SupposeX is a real Banach space andF, K :X →X are accretive maps

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http://jipam.vu.edu.au/

Volume 4, Issue 5, Article 92, 2003

ITERATIVE SOLUTION OF NONLINEAR EQUATIONS OF HAMMERSTEIN TYPE

H. ZEGEYE BAHIRDARUNIVERSITY,

P. O. BOX. 859, BAHIRDAR, ETHIOPIA. habz@ictp.trieste.it

Received 10 August, 2003; accepted 05 September, 2003 Communicated by R. Verma

ABSTRACT. SupposeX is a real Banach space andF, K :X →X are accretive maps. Under different continuity assumptions onFandKsuch that the inclusion0 =u+KF uhas a solution, iterative methods are constructed which converge strongly to such a solution. No invertibility assumption is imposed onKand the operatorsKandFneed not be defined on compact subsets ofX. Our method of proof is of independent interest.

Key words and phrases: Accretive operators, Uniformly smooth spaces, Duality maps.

2000 Mathematics Subject Classification. 47H04, 47H06, 47H30, 47J05, 47J25.

1. INTRODUCTION

LetX be a real normed linear space with dualX^∗. For 1 < q < ∞, we denote by J_q the generalized duality mapping fromXto2^X^∗ defined by

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

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

A mapA with domain D(A) ⊆ X is said to be accretive if for every x, y ∈ D(A)there existsj(x−y)∈J(x−y)such that

hAx−Ay, j(x−y)i ≥0.

Ais said to bem−accretive if it is accretive andR(I+λA) (range of (I+λA)) = X,for all λ >0,whereIis 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||,

ISSN (electronic): 1443-5756

125-03

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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 the class of pseudocontractive mappings. A mappingT : X → X is said to be pseudocontractive if and only ifA := 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. If X 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,

whereAis 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 ifAis locally Lipschitzian and accretive then A ism−accretive which immediately implies that the equationx+Ax = hhas a solutionx^∗ ∈ D(A)for anyh ∈ X. This result was subsequently 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). Consequently, consid- erable 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 general- ization of equation (1.2) is the so-called equation of Hammerstein type (see e.g., [18]), where a nonlinear integral 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),

wheredyis aσ-finite measure on the measure spaceΩ; the real kernelKis defined onΩ×Ω, f is a real-valued function defined onΩ× <and is, in general, nonlinear andhis a given function onΩ. If we now define an operatorK 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), thenA := I +KF need not be accretive. Inter- est 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]). Equa- tions 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 ofA is 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 recursion formulas obtained involvedK⁻¹ and 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 subsets D of X 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 wheneverKandF are, and whose zeros are solutions of equation (1.4).

Moreover, the operatorsK andF need not be defined on a compact or angle-bounded subset ofX. Furthermore, our method which does not involveK⁻¹provides an explicit algorithm for the computation of solutions of equation (1.4).

2. PRELIMINARIES

LetX be a real normed linear space of dimension≥2. The modulus of smoothness ofXis defined by:

ρX(τ) := sup

kx+yk+kx−yk

2 −1 :kxk= 1,kyk=τ

; τ > 0.

It is well known that ρ_X(τ) ≤ τ ∀τ > 0(see e.g., [26]). If ρ_X(τ) > 0 ∀τ > 0, then X is said to be smooth. If there exist a constant c > 0and a real number 1 < q < ∞, such that ρ_X(τ)≤cτ^q, thenX is said to beq-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+Dmaxn

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

o

ρ_X(||y||),

for every x, y ∈ X, where D and care positive constants (see e.g., [26]). Typical examples of such uniformly smooth spaces are the Lebesgue L_p, the sequence`_p and the Sobolev W_p^m spaces for1< p < ∞. Moreover, we have

(2.2) ρlp(τ) = ρL^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.

(1) X is uniformly smooth.

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

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

Lemma 2.2. (see, e.g., [13]). LetXbe a normed linear space andJbe the normalized duality map onE. Then for any givenx, y ∈X,the following inequality holds:

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

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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 constantλ ∈(0,1).Assume that Ax = 0 has a solution x^∗ ∈ X. Define A_ε : X → X by A_εx := x−εAx for x ∈ X whereε := ¹₂n

λ 1+L(3+L−λ)

o

. For arbitrary x₀ ∈ X,define the Picard sequence{x_n}inX by x_n+1 =A_εx_n, n ≥ 0. Then,{x_n}converges strongly tox^∗ with||x_n+1−x^∗|| ≤ δⁿ||x₁ −x^∗||

whereδ:= 1−¹₂λε

∈(0,1).Moreover,x^∗ is unique.

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

(i) limα_n= 0;

(ii) P

α_n =∞,

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

We note that Theorem 2.4 is Theorem 3.6 of [13] withA φ-strongly accretive mapping.

