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volume 4, issue 5, article 92, 2003.

Received 10 August, 2003;

accepted 05 September, 2003.

Communicated by:R. Verma

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Journal of Inequalities in Pure and Applied Mathematics

ITERATIVE SOLUTION OF NONLINEAR EQUATIONS OF HAMMERSTEIN TYPE

H. ZEGEYE

Bahir Dar University, P. O. Box. 859, Bahir Dar, Ethiopia.

EMail:habz@ictp.trieste.it

c

2000Victoria University ISSN (electronic): 1443-5756 125-03

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Iterative Solution of Nonlinear Equations of Hammerstein Type

H. Zegeye

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J. Ineq. Pure and Appl. Math. 4(5) Art. 92, 2003

http://jipam.vu.edu.au

Abstract

SupposeXis a real Banach space and F, K :X → Xare accretive maps.

Under different continuity assumptions onF andKsuch that the inclusion0 = u+KF u has a solution, iterative methods are constructed which converge strongly to such a solution. No invertibility assumption is imposed onKand the operatorsKandF need not be defined on compact subsets ofX. Our method of proof is of independent interest.

2000 Mathematics Subject Classification:47H04, 47H06, 47H30, 47J05, 47J25 Key words: Accretive operators, Uniformly smooth spaces, Duality maps.

1. Introduction

LetX be a real normed linear space with dualX^∗. For1< q <∞, we denote byJqthe generalized duality mapping fromX to2^X^∗ defined by

J_q(x) := {f^∗ ∈X^∗ :hx, f^∗i=||x||||f^∗||,||f^∗||=||x||^q−1},

where h·,·idenotes the generalized duality pairing. If q = 2, J_q = J₂ and is denoted byJ. IfX^∗is strictly convex, thenJ_qis single-valued (see e.g., [25]).

A mapAwith domainD(A)⊆X is said to be accretive if for everyx, y ∈ D(A)there existsj(x−y)∈J(x−y)such that

hAx−Ay, j(x−y)i ≥0.

A 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 accretive if for everyx, y ∈D(A)there existj(x−y)∈J(x−y)and a strictly increasing functionφ: [0,∞)→[0,∞),φ(0) = 0such that

hAx−Ay, j(x−y)i ≥φ(||x−y||)||x−y||,

and it is strongly accretive if for each x, y ∈ D(A), there exist j(x− y) ∈ J(x−y)and a constantk ∈(0,1)such that

hAx−Ay, j(x−y)i ≥k||x−y||².

Clearly, every strongly accretive map isφ-strongly accretive and everyφ-strongly accretive map is accretive. Closely related to the class of accretive mappings is

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the class of pseudocontractive mappings. A mapping T : X → X is said to be pseudocontractive if and only if A := I −T is accretive. One can easily show that the fixed point of pseudocontractive mapping T is the zero of accretive mapping A := I −T. IfX is a Hilbert space, accretive operators are also called monotone. The accretive mappings were introduced independently in 1967 by Browder [3] and Kato [20]. Interest in such mappings stems mainly from their firm connection with equations of evolution. It is known (see e.g., [28]) that many physically significant problems can be modelled by initial-value problems of the form

(1.1) x⁰(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. One of the fundamental results in the theory of accretive operators, due to Browder [4], states that if A is locally Lipschitzian and accretive then Ais m−accretive which immediately implies that the equation x+Ax =h has a solutionx^∗ ∈ D(A)for anyh ∈ X. This result was subse- quently generalized by Martin [22] to the continuous accretive operators. If in (1.1),x(t)is independent oft, then (1.1) reduces to

(1.2) Au= 0,

whose solutions correspond to the equilibrium points of the system (1.1). Con- sequently, considerable research efforts have been devoted, especially within the past 20 years or so, to methods of finding approximate solutions (when they exist) of equation (1.2). One important generalization of equation (1.2) is the

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so-called equation of Hammerstein type (see e.g., [18]), where a nonlinear inte- gral equation of Hammerstein type is one of the form:

