On Newton-Kantorovich Method for Solving the Nonlinear Operator Equation

Research Article

On Newton-Kantorovich Method for Solving the Nonlinear Operator Equation

Hameed Husam Hameed,

^1,2

Z. K. Eshkuvatov,

^3,4

Anvarjon Ahmedov,

^4,5

and N. M. A. Nik Long

^1,4

1Department of Mathematics, Faculty of Science, Universiti Putra Malaysia (UPM), Selangor, Malaysia

2Technical Institute of Alsuwerah, The Middle Technical University, Baghdad, Iraq

3Faculty of Science and Technology, Universiti Sains Islam Malaysia (USIM), Negeri Sembilan, Malaysia

4Institute for Mathematical Research, Universiti Putra Malaysia (UPM), Selangor, Malaysia

5Department of Process and Food Engineering, Faculty of Engineering, Universiti Putra Malaysia (UPM), Selangor, Malaysia

Correspondence should be addressed to Z. K. Eshkuvatov; zainidin@upm.edu.my Received 19 July 2014; Accepted 6 October 2014

Academic Editor: Gaohang Yu

Copyright © 2015 Hameed Husam Hameed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We develop the Newton-Kantorovich method to solve the system of2×2nonlinear Volterra integral equations where the unknown function is in logarithmic form. A new majorant function is introduced which leads to the increment of the convergence interval.

The existence and uniqueness of approximate solution are proved and a numerical example is provided to show the validation of the method.

1. Introduction

Nonlinear phenomenon appears in many scientific areas such as physics, fluid mechanics, population models, chem- ical kinetics, economic systems, and medicine and can be modeled by system of nonlinear integral equations. The difficulty lies in finding the exact solution for such system.

Alternatively, the approximate or numerical solutions can be sought. One of the well known approximate method is Newton-Kantorovich method which reduces the nonlinear into sequence of linear integral equations. The the approxi- mate solution is then obtained by processing the convergent sequence. In 1939, Kantorovich [1] presented an iterative method for functional equation in Banach space and derived the convergence theorem for Newton method. In 1948, Kantorovich [2] proved a semilocal convergence theorem for Newton method in Banach space, later known as the Newton-Kantorovich method. Uko and Argyros [3] proved a weak Kantorovich-type theorem which gives the same conclusion under the weaker conditions. Shen and Li [4] have

established the Kantorovich-type convergence criterion for inexact Newton methods, assuming that the first derivative of an operator satisfies the Lipschitz condition. Argyros [5]

provided a sufficient condition for the semilocal convergence of Newton’s method to a locally unique solution of a nonlinear operator equation. Saberi-Nadjafi and Heidari [6] introduced a combination of the Newton-Kantorovich and quadrature methods to solve the nonlinear integral equation of Urysohn type in the systematic procedure. Ezquerro et al. [7] studied the nonlinear integral equation of mixed Hammerstein type using Newton-Kantorovich method with majorant principle.

Ezquerro et al. [8] provided the semilocal convergence of Newton method in Banach space under a modification of the classic conditions of Kantorovich. There are many methods of solving the system of nonlinear integral equations, for example, product integration method [9], Adomian method [10], RBF network method [11], biorthogonal system method [12], Chebyshev wavelets method [13], analytical method [14], reproducing kernel method [15], step method [16], and single term Wlash series [17]. In 2003, Boikov and Tynda [18]

Volume 2015, Article ID 219616, 12 pages http://dx.doi.org/10.1155/2015/219616

implemented the Newton-Kantorovich method to the follow- ing system:

𝑥 (𝑡) − ∫^𝑡

𝑦(𝑡)ℎ (𝑡, 𝜏) 𝑔 (𝜏) 𝑥 (𝜏) 𝑑𝜏 = 0,

∫^𝑡

𝑦(𝑡)𝑘 (𝑡, 𝜏) [1 − 𝑔 (𝜏)] 𝑥 (𝜏) 𝑑𝜏 = 𝑓 (𝑡) ,

(1)

where0 < 𝑡₀ ≤ 𝑡 ≤ 𝑇,𝑦(𝑡) < 𝑡, and the functionsℎ(𝑡, 𝜏), 𝑘(𝑡, 𝜏) ∈ 𝐶_{[𝑡0,𝑇]×[𝑡0,𝑇]},𝑓(𝑡),𝑔(𝑡) ∈ 𝐶_[𝑡0,𝑇], and(0 < 𝑔(𝑡) < 1).

