Superconvergence of the Iterated Galerkin Methods

for Hammerstein Equations

Hideaki Kaneko

Department of Mathematics and Statistics

Old Dominion University Norfolk, Virginia 23529-0077, USA

and Yuesheng Xu^{∗}

Department of Mathematics North Dakota State University Fargo, North Dakota 58105, USA

∗This author is partially supported by NASA under grant NAG3-1312

SUPERCONVERGENCE OF THE ITERATED GALERKIN METHODS FOR HAMMERSTEIN EQUATIONS

H. Kaneko and Y. Xu

Abstract In this paper, the well known iterated Galerkin method and iterated Galerkin-Kantorovich regularization method for approximating the solution of Fredholm integral equations of the second kind are generalized to Hammerstein equations with smooth and weakly singular kernels. The order of convergence of the Galerkin method and those of superconvergence of the iterated methods are analyzed. Numerical examples are presented to illustrate the superconvergence of the iterated Galerkin approximation for Hammerstein equations with weakly singular kernels.

Key words: the iterated Galerkin method, the iterated Galerkin-Kantorovich regularization, Hammerstein equations with weakly singular kernels, superconvergence. Mathematics Subject Classification (1990): 65B05, 45L10.

1 Introduction

In this paper, we consider the following Hammerstein equation

x(t) − ∫_{0}^{1} k(t, s)ψ(s, x(s))ds = f(t), 0 ≤ t ≤ 1,  (1.1)

where k, f and ψ are known functions and x is the function to be determined. Define k_{t}(s) ≡ k(t, s) for t, s ∈ [0, 1] to be the t section of k. We assume throughout this paper unless stated otherwise, the following conditions on k, f and ψ:

1. lim_{t→τ} ∥k_{t} − k_{τ} ∥_{∞} = 0, τ ∈ [0, 1];

2. M ≡ sup_{t} ∫_{0}^{1} |k(t, s)|ds < ∞;

3. f ∈ C[0, 1];

4. ψ(s, x) is continuous in s ∈ [0, 1] and Lipschitz continuous in x ∈ (−∞, ∞), i.e., there exists a constant C_{1} > 0 for which

|ψ(s, x_{1}) − ψ(s, x_{2})| ≤ C_{1}|x_{1} − x_{2}|, for all x_{1}, x_{2} ∈ (−∞, ∞);

5. the partial derivative ψ^{(0,1)} of ψ with respect to the second variable exists and is Lipschitz continuous, i.e., there exists a constant C_{2} > 0 such that

|ψ^{(0,1)}(t, x_{1}) − ψ^{(0,1)}(t, x_{2})| ≤ C_{2}|x_{1} − x_{2}|, for all x_{1}, x_{2} ∈ (−∞, ∞);  (1.2)

6. for x ∈ C[0, 1], ψ(., x(.)), ψ^{(0,1)}(., x(.)) ∈ C[0, 1].

Additional assumptions will be given in Sections 2, 3 and 4 when they are needed.

Numerical methods for approximating the solutions of the Hammerstein equations have been studied extensively in the literature. A variation of Nyström’s method was proposed by Lardy (1981). A new collocation type method was presented by Kumar and Sloan (1987) and its superconvergence properties were obtained by Kumar (1987). Two different discrete collocation methods were proposed by Kumar (1988) and Atkinson and Flores (1991). A degenerate kernel scheme was intoduced by Kaneko and Xu (1991) for equations (1.1) with smooth kernels. A product integration method and a collocation method were used to solve Hammerstein equations with weakly singular kernels, and certain superconvergence properties of the approximate solutions were discovered by Kaneko, Noren and Xu (1992). A nice review paper by Atkinson (1992) is recommended to the readers who require more information on the numerical treatments of Hammerstein equations. Some theoretical results about Hammerstein equations may be found in a book by Zeidler (1990). The purpose of this paper is to investigate the superconvergence property of the iterated Galerkin method and iterated Kantorovich Galerkin method for the solution of the Hammerstein equation (1.1). The iterated method may be viewed as a nonlinear transformation (iteration) that accelerates the convergence of the approximate solutions obtained from the Galerkin approximation. The general theory of the acceleration of convergence of a sequence by linear or nonlinear transformations was studied by Wimp (1981) and Delahaye (1988), and in the references cited there.

For the Fredholm integral equations of the second kind, the Galerkin and the iterated Galerkin methods have been investigated by many authors, e.g., see Graham (1982), Graham, Joe and Sloan (1985), Sloan (1976), Sloan (1990), Sloan and Thomee (1985) and Vainikko, Pedas and Uba (1984). In these papers that deal with the iterated Galerkin method, it has been shown that under some suitable conditions the iterated Galerkin method gives a rate of convergence that is faster than the rate obtainable by the Galerkin method, a phenomenon commonly known as superconvergence.

The order of convergence for Galerkin approximation for the solutions of Hammerstein equations with weakly singular kernels can be obtained by a direct extension of the corresponding result in the Fredholm case. However, it does not seem to be available in the literature. Hence, we include the results in Section 2 for completeness. A substantial number of proofs of the theorems in Section 2 will be omitted since they are straightforward and follow from the work of Vainikko (1967) and from that of Atkinson and Potra (1987). In the latter paper, the reader can find the general theory of the Galerkin and the iterated Galerkin methods for the equation x = Kx where K is a completely continuous operator of a domain in a Banach space into itself. Our present approach and results are different from those of Atkinson and Potra (1987) in a number of ways. For instance, we establish an estimate of improvement that we can expect when the iterated Galerkin scheme is applied to the weakly singular Hammerstein equations. This will be done in Section 3. Several related results on superconvergence are also established in Section 3. In Section 3, we deal with equations with weakly singular kernels and “nice” forcing terms, while in Sections 4, we tackle equations with both singular kernels and singular forcing terms by employing the classical Kantorovich regularization technique. We extend the results of the iterated Galerkin method to the iterated Galerkin-Kantorovich regularization method. Numerical examples are given in Section 5 to illustrate the theoretical estimates.

2 The Galerkin Methods for Hammerstein Equations

In this section, we develop the Galerkin method for Hammerstein equations and establish the order of convergence. Results concerning the Galerkin approximation using spline functions for the solutions of equation (1.1) with smooth and weakly singular kernels are presented.

Let n be a positive integer and {X_{n}} be a sequence of finite dimensional subspaces of C[0, 1] such that for any x ∈ C[0, 1] there exists a sequence {x_{n}}, x_{n} ∈ X_{n}, for which

∥x_{n} − x∥_{∞} → 0 as n → ∞.  (2.1)

Let P_{n}: L_{2}[0, 1] → X_{n} be an orthogonal projection for each n. We assume that the projection P_{n} when restricted to C[0, 1] is uniformly bounded, i.e.

P := sup_{n} ∥P_{n}|_{C[0,1]}∥_{∞} < ∞.  (2.2)

Then from (2.1) and (2.2), it follows that for each x ∈ C[0, 1],

∥P_{n}x − x∥_{∞} → 0, as n → ∞.  (2.3)

Now let

(KΨ)(x)(t) ≡ ∫_{0}^{1} k(t, s)ψ(s, x(s))ds.

With this notation, equation (1.1) takes the following operator form

x − KΨx = f.  (2.4)

In many interesting cases, equation (1.1) allows multiple solutions. Hence it is assumed for the remaining of this paper that we are treating a solution x_{0} of equation (1.1) that is isolated.

Let {φ_{nj}}_{j=1}^{n} be a set of linearly independent functions that spans X_{n}. The Galerkin method is to find

x_{n} = ∑_{j=1}^{n} b_{nj}φ_{nj}

that satisfies

x_{n} − P_{n}KΨx_{n} = P_{n}f.  (2.5)

Equivalently one is required to find b_{nj}’s that satisfy the system of nonlinear equations described by

∑_{j=1}^{n} b_{nj} < φ_{nj}, φ_{ni} > − < ∫_{0}^{1} k(t, s)ψ(s, ∑_{j=1}^{n} b_{nj}φ_{nj}(s))ds, φ_{ni} >=< f, φ_{ni} >, 1 ≤ i ≤ n, (2.6)

where < ., . > denotes the inner product in L_{2}.

