Numerical solutions of Hammerstein equations

H. Kaneko & R. D. Noren Department of Mathematics and Statistics Old Dominion University Norfolk, Virginia 23529-0077 USA

Abstract

In this chapter, we survey recent results on the numerical solutions of the Hammerstein equations. Hammerstein equations arise naturally in connection with the Laplace equation with a certain class of nonlinear boundary conditions. The Hammerstein equations with smooth as well as weakly singular kernels will be treated.

1 Introduction

In this chapter, we make a report on some of the recently obtained numerical methods for finding an approximate solution of the Hammerstein equation,

x(t) − ∫_{0}^{1} k(t, s)ψ(s, x(s)) ds = f(t),  t ∈ [0, 1],  (1)

where k,f and ψ are known functions and x is the solution to be determined. In the past two decades, there has been considerable interest in the numerical analysis of solutions of integral equations. A comprehensive survey of numerical methods for the solution of the Fredholm integral equation of the second kind was made by Atkinson [1] in 1976. The survey encompasses most of the standard numerical techniques available to the practitioners and it is a prerequisite for the current article. Since the publication of Atkinson’s survey, numerous new research articles have appeared. In particular, many interesting numerical techniques that deal with the weakly singular Fredholm integral equations have been established, e.g.

[2],[3], . Also a substantial number of results obtained for the Fredholm equations were generalized to nonlinear Hammerstein equations. The onedimensional Hammerstein equation arises naturally in connection with the Laplace equations with nonlinear boundary conditions that are proposed in R^{2}. The purpose of this survey article is to update readers, mathematicians and engineers alike, on these new advances that have occurred in the area of Hammerstein equations. The majority of the materials presented in this article are taken primarily from the recent collaborative work done by Yuesheng Xu and the present authors. We point out that Vainikko [4] recently wrote a monograph on the numerical methods for multidimensional weakly singular Fredholm integral equations of the second kind. The methods described in Vainikko’s monograph are primarily based upon an approximation using irregular grids which correspond to the materials presented in Section 6 of the present work. We feel that many of the results in [5] can be extended to multidimensional weakly singular Hammerstein equations. We also feel that most of the methods described in this article in relation to one dimensional equations can be extended to hold for multidimensional Hammerstein equations.

This article consists of two parts. The first part, which is comprised of Sections 2, 3 and 4, is concerned with Hammerstein equations with kernels that are smooth. We begin this part in Section 2 by discussing the degenerate kernel method. The degenerate kernel method is a classical technique for approximating the solutions of Fredholm equations. This well known method was generalized to Hammerstein equations in [5]. In Section 3, the collocation and Galerkin methods for Hammerstein equations will be presented. The topic of the superconvergence of the iterates of the numerical solutions of these methods is taken up in Section 4. The second part is devoted to the numerical analysis of the weakly singular Hammerstein equations. When dealing with the weakly singular equations, one must be concerned with the nature of the regularity of the solution, since, without this knowledge, no numerical method would be successful. To achieve an optimal convergence rate of numerical solutions, it is imperative that we have a proper understanding of the kind of singularity that a solution possesses. We discuss this issue in Section 5. When solving numerically the weakly singular equations, one is required to evaluate a large number of weakly singular integrals. In the event of weakly singular kernel of convolution type, the technique of product-integration appears to have been a popular choice to approximate such integrals. The work of Piessens and Branders [6] and those of Sloan [7] and of Sloan and Smith [8] should be mentioned in this context. The critical recursion formulae in their product-integration methods, however, do not hold when integrands are altered slightly. Kaneko and Xu [9], on the other hand, established the Gauss-type numerical quadratures that can be applied to a wider variety of weakly singular integrals. A review of these quadrature schemes is given in Section 6. Also in this section, the collocation method for the weakly singular Hammerstein equation is discussed, extending the results of Section 3. The method of the singularity preserving Galerkin method is discussed in Section 7. Part 1

2 The degenerate kernel method

In this section, we are concerned with the problem of finding an approximate solution of Eqn. (1) by the degenerate kernel method. The existence of a unique solution is guaranteed by an application of the Banach contraction principle under the following assumptions:

1. k ∈ C([0, 1] × [0, 1])

2. ψ ∈ C([0, 1] × (−∞, ∞)) and

{∫_{0}^{1} |ψ(s, x(s))|^{2}ds}^{1/2} ≤ A∥x∥_{2},

where x ∈ L_{2}[0, 1] and A is a constant independent of x,

3. ψ satisfies the Lipschitz condition |ψ(t, x_{1}) − ψ(t, x_{2})| ≤ B|x_{1} − x_{2}| where x_{1}, x_{2} ∈ (−∞, ∞) and B is a constant independent of x_{1} and x_{2},

4. k is bounded by |k(t, s)| < C with BC < 1

As described in [5], the degenerate kernel method presented in this section can be applied to Hammerstein equations with multiple solutions. Hence, the conditions above, which guarantee the global uniqueness of a solution, can be relaxed for an application of the method. Suppose that k_{n} is an approximation of the kernel k in Eqn. (1) that has the following form,

k_{n}(t, s) = ∑_{i=1}^{n} B_{i}(t)C_{i}(s),  (2)

where {B_{i}}_{i=1}^{n} is a linearly independent set of functions in C[0, 1] and {C_{i}}_{i=1}^{n} is a set of functions from C[0, 1]. We assume that

{∫_{0}^{1} ∫_{0}^{1} |k_{n}(t, s) − k(t, s)|^{2}dtds}^{1/2} ⟶ 0  as n → ∞.  (3)

In the degenerate kernel method, an approximate solution x_{n} is obtained by solving the following equation,

x_{n}(t) − ∫_{0}^{1} k_{n}(t, s)ψ(s, x_{n}(s)) ds = f(t),  t ∈ [0, 1].  (4)

Replacing k_{n} by the expression of the right hand side in Eqn. (2), we obtain

x_{n}(t) − ∑_{i=1}^{n} B_{i}(t) ∫_{0}^{1} C_{i}(s)ψ(s, x_{n}(s))ds = f(t),  t ∈ [0, 1]. (5)

Let

c_{i} = ∫_{0}^{1} C_{i}(s)ψ(s, x_{n}(s))ds.

Then from Eqn. (5),

x_{n}(t) = f(t) + ∑_{i=1}^{n} c_{i}B_{i}(t),  (6)

where c_{i}’s are constants that can be determined by solving the following set of nonlinear equations.

c_{j} = ∫_{0}^{1} C_{j}(t)ψ(t, f(t) + ∑_{i=1}^{n} c_{i}B_{i}(t))dt,  for j = 1, 2, …, n. (7)

The accuracy of approximation of the degenerate kernel method depends upon the degree of approximation that k_{n} makes for k in Eqn. (3). The following theorem from [5] summarizes this point.

Theorem 2.1 Let k_{n} ∈ C([0, 1] × [0, 1]) satisfy Eqn. (2). Then equation (4) has a unique solution x_{n} ∈ L_{2}[0, 1] for all sufficiently large n. Moreover,

∥x − x_{n}∥_{2} ≤ \frac{A∥x_{n}∥_{2}}{1 − BC} ∥k − k_{n}∥_{2}.

Examples: (A) Consider

x(t) − ∫_{0}^{1} tsx^{2}(s)ds = \frac{3}{4}t, t ∈ [0, 1].