In 2010, Eshkuvatov et al. [19] used the Newton-Kantorovich hypothesis to solve the system of nonlinear Volterra integral equation of the form

𝑥 (𝑡) − ∫^𝑡

𝑦(𝑡)ℎ (𝑡, 𝜏) 𝑥²(𝜏) 𝑑𝜏 = 0,

∫^𝑡

𝑦(𝑡)𝑘 (𝑡, 𝜏) 𝑥²(𝜏) 𝑑𝜏 = 𝑓 (𝑡) ,

(2)

where 𝑥(𝑡) and 𝑦(𝑡) are unknown functions defined on [𝑡₀, ∞), 𝑡₀ > 0, andℎ(𝑡, 𝜏), 𝑘(𝑡, 𝜏) ∈ 𝐶_{[𝑡0,∞]×[𝑡0,∞]},𝑓(𝑡) ∈ 𝐶_[𝑡0,∞]. In 2010, Eshkuvatov et al. [20] developed the modified Newton-Kantorovich to obtain an approximate solution of system with the form

𝑥 (𝑡) − ∫^𝑡

𝑦(𝑡)𝐻 (𝑡, 𝜏) 𝑥^𝑛(𝜏) 𝑑𝜏 = 0,

∫^𝑡

𝑦(𝑡)𝐾 (𝑡, 𝜏) 𝑥^𝑛(𝜏) 𝑑𝜏 = 𝑓 (𝑡) ,

(3)

where0 < 𝑡₀ ≤ 𝑡 ≤ 𝑇, 𝑦(𝑡) < 𝑡, and the functions𝐻(𝑡, 𝜏), 𝐾(𝑡, 𝜏) ∈ 𝐶_{[𝑡0,∞]×[𝑡0,∞]},𝑓(𝑡) ∈ 𝐶_[𝑡0,∞], and the unknown functions𝑥(𝑡) ∈ 𝐶_[𝑡0,∞],𝑦(𝑡) ∈ 𝐶¹_[𝑡0,∞],𝑦(𝑡) < 𝑡.

In this paper, we consider the systems of nonlinear integral equation of the form

𝑥 (𝑡) − ∫^𝑡

𝑦(𝑡)ℎ (𝑡, 𝜏)log|𝑥 (𝜏)| 𝑑𝜏 = 𝑔 (𝑡) ,

∫^𝑡

𝑦(𝑡)𝑘 (𝑡, 𝜏)log|𝑥 (𝜏)| 𝑑𝜏 = 𝑓 (𝑡) ,

(4)

where0 < 𝑡₀ ≤ 𝑡 ≤ 𝑇,𝑦(𝑡) < 𝑡,𝑥(𝑡) ̸= 0,ℎ(𝑡, 𝜏), ℎ_𝜏(𝑡, 𝜏), 𝑘(𝑡, 𝜏), 𝑘_𝜏(𝑡, 𝜏) ∈ 𝐶(𝐷) and the unknown functions𝑥(𝑡) ∈ 𝐶[𝑡₀, 𝑇],𝑦(𝑡) ∈ 𝐶¹[𝑡₀, 𝑇]to be determined, and𝐷 = [𝑡₀, 𝑇] × [𝑡₀, 𝑇].

The paper is organized as follows, inSection 2, Newton- Kantorovich method for the system of integral equations(4) is presented. Section 3deals with mixed method followed by discretizations. In Section 4, the rate of convergence of the method is investigated. Lastly,Section 5demonstrates the numerical example to verify the validity and accuracy of the proposed method, followed by the conclusion inSection 6.

2. Newton-Kantorovich Method for the System

Let us rewrite the system of nonlinear Volterra integral equa- tion(4)in the operator form

𝑃 (𝑋) = (𝑃₁(𝑋) , 𝑃₂(𝑋)) = 0, (5)

where𝑋 = (𝑥(𝑡), 𝑦(𝑡))and

𝑃₁(𝑋) = 𝑥 (𝑡) − ∫^𝑡

𝑦(𝑡)ℎ (𝑡, 𝜏)log|𝑥 (𝜏)| 𝑑𝜏 − 𝑔 (𝑡) , 𝑃₂(𝑋) = ∫^𝑡

𝑦(𝑡)𝑘 (𝑡, 𝜏)log|𝑥 (𝜏)| 𝑑𝜏 − 𝑓 (𝑡) .