We next estimate the error of the Galerkin approximate solutions to the exact solutions. For notation convenience, we introduce operators T̂ and T_{n} by letting

T̂x ≡ f + KΨx  (2.9)

and

T_{n}x_{n} ≡ P_{n}f + P_{n}KΨx_{n}  (2.10)

so that equations (2.4) and (2.5) can be written respectively as x = T̂x and x_{n} = T_{n}x_{n}. A proof of the following theorem can be made by directly applying Theorem 2 of Vainikko (1967). The paper of Atkinson and Potra (1987) is also useful in this connection.

Theorem 2.1 Let x_{0} ∈ C[0, 1] be an isolated solution of equation (2.4). Assume that 1 is not an eigenvalue of the linear operator (KΨ)′(x_{0}), where (KΨ)′(x_{0}) denotes the Fréchet derivative of KΨ at x_{0}. Then the Galerkin approximation equation (2.5) has a unique solution x_{n} ∈ B(x_{0}, δ) for some δ > 0 and for sufficiently large n. Moreover, there exists a constant 0 < q < 1, independent of n, such that

\frac{α_{n}}{1 + q} ≤ ∥x_{n} − x_{0}∥_{∞} ≤ \frac{α_{n}}{1 − q} ,  (2.11)

where α_{n} ≡ ∥(I − T′_{n}(x_{0}))^{−1}(T_{n}(x_{0}) − T̂(x_{0}))∥_{∞}. Finally,

E_{n}(x_{0}) ≤ ∥x_{n} − x_{0}∥_{∞} ≤ CE_{n}(x_{0}),  (2.12)

where C is a constant independent of n and E_{n}(x_{0}) = inf_{u∈X_{n}} ∥x_{0} − u∥_{∞}.

We denote by W_{p}^{m}[0, 1], 1 ≤ p ≤ ∞, the Sobolev space of functions g whose m-th generalized derivative g^{(m)} belongs to L_{p}[0, 1]. The space W_{p}^{m}[0, 1] is equipped with the norm

∥g∥_{W_{p}^{m}} ≡ ∑_{k=0}^{m} ∥g^{(k)}∥_{p}.

We now specify the finite dimensional subspace X_{n}. For any positive integer n, let

Π_{n} : 0 = t_{0} < t_{1} < ... < t_{n−1} < t_{n} = 1

be a partition of [0, 1]. Let r and ν be nonnegative integers satisfying 0 ≤ ν < r. Let S_{r}^{ν}(Π_{n}) denote the space of splines of order r, continuity ν, with knots at Π_{n}, that is

S_{r}^{ν}(Π_{n}) = {x ∈ C^{ν}[0, 1] : x|_{[t_{i},t_{i+1}]} ∈ P_{r−1}, for each i = 0, 1, …, n − 1}

where P_{r−1} denotes the space of polynomials of degree ≤ r − 1. We assume that the sequence of partitions Π_{n} of [0, 1] satisfies the condition that there exists a constant C > 0, independent of n, with the property:

\frac{max_{1≤i≤n}(t_{i} − t_{i−1})}{min_{1≤i≤n}(t_{i} − t_{i−1})} ≤ C, for all n.  (2.14)

It is known from de Boor (1976) and Douglas, Dupont and Wahlbin (1975) that condition (2.14) implies that the Galerkin projections P_{n} are uniformly bounded. In addition, it is also well known from Demko (1976) and De Vore (1976) that if 0 ≤ ν < r, 1 ≤ p ≤ ∞, m ≥ 0 and x ∈ W_{p}^{m}, then for each n ≥ 1, there exists u_{n} ∈ S_{r}^{ν}(Π_{n}) such that

∥x − u_{n}∥_{p} ≤ Ch^{μ}∥x∥_{W_{p}^{μ}},  (2.15)

where μ = min{m, r} and h = max_{1≤i≤n}(t_{i} −t_{i−1}). Using Theorem 2.1 and the inequalities (2.12) and (2.15), we obtain the following theorem.

Theorem 2.2 Let x_{0} be an isolated solution of equation (1.1) and let x_{n} be the solution of equation (2.5) in a neighborhood of x_{0}. Assume that 1 is not an eigenvalue of (KΨ)′(x_{0}). If x_{0} ∈ W_{∞}^{l} (0 ≤ l ≤ r), then

∥x_{0} − x_{n}∥_{∞} = O(h^{μ}),

where μ = min{l, r}. If x_{0} ∈ W_{p}^{l} (0 < l ≤ r, 1 ≤ p < ∞), then

∥x_{0} − x_{n}∥_{∞} = O(h^{ν}),

where ν = min{l − 1, r}.

We remark that a similar result of Galerkin’s method for Urysohn equations was obtained by Atkinson and Potra (1987). Hence, Theorem 2.2 may be derived by specializing their result to Hammerstein equations.

In the remaining portion of this section, we investigate the order of convergence of the Galerkin method for Hammerstein equations with weakly singular kernels. For this purpose, we define some necessary notation. For any ϵ ∈ R, let [0, 1]_{ϵ} = {t ∈ [0, 1] : t + ϵ ∈ [0, 1]}. Let Δ_{h} denote the forward difference operator with step size h. For α > 0 and 1 ≤ p ≤ ∞, we define the Nikol’skii space N_{p}^{α}[0, 1] by

N_{p}^{α}[0, 1] = \{x ∈ L_{p}[0, 1] : |x|_{α,p} := sup_{h≠0} \frac{1}{|h|^{α_{0}}} ∥Δ_{h}^{2}x^{[α]}∥_{L_{p}[0,1]_{2h}} < ∞\},  (2.16)

where [α] is an integer and 0 < α_{0} ≤ 1 are chosen so that α = [α] + α_{0}. Clearly, N_{p}^{α}[0, 1] is a Banach space with the norm ∥x∥_{α,p} = ∥x∥_{p} + |x|_{α,p}. We remark that the function t^{α−1} is in N_{1}^{α}[0, 1] but is not in N_{1}^{β}[0, 1], for any β > α, and log t ∈ N_{1}^{1}[0, 1]. It is known from Graham (1982) that

N_{p}^{m+ϵ}[0, 1] ⊆ W_{p}^{m}[0, 1] ⊆ N_{p}^{m}[0, 1] ⊆ N_{p}^{m−ϵ}[0, 1],  (2.17)

for m ∈ N, 0 < ϵ < 1, and 1 ≤ p ≤ ∞; and

N_{p}^{α}[0, 1] ⊆ N_{q}^{β}[0, 1],  (2.18)

for α > 0, 1 ≤ p ≤ q ≤ ∞ and β = α − (1/p − 1/q) > 0. We consider Hammerstein equations with kernels given by

k(t, s) = m(t, s)k(t − s), t, s ∈ [0, 1],  (2.19)

with k ∈ N_{1}^{α}[0, 1] for some 0 < α < 1 and m ∈ C^{2}([0, 1] × [0, 1]), and ψ as defined in the previous section.

Again, we let X_{n} = S_{r}^{ν}(Π_{n}). When no further conditions are made on the partition Π_{n} other than the one given by (2.14), the next theorem gives the best possible order of convergence of the Galerkin approximation to the solution of equation (1.1) with a weakly singular kernel defined by (2.19).

Theorem 2.3 Let x_{0} be an isolated solution of equation (1.1) with a kernel given by (2.19). Assume that 1 is not an eigenvalue of (KΨ)′(x_{0}). If f ∈ N_{1}^{β+1}[0, 1] for some 0 < β < 1, then

∥x_{0} − x_{n}∥_{∞} = O(h^{γ}),

with γ = min{α, β}.