This equation possesses multiple solutions. They are x_{1}(t) = t and x_{2}(t) = 3t. The kernel is already degenerate. Thus we take B_{1}(t) = t and C_{1}(s) = s. The solutions to c^{2} − \frac{5}{2} + \frac{9}{16} = 0 are used in x_{n}(t) = \frac{3}{4}t + ct to obtain the exact solutions.

(B) Consider

x(t) − ∫_{0}^{1} e^{ts}e^{−x^{2}(s)}ds = \sqrt{t} − \frac{e^{t−1} − 1}{t − 1}, t ∈ [0, 1].

The kernel can be approximated by several different methods. For instance,

e^{ts} ∼ 1 + ts + \frac{t^{2}s^{2}}{2!} + ⋯ + \frac{t^{n}s^{n}}{n!}

corresponding to the Taylor approximation, or

e^{ts} ∼ ∑_{i=0}^{n} ∑_{j=0}^{n} e^{t_{i}s_{j}}B_{i}(t)C_{j}(s)

where B_{i} and C_{j} linear splines with respective knots 0 ≤ t_{1} < t_{2} < ⋯ < t_{n} ≤ 1 and 0 ≤ s_{1} < s_{2} < ⋯ < s_{n} ≤ 1. We obtain the following results for each of the two approximation schemes described above. The errors are computed in ∥·∥_{2} norm. Table 1.

Errors

n Taylor  spline
2 7.218e-2 6.203e-2
3 9.847e-3 1.553e-2
4 2.013e-3 3.768e-3

3 The projection methods

In this section, the collocation and Galerkin methods for Hammerstein equations will be described as two special cases of the projection method. First, we discuss the Galerkin method.

3.1 The Galerkin Method

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 → ∞.  (8)

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

P := sup_{n} ∥P_{n}^{G}|_{C[0,1]}∥_{∞} < ∞.  (9)

Then from Eqns. (8) and (9), it follows that for each x ∈ C[0, 1],

∥P_{n}^{G}x − x∥_{∞} → 0, as n → ∞.  (10)

If we put

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

then Eqn. (1) takes the following operator form

x − KΨx = f.  (11)

As indicated in the previous section, Eqn. (11) may admit multiple solutions. Hence it is assumed for the remainder of this paper that we are treating a solution x_{0} of Eqn. (11) 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}^{G}KΨx_{n} = P_{n}^{G}f.  (12)

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

\begin{aligned}∑_{j=1}^{n} b_{nj} < φ_{nj}, φ_{ni} > & − < ∫_{0}^{1} k(·, s)ψ(s, ∑_{j=1}^{n} b_{nj}φ_{nj}(s))ds, φ_{ni} > \\ & =< f, φ_{ni} >, 1 ≤ i ≤ n, \end{aligned}

(13)

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

Now we let

T x ≡ f + KΨx  (14)

and

T_{n}^{G}x_{n} ≡ P_{n}^{G}f + P_{n}^{G}KΨx_{n}  (15)

so that Eqns. (11) and (12) can be written respectively as x = T x and x_{n} = T_{n}^{G}x_{n}. A proof of the following theorem can be made by directly applying Theorem 2 of Vainikko [10]. The paper of Atkinson and Potra [11] is also useful in this context.

Theorem 3.1 Let x_{0} ∈ C[0, 1] be an isolated solution of Eqn. (11). 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 Eqn. (12) 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} ,  (16)

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

E_{n}(x_{0}) ≤ ∥x_{n} − x_{0}∥_{∞} ≤ CE_{n}(x_{0}),  (17)

where C is a constant independent of n and E_{n}(x_{0}) = inf_{u∈S_{n}^{r}} ∥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  (18)

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.  (19)

It is known from de Boor [12] and Douglas, Dupont and Wahlbin [13] that condition (19) implies that the Galerkin projections P_{n} are uniformly bounded. In addition, it is also well known from Demko [14] and De Vore [15] 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}},  (20)

where μ = min{m, r} and h = max_{1≤i≤n}(t_{i} − t_{i−1}). Using Theorem 3.1 and the inequalities (17) and (20), we obtain the following theorem.

Theorem 3.2 Let x_{0} be an isolated solution of Eqn. (11) and let x_{n} be the solution of Eqn. (12) 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 the Galerkin method for Urysohn equations was obtained by Atkinson and Potra [11]. Hence, Theorem 3.2 may be derived by specializing their result to Hammerstein equations.

3.2 The Collocation Method Let Π_{n} denote the partition of [0, 1] defined

in Eqn. (18). For each positive integer r, define a sequence of points {ξ_{i}}_{i=0}^{r} such that 0 ≤ ξ_{0} < ξ_{1} < ⋯ < ξ_{r} ≤ 1. Also

t_{ij} ≡ t_{i} + ξ_{j}(t_{i+1} − t_{i}),  i = 0, 1, …, n − 1; j = 0, 1, … r,  (21)

so that

t_{i} ≤ t_{i0} < t_{i1} < ⋯ < t_{ir} ≤ t_{i+1}, i = 0, 1, …, n − 1.

In the collocation method, the approximate solution is constructed in S_{r}^{ν}(Π_{n}) by the following strategy. For each i = 0, 1, …, n − 1, let l_{ij} denote the Lagrange fundamental polynomial for the knots {t_{ij}}_{j=0}^{r} defined by

l_{ij}(s) = ∏_{l=1,l≠j}^{r} \frac{s − t_{il}}{t_{ij} − t_{il}} , t_{i} ≤ s ≤ t_{i+1}.

We seek the approximate solution x_{n} ∈ S_{r}^{ν} in the form x_{n}(t) = ∑_{j=0}^{r} a_{ij}l_{ij}(t), for t_{i} ≤ t ≤ t_{i+1}, i = 0, 1, …, n − 1 by solving the following set of nonlinear equations for a_{ij},

a_{ij} − ∑_{p=0}^{n−1} ∫_{t_{p}}^{t_{p+1}} k(t_{ij}, s)ψ(s, ∑_{k=0}^{r} a_{pk}l_{pk}(s))ds = f(t_{ij}),  (22)

for i = 0, 1, …, n − 1 and j = 0, 1, …, r. The interpolation projector P_{n}^{C}: C[0, 1] → S_{r}^{ν} is defined by

P_{n}^{C}φ(t) = ∑_{k=0}^{r} φ(t_{ik})l_{ik}(t),  (23)

for φ ∈ C[0, 1] and for t_{i} ≤ t ≤ t_{i+1}, i = 0, 1, …, n − 1. Using the notations introduced above, Eqn. (22) can be described symbolically as

x_{n} − P_{n}^{C}KΨx_{n} = P_{n}^{C}f.  (24)

Under the assumption that h ≡ max_{0≤i≤n−1}(t_{i+1} − t_{i}) → 0 as n → ∞, we have

∥P_{n}^{C}φ − φ∥_{∞} → 0,  as n → ∞.  (25)

The following theorem which proves the existence of the collocation solution is analogous to Theorem 3.1. We list it here for purpose of completeness. Define T_{n}^{C}x_{n} ≡ P_{n}^{C}KΨx_{n} + P_{n}^{C}f.