(6)

To solve(5)we use initial iteration of Newton-Kantorovich method which is of the form

𝑃^󸀠(𝑋₀) (𝑋 − 𝑋₀) + 𝑃 (𝑋₀) = 0, (7)

where𝑋₀ = (𝑥₀(𝑡), 𝑦₀(𝑡))is the initial guess and𝑥₀(𝑡) and 𝑦₀(𝑡) can be any continuous functions provided that 𝑡₀ <

𝑦(𝑡) < 𝑡and𝑥(𝑡) ̸= 0.

The Frechet derivative of𝑃(𝑋)at the point𝑋₀is defined as

𝑃^󸀠(𝑋₀) 𝑋

= (lim

𝑠 → 0

1

𝑠[𝑃₁(𝑋₀+ 𝑠𝑋) − 𝑃₁(𝑋)] ,

𝑠 → 0lim 1

𝑠[𝑃₂(𝑋₀+ 𝑠𝑋) − 𝑃₂(𝑋)])

= (lim

𝑠 → 0

1

𝑠[𝑃₁(𝑥₀+ 𝑠𝑥, 𝑦₀+ 𝑠𝑦) − 𝑃₁(𝑥₀, 𝑦₀)] ,

𝑠 → 0lim 1

𝑠[𝑃₂(𝑥₀+ 𝑠𝑥, 𝑦₀+ 𝑠𝑦) − 𝑃₂(𝑥₀, 𝑦₀)])

= (lim

𝑠 → 0[𝜕𝑃₁(𝑥₀, 𝑦₀)

𝜕𝑥 𝑠𝑥 +𝜕𝑃₁(𝑥₀, 𝑦₀)

𝜕𝑦 𝑠𝑦

+1 2(𝜕²𝑃₁

𝜕𝑥² (𝑥₀+ 𝜃𝑠𝑥, 𝑦₀+ 𝛿𝑠𝑦) 𝑠²𝑥² + 2𝜕²𝑃₁

𝜕𝑥𝜕𝑦(𝑥₀+ 𝜃𝑠𝑥, 𝑦₀+ 𝛿𝑠𝑦) 𝑠²𝑥𝑦 + 𝜕²𝑃₁

𝜕𝑦² (𝑥₀+ 𝜃𝑠𝑥, 𝑦₀+ 𝛿𝑠𝑦) 𝑠𝑦²)] ,

𝑠 → 0lim 1 𝑠[𝜕𝑃₂

𝜕𝑥 (𝑥₀, 𝑦₀) 𝑠𝑥 +𝜕𝑃₂

𝜕𝑦 (𝑥₀, 𝑦₀) 𝑠𝑦 +1

2(𝜕²𝑃₂

𝜕𝑥² (𝑥₀+ 𝜃𝑠𝑥, 𝑦₀+ 𝛿𝑠𝑦) 𝑠²𝑥² + 2𝜕²𝑃₂

𝜕𝑥𝜕𝑦(𝑥₀+ 𝜃𝑠𝑥, 𝑦₀+ 𝛿𝑠𝑦) 𝑠²𝑥𝑦 +𝜕²𝑃₂

𝜕𝑦² (𝑥₀+ 𝜃𝑠𝑥, 𝑦₀+ 𝛿𝑠𝑦) 𝑠𝑦²)])

= (𝜕𝑃₁(𝑥₀, 𝑦₀)

𝜕𝑥 𝑥 + 𝜕𝑃₁(𝑥₀, 𝑦₀)

𝜕𝑦 𝑦,

𝜕𝑃₂(𝑥₀, 𝑦₀)

𝜕𝑥 𝑥 + 𝜕𝑃₂(𝑥₀, 𝑦₀)

𝜕𝑦 𝑦) .