3. MAINRESULTS

Lemma 3.1. Forq >1, letXbe a real uniformly smooth Banach space. LetE :=X×Xwith norm

||z||_E :=

||u||^q_X +||v||^q_X¹_q ,

for arbitraryz = [u, v] ∈ E. LetE^∗ := X^∗ ×X^∗ denote the dual space of E. For arbitrary x= [x₁, x₂]∈E define the mapj_q^E :E →E^∗by

j_q^E(x) =j_q^E[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) E is 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 that xandysatisfy condition (2) of Theorem 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 +g(||y₁||) +g(||y₂||) +q

n

y1, j_q^X(x1) +

y2, j_q^X(x2)o ,

whereg is continuous, strictly increasing and a convex function (using (2) of Theorem 2.1, sinceXis 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.

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(b) For arbitraryx= [x₁, x₂]∈E,letj_q^E(x) = j_q^E[x₁, x₂] =ψ_q. Thenψ_q = [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∗ .

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

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

(i) For each u₁, 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 each u₁, 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×X with 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 each z₁, z₂ ∈ E there exist j^E(z₁−z₂)∈J^E(z₁−z₂)and a strictly increasing functionφ: [0,∞)→[0,∞)withφ(0) = 0 such that

T z₁−T z₂, j^E(z₁−z₂)

≥φ(||z₁−z₂||)||z₁−z₂||.

Proof. Defineφ : [0,∞) → [0,∞)byφ(t) := min{r₁, r₂}t for eacht ∈ [0,∞).Clearly, φis a strictly increasing function withφ(0) = 0.Furthermore, observe that forz₁ = (u₁, v₁)and z₂ = (u₂, v₂)arbitrary elements in E we have

z₁, j^E(z₂)

= hu₁, j(u₂)i+hv₁, j(v₂)i.Thus

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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 Lemma 2.2) 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. 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 Lemma 3.2.

Lemma 3.3. SupposeXis a real uniformly smooth Banach space. LetF, K :X →Xbe 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||;

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(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||;

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

Let E andT be defined as in Lemma 3.2. Then for each z₁, z₂ ∈ E there exists a strictly increasing functionφ: [0,∞)→[0,∞)withφ(0) = 0such that

T z₁−T z₂, j^E(z₁−z₂)

≥φ(||z₁−z₂||)||z₁−z₂||.

Proof. Defineφ : [0,∞) → [0,∞)by φ(t) := min{r₁, r₂}t for each t ∈ [0,∞).Thus as in the proof of Lemma 3.2 we have thatφis a strictly increasing function withφ(0) = 0and for z₁ = (u₁, v₁)andz₂ = (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. SinceX is uniformly smooth for eachx, y ∈X by (2.1) we have that

||x+y||² ≤ ||x||²+ 2hy, j(x)i+Dmax{||x||+||y||,c

2}ρ_X(||y||)

≤ ||x||²+ 2hy, j(x)i+D{||x||+||y||+ c

2}ρ_X(||y||)

≤ ||x||²+ 2hy, j(x)i+D

||x||||y||+||y||²+ c

2ρ_X(||y||) (sinceρ_X(||y||)≤ ||y||)

≤ ||x||²+ 2hy, j(x)i+D ||x|²

2 +||y||²

2 +||y||²+ac 2 ||y||^q

(sinceρ_X(||y||)≤a||y||^q by 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

.

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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

≥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||,

completing the proof of Lemma 3.3.

3.1. Convergence Theorems for Lipschitz Maps.

Remark 3.4. IfKandF are Lipschitz single-valued maps with Lipschitz constantsL_KandL_F respectively, thenT is a Lipschitz map with constantL :=

d max{L²_F + 1, L²_K + 1}¹₂ for

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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

≤

L_F||u₁−u₂||+||v₁−v₂||2

+

||u₁−u₂||+L_K||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 3.5. Let X be real Banach space. Let F, K : X → X be Lipschitzian maps with Lipschitz constantsL_K andL_F, respectively such that the following conditions hold:

(i) For each u1, 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||;

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

(iii) φ_i(t) ≥ (2 + r_i)t for all t ∈ (0,∞) and for some r_i > 0, i = 1,2 and let γ :=

min{r₁, r₂}.

Assume thatu+KF u= 0has a solutionu^∗inXand letE :=X×Xand||z||²_E =||u||²_X+

||v||²_X forz = (u, v)∈Eand define the mapT :E →EbyT z :=T(u, v) = (F u−v, Kv+u).