(1.3) u(x) +

Z

Ω

K(x, y)f(y, u(y))dy =h(x),

where dy is a σ-finite measure on the measure space Ω; the real kernel K is defined on Ω ×Ω, f is a real-valued function defined on Ω × < and is, in general, nonlinear andhis a given function onΩ. If we now define an operator K by

Kv(x) :=

Z

Ω

K(x, y)v(y)dy; x∈Ω,

and the so-called superposition or Nemytskii operator byF u(y) := f(y, u(y)) then, the integral equation (1.3) can be put in operator theoretic form as follows:

(1.4) u+KF u= 0,

where, without loss of generality, we have taken h ≡ 0. We note that if K is an arbitrary accretive map (not necessarily the identity), then A := I +KF need not be accretive. Interest in equation (1.4) 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 (1.4) (see e.g., [23, Chapter IV]). Equations of Hammerstein type play a crucial role in the theory of optimal control systems (see e.g., [17]). Several existence and uniqueness theorems have been proved for equations of the Hammerstein type (see e.g., [2,5,6,8,15]).

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For the iterative approximation of solutions of equation (1.2), the accretivity/

monotonicity ofAis crucial. The Mann iteration scheme (see e.g., [21]) and the Ishikawa iteration scheme (see e.g., [19]) have successfully been employed (see e.g., [7,10,11,12,13,14,16,19,21,24,27]). Attempts to apply these schemes to equation (1.4) have not provided satisfactory results. In particular, the recur- sion formulas obtained involvedK⁻¹and this is not convenient in applications.

Part of the difficulty is, as has already been noted, the fact that the composition of two accretive operators need not be accretive. In the special case in which the operators are defined on subsetsDofX which are compact (or more generally, angle-bounded (see e.g., [1]), Brèzis and Browder [1] have proved the strong convergence of a suitably defined Galerkin approximation to a solution of (1.4).

It is our purpose in this paper to use the method introduced in [12] which contains an auxiliary operator, defined in terms of K and F in an arbitrary real Banach space which, under certain conditions, is accretive whenever K and F are, and whose zeros are solutions of equation (1.4). Moreover, the operatorsKandF need not be defined on a compact or angle-bounded subset of X. Furthermore, our method which does not involveK⁻¹ provides an explicit algorithm for the computation of solutions of equation (1.4).

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2. Preliminaries

LetXbe a real normed linear space of dimension≥2. The modulus of smooth- ness ofXis defined by:

ρ_X(τ) := sup

kx+yk+kx−yk

2 −1 :kxk= 1,kyk=τ

; τ >0.

It is well known thatρ_X(τ)≤ τ ∀τ >0(see e.g., [26]). Ifρ_X(τ)>0 ∀τ >0, thenX is said to be smooth. If there exist a constantc > 0and a real number 1 < q < ∞, such thatρ_X(τ)≤ cτ^q, thenX is said to be q-uniformly smooth.

A Banach spaceXis called uniformly smooth iflimτ→0 ρX(τ)

τ = 0.IfEis a real uniformly smooth Banach space, then

(2.1) ||x+y||² ≤ ||x||²+ 2hy, j(x)i+Dmax n

||x||+||y||,c 2

o

ρX(||y||), for everyx, y ∈X, whereDandcare positive constants (see e.g., [26]). Typical examples of such uniformly smooth spaces are the Lebesgue L_p, the sequence

`p and the SobolevW_p^m spaces for1< p <∞. Moreover, we have (2.2) ρ_l_p(τ) = ρ_L^p(τ) =ρ_W_m^p(τ)≤







1

pτ^p, if 1<2< p;

p−1

2 τ², if p≥2,

∀τ > 0(see e.g., [26]).

In the sequel we shall need the following results.

Theorem 2.1. [25]. Letq >1andX be a real smooth Banach space. Then the following are equivalent.

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1. X is uniformly smooth.

