)≥0 for eachξ∈Ax, η∈Ay. (1.2) IfXis a Hilbert space, accretive operators are also calledmonotone. The accretive mappings were introduced independently in 1967 by Browder [6] and Kato [24].

Interest in such mappings stems mainly from their firm connection with equations of evolution. It is known (see, e.g., [33]) that many physically significant problems can be modelled by initial-value problems of the form

x(t) +Ax(t)=0, x(0)=x0, (1.3)

2000 Mathematics Subject Classification: 47H06, 47H15, 47H17, 47J25 URL:http://dx.doi.org/10.1155/S1085337503209052

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whereAis an accretive operator in an appropriate Banach space. Typical examples, where such evolution equations occur, can be found in the heat, wave or Schr¨odinger equations. If in (1.3),x(t) is independent oft, then (1.3) reduces to

Au=0, (1.4)

whose solutions correspond to the equilibrium points of system (1.3). Conse- quently, considerable research eﬀorts have been devoted, especially within the past twenty years or so, to methods of finding approximate solutions (when they exist) of (1.4), and hence,

u+Au=0. (1.5)

One important generalization of (1.5) is the so-calledequation of Hammerstein type(see, e.g., [22]) where a nonlinear integral equation of Hammerstein type is one of the form

u(x) +

Ωκ(x, y)fy,u(y)dy=h(x), (1.6) wheredyis aσ-finite measure on the measure spaceΩ. The real kernelκis defined onΩ×Ω,f is a real-valued function defined onΩ× and is, in general, nonlinear, andhis a given function onΩ. Now if we define an operatorKby

Kv(x) :=

Ωκ(x, y)v(y)dy, x∈Ω, (1.7) and the so-calledsuperposition orNemytskiioperator by Fu(y) := f(y,u(y)), then the integral equation (1.6) can be put in operator theoretic form as follows:

u+KFu=0, (1.8)

where, without loss of generality, we have takenh≡0. Now it is obvious that equationu+Au=0 is a very special case of (1.8) in whichK=I (the identity operator onX) and A:=F. Interest in (1.8) stems mainly from the fact that several problems arising in diﬀerential equations, for instance, elliptic bound- ary value problems whose linear parts possess Greens functions can, as a rule, be transformed into form (1.8) (see, e.g., [27, Chapter IV]). Equations of Hammer- stein type play a crucial role in the theory of optimal control systems (see, e.g., [21]). Several existence and uniqueness theorems have been proved for equations of the Hammerstein type (see, e.g., [3,5,7,8,19,10]).

For the iterative approximation of solutions of (1.4) and (1.5), themonotonic- ity/accretivityofAis crucial. The Mann iteration scheme (see, e.g., [26]) and the Ishikawa iteration scheme (see, e.g., [23]) have successfully been employed (see, e.g., [1,9,10,11,12,13,14,15,16,17,18,19,20,21,23,25,27,28,29,30, 31,32,33,34]). Attempts to apply these methods to (1.8) have not provided satisfactory results. In particular, the recursion formulas obtained involvedK⁻¹

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(see, e.g., [12,15,28]) and this is not convenient in applications. Part of the dif- ficulty 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 subsetsDof Xwhich are compact (or more generally, angle-bounded), Br´ezis and Browder [2] have proved the strong convergence of a suitably defined Galerkin approximation to a solution of (1.8) (see also [4]).

It is our purpose in this paper to introduce a new method that contains an auxiliary operator, defined in an appropriate real Banach space in terms ofK andF, which under certain conditions, is accretive wheneverK andFare, and whose zeros are solutions of (1.8). Moreover, the operatorsKandFneed not be defined on compact or angle-bounded subset ofX. Furthermore, our method which does not involveK⁻¹provides an explicit algorithm for the computation of solutions of (1.8).