LetLdenote the Lipschitz constant of T andε := ¹₂

γ 1+L(3+L−γ)

. Define the mapA_ε:E →E byA_εz :=z−εT z for eachz ∈E. For arbitraryz₀ ∈ E,define the Picard sequence{z_n}in E byz_n+1 :=A_εz_n, n ≥ 0. Then{z_n}converges strongly toz^∗ = [u^∗, v^∗]the unique solution of the equationT z = 0with||zn+1−z^∗|| ≤δⁿ||z1−z^∗||, whereδ := 1− ¹₂γε

∈(0,1).

Proof. Observe thatu^∗is a solution ofu+KF u= 0if and only ifz^∗ = [u^∗, v^∗]is a solution of T z = 0.HenceT z = 0has a solutionz^∗ = [u^∗, v^∗]inE. SinceT is Lipschitz and by Lemma 3.2 it is strongly accretive with constantγ (which, without loss of generality, we may assume is in(0,1)), the conclusion follows from Theorem 2.3.

Following the method of the proof of Theorem 3.5 and making use of Lemma 3.3 instead of Lemma 3.2 we obtain the following theorem.

Theorem 3.6. Let X be a real uniformly smooth Banach space. Let F, K : X → X be Lipschitzian maps with Lipschitz constants L_K and L_F, respectively such that conditions (i)- (iii) of Lemma 3.3 are satisfied and letγ := min{r1, r2}. Assume that u+KF u = 0has the solutionu^∗and setEandT as in Theorem 3.5. LetL, ε, Aε, and{zn}be defined as in Theorem 3.5. Then the conclusion of Theorem 3.5 holds.

3.2. Convergence Theorems for Uniformly Continuousφ-Strongly Accretive Maps.

Theorem 3.7. LetXbe a real normed linear space. LetF, K :X →X be uniformly continu- ous maps such that the following conditions hold:

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(i) For each u₁, 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||;

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.

Assume that0 = u+KF uhas a solutionu^∗inX. LetE :=X ×Xand||z||²_E =||u||²_X +

||v||²_X forz = (u, v)∈Eand define the mapT :E →EbyT z :=T(u, v) = (F u−v, u+Kv).

Then there exists a real number γ₀ > 0such 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 toz^∗ = [u^∗, v^∗], whereu^∗is the unique solution of0 = u+KF u.

Proof. SinceK andF are uniformly continuous maps we have thatT is a uniformly continuous map. Observe also thatu^∗ is the solution of 0 = u+KF uinX if and only ifz^∗ = [u^∗, v^∗]is a solution of0 = T zinE. Thus we obtain thatN(T) (null space of T) 6=∅. Also by Lemma 3.2,T isφ−strongly accretive. Therefore the conclusion follows from Theorem 2.4.

Following the method of proof of Theorem 3.7 and making use of Lemma 3.3 instead of Lemma 3.2 we obtain the following theorem.

Theorem 3.8. Let X be a real uniformly smooth Banach space. Let F, K : X → X be uniformly continuous maps such that conditions (i)-(iii) of Theorem 3.6 are satisfied. Assume that0 = u+KF uhas a solution u^∗ inX. Let E, T and{z_n}be defined as in Theorem 3.7.

Then, the conclusion of Theorem 3.7 holds.

Remark 3.9. We note that for the special case in which the real Banach spaceXisq−uniformly smooth using the above method, the author and Chidume [12] proved the following theorem.

Theorem 3.10. [12]. Let X be a real q-uniformly smooth Banach space. Let F, K : X → X be Lipschitzian maps with positive constants L_K and L_F respectively 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 setE andT as in Theorem 3.5. LetLbe a Lipschitz constant ofT andε, A_ε andz_n be defined as in Theorem 3.5. Then{z_n}converges strongly to z^∗ = [u^∗, v^∗] the unique solution of the equation T z = 0 with ||z_n+1 −z^∗|| ≤ δⁿ||z₁−z^∗||, whereu^∗is the solution of the equationu+KF u= 0andδ := 1− ¹₂γε

∈(0,1).

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The cases for Hilbert spaces andL_p spaces(1 < p <∞)are easily deduced from Theorem 3.10. The theorems proved in this paper are analogues of the theorems in [12] for the more general real Banach spaces considered here.

3.3. Explicit Algorithms.

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

(a) For Lipschitz operators (Theorem 3.5 and Theorem 3.6) with initial valuesu₀, v₀ ∈ X, define the sequences{u_n}and{v_n}inX as follows:

u_n+1 =u_n−ε

F u_n−v_n

; v_n+1 =v_n−ε

Kv_n+u_n .

Thenu_n → u^∗ inX, the unique solutionu^∗ of0 = u+KF u,whereεis as defined in Theorem 3.5.

(b) For uniformly continuous operators (Theorem 3.7 and Theorem 3.8) with initial values u₀, v₀ ∈X, define the sequences{u_n}and{v_n}inXas 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 Theorem 3.7.

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