2. There exists a continuous, strictly increasing and convex functiong :R⁺→ R⁺, such that for everyx, y ∈B_rfor somer >0we get

(2.3) ||x+y||^q ≤ ||x||^q+qhy, j_q(x)i+g(||y||).

Lemma 2.2. (see, e.g., [13]). Let X be a normed linear space andJ be the normalized duality map on E. Then for any given x, y ∈ X, the following inequality holds:

||x+y||² ≤ ||x||²+ 2hy, j(x+y)i, ∀j(x+y)∈J(x+y).

Theorem 2.3. [9]. Let X be a real Banach space,A : X → X be a Lipschitz and strongly accretive map with Lipschitz constantL >0and strong accretivity constant λ ∈ (0,1). Assume that Ax = 0 has a solution x^∗ ∈ X. Define A_ε :X →X byA_εx:=x−εAxforx∈X whereε:= ¹₂n

λ 1+L(3+L−λ)

o . For arbitraryx₀ ∈X,define the Picard sequence{x_n}inXbyx_n+1 =A_εx_n, n≥ 0. Then,{xn}converges strongly tox^∗ with||xn+1−x^∗|| ≤δⁿ||x1−x^∗||where δ := 1− ¹₂λε

∈(0,1).Moreover,x^∗ is unique.

Theorem 2.4. [13] Let X be a real normed linear space. LetA : X → X be uniformly continuous φ−strongly accretive mapping. Assume 0 = Ax has a solutionx^∗ ∈ X. Then, there exists a real number γ₀ > 0such that if the real sequence{α_n} ⊂[0, γ₀]satisfies the following conditions:

(i) limα_n = 0;

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(ii) P

α_n=∞,

then for arbitraryx₀ ∈X the sequence{x_n}, defined by xn+1 :=xn−αnAxn, n≥0, converges strongly tox^∗, the unique solution of0 = Ax.

We note that Theorem2.4is Theorem 3.6 of [13] withA φ-strongly accretive mapping.

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3. Main Results

Lemma 3.1. For q > 1, letX be a real uniformly smooth Banach space. Let E :=X×Xwith norm

||z||_E :=

||u||^q_X +||v||^q_X¹_q ,

for arbitrary z = [u, v] ∈ E. LetE^∗ := X^∗×X^∗ denote the dual space ofE.

For arbitraryx= [x₁, x₂]∈E define the mapj_qÊ :E →E^∗by j_qÊ(x) =j_qÊ[x₁, x₂] := [j_q^X(x₁), j_q^X(x₂)],

so that for arbitraryz₁ = [u₁, v₁], z₂ = [u₂, v₂]inE the duality pairingh·,·iis given by

z₁, j_q^E(z₂)

=

u₁, j_q^X(u₂) +

v₁, j_q^X(v₂) . Then

(a) Eis uniformly smooth;

(b) j_q^E 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) of Theorem2.1. 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 +g(||y₁||) +g(||y₂||) +qn

y₁, j_q^X(x₁) +

y₂, j_q^X(x₂)o ,

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wheregis continuous, strictly increasing and a convex function (using (2) of Theorem2.1, sinceX is uniformly smooth). It follows that

||x+y||^q_E ≤ ||x||^q_E+q

y, j_q^E(x)

+g⁰(||y||),

whereg⁰(||y||) := g(||y₁||)+g(||y₂||). So, the result follows from Theorem 2.1.

(b) For arbitrary x = [x₁, x₂] ∈ E, let j_q^E(x) = j_q^E[x₁, x₂] = ψ_q. Then ψq = [j_q^X(x1), j_q^X(x2)]inE^∗.Observe that forp >1such that ¹_p + ¹_q = 1,

||ψ_q||_E^∗ =

||[j_q^X(x₁), j_q^X(x₂)]||¹_p

=

||j_q(x₁)||^p_X∗+||j_q(x₂)||^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−1_X . Hence,||ψ_q||_E^∗ =||x||^q−1_E . Furthermore,

hx, ψ_qi=

[x₁, x₂],[j_q^X(x₁), j_q^X(x₂)]

=

x1, j_q^X(x1) +

x2, j_q^X(x2)

=||x1||^q_X +||x2||^q_X

= (||x₁||^q_X +||x₂||^q_X)¹^q

||x₁||^q_X +||x₂||^q_X^q−1_q

=||x||_E· ||ψ||^q−1_E∗ .