2. Preliminaries

LetX be a real normed linear space of dimension≥2. Themodulus of smoothness ofX is defined by

ρ_X(τ) :=sup x+y+x−y

2 ⁻1 :x =1,y =τ

, τ >0. (2.1) If there exist a constantc >0 and a real number 1< q <∞, such thatρX(τ)≤cτ^q, thenX is said to beq-uniformly smooth. Typical examples of such spaces are the LebesgueLp, the sequencep, and the SobolevW_p^mspaces for 1< p <∞where

L_porl_porW^m_p =



2-uniformly smooth if 2≤p <∞;

p-uniformly smooth if 1< p <2. (2.2) A Banach spaceXis calleduniformly smoothif lim_τ_→0ρX(τ)/τ=0. A multivalued mapA is said to be m-accretive if it is accretive and R(I+λA) (range of (I+λA))=X, for allλ >0, whereI is the identity mapping.Ais said to beφ- strongly accretiveif for everyx, y∈D(A), there existjq(x−y)∈Jq(x−y) and a strictly increasing functionφ: [0,∞)→[0,∞),φ(0)=0 such that

ξ−η, j_q(x−y)≥φx−y

x−y^q⁻¹, (2.3) for eachξ∈Ax,η∈Ay, and it is strongly accretive if for eachx, y∈D(A), there existj_q(x−y)∈J_q(x−y) and a constantk∈(0,1) such that

ξ−η, jq(x−y)≥kx−y^q for eachξ∈Ax, η∈Ay. (2.4) Let CB(X) be a family of all nonempty closed bounded subsets ofX. A multivalued mappingA:X→CB(X) is said to beuniformly continuousif for every given ε >0, there exists aδ >0 such that for any givenx, y∈Xwithx−y< δ, we

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haveH(Ax,Ay)< εwhereHis the Hausdorﬀmetric on CB(X), that is, for any givenD,F∈CB(X),

H(D,F) :=max

sup

x∈D

inf

y∈Fd(x, y),inf

x∈D

sup

y∈Fd(x, y)

. (2.5)

In the sequel, we will need the following results.

Theorem2.1 [32]. Letq >1andXbe a real Banach space. Then the following are equivalent:

(1)Xisq-uniformly smooth;

(2)there exists a constantd_q>0such that for allx, y∈X

x+y^q≤ x^q+qy, j_q(x)+d_qy^q; (2.6) (3)there exists a constantc_q>0such that for allx, y∈Xandλ∈[0,1]

(1−λ)x+λy^q≥(1−λ)x^q+λy^q−wq(λ)cqx−y^q, (2.7) wherewq(λ)=λ^q(1−λ) +λ(1−λ)^q.

Theorem2.2 [17]. LetXbe a real uniformly smooth Banach space. LetA:X→X be a boundedφ-strongly accretive map. Assume0=Ax has a solution x^∗∈X.

Then, there exists a real numberγ0>0such that if the real sequence{α_n} ⊂[0,γ0] satisfies the following conditions:(i) limαn=0;(ii)αn= ∞, then for arbitrary x0∈Xthe sequence{xn}, defined by

x_n+1:=x_n−α_nAx_n, n≥0, (2.8) converges strongly tox^∗, the unique solution ofAx=0.

Theorem2.3 [11]. Let X be an arbitrary real Banach space. LetA:X→X be a Lipschitz and strongly accretive map with Lipschitz constantL >0 and strong accretivity constantλ∈(0,1). Assume thatAx=0has a solutionx^∗∈X. Define A_ε:X→X by A_εx:=x−εAx for x∈X where ε:=1/2{λ/(1 +L(3 +L−λ))}. For arbitraryx0∈X, define the Picard sequence{xn}inXbyxn+1=Aεxn,n≥0.

Then,{xn}converges strongly tox^∗withxn+1−x^∗ ≤δⁿx1−x^∗whereδ:= (1−1/2λε)∈(0,1). Moreover,x^∗is unique.

