Newton-Kantorovich and Smale Uniform Type Convergence Theorem for a Deformed Newton Method in Banach Spaces

Lemma 1. If η ≤ β, then the function g(t) has two positive real roots r₁, r₂ (0 < r₁ ≤ α ≤ r₂ < R).

Proof. Because g(0) = η > 0,  g(R⁺) > 0, and g^′′(t) = L(t) > 0, we know that g(t) is the convex function for t ∈ (0, R). Hence, α is a unique positive root of $$. So, the necessary and sufficient condition that g(t) has two positive roots for t ∈ (0, R) is that the minimum of g(t) satisfies the condition g(α) ≤ 0, which holds for η ≤ β. This completes the proof of Lemma 1.

Lemma 2. Suppose the sequences {t_n}, {s_n} are generated by (6). Then, for η ≤ β, the sequences {t_n}, {s_n} are increasing and converge to the minimum positive root of g(t), and

Proof. Denote

On [0, r₁), we know g(t) > 0, g^′(t) < 0, g^′′(t) > 0, and g^′′(t) is increasing. Denoting y = (x + U(x))/2 = x − g(x)/2g^′(x), then

Therefore, U(x), V(x) are increasing on [0, r₁]. Thus, for x ∈ [0, r₁), U(x) < U(r₁) = r₁, V(x) < V(r₁) = r₁. Moreover,

hence we can easily prove Lemma 2 by the induction.

Suppose X and Y are the Banach spaces, Ω ⊂ X is an open convex subset, F : Ω ⊂ X → Y has the second-order Fréchet derivative, F^′(x₀) ⁻¹ exists for x₀ ∈ Ω, and the following conditions hold:

where ρ(x) = ∥x − x₀∥ and $$.

Lemma 3. Suppose F satisfies (11) and ∥x − x₀∥ < α. Then F^′(x) ⁻¹ exists, and

Proof. Firstly, by the conditions (11), we know that

Secondly, we know g^′(t) < 0 for t < α. Hence

By Banach Theorem, we know F^′(x) ⁻¹ exists, and

This completes the proof of Lemma 3.

Lemma 4. Suppose X and Y are Banach spaces, Ω is an open convex of the Banach space X, F : Ω ⊂ X → Y has the second-order Fréchet derivative, and the sequences {x_n}, {y_n} are generated by (5). Then, for any natural number n, the following formula holds:

Proof. By (5), we have

Hence

This completes the proof of Lemma 4.

Theorem 5. Suppose X and Y are Banach spaces, Ω ⊂ X is an open convex subset, F : Ω ⊂ X → Y satisfies condition (11), η ≤ β, and

Then the sequence {x_n} _n≥0 generated by (5) is well defined, $$, and converges to the unique solution x^* in S(x₀,   α) and

Proof. By induction, we can prove that the following formulae hold:

In fact, by Lemma 2 we know t_n < r₁ for any natural number n. It is easy to prove that for n = 0 the above formulae hold. Suppose the above formulae also hold for n > 0. Then

By Lemma 3, we get

By Lemmas 3 and 4 and the fact that −g^′(t) ⁻¹, g^′′(t) are positive and increasing on [0,   α), we have

Hence we get

By Lemma 3, we get

Moreover, we have

Hence, the sequence {x_n} _n≥0 generated by (5) is well defined, $$, and {x_n} converges to the solution $$ of (1).

Now we prove the uniqueness. Suppose y^* is also a solution of (1) on S(x₀,   α). We know that g^′(t) < 0 for t ∈ [0, α). Then

By Banach Theorem, we know the inverse of $$ exists and

hence we get y^* = x^*. This completes the proof of the uniqueness of the solution of (1).

For m > n, we know that

When m → ∞, we get

This completes the proof of Theorem 5.

Corollary 6. Suppose X and Y are the Banach spaces, Ω is an open convex subset of the Banach space X, F : Ω ⊂ X → Y has the second-order Fréchet derivative, F^′(x₀) ⁻¹ exists for x₀ ∈ Ω, and the following conditions hold:

Then the sequence {x_n} _n≥0 generated by (5) is well defined, $$, and {x_n} converges to the unique solution x^* on S(x₀, α) of (1), where r₁ ≤ r₂ are two positive roots of g(t) = (1/6)Kt³ + (1/2)γt² − t + η.

Corollary 7 (see [10].)Suppose X and Y are Banach spaces, Ω is an open convex subset of the Banach space X, F : Ω ⊂ X → Y has the third-order Fréchet derivative, F^′(x₀) ⁻¹ exists for x₀ ∈ Ω, and the following conditions hold:

Then the sequence {x_n} _n≥0 generated by (5) is well defined, $$, and {x_n} converges to the unique solution x^* of (1) on $$, where

are two positive roots of the equation g(t) = η − t + γt²/(1 − γt).

Example 1. Consider the equation

We choose the initial point x₀ = 0, Ω = [−1,1]; then

Hence, by Corollary 6, the sequence {x_n} _n≥0 generated by (5) is well defined, and {x_n} converges to the solution x^* of (36).

Now, we will analyze errors ∥x_n − x^*∥ by Corollary 6 (see Table 1). In this case, we take x₀ = 0; then r₁ = 0.462598422⋯.

Example 2. Consider the system of equation [18] F(u, v) = 0, where

Then, we have

We choose x₀ = (u₀, v₀) = (1.75,1.75) and Ω = {x∣∥x − x₀∥ ≤ 1.75}. We take the max-norm in R² and the norm ∥A∥ = max {|a₁₁| + |a₁₂|, |a₂₁| + |a₂₂|} for $$. Define the norm of a bilinear operator B on R² by

where u = (u₁, u₂) ^T and

Then we get the following results:

This means that the hypotheses of Corollary 6 are satisfied.

Now, we will analyze errors ∥x_n − x^*∥ by Corollary 6 (see Table 2). In this case, we take x₀ = (u₀, v₀) = (1.75,1.75); then r₁ = 1.125.

Example 3. Consider the following integral equations:

and the space X = C[0,1] with the norm

This equation arises in the theory of radiative transfer and neutron transport and in the kinetic theory of gases. Define the operator F on X by

Then, for x₀ = 1, we obtain

This means that the hypotheses of Corollary 7 are satisfied.

Now, we will analyze errors ∥x_n − x^*∥ by Corollary 7 (see Table 3). In this case, we take x₀ = 1; then r₁ = 0.289222⋯.