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Hence, j_q^E is a single-valued (sinceE is uniformly smooth) duality mapping onE.

Lemma 3.2. SupposeX is a real normed linear space. LetF, K :X → Xbe maps such that the following conditions hold:

(i) For eachu₁, u₂ ∈ X there exist j(u₁ −u₂) ∈ J(u₁−u₂) and a strictly increasing functionφ1 : [0,∞)→[0,∞), φ1(0) = 0such that

hF u₁−F u₂, j(u₁ −u₂)i ≥φ₁(||u₁−u₂||)||u₁−u₂||;

(ii) For eachu₁, u₂ ∈ X there exist j(u₁ −u₂) ∈ J(u₁−u₂) and a strictly increasing functionφ₂ : [0,∞)→[0,∞), φ₂(0) = 0such that

hKu₁−Ku₂, j(u₁−u₂)i ≥φ₂(||u₁−u₂||)||u₁−u₂||;

(iii) φ_i(t)≥(2 +r_i)tfor allt∈(0,∞)and for somer_i >0, i = 1,2.

LetE :=X×Xwith norm||z||²_E =||u||²_X +||v||²_X forz = (u, v)∈E and define a map T : E → E by T z := T(u, v) = (F u−v, u+Kv). Then for eachz₁, z₂ ∈Ethere existj^E(z₁−z₂)∈J^E(z₁−z₂)and a strictly increasing functionφ: [0,∞)→[0,∞)withφ(0) = 0such that

T z₁−T z₂, j^E(z₁ −z₂)

≥φ(||z₁−z₂||)||z₁ −z₂||.

Proof. Define φ : [0,∞) → [0,∞) by φ(t) := min{r₁, r₂}t for each t ∈ [0,∞).Clearly,φ is a strictly increasing function withφ(0) = 0.Furthermore,

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observe that forz₁ = (u₁, v₁)andz₂ = (u₂, v₂)arbitrary elements inEwe have z1, j^E(z2)

=hu1, j(u2)i+hv1, j(v2)i.Thus we have the following estimates:

T z₁−T z₂, j^E(z₁−z₂)

=hF u₁−F u₂−(v₁−v₂), j(u₁−u₂)i

+hKv₁−Kv₂ + (u₁−u₂), j(v₁−v₂)i

=hF u₁−F u₂, j(u₁−u₂)i − hv₁−v₂, j(u₁−u₂)i

+hKv1−Kv2, j(v1−v2)i+hu1−u2, j(v1−v2)i

≥φ₁(||u₁−u₂||)||u₁−u₂||+φ₂(||v₁−v₂||)||v₁−v₂||

− hv₁−v₂, j(u₁−u₂)i+hu₁−u₂, j(v₁ −v₂)i. (3.1)

SinceX is an arbitrary real normed linear space, for eachx, y ∈ X andj(x+ y)∈J(x+y)(by Lemma2.2) we have that

||x+y||² ≤ ||x||²+ 2hy, j(x+y)−j(x)i+ 2hy, j(x)i

≤ ||x||²+ 2||y||||j(x+y)−j(x)||+ 2hy, j(x)i

≤ ||x||²+ 2||y||

||x+y||+||x||

+ 2hy, j(x)i

≤ ||x||²+ 2

||y||²

2 + ||x+y||²

2 + ||y||²

2 +||x||² 2

+ 2hy, j(x)i

= 2||x||²+ 2||y||²+||x+y||² + 2hy, j(x)i. Thus we gethy, j(x)i ≥ −||x||²− ||y||².