3. Main results

Lemma3.1. Forq >1, letXbe a realq-uniformly smooth Banach space. LetE:= X×Xwith norm

z_E:=

u^q_X+v^q_X1/q

, (3.1)

for arbitraryz=[u,v]∈E. Let E^∗:=X^∗×X^∗ denote the dual space ofE. For

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arbitraryx=[x1,x2]∈E, define the map jÊ_q :E→E^∗ by jÊ_q(x)= j_qÊ[x1,x2] := [j_q^X(x1), j^X_q(x2)], so that for arbitraryz1=[u1,v1],z2=[u2,v2]inEthe duality pairing·,·is given by

z1, j_q^Ez2

=

u1, j_q^Xu2

+v1, j_q^Xv2

. (3.2)

Then,

(a)Eisq-uniformly smooth;

(b) j^E_q is a single-valued duality mapping onE.

Proof. (a) Letx=[x1,x2], y=[y1, y2] be arbitrary elements ofE. It suﬃces to show thatxandysatisfy condition (2) ofTheorem 2.1. We compute as follows:

x+y^q_E=x1+y1,x2+y2^q

E=x1+y1^q

X+x2+y2^q

X

≤x1^q

X+x2^q

X+dqy1^q

X+y2^q

X

+qy1, j_q^Xx1

+y2, j_q^Xx2

(3.3)

for some constants dq>0 (using (2) of Theorem 2.1 sinceX is q-uniformly smooth). It follows that

x+y^q_E≤ x^q_E+qy, j_q^E(x)+dqy^q_E. (3.4) So, the result follows fromTheorem 2.1. SinceEisq-uniformly smooth, it is smooth and so any duality mapping onEis single-valued.

(b) For arbitrary x=[x1,x2]∈E, let j_q^E(x)= j_q^E[x1,x2]=ψq. Then ψq= [j_q^X(x1), j^X_q(x2)] inE^∗. Observe that forp >1 such that 1/ p+ 1/q=1,

ψ_q_E∗=j_q^Xx1

, j_q^Xx2¹^{/ p}=j_qx1^p

X^∗+j_qx2^p

X^∗

1/ p

=x1⁽^q⁻¹⁾^p

X +x2⁽^q⁻¹⁾^p

X

1/ p

=x1^q

X+x2^q

X

(q−1)/q

= x^q_X⁻¹.

(3.5)

Hence,ψ_qE^∗= x^q_E⁻¹. Furthermore, x,ψ_q=

x1,x2

,j_q^Xx1

, j_q^Xx2

=

x1, j_q^Xx1

+x2, j^X_qx2

=x1^q

X+x2^q

X=x1^q

X+x2^q

X

1/qx1^q

X+x2^q

X

(q−1)/q

= xE· ψ^q_E⁻^∗¹.

(3.6)

Hence,j_q^Eis a single-valued duality mapping onE.

Lemma3.2. LetXbe a realq-uniformly smooth Banach space. LetF,K:X→Xbe maps withD(K)=F(X)=Xsuch that the following conditions hold:

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(i)for eachu1,u2∈D(F), there exists a strictly increasing functionφ1: [0,∞)

→[0,∞), φ1(0)=0such that Fu1−Fu2, jq

u1−u2

≥φ1u1−u2u1−u2^q⁻¹; (3.7) (ii)for eachu1,u2∈D(K), there exists a strictly increasing functionφ2: [0,∞)

→[0,∞),φ2(0)=0such that Ku1−Ku2, jq

u1−u2

≥φ2u1−u2u1−u2^q⁻¹; (3.8) (iii)φi(t)≥(d+ri)tfor allt∈[0,∞)and for someri>0,i=1,2whered:= q⁻¹(1 +dq−c⁻¹2^q⁻¹);c=max{1,cq}anddq,cqare the constants appear- ing in inequalities (2.6) and (2.7), respectively.