Theorem 3.3 Let x_{0} ∈ C[0, 1] be an isolated solution of Eqn. (11). 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 collocation approximation Eqn. (24) 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} ,  (3.19)

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

E_{n}(x_{0}) ≤ ∥x_{n} − x_{0}∥_{∞} ≤ CE_{n}(x_{0}),  (3.20)

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

To obtain the convergence and the rate of convergence of the collocation approximation, we argue from Eqn. (26) as follows; Proof:

∥x_{n} − x_{0}∥_{∞} ≤ \frac{α_{n}}{1−q}
= ^{\frac{∥(I−T_{n}^{C′} (x_{0}))^{−1}(T_{n}^{C}(x_{0})−T (x_{0}))∥_{∞}}{1−q}}
≤ ^{\frac{∥(I−T_{n}^{C′} (x_{0}))^{−1}∥_{∞}∥(T_{n}^{C}(x_{0})−T (x_{0}))∥_{∞}}{1−q}}
= C∥P_{n}^{C}KΨ(x_{0}) − KΨ(x_{0}) + P_{n}^{C}f − f∥_{∞}
= C∥P_{n}^{C}(x_{0}) − x_{0}∥_{∞},

where C is a constant independent of n. From this and Eqn. (25) along with (20), if x_{0} ∈ W_{∞}^{r}, then

∥x_{0} − x_{n}∥_{∞} = O(h^{r}).  (3.21)

4 Superconvergence of the iterates

In this section, we review some results concerning the superconvergence of the iterated Galerkin and the iterated collocation methods for Hammerstein equations. The results presented here are taken from the recent papers of Kaneko and Xu [16], and Kaneko, Noren and Padilla [17]. The superconvergence phenomena of the iterates for the Fredholm equations was originally studied by Sloan [18] and it was extended by him and by his collaborators [19-26]. Some of their results are generalized in this section to hold for the Hammerstein equation. The discussion on the iterated Galerkin method is given below. A discussion that is pertinent to the iterated collocation method for Hammerstein equations is quite similar. Therefore only the points that distinguish the iterated collocation method from that of the Galerkin method will be given here.

Throughout this section, in addition to the four assumptions that are described at the beginning of Section 2, we also assume the following two additional conditions; 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} ∈ (−∞, ∞);

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

We assume that x_{n} is the unique solution of Eqn. (12) in the sphere B(x_{0}, δ) for some δ > 0. Define

x_{n}^{I} = f + KΨx_{n}.  (29)

Applying P_{n}^{G} to the both sides of (29), we obtain

P_{n}^{G}x_{n}^{I} = P_{n}^{G}f + P_{n}^{G}KΨx_{n}.  (30)

From Eqns. (30) and (12), we see that

P_{n}^{G}x_{n}^{I} = x_{n}.  (31)

Hence the function x_{n}^{I} satisfies the following new Hammerstein equation

x_{n}^{I} = f + KΨP_{n}^{G}x_{n}^{I}.  (32)

By letting S_{n}^{G} ≡ f + KΨP_{n}^{G}, we may rewrite Eqn. (32) as x_{n}^{I} = S_{n}^{G}x_{n}^{I}. The following two lemmas are instrumental to Theorem 4.3 below which proves the superconvergence of the iterated Galerkin method.

Lemma 4.1 Let x_{0} ∈ C[0, 1] be an isolated solution of Eqn. (11). Assume that 1 is not an eigenvalue of (KΨ)′(x_{0}). Then for sufficiently large n, the operators I −(S_{n}^{G})′(x_{0}) are invertible and there exists a constant L > 0 such that

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

Lemma 4.2 Let x_{0} ∈ C[0, 1] be an isolated solution of Eqn. (11) and x_{n} be the unique solution of Eqn. (12) 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}^{I} defined by the iterated scheme Eqn. (29) is the unique solution of Eqn. (32) in the sphere B(x_{0}, δ), for some δ > 0. Moreover, there exists a constant 0 < q < 1, independent of n, such that

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

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

∥x_{n}^{I} − x_{0}∥_{∞} ≤ CE_{n}(x_{0}).

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}),  (33)

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}^{G}x_{0}(s), P_{n}^{G}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 4.3 Let x_{0} ∈ C[0, 1] be an isolated solution of Eqn. (11) and x_{n} be the unique solution of Eqn. (12) in the sphere B(x_{0}, δ) for some δ > 0. Let x_{n}^{I} be defined by the iterated scheme Eqn. (29). Assume that 1 is not an eigenvalue of (KΨ)′(x_{0}). Then, for all 1 ≤ p ≤ ∞,

\begin{aligned}∥x_{0} − x_{n}^{I}∥_{∞} ≤ & C\{∥x_{0} − P_{n}^{G}x_{0}∥_{∞}^{2} \\ & + sup_{0≤t≤1} inf_{u∈X_{n}} ∥k(t, ·)ψ^{(0,1)}(·, x_{0}(·)) − u∥_{q}∥x_{0} − P_{n}^{G}x_{0}∥_{p}\}, \end{aligned}

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

Proof: From Eqns. (11) and (32) we have

x_{0} − x_{n} = K(Ψx_{0} − ΨP_{n}^{G}x_{n}^{I})
= K(Ψx_{0} − ΨP_{n}^{G}x_{0}) + K(ΨP_{n}^{G}x_{0} − ΨP_{n}^{G}x_{n}^{I})
= K(Ψx_{0} − ΨP_{n}^{G}x_{0})(G_{n}P_{n}^{G}(x_{0} − x_{n}^{I}))(t).  (34)

By using assumption (5) and the fact 0 < θ < 1, we have, for all x ∈ C[0, 1],

\begin{aligned}∥(G_{n}x) − (Gx)∥_{∞} & ≤ sup_{0≤t≤1} ∫_{0}^{1} |k(t, s)|ds∥x∥_{∞}× \\ & (∥P_{n}^{G}x_{0} − x_{0}∥_{∞} + ∥P_{n}^{G}∥_{∞}∥x_{n}^{I} − x_{0}∥_{∞}). \end{aligned}

Consequently, by Eqn. (10) and Lemma 4.2,

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

for some constant M > 0. Hence G_{n} → G in the norm of C[0, 1] as n → ∞. Moreover, for each x ∈ C[0, 1],

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

where

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

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

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

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

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

It can be shown that {G_{n}P_{n}^{G}} is collectively compact. Since G = (KΨ)′(x_{0}) is compact and (I − G)^{−1} exists, it follows from the theory of collectively compact operators [1] that (I − G_{n}P_{n}^{G})^{−1} exists and is uniformly bounded for sufficiently large n. By Eqn. (34), we have the following estimate

sup_{0≤t≤1} |(x_{0} − x_{n}^{I})(t)| ≤ C sup_{0≤t≤1} |K(Ψx_{0} − ΨP_{n}^{G}x_{0})(t)|.  (35)

Next, we estimate the function d(t) ≡ |K(Ψx_{0} − ΨP_{n}^{G}x_{0})(t)|. Using Eqn. (33) with y = P_{n}^{G}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}^{G}x_{0}(s), θ)(x_{0}(s) − P_{n}^{G}x_{0}(s))ds| .

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

Now, by assumption (5), we have

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

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

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

Therefore,

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

This proves the theorem. □

The next theorem is concerned with the case X_{n} = S_{r}^{ν}(Π_{n}) where Π_{n} satisfies Eqn. (19).

