Strong convergence theorem for integral equations of Hammerstein type in Hilbert spaces
Introduction
Let E be a real normed space and let . E is said to have a Gâteaux differentiable norm (and E is called smooth) if the limitexists for each is said to have a uniformly Gâteaux differentiable norm if for each the limit is attained uniformly for . Further, E is said to be uniformly smooth if the limit exists uniformly for . The modulus of smoothness of E is defined by
E is equivalently said to be smooth if . Let is said to be q-uniformly smooth (or to have a modulus of smoothness of power type ) if there exists such that . Hilbert spaces, spaces, , and the Sobolev spaces, , are q-uniformly smooth. Hilbert spaces are 2-uniformly smooth whileLet E be a real normed space and let denote the generalized duality mapping from E into given bywhere denotes the dual space of E and denotes the generalized duality pairing. It is well known (see, for example, Xu [24]) that if . For , the mapping from E to is called normalized duality mapping. It is well known that if E is uniformly smooth, then J is single-valued (see, e.g., [24], [25]). In the sequel, we shall denote the single-valued normalized duality mapping by j.
A mapping is said to be accretive if , there exists such thatFor some real number , A is called -strongly accretive if , there exists such that
In Hilbert spaces, accretive operators are called monotone. The accretive operators were introduced independently in 1967 by Browder [6] and Kato [18]. Interest in such mappings stems from their firm connection with equations of evolution. For more on accretive/monotone mappings and connections with evolution equations, the reader may consult any of the following references [1], [2], [11], [14], [21].
A nonlinear integral equation of Hammerstein type (see, e.g., [17]) is one of the formwhere dy is a -finite measure on the measure space ; the real kernel k is defined on is a real-valued function defined on and is, in general, nonlinear and h is a given function on . If we now define an operator K byand the so-called superposition or Nemyskii operator by then, the integral equation (1.1) can be put in the operator theoretic form as follows:where, without loss of generality, we have taken .
Interest in Eq. (1.2) stems mainly from the fact that several problems that arise in differential equations, for instance, elliptic boundary value problems whose linear parts possess Green’s functions can, as a rule, be transformed into the form (1.2). Among these, we mention the problem of the forced oscillations of finite amplitude of a pendulum (see, e.g., [20, Chapter IV]). Example 1.1 The amplitude of oscillation is a solution of the problemwhere the driving force is periodical and odd. The constant depends on the length of the pendulum and on gravity. Since the Green’s function for the problemis the triangular functionproblem (1.3) is equivalent to the nonlinear integral equationIfthen (1.4) can be written as the Hammerstein equationwhere .
Equations of Hammerstein type play a crucial role in the theory of optimal control systems and in automation and network theory (see, e.g., [16]). Several existence and uniqueness theorems have been proved for equations of the Hammerstein type (see, e.g., [3], [4], [5], [6], [7], [8], [10], [15]). The Mann iteration scheme (see, e.g., [19]) has successfully been employed (see, e.g., the recent monographs of Berinde [1] and Chidume [11]). 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.
Recently, Chidume and Ofoedu [13] introduced a coupled explicit iterative scheme and proved the following strong convergence theorem for approximation of solution of a nonlinear integral equation of Hammerstein type in 2-uniformly smooth real Banach space. In particular, they proved the following theorem. Theorem 1.2 Chidume and Ofoedu, [13] Let E be a 2-uniformly smooth real Banach space. Let be bounded and accretive mappings. Let and be sequences in E defined iteratively from arbitrary bywhere and are real sequences in such that and . Suppose that has a solution in E. Then, there exist real constants and a set such that if and , for some and (where ), the sequence converges strongly to .
We make the following remarks about Theorem 1.2.
- 1.
The significance of Theorem 1.2 is not clear because of the requirement that has a solution and that . In fact, the set is referred to in the statement of Theorem 1.2, but without knowing what is in the statement of theorem and the definition of is given in the proof of Theorem 1.2.
- 2.
The requirement that seems to be a strong assumption.
- 3.
We suspect the result itself will never find serious applications or be of much interest (except to the authors) unless some examples are given.
It is our purpose in this paper to sharpen Theorem 1.2 with a new explicit iteration scheme in the settings of real Hilbert spaces without the requirement that has a solution and that . Furthermore, we give some examples so that our result will find serious applications and be of much interest to the readers. Also, the conditions imposed on the iteration parameters in Theorem 1.2 are too strong compared to the conditions we shall impose on our iteration parameters in Section 3. Thus, our result improves significantly on the result of Chidume and Ofoedu [13] in the settings of real Hilbert spaces.
Section snippets
Preliminaries
We shall make use of the following lemmas in the sequel. Lemma 2.1 Let H be a real Hilbert space. Thenfor all . Lemma 2.2 Let be a sequence of nonnegative real numbers satisfying the following relation:where and satisfy the conditions: ; , . Then, as .see, e.g., [2], [23]
Lemma 2.3 Shioji and Takahashi, [22]
Let be such that for all Banach limits . If , then .
Lemma 2.4 Xu, [24]
Let H
Main results
Lemma 3.1 Let H be a real Hilbert space. Let be a bounded, coercive and maximal monotone mapping. Let be a bounded and maximal monotone mapping. Let and be sequences in E defined iteratively from arbitrary bywhere is a real sequence in such that . Suppose that is a solution to . Then, the sequences and are bounded. Proof Let with the norm
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