LetE:=X×Xwith normz^q_E= u^q_X+v^q_Xforz=(u,v)∈Eand define a mapT:E→2^EbyTz:=T(u,v)=(Fu−v,u+Kv). Then for eachz1,z2∈E, there exists a strictly increasing functionφ: [0,∞)→[0,∞)withφ(0)=0such that

Tz1−Tz2, j^E_qz1−z2

≥φz1−z2z1−z2^q⁻¹. (3.9) Proof. Defineφ: [0,∞)→[0,∞) byφ(t) :=min{r1,r2}tfor eacht∈[0,∞). Ob- serve thatφis a strictly increasing function withφ(0)=0. Furthermore, forq >

1,z1=(u1,v1) andz2=(u2,v2) arbitrary elements inE, we havez1, j_q^E(z2) = u1, jq(u2)+v1, jq(v2). Thus, we have the following estimates:

Tz1−Tz2, j_q^Ez1−z2

=

Fu1−Fu2− v1−v2

, jq

u1−u2 +Kv1−Kv2+u1−u2

, jq

v1−v2

=

Fu1−Fu2, jq

u1−u2

−

v1−v2, jq

u1−u2

+Kv1−Kv2, jq

v1−v2

+u1−u2, jq

v1−v2

≥φ1u1−u2u1−u2^q⁻¹+φ2v1−v2v1−v2^q⁻¹

−

v1−v2, j_qu1−u2

+u1−u2, j_qv1−v2

.

(3.10)

Since X is real q-uniformly smooth, inequality (2.7) holds for each x, y∈X.

Settingλ=1/2 in this inequality yields the following estimate:

x+y^q+x−y^q≥c⁻¹2^q⁻¹x^q+y^q

, (3.11)

wherec=max{1,c_q}. Furthermore, from inequality (2.6), replacingyby−y, we obtain the following inequality:

−

y, jq(x)≥q⁻¹x−y^q− x^q−dqy^q

. (3.12)

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Using (3.10), (3.12), (2.6), and (3.11), we obtain the following estimates:

≥φ1u1−u2u1−u2^q⁻¹+φ2v1−v2v1−v2^q⁻¹ +q⁻¹v1−v2−

u1−u2^q−u1−u2^q−d_qv1−v2^q +q⁻¹v1−v2+u1−u2^q−v1−v2^q−dqu1−u2^q

≥φ1u1−u2u1−u2^q⁻¹+φ2v1−v2v1−v2^q⁻¹ +q⁻¹c⁻¹2^q⁻¹u1−u2^q+v1−v2^q

−q⁻¹1 +d_qu1−u2^q+1 +d_qv1−v2^q

≥

φ1u1−u2−du1−u2u1−u2^q⁻¹ +φ2v1−v2−dv1−v2v1−v2^q⁻¹

≥minr1,r2u1−u2^q+v1−v2^q

=minr1,r2z1−z2·z1−z2^q⁻¹

=φz1−z2z1−z2^q⁻¹,

(3.13)

completing the proof ofLemma 3.2.

Corollary3.3. LetXbe a realq-uniformly smooth Banach space. LetF,K:X→ Xbe maps withD(K)=F(X)=Xsuch that the following conditions hold:

(i)for eachu1,u2∈D(F), there existsα >0such that Fu1−Fu2, jq

u1−u2

≥αu1−u2^q; (3.14) (ii)for eachu1,u2∈D(K), there existsβ >0such that

Ku1−Ku2, j_qu1−u2

≥βu1−u2^q; (3.15) (iii)α,β > d:=q⁻¹(1 +d_q−c⁻¹2^q⁻¹)andγ:=min{α−d,β−d}wherecand

dqare as in (3.11) and (2.6), respectively.

LetEandTbe defined as inLemma 3.2. Then, forz1,z2∈E, we have that Tz1−Tz2, j_q^Ez1−z2

≥γz1−z2^q. (3.16) Proof. Letα,β, andγbe real constants satisfying (iii), then following precisely the method of proof ofLemma 3.2, we get the required result.

Corollary3.4. LetX=Hbe a real Hilbert space. LetF,K:H→Hbe maps with D(K)=F(X)=X such that conditions (i) and (ii) ofCorollary 3.3are satisfied.

Letα,β >0,E, andTbe defined as inCorollary 3.3. Then, forz1,z2∈E, we have

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that

≥γz1−z2^q, (3.17) whereγ:=min{α,β}.

Proof. Since, for Hilbert spaces, the duality mappingj_q^Eis the identity map,q= 2,dq=1,c=1, the result follows fromCorollary 3.3.