Theorem 4.4 Let x_{0} ∈ W_{p}^{l} (0 < l ≤ r) be an isolated solution of Eqn. (11), x_{n} be the unique solution of Eqn. (12) in B(x_{0}, δ), for some δ > 0. Let x_{n}^{I} be defined by the iterated scheme Eqn. (29). Assume that 1 is not an eigenvalue of (KΨ)′(x_{0}). Assume also that for all t ∈ [0, 1], k_{t}(.)ψ^{(0,1)}(., x_{0}(.)) ∈ W_{q}^{m}(0 ≤ m ≤ r). Then

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

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

If, in Eqn. (29), x_{n} denotes the collocation solution, then the corresponding x_{n}^{I} satisfies

x_{n}^{I} = f + KΨP_{n}^{C}x_{n}^{I}.  (36)

The following theorem for the superconvergence of the iterated collocation method is proved in [17].

Theorem 4.5 Let x_{0} ∈ C[0, 1] be an isolated solution of equation Eqn.

(11) and x_{n} be the unique solution of Eqn. (24) in the sphere B(x_{0}, δ_{1}). Let x_{n}^{I} be defined by the iterated scheme Eqn. (29). Assume that 1 is not an eigenvalue of (KΨ)′(x_{0}). Assume that x_{0} ∈ W_{1}^{l} (0 < l ≤ 2r) and g_{t} ∈ W_{1}^{m} (0 < m ≤ r) with ∥g_{t}∥_{W_{1}^{m}} bounded independently of t. Then

∥x_{0} − x_{n}^{I}∥_{∞} = O(h^{γ}), where γ = min{l, r + m}.

Proof: The first part of the proof for this theorem given in [17] follows the same way as that of Theorem 4.3 up to equation Eqn. (35). Thereafter, we invoke to the following four known inequalities. Let ψ_{n} ∈ S_{l}^{0}(Π_{n}) be such that

∑_{i=1}^{n} ∥(x_{0} − ψ_{n})^{(j)}∥_{W_{1}^{m}(I_{i})} ≤ ch^{l−j}∥x_{0}∥_{W_{1}^{l}},  0 ≤ j ≤ l,  (37)
max_{1≤i≤n} ∥ψ_{n}^{(j)}∥_{W_{∞}^{m}(I_{i})} ≤ c∥x_{0}∥_{W_{1}^{l}},  j ≥ 0.  (38)

Also for each t ∈ [0, 1], there exists φ_{n,t} ∈ S_{m}^{0}(Π_{n}) such that

∑_{i=1}^{n} ∥(g_{t} − φ_{n,t})^{(j)}∥_{W_{1}^{m}(I_{i})} ≤ ch^{m−j}K_{m},  0 ≤ j ≤ m,  (39)

max_{1≤i≤n} ∥φ_{n,t}^{(j)}∥_{W_{∞}^{m}(I_{i})} ≤ cK_{m},  j ≥ 0,  (40)

where K_{m} = sup_{0≤t≤1} ∥k_{t}∥_{W_{1}^{m}} < ∞. Now for t ∈ [0, 1] we have

\begin{aligned}K(Ψx_{0} − ΨP_{n}x_{0})(t) & = (g_{t} − φ_{n,t}, x_{0} − P_{n}x_{0}) + (φ_{n,t}, (I − P_{n})(x_{0} − ψ_{n})) \\ & +(φ_{n,t}, (I − P_{n})ψ_{n}). \end{aligned}

(41)

Using Eqns. (37)-(40) along with the arguments from [19] (p.362) we can show that each of the three terms is bounded by ch^{γ} uniformly in t. This completes the proof. □

Examples supporting the results given in Theorems 4.3 and 4.5 can be found in [16],[17].

PART II - Weakly Singular Hammerstein Equations

5 Regularities of the solutions

In this section we are concerned with the regularity properties of the solution to Eqn. (1) when the kernel k(t, s) is weakly singular. In particular we assume that

k(s, t) = g_{α}(|s − t|)m(s, t),  (5.1)

where m ∈ C([0, 1] × [0, 1]). (further hypotheses on m appear later), and

g_{α}(|s − t|) = \{ \begin{matrix}|s − t|^{α−1}, & 0 < α < 1, \\ log |s − t|, & α = 1. \end{matrix}  (5.2)

We further assume that

ψ ∈ C([0, 1] × (−∞, ∞))  (5.3)
|ψ(t, y_{1}) − ψ(t, y_{2})| ≤ A|y_{1} − y_{2}|.  (5.4)

There is a large literature, e.g. [41][42], on existence of solution of Hammerstein equations. The following theorem is typical of such results.

Theorem 5.1 Let f ∈ C[0, 1] and assume

A · sup_{0≤α≤1} ∫_{0}^{1} |g_{α}(s − t)m(s, t)|dt < 1.

Then the Hammerstein equation with weakly singular kernel has a unique solution in C[0, 1].

Define for 0 < α ≤ 1 and nonnegative integer m the set C^{(m,α)}[0, 1] of all functions x ∈ C^{m}[0, 1] such that there exists constants A > 0 and B > 1 with

|x^{(m)}(s) − x^{(m)}(t)| ≤ A · \{ \begin{matrix}|s − t|^{α} & , 0 < α < 1, \\ |s − t| log(\frac{B}{|s−t|}) & , α = 1. \end{matrix}

Then x is called α-Hölder continuous. A nonstandard way of defining Hölder continuity for α = 1 should be noted here.

The proof of Theorem 5.3 below uses the following lemma in [14].

Lemma 5.2 Let m ∈ C^{1}([0, 1] × [0, 1]). Then (i) If x ∈ C[0, 1], Kx ∈ C^{(0,α)}[0, 1]; (ii) if x ∈ C^{(0,μ)}, 0 ≤ μ ≤ 1−α < 1, |m_{α}(s)−m_{α}(t)| ≤ |s−t|^{α+μ}, s, t ∈

[0, 1] where m_{α} ≡ ∫_{0}^{1} g_{α}(|s − t|)m(s, t)dt, then Kx ∈ C^{(0,α+μ)}[0, 1];

(iii) if x ∈ C^{(0,μ)}[0, 1],0 ≤ 1−α < μ ≤ 1, lim_{r→s} x(s)\frac{(m_{α}(s)−m_{α}(r))}{s−r},r ∈ [0, 1], exists for all s ∈ [0, 1], and continuous in s, then Kx ∈ C^{1}[0, 1] and \frac{d}{ds}Kx(s) = ∫_{0}^{1} \frac{∂}{∂s}\{g_{α}(|s − t|)m(s, t)\}[x(t) − x(s)]dt + x(s)\frac{d}{ds}m_{α}(s).

The next theorem generalizes the result in [26].

Theorem 5.3 Let n be a nonnegative integer, m ∈ C^{n+1}([0, 1] × [0, 1]), f ∈ C^{(0,α)}[0, 1] ∩ C^{n}(0, 1) and f_{i}(s) ≡ s^{i}(1 − s)^{i}f^{(i)}(s),i = 1, …, n be α-Hölder continuous on [0, 1]. For n = 0, 1, we assume that ψ ∈ C^{(0,1)}([0, 1]× (−∞, ∞)) and for n ≥ 2, we assume that ψ ∈ C^{(n−1)}([0, 1]×(−∞, ∞)). If x is any solution of (1.4), then it belongs to C^{n}(0, 1) ∩ C^{(0,α)}[0, 1]. Moreover, x_{i}(s) ≡ s^{i}(1 − s)^{i}ψ^{(i)}(s) belongs to C^{(0,∞)}[0, 1] for i = 1, 2, … n.