Proof: By Theorem 2.1, we have

∥x_{0} − x_{n}∥_{∞} ≤ C inf_{u∈S_{r,n}^{ν}(Π_{n})} ∥x_{0} − u∥_{∞}.  (2.20)

A similar proof to the one given for Theorem 3 (ii) of Graham (1982) shows that if f ∈ N_{1}^{β+1}[0, 1] then x_{0} ∈ N_{1}^{min{α+1,β+1}}[0, 1] ⊆ N_{∞}^{min{α,β}}[0, 1]. In addition, (2.17) implies that f ∈ W_{1}^{1}[0, 1]. Hence f is equal to an absolutely continuous function almost everywhere. Without loss of generality, we have f ∈ W_{1}^{1}[0, 1]∩C[0, 1]. It can be shown that x_{0} ∈ C[0, 1]. Thus, x_{0} ∈ N_{∞}^{γ}[0, 1]∩C[0, 1]. It was proved in Graham (1982) that if ϕ ∈ N_{∞}^{η}[0, 1] ∩ C[0, 1] for some 0 < η < 1, then there exists a spline v ∈ S_{r}^{ν}(Π_{n}) such that ∥ϕ − v∥_{∞} ≤ Ch^{η} where C is a constant independent of h. The result of this theorem follows immediately from (2.20) and the above argument. □

Now we consider a special form of (2.19). Namely we assume

k(t, s) = m(t, s)g_{α}(|t − s|),  (2.21)

where m ∈ C^{μ+1}([0, 1] × [0, 1]) and

g_{α}(s) = \{ \begin{matrix}s^{α−1}, & 0 < α < 1, \\ log s, & α = 1. \end{matrix}  (2.22)

With these kernels, certain regularities of the solutions of (1.1) are known. Let S be a finite set in [0, 1] and we define the function ω_{S}(t) = inf{|t − s| : s ∈ S}. A function x is said to be of T ype(α, k, S), for −1 < α < 0, if

|x^{(k)}(t)| ≤ C[ω_{S}(t)]^{α−k} t ∉ S,

and for α > 0, if the above condition holds and x ∈ Lip(α). Kaneko, Noren and Xu (1990) proved that if f is of T ype(β, μ, {0, 1}), then a solution of equation (1.1) is of T ype(γ, μ, {0, 1}), where γ = min{α, β}. In order to recover the optimal rate of convergence of numerical solutions, we define a partition Π_{n}^{γ} of [0, 1] corresponding to the regularity of a solution. The knots of this partition Π_{n}^{γ} are given by

\begin{matrix}t_{i} = (1/2)(2i/n)^{q}, & 0 ≤ i ≤ n/2, \\ t_{i} = 1 − t_{n−i}, & n/2 < i ≤ n, \end{matrix}  (2.23)

where q = \frac{r}{γ}. Let S_{r,n}^{ν,γ} = S_{r}^{ν}(Π_{n}^{γ}), with r = 1 and ν = 0, or r ≥ 2 and ν ∈ \{0, 1\}. The following theorem gives the order of convergence of the Galerkin approximations to the solution of Hammerstein equations with kernels defined by (2.21) and (2.22). It should be noted that the technique of approximating a solution of the type described above by elements from the nonlinear spline space has been used on many occasions in dealing with the weakly singular Fredholm integral equations. For example, Vainikko and Uba (1981) describe the collocation method, whereas in Vainikko, Pedas and Uba (1984) they describe the Galerkin method. Schneider (1981) on the other hand establishes the product-integration method based upon the idea of the nonlinear spline approximation with nonuniform knots.

Theorem 2.4 Let x_{0} be an isolated solution of (1.1) with kernels (2.21) and (2.22) and let x_{n} be the Galerkin approximation to x_{0}. Let m ∈ C^{μ+1}([0, 1] × [0, 1]), and f be of T ype(β, μ, {0, 1}). Assume that ψ ∈ C^{(0,1)}([0, 1]×(−∞, ∞)) for μ = 0, 1 and ψ ∈ C^{μ−1}([0, 1]×(−∞, ∞)) for μ ≥ 2. We also assume 1 is not an eigenvalue of (KΨ)′(x_{0}). Then

∥x_{0} − x_{n}∥_{∞} = O(\frac{1}{n^{r}}).

Proof: This follows from Theorem 2.1, the regularity of the solution x_{0}, and from the results of Rice (1969). □

3 The Iterated Galerkin Method

In this section, we study the superconvergence of the iterated Galerkin method for the Hammerstein equation (1.1). Generalizing the linear case we first define the iterated scheme. Assume that x_{0} is an isolated solution of (1.1). As in Section 2, let P_{n} be the orthogonal projection from L_{2}[0, 1] onto X_{n} with conditions (2.1) and (2.2) satisfied. Assume that x_{n} is the unique solution of (2.5) in the sphere B(x_{0}, δ) for some δ > 0. Define

x′_{n} = f + KΨx_{n}.  (3.1)

Applying P_{n} to the both sides of (3.1), we obtain

P_{n}x′_{n} = P_{n}f + P_{n}KΨx_{n}.  (3.2)

Comparing (3.2) with (2.5), we see that

P_{n}x′_{n} = x_{n}.  (3.3)

Upon substituting (3.3) into (3.1), we find that the function x′_{n} satisfies the following new Hammerstein equation

x′_{n} = f + KΨP_{n}x′_{n}.  (3.4)

By letting S_{n} ≡ f + KΨP_{n}, we may rewrite (3.4) as x′_{n} = S_{n}x′_{n}. We first study the invertibility of the linear operators I − S′_{n}(x_{0}) in the following theorem, which will be used to prove the main results of this section.

Lemma 3.1 Let x_{0} ∈ C[0, 1] be an isolated solution of (1.1). Assume that 1 is not an eigenvalue of (KΨ)′(x_{0}). Then for sufficiently large n, the operators I−S′_{n}(x_{0}) are invertible and there exists a constant L > 0 such that

∥(I − S′_{n}(x_{0}))^{−1}∥_{∞} ≤ L, for sufficeintly large n.

Proof: Recalling the definition of Fréchet derivatives S′_{n}(x_{0}) and T̂′(x_{0}), we have, for each x ∈ C[0, 1],

∥T̂′(x_{0})(x) − S′_{n}(x_{0})(x)∥_{∞} ≤ sup_{0≤t≤1} ∫_{0}^{1} |k(t, s)|ψ^{(0,1)}(s, x_{0}(s))|ds∥x − P_{n}x∥_{∞}

+C sup_{0≤t≤1} M∥P_{n}∥_{∞}∥x∥_{∞}∥x_{0} − P_{n}x_{0}∥_{∞}.

By (2.1), the last two terms can be made arbitrarily small as n → ∞. This implies that S′_{n}(x_{0}) → T̂′(x_{0}) pointwise in C[0, 1], as n → ∞. By Assumptions 1, 2, and 6, T̂′(x_{0}) is a compact operator in C[0, 1]. Notice that by Assumptions 5, 6 and condition (2.1), there exists a constant C > 0 such that

|ψ^{(0,1)}(s, P_{n}x_{0}(s))| ≤ C_{2}∥P_{n}x_{0} − x_{0}∥_{∞} + ∥ψ^{(0,1)}(., x_{0}(.))∥_{∞} ≤ C, for all n.

Therefore, ∥S′_{n}(x_{0})(x)∥_{∞} ≤ MCP ∥x∥_{∞}, and

|S′_{n}(x_{0})(x)(t) − S′_{n}(x_{0})(x)(t′)| ≤ CP ∥k_{t} − k_{t′}∥_{1}∥x∥_{∞}.