Replacingyby−ywe obtain− hy, j(x)i ≥ −||x||²− ||y||².Therefore,

− hv₁ −v₂, j(u₁−u₂)i ≥ −||u₁−u₂||²− ||v₁−v₂||² and hu₁−u₂, j(v₁ −v₂)i ≥ −||v₁−v₂||²− ||u₁−u₂||².

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Thus (3.1) and the above estimates give that T z₁−T z₂, j^E(z₁−z₂)

≥φ₁(||u₁−u₂||)||u₁−u₂||+φ₂(||v₁−v₂||)||v₁−v₂||

−2||u₁−u₂||²−2||v₁−v₂||²

≥

φ₁(||u₁−u₂||)−2||u₁−u₂||

||u₁−u₂||

+

φ₂(||v₁−v₂||)−2||v₁−v₂||

||v₁−v₂||

≥r₁||u₁−u₂||²+r₂||v₁−v₂||²

≥min{r₁, r₂}n

||u₁−u₂||²+||v₁−v₂||²o

= min{r₁, r₂}||z₁−z₂||²

=φ(||z₁−z₂||)||z₁−z₂||, completing the proof of Lemma3.2.

Lemma 3.3. Suppose X is a real uniformly smooth Banach space. LetF, K : X →X be maps such that the following conditions hold:

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

hF u1−F u2, j(u1 −u2)i ≥φ1(||u1−u2||)||u1−u2||;

(ii) For eachu₁, u₂ ∈Xthere exists a strictly increasing functionφ₂ : [0,∞)→ [0,∞), φ₂(0) = 0such that

hKu₁−Ku₂, j(u₁−u₂)i ≥φ₂(||u₁−u₂||)||u₁−u₂||;

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(iii) φ_i(t)≥(D+r_i)t+^acD₄ t^q−1,ρ_X(t)≤at^qfor allt∈(0,∞)and for some q > 1, a > 0 and r_i > 0, i = 1,2, where c and D are the constants appearing in inequality (2.1).

LetE and T be defined as in Lemma 3.2. Then for eachz1, z2 ∈ E there exists a strictly increasing function φ : [0,∞) → [0,∞)with φ(0) = 0 such that

T z1−T z2, j^E(z1 −z2)

≥φ(||z1−z2||)||z1 −z2||.

Proof. Define φ : [0,∞) → [0,∞) by φ(t) := min{r₁, r₂}t for each t ∈ [0,∞).Thus as in the proof of Lemma3.2we have thatφ is a strictly increasing function with φ(0) = 0and for z₁ = (u₁, v₁)and z₂ = (u₂, v₂) arbitrary elements inE we have the following estimate:

(3.2)

T z₁−T z₂, j^E(z₁−z₂)

=φ₁(||u₁−u₂||)||u₁−u₂||+φ₂(||v₁−v₂||)||v₁−v₂||

− hv₁−v₂, j(u₁−u₂)i+hu₁−u₂, j(v₁−v₂)i. SinceXis uniformly smooth for eachx, y ∈Xby (2.1) we have that

||x+y||² ≤ ||x||²+ 2hy, j(x)i+Dmax n

||x||+||y||,c 2

o

ρ_X(||y||)

≤ ||x||²+ 2hy, j(x)i+D n

||x||+||y||+ c 2

o

ρX(||y||)

≤ ||x||²+ 2hy, j(x)i+D

||x||||y||+||y||²+ c

2ρ_X(||y||) (since ρ_X(||y||)≤ ||y||)

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≤ ||x||²+ 2hy, j(x)i+D ||x|²

2 + ||y||²

2 +||y||²+ac 2 ||y||^q

(since ρ_X(||y||)≤a||y||^qby assumption forq >1anda >0)

≤

1 + D 2

||x||²+ 3D

2 ||y||²+ acD

2 ||y||^q+ 2hy, j(x)i, and hence

hy, j(x)i ≥ 1

2||x+y||²−1 2

1 + D

2

||x||²+3D

2 ||y||²+acD 2 ||y||^q

.