The proof requires the following lemma.

Lemma 5.4 (i) (s − t)\frac{∂}{∂s}g_{α}(|s − t|) = \{ \begin{matrix}(α − 1)g_{α}(|s − t|), & 0 < α < 1, \\ 1, & α = 1; \end{matrix}

(ii) \frac{∂}{∂s} ∫_{0}^{t} g_{α}(|s − y|)dy = g_{α}(s) − g_{α}(|s − t|);

(iii)

\frac{d}{ds}m_{α}(s) = m(s, 0)g_{α}(s) − m(s, 1)g_{α}(1 − s) + ∫_{0}^{1} \frac{∂k(s, t)}{∂s}g_{α}(|s − t|)dt

+ ∫_{0}^{1} \frac{∂m(s, t)}{∂t}g_{α}(|s − t|)dt.

Now we are ready to present a proof of Theorem 5.3. Proof: For n = 0, because of Lemma 5.2 (i), the result is true. For n = 1, multiplication by h(t) ≡ t(1 − t) gives

h(t)x(t) − h(t)KΨ(x)(t) = h(t)f(t),  0 ≤ t ≤ 1.  (5.5)

By assumption hf ∈ C^{(1,α)}[0, 1]. Note that h(t)ψ(x)(t) = KhΨ(x)(t) + ∫_{0}^{1} g_{α}(|s − t|)m(t, s)ψ(s, x(s))(t − s)(1 − t − s)ds. If we define this last term as K̂Ψ(x)(t), then (5.5) becomes

h(t)x(t) − KhΨ(x)(t) = h(t)f(t) + K̂Ψ(x)(t),  0 ≤ t ≤ 1.  (5.6)

The multiplication by (t−s) in the kernel K̂Ψ makes K̂Ψ(x) smoother than KΨ(x). For more information on this point see [24]. Hence hf + K̂Ψ(x) ∈ C^{1}[0, 1]. It follows from Lemma 5.2 that KhΨ(x) ∈ C^{(0,α)}[0, 1] and by (5.6) that hx ∈ C^{(0,α)}. In the remainder of the section, M denotes a constant whose value may change each time it appears. Now for t_{1}, t_{2} ∈ [0, 1],

|h(t_{1})ψ(t_{1}, x(t_{1})) − h(t_{2})ψ(t_{2}, x(t_{2}))|

≤ |h(t_{1})||ψ(t_{1}, x(t_{1})) − ψ(t_{2}, x(t_{2}))| + |h(t_{1})||ψ(t_{1}, x(t_{2})) − ψ(t_{2}, x(t_{2}))|

+|h(t_{1}) − h(t_{2})||ψ(t_{2}, x(t_{2}))|

≤ M|h(t_{1})(x(t_{1}) − x(t_{2}))| + M|h(t_{2})||t_{1} − t_{2}|

+|ψ(t_{2}, x(t_{2}))||t_{1} − t_{2}|

≤ M|h(t_{1})x(t_{1}) − h(t_{2})x(t_{2})| + M|x(t_{2})||h(t_{2}) − h(t_{1})|

+M|t_{1} − t_{2}|.
(5.7)

Since hx ∈ C^{(0,α)}[0, 1], we get hψ(x) ∈ C^{(0,α)}[0, 1]. If α > \frac{1}{2}, let μ = α. Then 0 ≤ 1 − α < μ ≤ 1 and hψ(x) ∈ C^{(0,α)}[0, 1]. If 0 < α ≤ \frac{1}{2}, then Lemma 5.4 (ii) yields hψ(x)\frac{dm_{α}(s)}{ds} ∈ C[0, 1], and then Lemma 5.2 (ii) implies KhΨ(x) ∈ C^{(0,2α)}[0, 1]. By (5.6), hx ∈ C^{(0,2α)}[0, 1]. The argument may be repeated to ensure the existence of μ with 0 ≤ 1 − α < μ ≤ 1 such that hΨ(x) ∈ C^{(0,μ)}[0, 1]. By Lemma 5.2 (iii), it follows that KhΨ(x) ∈ C^{1}[0, 1]. By application of (5.6), we get hx ∈ C^{1}[0, 1]. Finally, x ∈ C^{1}(0, 1) ∩ C[0, 1]. To complete the proof, integrate by parts in (1.1) to obtain

\begin{matrix}x(t) + ∫_{0}^{1} G_{α}(|t − s|)m(t, s) [\frac{∂ψ(s,x(s))}{∂s} + \frac{∂ψ(s,x(s))}{dx} \frac{dx(s)}{ds}] ds \\ + ∫_{0}^{1} G_{α}(|t − s|)\frac{∂m(t,s)}{∂s}ψ(s, x(s))ds = f(t) + G_{α}(1 − t)m(t, 1)ψ(1, x(1)) \end{matrix}

(5.8) where G_{α}(|s − t|) = ∫_{0}^{t} g_{α}(|s − y|)dy. Now the differentiation of equation (5.8) yields

\frac{dx(t)}{dt} − ∫_{0}^{1} g_{α}(|t − s|)m(t, s)\frac{∂ψ(s, x(s))}{∂x} \frac{dx(s)}{ds}ds = F (t)  (5.9)

where F (t) consists of 8 terms and it is easy to see, term by term, that h(t)F (t) ∈ C^{(0,α)}[0, 1]. Multiplying (5.9) by h and letting x_{1} = h(t)\frac{dx(t)}{dt}, m_{1}(t, s) = m(t, s)\frac{∂ψ(s,x(s))}{∂x} and

F_{1}Z(t) = h(t)F (t) + ∫_{0}^{1} g_{α}(|t − s|)m_{1}(t, s)(t − s)(1 − t − s)\frac{dx(s)}{ds}

we find that x_{1} satisfies the linear equation

x_{1}(t) − ∫_{0}^{1} g_{α}(|t − s|)m_{1}(t, s)x_{1}(s)ds = F_{1}(t),  0 ≤ t ≤ 1. (5.10)

It is easy to see that F_{1} ∈ C^{(0.α)}[0, 1]. By Lemma 5.2 (i), x_{1} ∈ C^{(0,α)}[0, 1]. Let x̂_{1}(t) = h(t)\frac{dx_{1}(t)}{dt}. A similar analysis to the one above show that x_{1} ∈ C^{(0,α)}[0, 1] and x_{1} ∈ C^{1}(0, 1). Noting that x̂_{1}(t) = h^{2}(t)\frac{d^{2}x(t)}{dt^{2}} +(1−2t)\frac{dx(t)}{dt}, we deduce that h^{2}(t)\frac{d^{2}x(t)}{dt^{2}} ∈ C^{(0,α)}[0, 1]. Moreover \frac{dx_{1}(t)}{dt} = h(t)\frac{dx^{2}(t)}{dt^{2}} +(1− 2t)\frac{dx(t)}{dt}. Hence \frac{d^{2}x}{dt^{2}} ∈ C(0, 1). This procedure can be repeated to prove that x ∈ C^{n}(0, 1). This completes the proof. □