(8) Hence,

𝑃^󸀠(𝑋₀) 𝑋 = (

𝜕𝑃₁

𝜕𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨(𝑥0,𝑦0)

𝜕𝑃₁

𝜕𝑦󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨(𝑥0,𝑦0)

𝜕𝑃₂

𝜕𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨(𝑥0,𝑦0)

𝜕𝑃₂

𝜕𝑦󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨(𝑥0,𝑦0)

) (𝑥𝑦). (9)

From(7)and(9)it follows that

𝜕𝑃₁

𝜕𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨(𝑥0,𝑦0)(Δ𝑥 (𝑡)) + 𝜕𝑃₁

𝜕𝑦󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨(𝑥0,𝑦0)(Δ𝑦 (𝑡))

= −𝑃₁(𝑥₀(𝑡) , 𝑦₀(𝑡)) ,

𝜕𝑃₂

𝜕𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨(𝑥₀,𝑦₀)(Δ𝑥 (𝑡)) + 𝜕𝑃₂

𝜕𝑦󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨(𝑥0,𝑦0)(Δ𝑦 (𝑡))

= −𝑃₂(𝑥₀(𝑡) , 𝑦₀(𝑡)) ,

(10)

whereΔ𝑥(𝑡) = 𝑥₁(𝑡) − 𝑥₀(𝑡),Δ𝑦(𝑡) = 𝑦₁(𝑡) − 𝑦₀(𝑡), and(𝑥₀(𝑡), 𝑦₀(𝑡))is the initial given functions. To solve(10)with respect toΔ𝑥andΔ𝑦we need to compute all partial derivatives:

𝜕𝑃₁

𝜕𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨(𝑥₀,𝑦₀)=lim

𝑠 → 0

1

𝑠(𝑃₁(𝑥₀+ 𝑠𝑥, 𝑦₀) − 𝑃₁(𝑥₀, 𝑦₀))

=lim

𝑠 → 0

1 𝑠[𝑠𝑥 (𝑡)

− ∫^𝑡

𝑦0(𝑡)ℎ (𝑡, 𝜏) (log󵄨󵄨󵄨󵄨𝑥⁰(𝜏) + 𝑠𝑥 (𝜏)󵄨󵄨󵄨󵄨

− log󵄨󵄨󵄨󵄨𝑥⁰(𝜏)󵄨󵄨󵄨󵄨) 𝑑𝜏]

= 𝑥 (𝑡) − ∫^𝑡

𝑦0(𝑡)ℎ (𝑡, 𝜏) 𝑥 (𝜏) 𝑥₀(𝜏)𝑑𝜏,

𝜕𝑃₁

𝜕𝑦󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨(𝑥0,𝑦0)=lim

𝑠 → 0

1

𝑠(𝑃₁(𝑥₀, 𝑦₀+ 𝑠𝑦) − 𝑃₁(𝑥₀, 𝑦₀))

=lim

𝑠 → 0

1

𝑠[∫^{𝑦0(𝑡)+𝑠𝑦(𝑡)}

𝑦₀(𝑡) ℎ (𝑡, 𝜏)log󵄨󵄨󵄨󵄨(𝑥⁰(𝜏))󵄨󵄨󵄨󵄨 𝑑𝜏]

= ℎ (𝑡, 𝑦₀(𝑡))log󵄨󵄨󵄨󵄨𝑥⁰(𝑦₀(𝑡))󵄨󵄨󵄨󵄨 𝑦 (𝑡) ,

(11) and in the same manner we obtain

𝜕𝑃₂

𝜕𝑥󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨^(𝑥0,𝑦₀)= ∫^𝑡

𝑦₀(𝑡)𝑘 (𝑡, 𝜏) 𝑥 (𝜏) 𝑥₀(𝜏)𝑑𝜏,

𝜕𝑃₂

𝜕𝑦󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨(𝑥0,𝑦0)= −𝑘 (𝑡, 𝑦₀(𝑡))log󵄨󵄨󵄨󵄨𝑥⁰(𝑦₀(𝑡))󵄨󵄨󵄨󵄨 𝑦 (𝑡) .