Replacingyby−ywe obtain

− hy, j(x)i ≥ 1

2||x−y||²− 1 2

1 + D

2

||x||²+ 3D

2 ||y||² +acD 2 ||y||^q

. Thus (3.2) and the above estimates give that

T z₁−T z₂, j^E(z₁−z₂)

≥φ₁(||u₁ −u₂||)||u₁−u₂||+φ₂(||v₁−v₂||)||v₁−v₂||

+ 1 2

||u₁−u₂−(v₁−v₂)||²−

1 + D 2

||u₁−u₂||²

− 3D

2 ||v₁−v₂||²− acD

2 ||v₁−v₂||^q

+1 2

||u₁−u₂+v₁−v₂||²−

1 + D 2

||v₁−v₂||²

− 3D

2 ||u₁−u₂||²− acD

2 ||u₁−u₂||^q

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≥φ₁(||u₁ −u₂||)||u₁−u₂||+φ₂(||v₁−v₂||)||v₁−v₂||

+1

2 ||u₁−u₂−(v₁−v₂)||²+||u₁−u₂+v₁−v₂||²

− 1 2

(1 + 2D)||u1−u2||²+ acD

2 ||u1−u2||^q

− 1 2

(1 + 2D)||v1−v2||²+ acD

2 ||v1−v2||^q

. (3.3)

Since for allx, y ∈X, x6=y,

x+y 2

2

≤ 1

2(||x||²+||y||²) we have that

||(u₁−u₂)−(v₁−v₂)||²+||(u₁−u₂) + (v₁−v₂)||² ≥ ||u₁−u₂||²+||v₁−v₂||². Then (3.3) becomes

T z₁−T z₂, j^E(z₁−z₂)

≥φ₁(||u₁−u₂||)||u₁−u₂|| −

D||u₁−u₂||² + acD

4 ||u₁−u₂||^q

+φ₂(||v₁−v₂||)||v₁−v₂||

−

D||v₁−v₂||²+ acD

4 ||v₁−v₂||^q

≥r₁||u₁−u₂||²+r₂||v₁−v₂||²

≥min{r₁, r₂}

||u₁−u₂|²+||v₁−v₂||²

= min{r1, r2}||z1−z2||² =φ(||z1−z2||)||z1 −z2||,

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completing the proof of Lemma3.3.

3.1. Convergence Theorems for Lipschitz Maps

Remark 3.1. IfK andF are Lipschitz single-valued maps with Lipschitz con- stants L_K andL_F respectively, thenT is a Lipschitz map with constant L :=

d max{L²_F + 1, L²_K+ 1}¹₂

for some constantd >0.Indeed, ifz₁ = (u₁, v₁), z₂ = (u₂, v₂)inE then we have that

||T z₁ −T z₂||²

=||(F u₁−F u2)−(v1−v2)||²+||u₁−u2+Kv1−Kv2||²

≤

LF||u1−u2||+||v1−v2||2

+

||u1−u2||+LK||v1−v2||2

≤d

L²_F||u₁−u₂||²+||v₁−v₂||²+||u₁−u₂||²+L²_K||v₁−v₂||² for somed >0

≤dmax{L²_F + 1, L²_K+ 1}

||u₁−u₂||²+||v₁−v₂||²

=dmax{L²_F + 1, L²_K+ 1}||z1−z2||².

Thus||T z₁−T z₂|| ≤L||z₁−z₂||. Consequently, we have the following theorem.