In the remainder of this section, we consider the Hammerstein equation with logarithmic singularity because of its important applicability to a class of boundary value problems and its application to the singularity preserving Galerkin scheme that will be discussed in Section 7. We consider

x(t) − ∫_{0}^{1} log |t − s|m(t, s)ψ(s, x(s))ds = f(t),  0 ≤ t ≤ 1 (5.11)

(see (1.1) also). With

KΨx(t) ≡ ∫_{0}^{1} log |t − s|m(t, s)ψ(s, y(s))ds.  (5.12)

Then equation (5.11) can be written in operator form as

x − KΨx = f.  (5.13)

We let W = W_{n} be the linear space spanned by the functions t^{i} log^{j} t, (1 − t)^{i} log^{j}(1−t); i, j = 1, 2, ..., n−1. Throughout the remainder of this section, we assume the following conditions:

m ∈ C^{2n}([0, 1] × [0, 1]), n ≥ 1, m ∈ C^{1}([0, 1] × [0, 1]), n = 0.  (5.14)

ψ ∈ C^{2n+1}(R × R)
f ∈ W ⊕ W_{2}^{n}.  (5.15)

We define

Ky(t) ≡ ∫_{0}^{1} log |t − s|m(t, s)y(s)ds.  (5.16)

Also let u_{1}(t) = t^{p} log^{q} t, and u_{2}(t) = (1 − t)^{p} log^{q}(1 − t), where p, q ≥ 1 are integers. First we quote the following result (lemma 4.4(2)) from [5].

Lemma 5.5 Let f ∈ W_{2}^{n−1} and assume m ∈ C^{n+1}([0, 1] × [0, 1]). Then,

(Kf)(t) = ∑_{j=1}^{n−1} [c_{j}t^{j} log t + d_{j}(1 − t)^{j} log(1 − t)] + v_{n}(t),

(Ku_{1})(t) = ∑_{j=p+1}^{n−1} ∑_{i=1}^{q+1} c_{ij}t^{j}(log t)^{i} + ∑_{j=q+1}^{n−1} d_{j}(1 − t)^{j} log(1 − t) + v_{n}(t),

and

(Ku_{2})(t) = ∑_{j=p+1}^{n−1} ∑_{i=1}^{q+1} c_{ij}(1 − t)^{j}(log(1 − t))^{i} + ∑_{j=q+1}^{n−1} d_{j}t^{j} log t + v_{n}(t).

We also need the following lemmas from [20].

Lemma 5.6 If u_{1}(t) = t^{p} log^{q} t, u_{2}(t) = (1−t)^{r} log^{u}(1−t), where p, q, r, u ≥ 1 are integers, then u_{1}u_{2} ∈ W ⊕ W_{2}^{n}.

Lemma 5.7 A product of an W_{2}^{n} function with a function in W is in W_{2}^{n}⊕ W.

Lemma 5.8 The operator KΨ maps W ⊕ W_{2}^{n} into W ⊕ W_{2}^{n+1}.

The next theorem characterizes the solution of (5.10).

Theorem 5.9 Suppose the conditions (5.4)-(5.6) hold and x is an isolated solution of (5.1). Then there are constants a_{ij} and b_{ij}, for i, j = 1, 2, ..., n− 1, and there is a function v_{n} in W_{2}^{n} such that

x(t) = ∑_{i=1}^{n−1} ∑_{j=1}^{n−1}[a_{ij}t^{i} log^{j} t + b_{ij}(1 − t)^{i} log^{j}(1 − t)] + v_{n}(t).  (5.17)

Proof: For n = 0, this follows from Lemma 5.8 with n = 0. Assume that the result holds for n = k, that is, if f ∈ W_{2}^{k} ⊕ W, then (5.13) holds with n = k. Say x = w_{k} + v_{k}, where v_{k} ∈ W_{2}^{k}, w_{k} = ∑_{i=1}^{k−1} ∑_{j=1}^{k−1}[a_{ij}t^{i} log^{j} t + b_{ij}(1 − t)^{i} log^{j}(1 − t)].

Now consider the case n = k + 1 and suppose f ∈ W_{2}^{k+1} ⊕ W, here of course W = W_{k+1}.

Since x = w_{k} + v_{k} we write x = KΨx + f = KΨ(w_{k} + v_{k}) + f. From Lemma 5.3, KΨ(w_{k} + v_{k}) ∈ W ⊕ W_{2}^{k+1}. The proof is complete. □

6 Gauss-Type Quadarture Schemes

In this section, the Gauss-type quadrature schemes recently developed by Kaneko and Xu [17] are reviewed. The utility of the quadrature schemes becomes apparent in light of the analysis of the previous section concerning the weakly singular Hammerstein equations. Let S be a subset of [0, 1] containing a finite number of points. Define a function ω_{S} by

ω_{S}(x) = inf{|x − t|: t ∈ S}.  (6.1)

For α > −1 and a nonnegative integer k, we say that f belongs to T ype(α, k, S) if

|f^{(k)}(x) ≤ C[ω_{S}(x)]^{(α−k)},  x ∉ S, f ∈ C^{k}([0, 1] \ S).  (6.2)

For α > 0, this class of functions was introduced by Rice [43]. For example, f(x) = x^{α} and f(x) = sin(x^{α}) for α > −1 belong to T ype(α, k, {0}) for each nonnegative integer k; f(x) = log x belongs to T ype(0, k, {0}) whereas f(x) = x log x belongs to T ype(1, k, {0}) for each nonnegative integer k. Now we consider the following integral and the Gauss-type quadrature schemes to approximate it;

I(f) ≡ ∫_{0}^{1} ρ(x)f(x) dx,  (6.3)

where f ∈ T ype(α, 2k, S) with α > −1 and ρ is some weight function.

In particular, for our first case, we take f ∈ T ype(α, 2k, {0}) ∩ C[0, 1] with α > 0 and ρ ∈ L_{1}[0, 1]. We let q = \frac{2k}{α} and for a positive integer n, let π_{α} be a partition of [0, 1] given by

\begin{aligned}t_{0} & = 0 \\ t_{j} & = (j/n)^{q},  j = 1, 2, …, n. \end{aligned}  (6.4)

On each [t_{i}, t_{i+1}], 1 ≤ i ≤ n − 1, let

t_{i} ≤ u_{1}^{(i)} < u_{2}^{(i)} < ⋯ < u_{k}^{(i)} ≤ t_{i+1}  (6.5)

be the k zeros of the orthogonal polynomial of degree k with respect to the weight function ρ transformed into [t_{i}, t_{i+1}]. The function f is now approximated by the following piecewise polynomial S_{k} with knots defined by (6.4); S_{k}(x) is the linear interpolant of f at t_{0} and t_{1} for x ∈ [t_{0}, t_{1}] and S_{k}(x) is the Lagrange interpolant of degree k − 1 to f at {u_{j}^{(i)}}_{j=1}^{k} for x ∈ [t_{i}, t_{i+1}] and for i = 1, 2, …, n−1. I(f) is then approximated by I(S_{k}).

