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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 169751, 12 pages
Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type
1Mathematics Institute, African University of Sciences and Technology, Abuja, Nigeria
2Mathematics Department, Gaston Berger University, Saint Louis, Senegal
Received 29 November 2011; Accepted 10 January 2012
Academic Editors: F. Arandiga and J. Cui
Copyright © 2012 C. E. Chidume and N. Djitté. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Suppose that is a real Hilbert space and are bounded monotone maps with . Let denote a solution of the Hammerstein equation . An explicit iteration process is shown to converge strongly to . No invertibility or continuity assumption is imposed on and the operator is not restricted to be angle-bounded. Our result is a significant improvement on the Galerkin method of Brézis and Browder.
Let be a real normed linear space with dual . For , we denote by the generalized duality mapping from to defined by where denotes the generalized duality pairing. is denoted by . If is strictly convex, then is single-valued. A map with domain in a normed linear space is said to be strongly accretive if there exists a constant such that for every , there exists such that If , is said to be accretive. If is a Hilbert space, accretive operators are called monotone. The accretive mappings were introduced independently in 1967 by Browder  and Kato . Interest in such mappings stems mainly from their firm connection with equations of evolution. It is known (see, e.g., Zeidler ) that many physically significant problems can be modelled by initial-value problems of the form where 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), is independent of , then (1.3) reduces to 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 , states that the initial value problem (1.3) is solvable if is locally Lipschitzian and accretive on . Utilizing the existence result for (1.3), Browder  proved that if is locally Lipschitzian and accretive on , then is -accretive, that is, , where denotes the range of . Clearly, a consequence of this is that the equation has a solution. One important generalization of (1.5) is the so-called equation of Hammerstein type (see, e.g., Hammerstein ), where a nonlinear integral equation of Hammerstein type is one of the form: where is a -finite measure on the measure space ; the real kernel is defined on , is a real-valued function defined on and is, in general, nonlinear and is a given function on . If we now define an operator by and the so-called superposition or Nemytskii operator by then, the integral equation (1.6) can be put in operator theoretic form as follows: where, without loss of generality, we have taken .
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 , 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 ).
Several existence and uniqueness theorems have been proved for equations of the Hammerstein type (see e.g., Brézis and Browder [7–9], Browder , Browder et al. , Browder and Gupta , Cydotchepanovich , and De Figueiredo and Gupta ). For the iterative approximation of solutions of (1.4) and (1.5), the monotonicity/accretivity of is crucial. The Mann iteration scheme (see, e.g., Mann ) has successfully been employed (see, e.g., the recent monographs of Berinde  and Chidume ). The recurrence formulas used involved 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 of which are compact (or more generally, angle-bounded see e.g., Pascali and Sburlan  for definition), Brézis and Browder  have proved the strong convergence of a suitably defined Galerkin approximation to a solution of (1.8) (see also Brézis and Browder ).
It is our purpose in this paper to prove that an explicit coupled iteration process recently introduced by Chidume and Zegeye  which does not involve which is also required to be monotone converges strongly to a solution of (1.8) when and are bounded and monotone. Our new method of proof is also of independent interest.
In the sequel, we will need the followings results.
Lemma 2.1 (see Xu ). Let be a sequence of nonnegative real numbers satisfying the following relations: where , ; ; , , . Then, as .
Lemma 2.2 (see Chidume and Djitte, [19, Lemma 2.5]). Let be a real Hilbert space and be a map with . Suppose that is -accretive, that is, for all, ; for some . Then satisfies the range condition, that is, for all .
We now prove the following result.
Lemma 2.3. Let be a real Hilbert space and be maps with . Let and be the map defined by: Assume that and are monotones and satisfy the range condition. Then, is monotone and also satisfies the range condition.
Proof. On we have the natural norm and natural inner product given by:
Step 1. We prove that is monotone. Let , . We have and . So, . Therefore, using the fact that and are monotone, we obtain,
So, is monotone.Step 2. We show that for all , . In fact let such that . Since and are monotone and satisfy the range condition, then it is known that and are bijective and moreover, the resolvent of and the resolvent of are nonexpansive.
Let . Define by Using the fact that and are nonexpansive, we have, Therefore is a contraction. So, by the Banach fixed point theorem, has a unique fixed point , that is or equivalently, These imply . Therefore, .
By Lemma 2.2, it follows that satisfies the range condition. This completes the proof.