This implies that {S′_{n}(x_{0})} is collectively compact. It follows from the theory of collectively compact operators in Anselone (1971) and Atkinson (1976) that (I − S′_{n}(x_{0}))^{−1} exists for sufficiently large n and there exists a constant L > 0 such that ∥(I − S′_{n}(x_{0}))^{−1}∥ ≤ L for sufficiently large n. □

For simplicity, from Lemma 3.1 we assume without loss of generality that I − S′_{n}(x_{0}) is invertible for each n ≥ 1 and

L = sup{∥(I − S′_{n}(x_{0}))^{−1}∥_{∞} : n ≥ 1} < ∞.

Throughout the rest of this section, we assume without further mention that δ > 0 satisfies LC_{2}MP δ < 1 and δ_{1} is chosen so that C_{1}Mδ_{1} ≤ δ. The following lemma establishes that x′_{n} defined in (3.1) is a unique solution of (3.4) in some neighborhood of x_{0} and provides an error bound for x′_{n} approximating x_{0}.

Lemma 3.2 Let x_{0} ∈ C[0, 1] be an isolated solution of equation (1.1) and x_{n} be the unique solution of (2.5) in the sphere B(x_{0}, δ_{1}). Assume that 1 is not an eigenvalue of (KΨ)′(x_{0}). Then for sufficiently large n, x′_{n} defined by the iterated scheme (3.1) is the unique solution of (3.4) in the sphere B(x_{0}, δ). Moreover, there exists a constant 0 < q < 1, independent of n, such that

\frac{β_{n}}{1 + q} ≤ ∥x′_{n} − x_{0}∥_{∞} ≤ \frac{β_{n}}{1 − q} ,

where β_{n} = ∥(I − S′_{n}(x_{0}))^{−1}[S_{n}(x_{0}) − T̂(x_{0})]∥_{∞}. Finally,

∥x′_{n} − x_{0}∥_{∞} ≤ CE_{n}(x_{0}).

Proof: This follows easily using Lemma 2.1 and Theorem 2 of Vainikko (1967). □ One way to ensure a superconvergence of the iterated Galerkin method is to assume

∥(KΨ)′(x_{0})(I − P_{n})|_{C[a,b]}∥_{∞} → 0  as n → ∞.  (3.5)

In this case, using the identity (ref. Theorem 2.3 of Atkinson and Potra (1987))

(I − (KΨ)′(x_{0}))(x′_{n} − x_{0})
= [I − (KΨ)′(x_{0})(I − P_{n})][KΨ(x_{n}) − KΨ(x_{0}) − (KΨ)′(x_{0})(x_{n} − x_{0})]
−(KΨ)′(x_{0})(I − P_{n})((KΨ)′(x_{0}) − I)(x_{n} − x_{0}).

we obtain

∥x′_{n} − x_{0}∥_{∞} ≤ ∥(I − (KΨ)′(x_{0}))^{−1}∥_{∞} {∥I − (KΨ)′(x_{0})(I − P_{n})∥_{∞}

× sup_{0≤θ≤1} ∥(KΨ)′(x_{0} + θ(x_{n} − x_{0})) − (KΨ)′(x_{0})∥_{∞}∥x_{0} − x_{n}∥_{∞}

+∥(KΨ)′(x_{0})(I − P_{n})((KΨ)′(x_{0}) − I)(x_{n} − x_{0})∥_{∞} } .

This with (3.5) gives a superconvergence of x′_{n} to x_{0}. In the next theorem, we establish superconvergence of the iterated Galerkin method in a general setting. In establishing superconvergence of the iterates of the Fredholm equations, many authors assumed the condition ∥K(I − P_{n})∥ → 0 as n → ∞ with K being a compact linear operator (e.g., Theorem 5 of Graham (1982) and Theorem 3.1 of Sloan (1990)). In our current problem, this is equivalent to assuming condition (3.5). However, the next theorem is proved without assumption (3.5). First, we apply the mean-value theorem to ψ(s, y) to conclude

ψ(s, y) = ψ(s, y_{0}) + ψ^{(0,1)}(s, y_{0} + θ(y − y_{0}))(y − y_{0}),  (3.6)

where θ := θ(s, y_{0}, y) with 0 < θ < 1. The boundedness of θ is essential for the proof of the next theorem, although it may depend on s, y_{0}, y. Let

g(t, s, y_{0}, y, θ) = k(t, s)ψ^{(0,1)}(s, y_{0} + θ(y − y_{0})),

(G_{n}x)(t) = ∫_{0}^{1} g(t, s, P_{n}x_{0}(s), P_{n}x′_{n}(s), θ)x(s)ds,

and (Gx)(t) = ∫_{0}^{1} g_{t}(s)x(s)ds, where g_{t}(s) = k(t, s)ψ^{(0,1)}(s, x_{0}(s)).

Theorem 3.3 Let x_{0} ∈ C[0, 1] be an isolated solution of equation (1.1) and x_{n} be the unique solution of (2.5) in the sphere B(x_{0}, δ_{1}). Let x′_{n} be defined by the iterated scheme (3.1). Assume that 1 is not an eigenvalue of (KΨ)′(x_{0}). Then, for all 1 ≤ p ≤ ∞,

∥x_{0} − x′_{n}∥_{∞} ≤ C {∥x_{0} − P_{n}x_{0}∥_{∞}^{2} + sup_{0≤t≤1} inf_{u∈X_{n}} ∥k(t, .)ψ^{(0,1)}(., x_{0}(.)) − u∥_{q}∥x_{0} − P_{n}x_{0}∥_{p}} ,

where 1/p + 1/q = 1 and C is a constant independent of n.

Proof: Note that from equations (1.1) and (3.4) we have

x_{0} − x′_{n} = K(Ψx_{0} − ΨP_{n}x′_{n}) = K(Ψx_{0} − ΨP_{n}x_{0}) + K(ΨP_{n}x_{0} − ΨP_{n}x′_{n}).  (3.7)

Replacing y by P_{n}x′_{n} and y_{0} by P_{n}x_{0} in equation (3.6), the last term of (3.7) can be written as

K(ΨP_{n}x_{0} − ΨP_{n}x′_{n})(t) = (G_{n}P_{n}(x_{0} − x′_{n}))(t).

Equation (3.7) now becomes

x_{0} − x′_{n} = K(Ψx_{0} − ΨP_{n}x_{0}) + G_{n}P_{n}(x_{0} − x′_{n}).  (3.8)

By using condition (1.2) and the fact 0 < θ < 1, we have, for all x ∈ C[0, 1],

∥(G_{n}x) − (Gx)∥_{∞} ≤ sup_{0≤t≤1} ∫_{0}^{1} |k(t, s)|ds∥x∥_{∞}(∥P_{n}x_{0} − x_{0}∥_{∞} + ∥P_{n}∥_{∞}∥x′_{n} − x_{0}∥_{∞}).

Consequently, by assumption (2.1) and Lemma 3.2,

∥G_{n} − G∥_{∞} ≤ M(∥P_{n}x_{0} − x_{0}∥_{∞} + P ∥x′_{n} − x_{0}∥_{∞}) → 0 as n → ∞.

That is, G_{n} → G in the norm of C[0, 1] as n → ∞. Moreover, for each x ∈ C[0, 1],

sup_{0≤t≤1} |(GP_{n}x)(t) − (Gx)(t)| = sup_{0≤t≤1} | ∫_{0}^{1} g_{t}(s)[P_{n}x(s) − x(s)]ds| ≤ MM_{1}∥P_{n}x − x∥_{∞},

where

M_{1} = sup_{0≤t≤1} |ψ^{(0,1)}(t, x_{0}(t))| < +∞.

It follows that GP_{n} → G pointwise in C[0, 1] as n → ∞. Again since P_{n} is uniformly bounded, we have for each x ∈ C[0, 1],

∥G_{n}P_{n}x − Gx∥_{∞} ≤ ∥G_{n} − G∥_{∞}∥P_{n}∥_{∞}∥x∥_{∞} + ∥GP_{n}x − Gx∥_{∞}.