3.1. Convergence theorems for Lipschitz maps

Remark 3.5. IfK andF are Lipschitzian maps with positive constantsLK and LF, respectively, thenT is Lipschitzian map with constantL:=(dmax{L^q_F+ 1, L^q_K+ 1})¹^/q for some constantd >0. Indeed, ifz1=(u1,v1),z2=(u2,v2) inE, then we have that

Tz1−Tz2^q=Fu1−Fu2

−

v1−v2^q+u1−u2+Kv1−Kv2^q

≤

L_Fu1−u2+v1−v2^q+u1−u2+L_Kv1−v2^q

≤dL^q_Fu1−u2^q+v1−v2^q+u1−u2^q+L^q_Kv1−v2^q for somed >0

≤dmaxL^q_F+ 1,L^q_K+ 1u1−u2^q+v1−v2^q

=dmaxL^q_F+ 1,L^q_K+ 1z1−z2^q.

(3.18) Thus,Tz1−Tz2 ≤Lz1−z2.

Consequently, we have the following theorem.

Theorem3.6. LetXbe realq-uniformly smooth Banach space. LetF,K:X→Xbe Lipschitzian maps with positive constantsL_KandL_F, respectively such thatD(K)= F(X)=Xwith the following conditions:

(i)there existsα >0such that Fu1−Fu2, jq

u1−u2

≥αu1−u2^q, ∀u1,u2∈D(F); (3.19) (ii)there existsβ >0such that

Ku1−Ku2, jq

u1−u2

≥βu1−u2^q, ∀u1,u2∈D(K); (3.20) (iii)α,β > d:=q⁻¹(1 +d_q−c⁻¹2^q⁻¹)andγ:=min{α−d,β−d}.

Assume thatu+KFu=0has solutionu^∗, letE:=X×Xbe with normz^q_E= u^q_X+v^q_X forz=(u,v)∈E, and define the mapT:E→EbyTz:=T(u,v)= (Fu−v,Kv+u). Let L be Lipschitz constant of T and ε:=(1/2)(γ/(1 + L(3 +L−γ))). Define the mapA_ε:E→EbyA_εz:=z−εTzfor eachz∈E. For arbitraryz0∈E, define the Picard sequence{z_n}inEbyz_n+1:=A_εz_n,n≥0. Then

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{zn}converges strongly toz^∗=[u^∗,v^∗], the unique solution of the equationTz=0 withzn+1−z^∗ ≤δⁿz1−z^∗ wherev^∗=Fu^∗ andu^∗ is the solution of the equationu+KFu=0andδ:=(1−(1/2)γε)∈(0,1).

Proof. Observe thatu^∗is a solution ofu+KFu=0 if and only ifz^∗=[u^∗,v^∗] is a solution ofTz=0 forv^∗=Fu^∗. Hence,Tz=0 has a solutionz^∗=[u^∗,v^∗] in E. SinceTis Lipschitz, and byCorollary 3.3, it is strongly accretive with constant γ(which, without loss of generality we may assume, is in (0, 1)). The conclusion

follows fromTheorem 2.3.

Remark 3.7. SinceL^pspaces, 1< p <∞, areq-uniformly smooth spaces where q=min{2, p}, thencq=dq≥1 and is given by

cq=dq=







1 +b^q⁻¹

(1 +b)^q⁻¹, if 1< p <2, p−1, if 2≤p <∞,

(3.21)

wherebis the unique solution of the equation (q−2)t^q⁻¹+ (q−1)t^q⁻²−1=0, 0< t <1 (see, e.g., [32]).

As a consequence of Theorem 3.6 and Remark 3.7, we have the following corollaries.

Corollary 3.8. Suppose X=Lp(1< p <∞). Let F,K:X→X be Lipschitzian maps with positive constantsL_KandL_F, respectively, andD(K)=F(X)=Xwith conditions (i) and (ii) of Theorem 3.6. Supposeα,β > dandγ:=min{α−d,β− d}where

d:=









 1 2

p− 2 p−1

, if2≤p <∞,

q⁻¹

1 + 1 +b^q⁻¹ (1 +b)^q⁻¹⁻

(1 +b)^q⁻¹ 1 +b^q⁻¹ 2^q⁻¹

, if1< p <2.