Theorem 6.1 Let f ∈ T ype(α, 2k, {0})∩C[0, 1] with α > 0 and ρ ∈ L_{1}[0, 1] with ρ > 0 almost everywhere in [0, 1]. Then

|I(f) − I(S_{k})| = O(n^{−2k+1}).

Proof: Define E_{k,i}(f) = ∫_{t_{i}}^{t_{i+1}} ρ(x)[f(x) − S_{k}(x)] dx for i = 0, 1, …, n − 1. First,

|E_{k,0}(f)| ≤ ∫_{t_{0}}^{t_{1}} ρ(x)|f(x) − S_{k}(x)| dx
≤ ∫_{t_{0}}^{t_{1}} ρ(x)|f(x) − f(0)|dx + ∫_{t_{0}}^{t_{1}} ρ(x)|S_{k}(0) − S_{k}(x)|dx
≤ C ∫_{t_{0}}^{t_{1}} ρ(x)x^{α}dx + |S_{k}(0) − S_{k}(x)| ∫_{t_{0}}^{t_{1}} ρ(x)dx
≤ Ct_{1}^{α} + |f(0) − f(t_{1})| ∫_{t_{0}}^{t_{1}} ρ(x)dx
≤ Ct_{1}^{α} = Cn^{−2k},

where C denotes a generic constant that is independent of n. For i = 1, 2, …, n − 1, since f ∈ T ype(α, 2k, {0}), f ∈ C^{2k}[t_{i}, t_{i+1}]. Using the wellknown error formula for the Gaussian quadrature, there exists η_{i} ∈ [t_{i}, t_{i+1}] such that

|E_{k,i}(f)| = \frac{|f^{(2k)}(η_{i})|}{(2k)!} |∫_{t_{i}}^{t_{i+1}} ρ(x)(x − u_{1}^{(i)})^{2} ⋯ (x − u_{k}^{(i)})^{2}dx| . (6.6)

Assuming without loss of generality that α ≤ 2k, we obtain from (6.6)

|E_{k,i}(f)| ≤ C|η_{i}|^{α−2k}(t_{i+1} − t_{i})^{2k} ≤ Ct_{i}^{α−2k}(t_{i+1} − t_{i})^{2k}
= C(i^{q}t_{1})^{α−2k}[(i + 1)^{q} − i^{q}]^{2k}t_{1}^{2k}
≤ Cq^{2k}[(i + 1)^{2k(q−1)}i^{q(α−2k)}]t_{1}^{2k}
≤ Cq^{2k}2^{2k(q−1)}t_{1}^{α}
= Cn^{−2k}.

The second to the last inequality is obtained by noting (i + 1)^{q} − i^{q} ≤ q(i + 1)^{q−1}, whereas the last inequality is obtained by noting q(α − 2k) = \frac{2k}{α} (α − 2k) = 2k(1 − q) and

(i + 1)^{2k(q−1)}i^{q(α−2k)} = (\frac{i + 1}{i})^{2k(q−1)} ≤ 2^{2k(q−1)}.

Thus

|I(f) − I(S_{k})| = |∑_{i=0}^{n−1} E_{k,i}(f)| ≤ Cn^{−2k+1}.

This completes the proof.□

As the second special case, we consider f ∈ T ype(α, 2k, {0}) with α > −1 and ρ ∈ L_{∞}[0, 1]. For this case, we construct a piecewise polynomial S_{k} of degree k − 1 that approximates f by the following rule; S_{k}(x) = 0 for x ∈ [t_{0}, t_{1}) and S_{k}(x) is the Lagrange interpolant of degree k − 1 to f at {u_{j}^{(i)}}_{j=1}^{k} for x ∈ [t_{i}, t_{i+1}), i = 1, 2, …, n − 2 and for x ∈ [t_{n−1}, t_{n}] when i = n − 1. The following theorem can be proved in a similar way to the previous theorem.

Theorem 6.2 Let f ∈ T ype(α, 2k, {0}) with α > −1 and ρ ∈ L_{∞}[0, 1] be positive almost everywhere in [0, 1]. Then

|I(f) − I(S_{k})| = O(n^{−2k}).

We note that the case for ρ(x) ≡ 1 on [0, 1] is discussed in detail in [16] under the title of Gauss-Legendre-Type quadrature. It is also demonstrated in [16] that the quadrature is useful in approximating the integrals that must be evaluated in application of a numerical scheme to solve the weakly singular Fredholm equations of the second kind. An application of this quadrature to the weakly singular Hammerstein equation is now discussed. The collocation method for the weakly singular Hammerstein equation is mentioned here. We refer the reader to [17] for a discussion of the Galerkin method for the weakly singular equation.

Due to the type of its singularities at the end points of the solution of the weakly singular Hammerstein equation that were described in Theorem 5.3, one can not expect the collocation method (or the Galerkin method) to produce a numerical solution with optimal convergence rate unless this special regularity property of the solution is taken into account. What Theorem 5.3 reveals in the terminology of the newly defined class of functions T ype(α, k, S) is that if x is the solution of equation (1.1) with the kernel k defined by (5.1) and if the forcing function f ∈ T ype(β, k, {0, 1}), then x ∈ T ype(γ, k, {0, 1}), where γ = min{α, β}. Hence, to approximate x, we use a theorem of Rice from [43]. Namely, when S_{r}^{ν}(Π_{n}) is used as an approximating space, the optimal convergence rate of the collocation solution can be recovered by selecting the partition points by

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

where q ≡ \frac{r}{γ} is called the index of singularity. Between t_{i} and t_{i+1} for each i, the collocation points t_{ij} are selected according to (3.14). For convenience, we recall here the collocation equation (3.15).

a_{ij} − ∑_{p=0}^{n−1} ∫_{t_{p}}^{t_{p+1}} k(t_{ij}, s)ψ(s, ∑_{k=0}^{r} a_{pk}l_{pk}(s))ds = f(t_{ij}),

For p = i, j = 0, 1, …, r, we need to compute the integrals that contain in their integrands the functions that belong to T ype(α, k, {t_{p}, t_{p+1}}). The quadrature schemes described in this section now become useful for evaluating these integrals. We note that the examples given in [17], [18] and [19] were obtained by making use of the quadratures described in this section.

7 Singularity Preserving Galerkin Method

In this section, we discuss the method of singularity preserving Galerkin method for equation (1.1) with logarithmic kernel. As in Section 3, define the partition of [0, 1] as

Π_{m} : 0 = t_{0} < t_{1} < ... < t_{m} = 1.