(12)

So that from(10)–(12)it follows that Δ𝑥 (𝑡) − ∫^𝑡

𝑦0(𝑡)ℎ (𝑡, 𝜏)Δ𝑥 (𝜏) 𝑥₀(𝜏)𝑑𝜏

+ ℎ (𝑡, 𝑦₀(𝑡))log󵄨󵄨󵄨󵄨𝑥⁰(𝑦₀(𝑡))󵄨󵄨󵄨󵄨 Δ𝑦 (𝑡)

= ∫^𝑡

𝑦0(𝑡)ℎ (𝑡, 𝜏)log󵄨󵄨󵄨󵄨𝑥⁰(𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏 − 𝑥⁰(𝑡) + 𝑔 (𝑡) ,

∫^𝑡

𝑦0(𝑡)𝑘 (𝑡, 𝜏)Δ𝑥 (𝜏) 𝑥₀(𝜏)𝑑𝜏

− 𝑘 (𝑡, 𝑦₀(𝑡))log󵄨󵄨󵄨󵄨𝑥⁰(𝑦₀(𝑡))󵄨󵄨󵄨󵄨 Δ𝑦 (𝑡)

= − ∫^𝑡

𝑦0(𝑡)𝑘 (𝑡, 𝜏)log󵄨󵄨󵄨󵄨𝑥⁰(𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏 + 𝑓 (𝑡) .

(13)

Equation(13)is a linear, and, by solving it forΔ𝑥andΔ𝑦, we obtain(𝑥₁(𝑡), 𝑦₁(𝑡)). By continuing this process, a sequence of approximate solution(𝑥_𝑚(𝑡), 𝑦_𝑚(𝑡))can be evaluated from 𝑃^󸀠(𝑋₀) Δ𝑋_𝑚+ 𝑃 (𝑋_𝑚) = 0, (14) which is equivalent to the system

Δ𝑥_𝑚(𝑡) − ∫^𝑡

𝑦0(𝑡)ℎ (𝑡, 𝜏)Δ𝑥_𝑚(𝜏) 𝑥₀(𝜏) 𝑑𝜏 + ℎ (𝑡, 𝑦₀(𝑡))log󵄨󵄨󵄨󵄨𝑥⁰(𝑦₀(𝑡))󵄨󵄨󵄨󵄨 Δ𝑦^𝑚(𝑡)

= ∫^𝑡

𝑦0(𝑡)ℎ (𝑡, 𝜏)log󵄨󵄨󵄨󵄨𝑥⁰(𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏 − 𝑥⁰(𝑡) + 𝑔 (𝑡) ,

∫^𝑡

𝑦0(𝑡)𝑘 (𝑡, 𝜏)Δ𝑥_𝑚(𝜏) 𝑥₀(𝜏) 𝑑𝜏

− 𝑘 (𝑡, 𝑦₀(𝑡))log󵄨󵄨󵄨󵄨𝑥⁰(𝑦₀(𝑡))󵄨󵄨󵄨󵄨 Δ𝑦^𝑚(𝑡)

= − ∫^𝑡

𝑦₀(𝑡)𝑘 (𝑡, 𝜏)log󵄨󵄨󵄨󵄨𝑥⁰(𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏 + 𝑓 (𝑡) ,

(15)

whereΔ𝑥_𝑚(𝑡) = 𝑥_𝑚(𝑡)−𝑥_𝑚−1(𝑡)andΔ𝑦_𝑚(𝑡) = 𝑦_𝑚(𝑡)−𝑦_𝑚−1(𝑡), 𝑚 = 1, 2, 3, . . ..

Thus, one should solve a system of two linear Volterra integral equations to find each successive approximation.

Let us eliminateΔ𝑦(𝑡)from the system(13) by finding the expression ofΔ𝑦(𝑡)from the first equation of this system and substitute it in the second equation to yield

Δ𝑦 (𝑡) = 1 𝐻 (𝑡)[∫^𝑡

𝑦0(𝑡)ℎ (𝑡, 𝜏) [Δ𝑥 (𝜏)

𝑥₀(𝜏) +log󵄨󵄨󵄨󵄨𝑥⁰(𝜏)󵄨󵄨󵄨󵄨] 𝑑𝜏

− [Δ𝑥 (𝑡) + 𝑥₀(𝑡) − 𝑔 (𝑡)]] ,

𝐺 (𝑡) [∫^𝑡

𝑦₀(𝑡)ℎ (𝑡, 𝜏) [Δ𝑥 (𝜏)

𝑥₀(𝜏) +log󵄨󵄨󵄨󵄨𝑥⁰(𝜏)󵄨󵄨󵄨󵄨] 𝑑𝜏

− [Δ𝑥 (𝑡) + 𝑥₀(𝑡) − 𝑔 (𝑡)]]