Thus, G_{n}P_{n} → G pointwise in C[0, 1] as n → ∞. By Assumptions 2, 5, and 6, we see that there exists a constant C > 0 such that for all n

|ψ^{(0,1)}(s, P_{n}x_{0}(s) + θ(P_{n}x′_{n}(s) − P_{n}x_{0}(s)))| ≤ C_{2}∥P_{n}x_{0} − x_{0}∥_{∞} + θC_{2}P ∥x′_{n} − x_{0}∥_{∞} + M_{1} ≤ C.

By a proof similar to that for Lemma 3.1, we can show that {G_{n}P_{n}} is collectively compact. Since G = (KΨ)′(x_{0}) is compact and (I − G)^{−1} exists, it follows from the theory of collectively compact operators that (I − G_{n}P_{n})^{−1} exists and is uniformly bounded for sufficiently large n. By (3.8), we have the following estimate

sup_{0≤t≤1} |(x_{0} − x′_{n})(t)| ≤ C sup_{0≤t≤1} |K(Ψx_{0} − ΨP_{n}x_{0})(t)|.

Next, we estimate the function d(t) ≡ |K(Ψx_{0} − ΨP_{n}x_{0})(t)|. Using (3.6) with y = P_{n}x_{0} and y_{0} = x_{0}, we obtain, for 0 < θ < 1,

d(t) = |∫_{0}^{1} g(t, s, x_{0}(s), P_{n}x_{0}(s), θ)(x_{0}(s) − P_{n}x_{0}(s))ds| .

Note that ∫_{0}^{1} u(s)[x_{0}(s) − P_{n}x_{0}(s)]ds = 0, for all u ∈ X_{n}. Thus, for all u ∈ X_{n},

d(t) = |∫_{0}^{1}[g(t, s, x_{0}(s), P_{n}x_{0}(s), θ) − u(s)](x_{0}(s) − P_{n}x_{0}(s))ds|
≤ ∫_{0}^{1} |g(t, s, x_{0}(s), P_{n}x_{0}(s), θ) − g_{t}(s)|ds∥x_{0} − P_{n}x_{0}∥_{∞}
+ |∫_{0}^{1}[g_{t}(s) − u(s)](x_{0}(s) − P_{n}x_{0}(s))ds| .

Now, by condition (1.2), we have

∫_{0}^{1} |g(t, s, x_{0}, P_{n}x_{0}(s), θ) − g_{t}(s)|ds ≤ C_{1}θ ∫_{0}^{1} |k(t, s)|ds∥x_{0} − P_{n}x_{0}∥_{∞} ≤ C_{1}M∥x_{0} − P_{n}x_{0}∥_{∞}.

Moreover, for 1/p + 1/q = 1,

|∫_{0}^{1}[g_{t}(s) − u(s)][x_{0}(s) − P_{n}x_{0}(s)]ds| ≤ ∥g_{t} − u∥_{q}∥x_{0} − P_{n}x_{0}∥_{p}.

Therefore,

d(t) ≤ C_{1}M∥x_{0} − P_{n}x_{0}∥_{∞}^{2} + ∥g_{t} − u∥_{q}∥x_{0} − P_{n}x_{0}∥_{p}, for all u ∈ X_{n}.

Hence the desired result follows. □

In the next two theorems, we consider the case that X_{n} = S_{r}^{ν}(Π_{n}) where Π_{n} is an arbitrary partition of [0, 1] satisfying (2.14). First, we consider the case when both the kernels and the solutions of equation (1.1) are smooth.

Theorem 3.4 Let x_{0} ∈ W_{p}^{l} (0 < l ≤ r) be an isolated solution of (1.1), x_{n} be the unique solution of (2.5) in B(x_{0}, δ_{1}), and x′_{n} be defined by the iterated scheme (3.1). Assume that 1 is not an eigenvalue of (KΨ)′(x_{0}). Assume that for all t ∈ [0, 1], k_{t}(.)ψ^{(0,1)}(., x_{0}(.)) ∈ W_{q}^{m}(0 ≤ m ≤ r). Then

∥x_{0} − x′_{n}∥_{∞} = O(h^{μ+min{μ,ν}}),

where μ = min{l, r} and ν = min{m, r}.

Proof: Since the partition Π_{n} of [0, 1] satisfies condition (2.14), we conclude that

P := sup_{n} ∥P_{n}∥_{∞} < ∞.

Hence,

∥x_{0} − P_{n}x_{0}∥_{p} ≤ ∥x_{0} − P_{n}x_{0}∥_{∞} ≤ (1 + P) inf_{u∈S_{r}^{ν}(Π_{n})} ∥x_{0} − u∥_{∞} ≤ Ch^{μ}.

In addition,

sup_{0≤t≤1} inf_{u∈S_{r}^{ν}(Π_{n})} ∥k_{t}(.)ψ^{(0,1)}(., x_{0}(.)) − u∥_{q} ≤ Ch^{ν}.

The result of this theorem follows from Theorem 3.3 with X_{n} = S_{r}^{ν}(Π_{n}). □

We remark that Theorem 3.4 may be obtained from Theorem 5.2 of Atkinson and Potra (1987), Theorem 3.4 being a special case of Atkinson and Potra’s theorem to Hammerstein equations.

In the following theorem, we assume that k(t, s) is a kernel given by (2.19), i.e., k(t, s) = m(t, s)k(t − s), with k ∈ N_{1}^{α}[0, 1] for some 0 < α < 1 and m ∈ C^{2}([0, 1] × [0, 1]). Also, we assume that S_{r}^{ν}(Π_{n}) is such that ν ≥ 1.

Theorem 3.5 Let x_{0} be an isolated solution of equation (1.1) with kernels given by (2.19), x_{n} be the unique solution of equation (2.5) in B(x_{0}, δ_{1}), and x′_{n} be defined by iterated scheme (3.1). Assume that 1 is not an eigenvalue of (KΨ)′(x_{0}), f ∈ N_{1}^{β+1}[0, 1] for some 0 < β < 1, ψ^{(0,1)}(., x(.)) ∈ W_{1}^{1} for x ∈ W_{1}^{1}. Then

∥x_{0} − x′_{n}∥_{∞} = O(h^{2γ}),

with γ = min{α, β}.

Proof: Following the proof of Theorem 3.4, we have

∥x_{0} − P_{n}x_{0}∥_{∞} ≤ (1 + P) inf_{u∈S_{r}^{ν}(Π_{n})} ∥x_{0} − u∥_{∞}.  (3.9)

As stated in the proof of Theorem 2.4, we know that

x_{0} ∈ N_{∞}^{γ}[0, 1] ∩ C[0, 1] ∩ W_{1}^{1}.  (3.10)

Using (3.9) and an argument similar to the one used in the proof of Theorem 2.4, we obtain ∥x_{0} − P_{n}x_{0}∥_{∞} ≤ Ch^{γ}. Now, by Theorem 4(i) of Graham (1982), we find that there exists v_{t} ∈ S_{r}^{ν}(Π_{n}) such that ∥k_{t} − v_{t}∥_{1} = O(h^{α}). Since ν ≥ 1, it follows that S_{r}^{ν}(Π_{n}) ⊂ W_{1}^{1}. Thus, v_{t} ∈ W_{1}^{1}. From (3.10), x_{0} ∈ W_{1}^{1}. This yields that ψ^{(0,1)}(., x_{0}(.)) ∈ W_{1}^{1}. Consequently, v_{t}(.)ψ^{(0,1)}(., x_{0}(.)) ∈ W_{1}^{1}. The remark made before Theorem 2.2 implies that there exists u_{t} ∈ S_{r}^{ν}(Π_{n}) for which

∥v_{t}(.)ψ^{(0,1)}(., x_{0}(.)) − u_{t}(.)∥_{1} = O(h).

