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Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type with Lipschitz and Bounded Nonlinear Operators
Let E be a reflexive real Banach space with uniformly Gâteaux differentiable norm and F, K : be Lipschitz accretive maps with Suppose that the Hammerstein equation has a solution. An explicit iteration method is shown to converge strongly to a solution of the equation. No invertibility assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our theorems are significant improvements on important recent results (e.g., (Chiume and Djitte, 2012)).
Let be a real normed space and let . is said to have a Gâteaux differentiable norm if the limit exists for each . The normed space is said to have a uniformly Gâteaux differentiable norm if for each the limit is attained uniformly for . Furthermore, is said to be uniformly smooth if the limit exists uniformly for .
Let be a normed linear space with dimension greater than or equal to . The modulus of smoothness of is the function defined by In terms of the modulus of smoothness, the space is called uniformly smooth if and only if . is called -uniformly smooth if there exists a constant such that . (and ) spaces, are -uniformly smooth. In particular, 2-uniformly smooth if and -uniformly smooth if . It is easy to see that for , every -uniformly smooth real Banach space is uniformly smooth and thus has a uniformly Gâteaux differentiable norm.
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. We note that , for . It is known that if is a real Banach space with uniform Gâteaux differentiable norm, then the duality map is norm-to-weak* uniformly continuous on bounded subsets of .
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 the Schrödinger equations. If in (1.5), is independent of , then (1.5) reduces to whose solutions correspond to the equilibrium points of the system (1.5). Consequently, considerable research efforts have been devoted, especially within the past 40 years or so, to methods of finding approximate solutions (when they exist) of (1.6). An early fundamental result in the theory of accretive operators, due to Browder , states that the initial-value problem (1.5) is solvable if is locally Lipschitzian and accretive on . Utilizing the existing result for (1.5), Browder  proved that if is locally Lipschitzian and accretive on , then is -accretive that is, . Clearly, a consequence of this is that the following equation has a solution.
One important generalization of (1.7) 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 (1.8) can be put in operator theoretic form as follows: where without loss of generality, we have taken .
Interest in (1.10) 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 (1.10). Among these, we mention the problem of the forced oscillations of finite amplitude of a pendulum (see, e.g., Pascali and Sburlan , Chapter IV).
Example 1.1. We consider the problem of the pendulum.
where the driving force is odd. The constant , depends on the length of the pendulum and gravity. Since Green's function of the problem:
is given by:
it follows that problem (1.11) is equivalent to the nonlinear integral equation
If we set
and (1.14) can be written as
which is in the Hammerstein equation form:
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. , Bowder and Gupta , Chepanovich , and De Figueiredo and Gupta ).
For the iterative approximation of solutions of (1.6) and (1.7), 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 to be monotone. In the special case in which the operators are defined on subsets of which are compact, Brézis and Browder  proved the strong convergence of a suitably defined Galerkin approximation to a solution of (1.10) (see also Brézis and Browder ). In fact, they proved the following Theorem.
Theorem BB (Brézis and Browder). Let be a separable real Hilbert space and be a closed subspace of . Let be a bounded continuous monotone operator and be an angle-bounded and weakly compact mapping. For a given , consider the Hammerstein equation and its th Galerkin approximation given by where and , where the symbols have their usual meanings (see, ). Then, for each , the Galerkin approximation (1.20) admits a unique solution in and converges strongly in to the unique solution of (1.19).
Theorem BB is a 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 real Banach space.
We observe that the Galerkin method of Brézis and Browder is not iterative. The first attempt to construct an iterative method for the approximation of a solution of a Hammerstein equation, as far as we know, was made by Chidume and Zegeye  who constructed a sequence in the cartesian product and proved the convergence of the sequence to a solution of the Hammerstein equation. In subsequent papers, [18, 19], these authors were able to construct explicit coupled algorithms in the original space which converge strongly to a solution of the equation. Following this, Chidume and Djitte studied this explicit coupled algorithms and proved several strong convergence theorems (see, [20–22]).
Recently, Chidume and Djitte  introduced and studied a coupled explicit iterative process (see Theorem CD below) to approximate solutions of nonlinear equations of Hammerstein-type in real Hilbert spaces when the operators and are bounded and maximal monotone. They proved the following theorem.
Theorem CD (Chidume and Djitte, ). Let be a real Hilbert space and be bounded and maximal monotone operators. 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 .
It is Kown that spaces, , are Hilbert and spaces, , are not. So, the results of Chidume and Djitte  do not cover spaces, .