= ∫^𝑡

𝑦₀(𝑡)𝑘 (𝑡, 𝜏)Δ𝑥 (𝜏) 𝑥₀(𝜏)𝑑𝜏

− ∫^𝑡

𝑦₀(𝑡)𝑘 (𝑡, 𝜏)log󵄨󵄨󵄨󵄨𝑥⁰(𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏 + 𝑓 (𝑡) ,

(16) where 𝐺(𝑡) = 𝑘(𝑡, 𝑦₀(𝑡))/ℎ(𝑡, 𝑦₀(𝑡)) and 𝐻(𝑡) = 1/[ℎ(𝑡, 𝑦₀(𝑡))log|𝑥₀(𝑦₀(𝑡))|], and the second equation of(16)yields

Δ𝑥 (𝑡) − ∫^𝑡

𝑦0(𝑡)𝑘₁(𝑡, 𝜏)Δ𝑥 (𝜏)

𝑥₀(𝜏)𝜏 = 𝐹₀(𝑡) , (17) where

𝑘₁(𝑡, 𝜏) = ℎ (𝑡, 𝜏) −𝑘 (𝑡, 𝜏) 𝐺 (𝑡) , 𝐺 (𝑡) = 𝑘 (𝑡, 𝑦₀(𝑡))

ℎ (𝑡, 𝑦₀(𝑡)), 𝑘 (𝑡, 𝑦₀(𝑡)) ̸= 0 ∀𝑡 ∈ [𝑡₀, 𝑇] , 𝐹₀(𝑡) = ∫^𝑡

𝑦0(𝑡)𝑘₁(𝑡, 𝜏)log󵄨󵄨󵄨󵄨𝑥⁰(𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏 − 𝑥⁰(𝑡) + 𝑔 (𝑡) + 𝑓 (𝑡) 𝐺 (𝑡).

(18) In an analogous way,Δ𝑦_𝑚(𝑡)andΔ𝑥_𝑚(𝑡)can be written in the form

Δ𝑦_𝑚(𝑡)

= 1

𝐻 (𝑡)[∫^𝑡

𝑦0(𝑡)ℎ (𝑡, 𝜏)Δ𝑥_𝑚(𝜏) 𝑥₀(𝜏) 𝑑𝜏 + ∫^𝑡

𝑦𝑚−1(𝑡)ℎ (𝑡, 𝜏)log󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏

− Δ𝑥_𝑚(𝑡) − 𝑥_𝑚−1(𝑡) + 𝑔 (𝑡)] ,

(19)

Δ𝑥_𝑚(𝑡) − ∫^𝑡

𝑦0(𝑡)𝑘₁(𝑡, 𝜏)Δ𝑥_𝑚(𝜏)

𝑥₀(𝜏) 𝑑𝜏 = 𝐹_𝑚−1(𝑡) , (20)

where

𝐹_𝑚−1(𝑡) = ∫^𝑡

𝑦𝑚−1(𝑡)𝑘₁(𝑡, 𝜏)log󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏 − 𝑥^𝑚−1(𝑡) + 𝑔 (𝑡) +𝑓 (𝑡)

𝐺 (𝑡).

(21)

3. The Mixed Method (Simpson

and Trapezoidal) for Approximate Solution

At each step of the iterative process we have to find the solution of(18)and(20)on the closed interval[𝑡₀, 𝑇]. To do this the grid (𝜔) of points𝑡_𝑖 = 𝑡₀+ 𝑖ℎ,𝑖 = 1, 2, 3, . . . , 2𝑁, ℎ = (𝑇−𝑡₀)/2𝑁is introduced, and by the collocation method with mixed rule we require that the approximate solution satisfies(18)and(20). Hence

Δ𝑥_𝑚(𝑡₀) = −𝑥_𝑚−1(𝑡₀) + 𝑔 (𝑡₀) +𝑓 (𝑡₀)

𝐺 (𝑡₀), (22) Δ𝑥_𝑚(𝑡_2𝑖) − ∫^𝑡2𝑖

𝑦0(𝑡2𝑖)𝑘₁(𝑡_2𝑖, 𝜏)Δ𝑥_𝑚(𝜏) 𝑥₀(𝜏) 𝑑𝜏

= 𝐹_𝑚−1(𝑡_2𝑖) , 𝑖 = 1, 2, . . . , 𝑁.