(3.22)

Assume thatu+KFu=0has solutionu^∗and setEandTas inTheorem 3.6. Let L,ε,A_ε, and{z_n}be defined as inTheorem 3.6. Then{z_n}converges strongly to z^∗=[u^∗,v^∗]withz_n+1−z^∗ ≤δⁿz1−z^∗whereδ:=(1−(1/2)γε)∈(0,1), v^∗=Fu^∗andu^∗is the unique solution ofu+KFu=0.

Corollary3.9. LetX=Hbe a real Hilbert space. LetFandKbe as inCorollary 3.8. Supposeα,β >0andγ:=min{α,β}. Assume thatu+KFu=0has solution u^∗ and setEandT as inCorollary 3.8. LetL,ε,A_ε, and{z_n}be defined as in Corollary 3.8. Then {zn}converges strongly toz^∗=[u^∗,v^∗]withzn+1−z^∗ ≤ δⁿz1−z^∗where δ:=(1−(1/2)γε)∈(0,1), v^∗=Fu^∗ andu^∗is the unique solution ofu+KFu=0.

Proof. The proof follows fromCorollary 3.8withp=2.

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3.2. Convergence theorems for bounded maps

Theorem3.10. LetXbe a realq-uniformly smooth Banach space. LetF,K:X→X withD(K)=F(X)=Xbe bounded maps such that the following conditions hold:

(i)for eachu1,u2∈X, there exists a strictly increasing functionφ1: [0,∞)→ [0,∞),φ1(0)=0such that

Fu1−Fu2, jq u1−u2

≥φ1u1−u2u1−u2^q⁻¹; (3.23) (ii)for eachu1,u2∈X, there exists a strictly increasing functionφ2: [0,∞)→

[0,∞),φ2(0)=0such that Ku1−Ku2, jq

u1−u2

≥φ2u1−u2u1−u2^q⁻¹; (3.24) (iii)φi(t)≥(d+ri)tfor allt∈[0,∞)andi=1,2for someri>0anddis as in

Lemma 3.2.

Assume that0=u+KFuhas solutionu^∗inX. LetE:=X×X be with norm z^q_E= u^q_X+v^q_X for z=(u,v)∈Eand define the mapT:E→EbyTz:= T(u,v)=(Fu−v,u+Kv). Then there exists a real numberγ0>0such that, if the real sequence{α_n} ⊂[0,γ0]satisfies the following conditions:(i) lim_n_→∞α_n=0;

(ii)αn= ∞, then for arbitraryz0∈E, the sequence{zn}defined by

z_n+1:=z_n−α_nTz_n, n≥0, (3.25) converges strongly toz^∗=[u^∗,v^∗]wherev^∗=Fu^∗andu^∗is the unique solution of 0=u+KFu.

Proof. Observe that sinceKandFare bounded maps, we have thatTis bounded map. Observe also thatu^∗is the solution of 0=u+KFuinXif and only ifz^∗= [u^∗,v^∗] is a solution of 0=TzinEforv^∗=Fu^∗. Thus, we obtain thatN(T) (null space ofT)= ∅. Also byLemma 3.2,T isφ-strongly accretive. Therefore,

the conclusion follows fromTheorem 2.2.

Following the method of proof ofTheorem 3.10and making use ofCorollary 3.3, we obtain the following theorem.

Theorem3.11. LetXbe a realq-uniformly smooth Banach space. LetF,K:X→X withD(K)=F(X)=Xbe bounded maps such that the following conditions hold:

(i)for eachu1,u2∈D(F), there existsα >0such that Fu1−Fu2, j_qu1−u2

≥αu1−u2^q; (3.26) (ii)for eachu1,u2∈D(K), there existsβ >0such that

Ku1−Ku2, jq

u1−u2

≥βu1−u2^q; (3.27)