Theorem 3.4. LetXbe real Banach space. LetF, K :X →Xbe Lipschitzian maps with Lipschitz constants L_K andL_F, respectively such that the following conditions hold:

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(i) For eachu₁, u₂ ∈ X there exist j(u₁ −u₂) ∈ J(u₁−u₂) and a strictly increasing functionφ1 : [0,∞)→[0,∞), φ1(0) = 0such that

hF u₁−F u₂, j(u₁ −u₂)i ≥φ₁(||u₁−u₂||)||u₁−u₂||;

(ii) For eachu₁, u₂ ∈ X there exist j(u₁ −u₂) ∈ J(u₁−u₂) and a strictly increasing functionφ₂ : [0,∞)→[0,∞), φ₂(0) = 0such that

hKu₁−Ku₂, j(u₁−u₂)i ≥φ₂(||u₁−u₂||)||u₁−u₂||;

(iii) φi(t) ≥ (2 +ri)t for allt ∈ (0,∞)and for some ri > 0, i = 1,2and let γ := min{r₁, r₂}.

Assume thatu+KF u= 0has a solutionu^∗inXand letE :=X×X and

||z||²_E =||u||²_X +||v||²_X forz = (u, v)∈ E and define the mapT :E → Eby T z :=T(u, v) = (F u−v, Kv+u). LetLdenote the Lipschitz constant of T and ε := ¹₂

γ 1+L(3+L−γ)

. Define the map A_ε : E → E by A_εz := z −εT z for each z ∈ E. For arbitrary z₀ ∈ E, define the Picard sequence{z_n}inE by z_n+1 := A_εz_n, n ≥ 0. Then {z_n} converges strongly to z^∗ = [u^∗, v^∗]the unique solution of the equationT z= 0with||zn+1−z^∗|| ≤δⁿ||z1−z^∗||, where δ := 1− ¹₂γε

∈(0,1).

Proof. Observe thatu^∗is a solution ofu+KF u= 0if and only ifz^∗ = [u^∗, v^∗] is a solution ofT z= 0.HenceT z = 0has a solutionz^∗ = [u^∗, v^∗]inE. Since T is Lipschitz and by Lemma3.2it is strongly accretive with constantγ(which, without loss of generality, we may assume is in(0,1)), the conclusion follows from Theorem2.3.

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Following the method of the proof of Theorem3.4and making use of Lemma 3.3instead of Lemma3.2we obtain the following theorem.

Theorem 3.5. Let X be a real uniformly smooth Banach space. Let F, K : X →Xbe Lipschitzian maps with Lipschitz constantsL_KandL_F, respectively such that conditions (i)-(iii) of Lemma3.3are satisfied and letγ := min{r₁, r₂}.

Assume thatu+KF u= 0has the solutionu^∗ and setEandT as in Theorem 3.4. LetL, ε, A_ε, and{z_n}be defined as in Theorem3.4. Then the conclusion of Theorem3.4holds.

3.2. Convergence Theorems for Uniformly Continuous φ - Strongly Accretive Maps

Theorem 3.6. Let X be a real normed linear space. Let F, K : X → X be uniformly continuous maps such that the following conditions hold:

(i) For eachu₁, u₂ ∈ X there exist j(u₁ −u₂) ∈ J(u₁−u₂) and a strictly increasing functionφ₁ : [0,∞)→[0,∞), φ₁(0) = 0such that

hF u₁−F u₂, j(u₁ −u₂)i ≥φ₁(||u₁−u₂||)||u₁−u₂||;

(ii) For eachu1, u2 ∈ X there exist j(u1 −u2) ∈ J(u1−u2) and a strictly increasing functionφ₂ : [0,∞)→[0,∞), φ₂(0) = 0such that

hKu₁−Ku₂, j(u₁−u₂)i ≥φ₂(||u₁−u₂||)||u₁−u₂||;

(iii) φi(t)≥(2 +ri)tfor allt∈(0,∞)and for someri >0, i= 1,2.

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Assume that0 = u+KF uhas a solution u^∗ inX. Let E := X ×X and

||z||²_E =||u||²_X +||v||²_X forz = (u, v)∈ E and define the mapT :E → Eby T z := T(u, v) = (F u−v, u+Kv). Then there exists a real numberγ₀ > 0 such that if the real sequence{α_n} ⊂[0, γ₀]satisfies the following conditions

(a) limn→∞α_n = 0;

(b) P

α_n=∞,

then for arbitraryz₀ ∈Ethe sequence{z_n}, defined by z_n+1 :=z_n−α_nT z_n, n≥0,

converges strongly to z^∗ = [u^∗, v^∗], where u^∗ is the unique solution of 0 = u+KF u.

