Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type

Suppose that H 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.


Introduction
Let X be a real normed linear space with dual X * .For q > 1, we denote by J q the generalized duality mapping from X to 2 X * defined by where •, • denotes the generalized duality pairing.J 2 is denoted by J.If X * is strictly convex, then J q is single-valued.A map G with domain D G in a normed linear space X is said to be strongly accretive if there exists a constant k > 0 such that for every x, y ∈ D G , there exists j q x − y ∈ J q x − y such that Gx − Gy, j q x − y ≥ k x − y q . 1.2 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 x t Ax t 0, x 0 x 0 , 1.
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 Schr ödinger equations.If in 1.3 , x t is independent of t, then 1.3 reduces to Au 0, 1.4 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 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 and the so-called superposition or Nemytskii operator by Fu y : f y, u y then, the integral equation 1.6 can be put in operator theoretic form as follows: 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.

Preliminaries
In the sequel, we will need the followings results.Lemma 2.1 see Xu 18 .Let {a n } be a sequence of nonnegative real numbers satisfying the following relations: Lemma 2.2 see Chidume and Djitte, 19, Lemma 2.5 .Let H be a real Hilbert space and A : We now prove the following result.

Lemma 2.3. Let H 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:
Assume that F and K are monotones and satisfy the range condition.Then, T is monotone and also satisfies the range condition.

ISRN Mathematical Analysis
Proof.On E we have the natural norm • E and natural inner product •, • E given by:

2.3
Step 1.We prove that T is monotone.Let Therefore, using the fact that F and K are monotone, we obtain,

2.4
So, T is monotone.
Step 2. We show that R I E rT E for all r, 0 < r < 1.In fact let r 0 such that 0 < r 0 < 1.Since F and K are monotone and satisfy the range condition, then it is known that I r 0 F and I r 0 K are bijective and moreover, the resolvent J F r 0 : I r 0 F −1 of F and the resolvent Using the fact that J F r 0 and J K r 0 are nonexpansive, we have, Therefore G is a contraction.So, by the Banach fixed point theorem, G has a unique fixed point These imply I E r 0 T w * h.Therefore, R I E r 0 T E. By Lemma 2.2, it follows that T satisfies the range condition.This completes the proof.

Theorem 2.4 see Reich 20 . Let H 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 H for all r > 0. Let J t x : I tA −1 x, t > 0 be the resolvent of A, and assume that A −1 0 is nonempty.Then for each x ∈ H, lim t → ∞ J t x ∈ A −1 0 .

Main Results
Let H 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, ii K is bounded and monotone, that is, iii F and K satisfy the range condition.
With these assumptions, we prove the following theorem.
Theorem 3.1.Let H be a real Hilbert space.Let {u n } and {v n } be sequences in H defined iteratively from arbitrary points u 1 , v 1 ∈ H as follows: where {λ n } and {θ n } are sequences in 0, 1 satisfying the following conditions: Suppose that u KFu 0 has a solution in H.Then, there exists a constant d 0 > 0 such that if λ n ≤ d 0 θ n for all n ≥ n 0 for some n 0 ≥ 1, then the sequence {u n } converges to u * , a solution of u KFu 0.
, where z u, v .Define the sequence {w n } in E by: w n : u n , v n .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 {w n } converges to w * in E.
For this, let n 0 ∈ N, there exists r > 0 sufficiently large such that w 1 ∈ B w * , r/2 , w n 0 ∈ 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 M We split the proof in three steps.
Step 1.We first prove that the sequence {w n } is bounded in E. Indeed, it suffices to show that w n is in B for all n ≥ n 0 .The proof is by induction.By construction, w n 0 ∈ B. Suppose that w n ∈ B for n ≥ n 0 .We prove that w n 1 ∈ B. Assume for contradiction that w n 1 / ∈ B. Then, we have w n 1 − w * E > r.We compute as follows:

3.4
We have

3.5
Observing that 3.6 and using 3.1 , we obtain the following estimate:

3.7
Following the same argument, we also obtain

3.8
Thus, we obtain

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Therefore

3.12
So we obtain the following estimate:

3.13
Let d 0 r 2 /4M.Then using the induction assumptions, the fact that w 1 ∈ B w * , r/2 and λ n ≤ d 0 θ n , we obtain Step 2. We show that there exists a unique sequence z n x n , y n ∈ E such that Ky n x n 0, 3.16 and x n → x * , y n → y * , with x * KFx * 0 and y * Fx * .In fact, let T : 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 Theorem 2.4, with t 1/θ n and x u 1 , v 1 , we obtain that lim t → ∞ J t x ∈ T −1 0 implies that lim

3.18
Therefore, 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 {z n } is unique.Using 3.17 and Theorem 2.4, we have, lim z n ∈ T −1 0 .Let x n → x * and y n → y * .Then x * , y * ∈ T −1 0 .So, T x * , y * 0, that is, Therefore, y * Fx * and x * KFx * 0.

3.21
We have

3.22
From the boundness of {u n }, {v n }, and F, there exists 15 and the fact that F is monotone, we obtain

3.23
for some constant M 3 > 0. Using 3.16 and similar arguments, we obtain:

3.24
for some constant M 4 > 0. Therefore, we have the following estimate:

3.25
On the other hand, using the monotonicity of F and K we have

3.26
Using 3.15 and 3.16 , we observe that

3.27
Therefore, Using 3.25 and the boundness of {x n } and {y n }, we obtain that there exists C > 0 such that: Let {u n } and {v n } be sequences in H defined iteratively from arbitrary points u 1 , v 1 ∈ H as follows: where {λ n } and {θ n } are sequences in 0, 1 satisfying the following conditions: Suppose that u KFu 0 has a solution in H.Then, there exists a constant d 0 > 0 such that if λ n ≤ d 0 θ n for all n ≥ n 0 for some n 0 ≥ 1, then the sequence {u n } converges to u * , a solution of u KFu 0.
Let X 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 Ax, y − Ay, x ≤ 2α Ax, x Theorem BB.Let H 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 u n in C n and {u n } 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 Theorem 3.1 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.We verify that these choices satisfy, in particular, condition 3 of Theorem 3.1.In fact, using the fact that 1 x p ≤ 1 px, for x > −1 and 0 < p < 1, we have as n → ∞.

Remark 3 . 5 .
Theorem BB guarantees the existence of a sequence {u n } 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 Theorem 3.1 are λ n n 1 −a and θ n n 1 −b with 0 < b < a and a b < 1.
29Thus, by Lemma 2.1, w n 1 − z n → 0. Since z n → x * , y * , we obtain that w n → x * , y * .But since w n u n , v n , this implies that u n → u * and v n → v * .This completes the proof.
1/2Ay, y 1/2 , ∀x, y ∈ D A ,3.31where •, • 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, 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.
n : H → C n and F n P n FP * n : C n → H,