Therefore,

∥g_{t} − u_{t}∥_{1} = ∫_{0}^{1} |m(t, s)k(t − s)ψ^{(0,1)}(s, x_{0}(s)) − u_{t}(s)|ds
≤ ∫_{0}^{1} |m(t, s)k(t − s)ψ^{(0,1)}(s, x_{0}(s)) − v_{t}(s)ψ^{(0,1)}(s, x_{0}(s))|ds
+ ∫_{0}^{1} |v_{t}(s)ψ^{(0,1)}(s, x_{0}(s)) − u_{t}(s)|ds
≤ ∥k_{t} − v_{t}∥_{1}∥ψ^{(0,1)}(., x_{0}(.))∥_{∞} + ∥v_{t}(.)ψ^{(0,1)}(., x_{0}(.)) − u_{t}∥_{1}
= O(h^{α}) + O(h) = O(h^{α}).

Now, applying Theorem 3.3 with q = 1, p = ∞, and X_{n} = S_{r}^{ν}(Π_{n}), we conclude that

∥x_{0} − x′_{n}∥_{∞} ≤ C {∥x_{0} − P_{n}x_{0}∥_{∞}^{2} + inf_{u∈S_{r}^{ν}(Π_{n})} ∥g_{t} − u_{t}∥_{1}∥x_{0} − P_{n}x_{0}∥_{∞}}
= O(h^{α+γ}) + O(h^{2γ}) = O(h^{2γ}).

The proof is complete. □

Next, we apply Theorem 3.3 to equation (1.1) with kernels given by (2.21) and (2.22) and use X_{n} = S_{r}^{ν}(Π_{n}^{γ}) as approximate spaces, where S_{r}^{ν}(Π_{n}^{γ}) of splines with nonuniform knots is defined as in Section 2 such that r ≥ 2 and ν = 1.

Theorem 3.6 Let x_{0} be an isolated solution of (1.1) with weakly singular kernels given by (2.21) and (2.22). Let x_{n} be the unique solution of (2.5) in B(x_{0}, δ_{1}), and x′_{n} be defined by the iterated scheme (3.1). Assume that 1 is not an eigenvalue of (KΨ)′(x_{0}) and that the hypotheses of Theorem 2.4 are satisfied with μ ≥ 1. Also assume that ψ^{(0,1)}(·, x_{0}(·)) is of T ype(α, r, {0, 1}) for α > 0 whenever x_{0} is of the same type. Then

∥x_{0} − x′_{n}∥_{∞} = O(\frac{1}{n^{α+r}}).

Proof: The proof of this theorem is similar to that of Theorem 3.5. We apply Theorem 3.3 with q = 1, p = ∞ and X_{n} = S_{r}^{ν}(Π_{n}^{γ}). By Rice (1969), we have ∥x_{0} − P_{n}x_{0}∥_{∞} = O(\frac{1}{n^{r}}). It can be proved that there exists u ∈ S_{r}^{ν}(Π_{n}^{γ}) such that ∥g_{t} − u∥_{1} = O(\frac{1}{n^{α}}). From this, the result of this theorem follows. □

As the last application of Theorem 3.3, we consider equation (1.1) with kernels having singularity at the four corners of the square [0, 1]×[0, 1], a problem that arises from boundary integration for the harmonic Dirichlet problem in plane domains with corners (see Kress (1990)). In the following theorem, we assume k_{s}(t) = k(t, s) is of T ype(α, μ, {0, 1}), for α > 0, and k_{t}(s) = k(t, s) is of T ype(α, μ, \{0, 1\}), for α > −1, e.g., k(t, s) = m(t, s)\sqrt{t}, and k(t, s) = m(t, s)\frac{1}{\sqrt{1−s}}, etc., with m(t, s) smooth, and assume f is of T ype(β, μ, {0, 1}), for α, β > 0 and a positive integer μ. It is not difficult to prove that an isolated solution x_{0}, of the corresponding equation (1.1), is of T ype(γ, μ, {0, 1}), where γ = min{α, β} if α > 0 and γ = min{α + 1, β} if −1 < α < 0 by modifying the proofs of theorems in Kaneko, Noren and Xu (1990). We again let q = \frac{r}{γ} and define the Galerkin subspace S_{r}^{ν}(Π_{n}^{γ}) as in Section 2 with r = 1 and ν = 0, and r ≥ 2 and ν ∈ {0, 1}, where partition Π_{n}^{γ} is defined as in (2.23). The following theorem describes the order of convergence of the Galerkin approximation x_{n} and that of superconvergence of the iterated Galerkin approximation x′_{n}. To the best of our knowledge, this result is not known in the literature even for Fredholm integral equations of the second kind.

Theorem 3.7 Let x_{0} be an isolated solution of (1.1) with kernels of the type defined in the paragraph preceding this theorem. Let x_{n} be the unique solution of (2.5) in B(x_{0}, δ_{1}), and x′_{n} be defined by the iterated scheme (3.1). Assume that 1 is not an eigenvalue of (KΨ)′(x_{0}) and that f is of T ype(β, r, {0, 1}). Also assume that ψ^{(0,1)}(·, x_{0}(·)) is of T ype(γ, r, {0, 1}) whenever x_{0} is of the same type. Then,

∥x_{0} − x_{n}∥_{∞} = O(\frac{1}{n^{r}}),

and

∥x_{0} − x′_{n}∥_{∞} = O(\frac{1}{n^{2r}}).

Proof: We present the proof for case when α > 0, since the proof for the other case is similar. The proof of the first estimate is similar to that for Theorem 2.6. Thus, we omit the details. Since P_{n} in this theorem is defined to be the Galerkin projection from C[0, 1] onto S_{r}^{ν}(Π_{n}^{γ}), where γ = min\{α, β\}, and since x_{0} is of T ype(γ, r, \{0, 1\}), we have ∥x_{0} − P_{n}x_{0}∥_{∞} = O(\frac{1}{n^{r}}). Meanwhile, since k_{t}(s) = k(t, s) is of T ype(α, r, {0, 1}) and γ ≤ α, we find that k_{t}(s) = k(t, s) is also of T ype(γ, r, {0, 1}). By the assumption on ψ^{(0,1)}, we conclude that ψ^{(0,1)}(., x_{0}(.)) is of T ype(γ, r, {0, 1}). Hence, k(t, .)ψ^{(0,1)}(., x_{0}(.)) is of T ype(γ, r, {0, 1}). It follows that

inf_{u∈S_{r}^{ν}(Π_{n}^{γ})} ∥k(t, .)ψ^{(0,1)}(., x_{0}(.)) − u∥_{1} = O(\frac{1}{n^{r}}).

Therefore, the result of this theorem follows from Theorem 3.3. The proof is complete. □

4 The Iterated Galerkin-Kantorovich Method

In this section, we extend the classical Kantorovich regularization and the iterated Galerkin-Kantorovich method for Fredholm integral equations of the second kind to Hammerstein equations. These extensions will be made on equations with both singular kernels and singular forcing terms. The superconvergence of the corresponding iterated solution is also investigated.

In equation (2.4) we put

z = KΨx  (4.1)

so that

x = f + z.  (4.2)

Upon applying KΨ on both sides of (4.2), we obtain

z = KΨ(f + z).  (4.3)

Now we define operators by Ψ_{0}(x)(t) ≡ ψ(t, x(t)), and

Ψ_{1}(x)(t) ≡ Ψ_{0}(f + x)(t) − Ψ_{0}(f)(t).  (4.4)

In addition, define f_{1} by

f_{1}(t) ≡ KΨ_{0}(f)(t) = ∫_{a}^{b} k(t, s)ψ(s, f(s))ds.  (4.5)

From (4.4), we have KΨ_{0}(f + z)(t) = KΨ_{1}(z)(t) + KΨ_{0}(f)(t) so that (4.3) becomes

z − KΨ_{1}(z) = KΨ_{0}(f) ≡ f_{1}.  (4.6)

Equation (4.6) will be called the “regularized” equation for the original Hammerstein equation (1.1). It is interesting to note that

|Ψ_{1}(x_{1})(t) − Ψ_{1}(x_{2})(t)| = |Ψ_{0}(f + x_{1})(t) − Ψ_{0}(f + x_{2})(t)| ≤ C_{1}|x_{1}(t) − x_{2}(t)|.