Theorem 2.4 (see Reich ). Let be a real Hilbert space. Let be monotone with and suppose that satisfies the range condition: for all . Let , be the resolvent of , and assume that is nonempty. Then for each , .
3. Main Results
Let be a real Hilbert space and be maps with such that the following conditions hold:(i) is bounded and monotone, that is, (ii) is bounded and monotone, that is, (iii) and satisfy the range condition.
With these assumptions, we prove the following theorem.
Theorem 3.1. Let be a real Hilbert space. Let and be sequences in defined iteratively from arbitrary points as follows: where and are sequences in satisfying the following conditions:(1),(2), ,(3). Suppose that has a solution in . Then, there exists a constant such that if for all for some , then the sequence converges to , a solution of .
Proof. Let with the norm , where . Define the sequence in by: . Let be a solution of , and . We observe that . It suffices to show that converges to in .
For this, let , there exists sufficiently large such that , , where denotes the ball of center and radius . Define . Since and are bounded, we set and . Let . We split the proof in three steps.
Step 1. We first prove that the sequence is bounded in . Indeed, it suffices to show that is in for all . The proof is by induction. By construction, . Suppose that for . We prove that . Assume for contradiction that . Then, we have . We compute as follows: We have Observing that and using (3.1), we obtain the following estimate: Following the same argument, we also obtain Thus, we obtain Using we have Therefore So we obtain the following estimate: Let . Then using the induction assumptions, the fact that and , we obtain a contradiction. Therefore, . Thus by induction, is bounded and so are and .Step 2. We show that there exists a unique sequence such that and , , with and .
In fact, let be defined by for all . Using the fact that and are monotone and satisfy the range condition, it follows from Lemma 2.3 that is monotone and also satisfies the range condition.
Applying Theorem 2.4, with and , we obtain that implies that Set . Then , for all . So we have, Therefore, Since is monotone and satisfies the range condition, then it is known that is bijective for every . So, the sequence is unique. Using (3.17) and Theorem 2.4, we have, . Let and . Then . So, , that is, Therefore, and .Step 3. We show that , where and .Claim 1. as . We compute as follows: We have From the boundness of , , and , there exists such that . Using (3.15) and the fact that is monotone, we obtain for some constant . Using (3.16) and similar arguments, we obtain: for some constant . Therefore, we have the following estimate: On the other hand, using the monotonicity of and we have Using (3.15) and (3.16), we observe that Therefore, Using (3.25) and the boundness of and , we obtain that there exists such that: Thus, by Lemma 2.1, . Since , we obtain that . But since , this implies that and . This completes the proof.
Corollary 3.2. Let be a real Hilbert space and be maps with such that the following conditions hold:(i) and are Lipschitz and monotone,(ii) and satisfy the range condition. Let and be sequences in defined iteratively from arbitrary points as follows: where and are sequences in satisfying the following conditions:(1),(2), ,(3). Suppose that has a solution in . Then, there exists a constant such that if for all for some , then the sequence converges to , a solution of .
Let be a real Banach space with dual space and let be a monotone linear operator. The mapping is said to be angle-bounded with constant if where denotes the duality pairing between elements of and those of . The class of angle-bounded operators is a subclass of the class of monotone operators. The angle-boundness of with corresponds to the symmetry of , that is, (See Pascali and Sburlan [5, Chapter IV, page 189]).
Let be a separable real Hilbert space and be a closed subspace of . For a given , consider the Hammerstein equation: and its th Galerkin approximation given by where and , where the symbols have their usual meanings (see  for the meaning of the symbols). Under this setting, Brézis and Browder (see ) proved the following approximation theorem.
Theorem BB. Let be a separable real Hilbert space. Let be a bounded continuous monotone operator and be an angle-bounded and weakly compact mapping. Then, for each , the Galerkin approximation (3.34) admits a unique solution in and converges strongly in to the unique solution 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 is bounded and monotone, the authors also required to be continuous. Furthermore, the operator 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 and are only assumed to be bounded and monotone and satisfy the range condition. We remark that continuity of the monotone map implies that is -accretive (see Martin ) and it is known that -accretive implies range condition.
Remark 3.5. Theorem BB guarantees the existence of a sequence 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 and with and .
We verify that these choices satisfy, in particular, condition (3) of Theorem 3.1. In fact, using the fact that , for and , we have as .
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