Here we assume no condition on the partition points except that with

h = max_{1≤i≤m}(t_{i} − t_{i−1}),

we assume h → 0 as m → ∞. For convenience of notation, in this section, we denote the space S_{r}^{ν} of splines of order r with continuity ν by S_{h}^{r,ν}. Let d denote the dimension of S_{h}^{r,ν} and suppose that {B_{i}}_{i=1}^{d} is the B-splines basis for S_{h}^{r,ν}. We let

V_{h}^{r,ν} ≡ W ⊕ S_{h}^{r,ν}  (7.1)

where W is defined in Section 5. We denote the orthogonal projection of L_{2}[0, 1] onto V_{h}^{r,ν} by P_{h}^{G}. The singularity preserving Galerkin method for approximating the solution of equation (1.1) requires the solution y_{h} ∈ V_{h}^{r,ν} to satisfy the following equation:

y_{h} − P_{h}^{G}KΨy_{h} = P_{h}^{G}f.  (7.2)

More specifically, we need to find y_{h} in the form

y_{h}(s) = ∑_{i,j=1}^{n−1} α_{ij}s^{i} log^{j} s + ∑_{i,j=1}^{n−1} β_{ij}(1 − s)^{i} log^{j}(1 − s) + ∑_{i=1}^{d} γ_{i}B_{i}(s) (7.3)

where {α_{ij}, β_{ij}}_{i,j=1}^{n−1} and {γ_{i}}_{i=1}^{d} are found by solving the following system of nonlinear equations:

∑_{i,j=1}^{n−1} α_{ij}(s^{i} log^{j} s, s^{p} log^{q} s) + ∑_{i,j=1}^{n−1} β_{ij}((1 − s)^{i} log^{j}(1 − s), s^{p} log^{q} s)
+ ∑_{i=1}^{d} γ_{i}(B_{i}, s^{p} log^{q} s) − (KΨ(∑_{i,j=1}^{n−1} α_{ij}s^{i} log^{j} s
+ ∑_{i,j=1}^{n−1} β_{ij}(1 − s)^{i} log^{j}(1 − s) + ∑_{i=1}^{d} γ_{i}B_{i}), s^{p} log^{q} s)
= (f, s^{p} log^{q} s) p, q = 1, 2, …, n − 1
∑_{i,j=1}^{n−1} α_{ij}(s^{i} log^{j} s, (1 − s)^{p} log^{q}(1 − s)) + ∑_{i,j=1}^{n−1} β_{ij}((1 − s)^{i} log^{j}(1 − s),
(1 − s)^{p} log^{q}(1 − s)) + ∑_{i=1}^{d} γ_{i}(B_{i}, (1 − s)^{p} log^{q}(1 − s))
−(KΨ(∑_{i,j=1}^{n−1} α_{ij}s^{i} log^{j} s + ∑_{i,j=1}^{n−1} β_{ij}(1 − s)^{i} log^{j}(1 − s)
+ ∑_{i=1}^{d} γ_{i}B_{i}), (1 − s)^{p} log^{q}(1 − s))
= (f, (1 − s)^{p} log^{q}(1 − s)) p, q = 1, 2, …, n − 1
∑_{i,j=1}^{n−1} α_{ij}(s^{i} log^{j} s, B_{p}) + ∑_{i,j=1}^{n−1} β_{ij}((1 − s)^{i} log^{j}(1 − s), B_{p})
+ ∑_{i=1}^{d} γ_{i}(B_{i}, B_{p}) − (KΨ(∑_{i,j=1}^{n−1} α_{ij}s^{i} log^{j} s
+ ∑_{i,j=1}^{n−1} b_{i,j}(1 − s)^{i} log^{j}(1 − s) + ∑_{i=1}^{d} γ_{i}B_{i}), B_{p})
= (f, B_{p}) p = 1, 2, …, d

where (·, ·) denotes the usual inner product defined on L_{2}[0, 1]. Now let P_{h} be the orthogonal projection of L_{2}[0, 1] onto S_{h}^{r,ν}. Then we have

P_{h}v → v  for all v ∈ L_{2}[0, 1].  (7.4)

It is well known (e.g. [11]) that if g ∈ W_{2}^{n}, n ≥ 0, then for each h > 0, there exists ϕ_{h} ∈ S_{h}^{r,ν} such that

∥g − ϕ_{h}∥_{2} ≤ Ch^{min{n,r}}∥g∥_{W_{2}^{n}},  (7.5)

where C > 0 is a constant independent of h. By virtue of the fact that P_{h}u is the best L_{2} approximation of u from S_{h}^{r,n}, we see immediately that

∥P_{h}u − u∥_{2} ≤ ∥u − ϕ_{h}∥_{2} ≤ Ch^{min{r,n}}∥u∥_{W_{2}^{n}}, for all u ∈ W_{2}^{n}. (7.6)

The following lemma from [5] is useful in the sequel.

Lemma 7.1 Let X be a Banach space. Suppose that U_{1} and U_{2} are two subspaces of X with U_{1} ⊆ U_{2}. Assume that P_{1} : X → U_{1} and P_{2} : X → U_{2} are linear operators. If P_{2} is a projection, then

∥x − P_{2}x∥_{X} ≤ (1 + ∥P_{2}∥_{X})∥x − P_{1}x∥_{X} for all x ∈ X.

For convenience, we introduce operators T̂ and T_{h} by letting

T̂y ≡ f + KΨy  (7.7)

and

T_{h}y_{h} ≡ P_{h}^{G}f + P_{h}^{G}KΨy_{h}  (7.8)

so that equations (1.1) and (7.2) can be written respectively as y = T̂y and y_{h} = T_{h}y_{h}. The following theorem guarantees the existence of a solution of the singularity preserving Galerkin method (7.2) and describes the accuracy of its approximation.

Theorem 7.2 Let y ∈ L_{2}[0, 1] be an isolated solution of equation (1.1). Assume that 1 is not an eigenvalue of the linear operator (KΨ)′(y), where (KΨ)′(y) denotes the Fréchet derivative of KΨ at y. Then the singularity preserving Galerkin approximation equation (7.2) has a unique solution y_{h} such that ∥y − y_{h}∥_{2} < δ for some δ > 0 and for all 0 < h < h_{0} for some h_{0} > 0. Moreover, there exists a constant 0 < q < 1, independent of h, such that

\frac{α_{h}}{1 + q} ≤ ∥y − y_{h}∥_{2} ≤ \frac{α_{h}}{1 − q} ,  (7.9)

where α_{h} ≡ ∥(I − T′_{h}(y))^{−1}(T_{h}(y) − T̂(y))∥_{2}. Finally, if y = w + v with w ∈ W and v ∈ W_{2}^{n}, then

∥y − y_{h}∥_{2} ≤ Ch^{min{r,n}}∥v∥_{W_{2}^{n}},  0 < h < h_{0},  (7.10)

where C > 0 is a constant independent of h.

Proof: The existence of a unique solution y_{h} of equation (7.2) in the disk of radius δ about y and the inequalities in (7.7) can be proved using Theorem 2 of Vainikko [40]. To get (7.10), first we note from Lemma 7.1, for v ∈ L_{2}[0, 1],

∥P_{h}^{G}v − v∥_{2} ≤ (1 + ∥P_{h}^{G}∥_{2})∥P_{h}v − v∥_{2}.  (7.11)

Now, from (7.9),

\begin{aligned}∥y − y_{h}∥_{2} & ≤ \frac{α_{h}}{1−q} \\ & = \frac{1}{1−q} ∥(I − T′_{h}(y))^{−1}(T_{h}(y) − T̂(y))∥_{2} \\ & ≤ C∥P_{h}^{G}KΨy − KΨy + P_{h}^{G}f − f∥_{2} \\ & = C∥P_{h}^{G}y − y∥_{2}. \end{aligned}  (7.12)

where C is independent of h. Using the uniform boundedness of {P_{h}^{G}}, (7.11) and (7.12), we obtain

∥y − y_{h}∥_{2} ≤ Ch^{min{r,n}}∥v∥_{W_{2}^{n}}.

□

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