Thus, Ψ_{1} is also Lipschitz continuous with the same Lipschitz constant C_{1} as one for Ψ_{0}. Hence the solvability of equation (4.6) is guaranteed by the solvability of the original equation (1.1).

The Galerkin method described in Section 2 is now applied to equation (4.6). Namely, we find z_{n} ∈ X_{n} that satisfies

z_{n} − P_{n}KΨ_{1}z_{n} = P_{n}f_{1}.  (4.7)

The Galerkin-Kantorovich regularization solution for (1.1) is now given by

x_{n}^{K} = f + z_{n}.  (4.8)

Note that x_{n}^{K} inherits the singularity of f. From equations (4.2) and (4.8), we have x−x_{n}^{K} = z−z_{n}. Since z, z_{n} ∈ C[0, 1], we see that x − x_{n}^{K} ∈ C[0, 1], although neither x nor x_{n}^{k} may be in C[0, 1]. Denote T_{n}z_{n} ≡ P_{n}f_{1} + P_{n}KΨ_{1}z_{n} and T z ≡ f_{1} + KΨ_{1}z.

Theorem 4.1 Let x_{0} be an isolated solution of equation (1.1) such that z_{0} = KΨ_{0}x_{0} ∈ C[0, 1]. Assume that 1 is not an eigenvalue of the linear operator (KΨ_{1})′(z_{0}). Then equation (4.7) has a unique solution z_{n} ∈ B(z_{0}, δ) for some δ > 0 and for sufficiently large n. Moreover, there exists a constant 0 < q < 1, independent of n, such that

\frac{α_{n}}{1 + q} ≤ ∥x_{n}^{K} − x_{0}∥_{∞} ≤ \frac{α_{n}}{1 − q} ,  (4.9)

where x_{n}^{K} = f + z_{n} and

α_{n} = ∥(I − T′_{n}(z_{0}))^{−1}(T_{n}(z_{0}) − T (z_{0}))∥_{∞}.  (4.10)

Finally,

E_{n}(z_{0}) ≤ ∥x_{0} − x_{n}^{K}∥_{∞} ≤ CE_{n}(z_{0})  (4.11)

where E_{n}(z_{0}) = inf_{y∈X_{n}} ∥y − z_{0}∥_{∞} and C is a constant independent of n.

Proof: The inequalities (4.9) follows again from Theorem 2 of Vainikko (1967). Also it is noted that

z_{0} − z_{n} = x_{0} − x_{n}^{K}.  (4.12)

Since z_{n} ∈ X_{n}, (4.10) holds and E_{n}(z_{0}) ≤ ∥z_{0} − z_{n}∥_{∞} = ∥x_{0} − x_{n}^{K}∥_{∞}. This gives the first inequality in (4.11). Since T_{n}(z_{0}) − T (z_{0}) = P_{n}(f_{1} − KΨ_{1}z_{0}) − z_{0} = P_{n}z_{0} − z_{0}, we find

∥z_{n} − z_{0}∥_{∞} ≤ \frac{∥(I − T′_{n}(z_{0}))^{−1}∥_{∞}∥T_{n}(z_{0}) − T (z_{0})∥_{∞}}{1 − q} = \frac{∥(I − T′_{n}(z_{0}))^{−1}∥_{∞}}{1 − q} ∥P_{n}z_{0} − z_{0}∥_{∞}.

Also for u ∈ X_{n},

∥z_{0} − P_{n}z_{0}∥ = ∥z_{0} − u − P_{n}(z_{0} − u)∥ ≤ (1 + ∥P_{n}∥)∥z_{0} − u∥.

Therefore, we have ∥x_{0} − x_{n}^{K}∥ ≤ CE_{n}(z_{0}) where C is a constant independent of n. □

We next consider the iterated Kantorovich method and investigate its superconvergence property. Assume that z_{0} is an isolated solution of (4.6) and z_{n} is the unique solution of (4.7) in the sphere B(z_{0}, δ) for some δ > 0. Define

z′_{n} = KΨ_{1}(z_{n}) + f_{1},  (4.13)

and x_{n}^{K′} = f + z′_{n}. The element x_{n}^{K′} is called the iterated Galerkin-Kantorovich approximate solution of equation (1.1). Applying P_{n} to both sides of (4.13) gives

P_{n}z′_{n} = P_{n}KΨ_{1}(z_{n}) + P_{n}f_{1}.  (4.14)

Again, by using (4.7), we have P_{n}z′_{n} = z_{n}. Upon substituting this equation into (4.13), we find that z′_{n} satisfies the following new Hammerstein equation z′_{n} = KΨ_{1}P_{n}z′_{n} + f_{1}. In view of the fact that Ψ_{1} is Lipschitz continuous with the same Lipschitz constant as one for Ψ_{0}, the same proofs given for Theorems 3.1, 3.2 and 3.3 can be applied to S_{n} ≡ KΨ_{1}P_{n} + f_{1} to obtain the following theorem. Here δ_{1} is chosen as in Section 3. As in Theorem 3.3, the assumption that ∥(KΨ)′(x_{0})(I − P_{n})∥_{∞} → 0 as n → ∞ is no longer needed.

Theorem 4.2 Let x_{0} be an isolated solution of equation (1.1) such that z_{0} = KΨ_{0}x_{0} ∈ C[0, 1]. Let z_{n} be the unique solution of equation (4.7) in the sphere B(z_{0}, δ_{1}). Let x_{n}^{K′} be the corresponding iterated Galerkin- Kantorovich approximate solution. Assume that 1 is not an eigenvalue of (KΨ_{1})′(z_{0}). Then, for all 1 ≤ p ≤ ∞,

∥x_{0} − x_{n}^{K′}∥ ≤ C {∥z_{0} − P_{n}z_{0}∥_{∞}^{2} + sup_{0≤t≤1} inf_{u∈X_{n}} ∥k(t, ·)ψ_{1}^{(0,1)}(·, z_{0}(·)) − u∥_{q}∥z_{0} − P_{n}z_{0}∥_{p}} ,

where \frac{1}{p} + \frac{1}{q} = 1. □

Results parallel to Theorems 3.4 - 3.7, for smooth and weakly singular kernels can be obtained also by using Theorem 4.2, for the iterated Kantorovich method. The iterated Kantorovich regularization method for the Fredholm equations of the second kind was investigated by Sloan (1984).

5 Numerical Examples

In this section, some numerical examples are given to illustrate the theory established in the previous sections. Example 1: Consider

x(t) − ∫_{0}^{1} \frac{x^{2}(s)}{\sqrt{|t − s|}}ds = f(t),  0 ≤ t ≤ 1,  (5.1)

where f is selected so that x(t) = \sqrt{t} is the solution. The splines of orders 1 (q = 2) and 2 (q = 4) with knots defined by equation (2.23) in terms of q, are used in computations. To establish the Galerkin matrix, it is required to compute the integral of the form

∫_{t_{i−1}}^{t_{i}} ∫_{t_{j−1}}^{t_{j}} \frac{φ_{i}(s)φ_{j}(t)}{\sqrt{|t − s|}}dtds,  (5.2)

where φ′_{i}s are respective B-splines of the above mentioned spline space. It can be proved that φ_{i}(s) ∫_{t_{j−1}}^{t_{j}} \frac{φ_{j}(t)}{\sqrt{|t−s|}}dt belongs to T ype(\frac{1}{2}, k, \{t_{j−1}, t_{j}\}). Consequently, we employ the recently developed Gauss-type quadrature formula of Kaneko and Xu (1994) to approximate integrals (5.2). This brings to our concern the problem of the discrete Galerkin method for Hammerstein equations with weakly singular kernels. This will be dealt with in a future paper. In the ensuing data, e_{n} ≡ ∥x − x_{n}∥_{∞} and e′_{n} ≡ ∥x − x′_{n}∥_{∞} were approximated respectively by

max\{|x(\frac{i}{100}) − x_{n}(\frac{i}{100})| : i = 0, 1, ⋯, 100\}

and

max\{|x(\frac{i}{100}) − x′_{n}(\frac{i}{100})| : i = 0, 1, ⋯, 100\}.