Proof. SinceK andF are uniformly continuous maps we have thatT is a uniformly continuous map. Observe also thatu^∗is the solution of0 = u+KF uin X if and only ifz^∗ = [u^∗, v^∗]is a solution of0 = T zinE. Thus we obtain that N(T) (null space of T) 6= ∅. Also by Lemma3.2,T is φ−strongly accretive.

Therefore the conclusion follows from Theorem2.4.

Following the method of proof of Theorem3.6 and making use of Lemma 3.3instead of Lemma3.2we obtain the following theorem.

Theorem 3.7. Let X be a real uniformly smooth Banach space. Let F, K : X →Xbe uniformly continuous maps such that conditions (i)-(iii) of Theorem 3.5 are satisfied. Assume that0 = u+KF uhas a solutionu^∗ inX. Let E, T and {z_n}be defined as in Theorem 3.6. Then, the conclusion of Theorem3.6 holds.

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Remark 3.2. We note that for the special case in which the real Banach space X is q−uniformly smooth using the above method, the author and Chidume [12] proved the following theorem.

Theorem 3.8. [12]. Let X be a real q-uniformly smooth Banach space. Let F, K : X → X be Lipschitzian maps with positive constantsL_K and L_F re- spectively with the following conditions:

(i) There existsα >0such that

hF u1−F u2, jq(u1−u2)i ≥α||u1−u2||^q, ∀u1, u2 ∈D(F);

(ii) There existsβ >0such that

hKu₁−Ku₂, j_q(u₁−u₂)i ≥β||u₁−u₂||^q, ∀u₁, u₂ ∈D(K);

(iii) α, β > d:=q⁻¹(1 +d_q−c⁻¹2^q−1)andγ := min{α−d, β−d}whered_q andcare as in (3.2) and (2.1) of [12], respectively.

Assume thatu+KF u = 0has solutionu^∗ and setEandT as in Theorem 3.4. Let L be a Lipschitz constant of T and ε, A_ε and z_n be defined as in Theorem3.4. Then{z_n}converges strongly toz^∗ = [u^∗, v^∗]the unique solution of the equation T z = 0 with ||z_n+1 − z^∗|| ≤ δⁿ||z₁ −z^∗||, where u^∗ is the solution of the equationu+KF u= 0andδ:= 1−¹₂γε

∈(0,1).

The cases for Hilbert spaces andL_p spaces(1< p <∞)are easily deduced from Theorem 3.8. The theorems proved in this paper are analogues of the theorems in [12] for the more general real Banach spaces considered here.

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3.3. Explicit Algorithms

The method of our proofs provides the following explicit algorithms for com- puting the solution of the inclusion0 = u+KF uin the spaceX.

(a) For Lipschitz operators (Theorem3.4and Theorem3.5) with initial values u₀, v₀ ∈X, define the sequences{u_n}and{v_n}inX as follows:

u_n+1 =u_n−ε

F u_n−v_n

; vn+1 =vn−ε

Kvn+un

.

Thenu_n →u^∗ inX, the unique solutionu^∗ of0 = u+KF u,whereεis as defined in Theorem3.4.

(b) For uniformly continuous operators (Theorem 3.6and Theorem3.7) with initial values u₀, v₀ ∈ X, define the sequences {u_n} and {v_n} in X as follows:

u_n+1 =u_n−α_n(F u_n−v_n);

v_n+1 =v_n−α_n(Kv_n+u_n).

Thenu_n→u^∗ inX, the unique solutionu^∗of0 =u+KF u,whereα_nis as defined in Theorem3.6.

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References

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[9] C.E. CHIDUME, Nonexpansive mappings, generalizations and iterative algorithms, Accepted to appear in Nonlinear Analysis and Applications.

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