It is our purpose in this paper to prove convergence theorems in reflexive real Banach spaces that include all spaces, . In fact, it is proved in this paper, that an iteration process converges strongly in reflexive real Banach spaces with uniformly Gâteaux differentiable norm, to a solution of the Hammerstein equation (assuming existence) when the operators and are Lipschitz and accretive. These spaces include spaces, . This complements the results of Chidume and Djitte  to provide iterative methods for the approximation of solutions of the Hammerstein equation in all spaces, . Our method of proof is different and is also of independent interest.
In the sequel, we will need the followings results.
Lemma 2.1 (see, e.g., ). Let be a sequence of nonnegative numbers and a sequence such that . Let the recursive inequality be given where is a strictly increasing function such that it is positive on , . If . Then , as .
Lemma 2.2. There exists a map such that (1) is linear; that is, and ,(2) is positive: for every with , for all ,(3) is normalized: , where ,(4) is shift invariant: , for all , where is the shift operator defined by the above properties on the functional imply the following: (5) has norm one; thus, , for all , (6) extends on the subspace of the convergent sequences: (7)for any :
Remark 2.3. Functions as above are called Banach Limits.
Lemma 2.5. Let be a real normed linear space. Then, the following inequality holds:
3. Main Results
We now prove the following theorems.
Theorem 3.1. Let be a real Banach space and be Lipschitz and accretive maps with and Lipschitz constants and , respectively. Let and be sequences in defined iteratively from arbitrary points as follows: where , , and are sequences in satisfying the following conditions: (1), , (2). Suppose that has a solution in . Then the sequences and are bounded.
Proof. Let with the norm , where . Define the sequence in by: . Let be a solution of , set and . We observe that . We show that the sequence is bounded in .
For this, define Since and , there exists such that and for all . Let sufficiently large such that and . Define .
Claim 1. is in for all . The proof is by induction. By construction . Suppose that for some . We prove that . Assume for contradiction that . Then we have . We compute as follows: Using Lemma 2.5, We have Observing that and using the fact that is accretive, we obtain the following estimate: Using Schwartz's inequality and the fact that is Lipschitz, it follows that: Thus, we obtain Using the fact that we have Following the same argument, we also obtain Using the fact that we have which implies that Since and , we have By assumption, . So we have Hence Therefore, using the fact that and , it follows that . A contradiction. So . This prove the boundedness of the sequences and .
Theorem 3.2. Let be reflexive real Banach space with uniformly Gâteaux differentiable norm and be Lipschitz accretive mappings. Let and be sequences in defined iteratively from arbitrary by where , , and are real sequences in such that , and . Suppose that the equation has a solution . Then there exists a subset of such that if with , then the sequence converges strongly to .
Proof. Since, by Theorem 3.1, the sequences and are bounded and is bounded, there exists an such that and for all . Furthermore, the sets and are nonempty closed convex and bounded subsets of . Let be a Banach limit. Define the maps and by
Then for is weakly lower semicontinuous. Since reflexive, then
Now assume that and let . Then, by convexity of , we have that . Thus, . It follows from Lemma 2.5 that
Thus, taking Banach limits gives
This implies that . Furthermore, the fact that has a uniformly Gâteaux differentiable norm gives, as , that
Thus, for all , there exists such that for all and for all ,
which implies that
Using the same argument, we have
Moreover, since , , and are all bounded, then, from (3.18) and that there exist positive constants and such that
Thus, and .
Again, using the fact that has a uniformly Gâteaux differentiable norm, we obtain that Therefore, the sequences and satisfy the conditions of Lemma 2.4.
Hence, Define then . Moreover, Using Lemma 2.5, and (3.18), we have: Using the fact that and are Lipschitz and the sequences and are bounded, we obtain the following estimate: where for some . Hence, by Lemma 2.1, as . But and . This implies that as . This completes the proof.
Corollary 3.3. Let be a -uniformly real Banach spaces, and be Lipschitz accretive mappings. Let and be sequences in defined iteratively from arbitrary by where , , and are real sequences in such that , and . Suppose that the equation has a solution . Then there exists a set in such that if with , then the sequence converges strongly to .
Corollary 3.4. Let (or ) space () and be Lipschitz accretive mappings. Let and be sequences in defined iteratively from arbitrary by where , , and are real sequences in such that , and . Suppose that the equation has a solution . Then there exists a subset of such that if with , then the sequence converges strongly to .
Corollary 3.5. Let H be a real Hilbert space and be Lipschitz monotone mappings. Let and be sequences in defined iteratively from arbitrary by where , , and are real sequences in such that , and . Suppose that the equation has a solution . Then there exists a subset of such that if with , then the sequence converges strongly to .
Remark 3.6. Real sequences that satisfy the hypotheses of Theorems 3.1 are and , with and .
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