Data 1. (q = 2)
n  e_{n}  decay exp.  e′_{n}  decay exp.
16 1.60D − 2  3.01D − 3
32 7.26D − 3  1.14  9.10D − 4  1.73
64 3.34D − 3  1.12  2.88D − 4  1.66
128 1.64D − 3  1.03  9.50D − 5  1.60

Data 2. (q = 4)
n  e_{n}  decay exp.  e′_{n}  decay exp.
16 4.01D − 3  8.04D − 4
32 9.93D − 4  2.01  1.30D − 4  2.61
64 2.46D − 4  2.01  2.28D − 5  2.51
128 6.06D − 5  2.02  3.90D − 6  2.55

It can be seen clearly that the iterated Galerkin approximation has superconvergence by an order \frac{1}{2}. Example 2. To illustrate the use of theorem 3.7, we consider

x(t) − ∫_{0}^{1} \frac{x^{2}(s)}{\sqrt[3]{s}}ds = f(t),  0 ≤ t ≤ 1,  (5.3)

where f is selected so that x(t) = \sqrt{t} is the solution of equation (5.3). As in the first example, the splines of orders 1 and 2 are used. Since the solution is of T ype(\frac{1}{2}, k, \{0, 1\}) for any positive integer k, the partition is formed according to α = \frac{1}{2}.

Data 1. (q = 2)
n  e_{n}  decay exp.  e′_{n}  decay exp.
16 1.12D − 2  2.10D − 3
32 5.15D − 3  1.12  5.21D − 4  2.01
64 2.22D − 3  1.21  1.30D − 4  2.00
128 1.08D − 3  1.04  3.25D − 5  2.00
Data 2. (q = 4)
n  e_{n}  decay exp.  e′_{n}  decay exp.
16 3.12D − 3  5.12D − 4
32 7.53D − 4  2.05  3.05D − 5  4.07
64 1.74D − 4  2.11  1.85D − 6  4.04
128 4.26D − 5  2.03  1.14D − 7  4.02

The iteration process doubles the rate of convergence.

ACKNOWLEDGEMENT: The authors would like to thank Professor K. Atkinson and the referee who made many useful suggestions that improved this paper.

References

1. P. M. Anselone, Collectively compact operator approximation theory and applications to Integral Equations, Prentice-Hall, Englewood Cliffs, N.J., 1971.

2. K. E. Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, 1976.

3. K. E. Atkinson, A survey of numerical methods for solving nonlinear integral equations, J. Int. Equ. Appl. 4 (1992), 15-46.

4. K. E. Atkinson and J. Flores, The discrete collocation method for nonlinear integral equations, Report on Computational Mathematics, No. 10, the University of Iowa, 1991.

5. K. E. Atkinson, I. Graham and I. Sloan, Piecewise continuous collocation for integral equations, SIAM J. Numer. Anal. 20 (1983), 172-186.

6. K. E. Atkinson and F. Potra, Projection and iterated projection methods for nonlinear integral equations, SIAM J. Numer. Anal. 24 (1987), 1352-1373.

7. C. de Boor, A bound on the L_{∞} norm of L_{2}-approximation by splines in terms of a global mesh radio. Math. Comp. 30 (1976), 765-771.

8. J. P. Delahaye, Sequence Transformations, Springer-Verlag, Berlin Heidelberg, 1988.

9. S. Demko, Splines approximation in Banach function spaces, in Theory of Approximation with Applications (Law, A. G. & Sahney, B. N., Eds), pp. 146-154, New York, Academic Press, 1976.

10. R. A. De Vore, Degree of approximation, in Approximation Theory II (Lorentz, G. G., Chui, C. K. & Schumaker, L. L., Eds) pp. 117-161, New York, Academic Press, 1976.

11. J. Douglas, T. Dupont, and L. Wahlbin, Optimal L_{∞} error estimates for Galerkin approximations to solutions of two point boundary value problems, Math. Comp. 29 (1975), 475-483.

12. I. Graham, Galerkin Methods for second kind integral equations with singularities, Math. Comp. 39 (1982), 519- 533.

13. I. Graham, S. Joe and I. Sloan, Iterated Galerkin versus iterated collocation for integral equations of the second kind, IMA J. Numer. Anal. 5 (1985), 355-369.

14. H. Kaneko, R. Noren and Yuesheng Xu, Regularity of the solution of Hammerstein equations with weakly singular kernels, Int. Eqs. Op. Thy. 13 (1990), 660-670.

15. H. Kaneko, R. Noren and Yuesheng Xu, Numerical solutions for weakly singular Hammerstein equations and their superconvergence, J. Int. Eqs. Appl. 4 (1992), 391-407.

16. H. Kaneko and Yuesheng Xu, Degenerate kernel method for Hammerstein equations, Math. Comp. 56 (1991), 141-148.

17. H. Kaneko and Yuesheng Xu, Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind, Math. Comp. , 62 (1994), 739-753.

18. L. V. Kantorovich, Functional analysis and applied mathematics, Usp. Mat. Mauk. 3 (1948), 89-185. English Trans. N. B. S. report 1509 (1952).

19. R. Kress, A Nyström method for boundary integral equations in domains with corners, Numer. Math. 58 (1990), 145-161.

20. S. Kumar, A discrete collocation-type method for Hammerstein equation, SIAM J. Numer. Anal. 25 (1988), 328-341.

21. S. Kumar, Superconvergence of a collocation-type method for Hammerstein equations, IMA J. Numer. Anal. 7 (1987), 313-325.

22. S. Kumar and I. H. Sloan, A new collocation-type method for Hammerstein equations, Math. Comp. 48 (1987), 585-593.

23. L. J. Lardy, A variation of Nystrom’s method for Hammerstein equations, J. Int. Eqs. 3 (1981), 43-60.

24. J. Rice, On the degree of convergence of nonlinear spline approximation, in Approximation with Special Emphasis on Spline Functions, ed. I. J. Schoenberg, Academic Press, N.Y. (1969), 349-365.

25. C. Scheider, Product integration for weakly singular integral equations, Math. Comp.36 (1981), 207-213.

26. I. H. Sloan, Four variants of the Galerkin methods for integral equations of the second kind, IMA J. Numer. Anal. 4 (1984), 9-17.

27. I. H. Sloan, Improvement by iteration for compact operator equations, Math. Comp. 30 (1976), 758-764.

28. I. H. Sloan, Superconvergence, in Numerical Solution of integral equations, ed. M. A. Golberg, Plenum Press, N.Y., 1990, 35-70.

29. I. H. Sloan and V. Thomee, Superconvergence of the Galerkin iterates for Integral equations of the second kind, J. Int. Eqs. 9 (1985), 1-23. 22 (1981), 431-438.

30. G. Vainikko, Perturbed Galerkin method and general theory of approximate methods for nonlinear equations, Zh. Vychisl. Mat. Fiz. 7 (1967), 723-751. Engl. Translation, U.S.S.R. Comp. Math. and Math. Phys. 7, No. 4 (1967), 1-41.

31. G. Vainikko and P. Uba, A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel, J. Austral. Math. Soc. Series B, 22 (1981), 431-438.

32. G. Vainikko, A. Pedas and P. Uba, Methods of Solving Weakly Singular Integral Equations (in Russian), Tartu University, 1984.

33. J. Wimp, Sequence Transformations and their Applications, Academic Press, New York, 1981.

34. E. Zeidler, Nonlinear Functional Analysis and its Applications II/B, Springer-Verlag, New York, 1990.