(23)

On the grid (𝜔) we setV_2𝑖= 𝑦₀(𝑡_2𝑖), suct that

𝑡_V2𝑖 = {𝑡_V2𝑖, 𝑡₀≤ 𝑦₀(𝑡_2𝑖) < 𝑡_2𝑖−2,

𝑡_2𝑖, 𝑡_2𝑖−2≤ 𝑦₀(𝑡_2𝑖) < 𝑡_2𝑖. (24)

Consequently, the system(23)can be written in the form

Δ𝑥_𝑚(𝑡_2𝑖) − ∫^𝑡V2𝑖

𝑦0(𝑡2𝑖)𝑘₁(𝑡_2𝑖, 𝜏)Δ𝑥_𝑚(𝜏) 𝑥₀(𝜏) 𝑑𝜏

− ∑^𝑖−1

𝑗=V_2𝑖∫^𝑡2𝑗+2

𝑡2𝑗

𝑘₁(𝑡_2𝑖, 𝜏)Δ𝑥_𝑚(𝜏) 𝑥₀(𝜏) 𝑑𝜏

= 𝐹_𝑚−1(𝑡_2𝑖) , 𝑖 = 1, 2, . . . , 𝑁.

(25)

By computing the integral in(26)using tapezoidal formula on the first integrals and Simpson formula on the second integral, we consider two cases.

Case 1.WhenV_2𝑖 ̸= 2𝑖,𝑖 = 1, 2, . . . , 𝑁, then

Δ𝑥_𝑚(𝑡_2𝑖) = 𝐹_𝑚−1(𝑡_2𝑖) + 𝐴 (𝑖) + 𝐵 (𝑖) + 𝐶 (𝑖) 1 − ((𝑡_2𝑖− 𝑡_2𝑖−2) /6 ⋅ 𝑥₀(𝑡_2𝑖)) 𝑘₁(𝑡_2𝑖, 𝑡_2𝑖),

(26)

where

𝐴 (𝑖) = 0.5 (𝑡_V2𝑖− 𝑦₀(𝑡_2𝑖))

× [𝑘₁(𝑡_2𝑖, 𝑡_V2𝑖)Δ𝑥_𝑚(𝑡_V2𝑖)

𝑥₀(𝑡_V2𝑖) + 𝑘₁(𝑡_2𝑖, 𝑦₀(𝑡_2𝑖))

× Δ𝑥_𝑚(𝑡_V2𝑖) (𝑡_V2𝑖− 𝑦₀(𝑡_2𝑖)) (𝑡_V2𝑖− 𝑡_V2𝑖−2) (𝑥₀(𝑦₀(𝑡_2𝑖))) + 𝑘₁(𝑡_2𝑖, 𝑦₀(𝑡_2𝑖))

×Δ𝑥_𝑚(𝑡_V2𝑖−2) (𝑦₀(𝑡_2𝑖) − 𝑡_V2𝑖−2) (𝑡_V2𝑖− 𝑡_V2𝑖−2) (𝑥₀(𝑦₀(𝑡_2𝑖))) ] , 𝐵 (𝑖) = ^𝑖−2∑

𝑗=V2𝑖

(𝑡_2𝑗+2− 𝑡_2𝑗) 6

× [𝑘₁(𝑡_2𝑖, 𝑡_2𝑗)Δ𝑥_𝑚(𝑡_2𝑗) 𝑥₀(𝑡_2𝑗)

+ 4𝑘₁(𝑡_2𝑖, 𝑡_2𝑗+1)Δ𝑥_𝑚(𝑡_2𝑗+1) 𝑥₀(𝑡_2𝑗+1) + 𝑘₁(𝑡_2𝑖, 𝑡_2𝑗+2)Δ𝑥_𝑚(𝑡_2𝑗+2)

𝑥₀(𝑡_2𝑗+2) ] , 𝐶 (𝑖) =(𝑡_2𝑖− 𝑡_2𝑖−2)

6 [𝑘₁(𝑡_2𝑖, 𝑡_2𝑖−2)Δ𝑥_𝑚(𝑡_2𝑖−2) 𝑥₀(𝑡_2𝑖−2) + 4𝑘₁(𝑡_2𝑖, 𝑡_2𝑖−1)Δ𝑥_𝑚(𝑡_2𝑖−1)

𝑥₀(𝑡_2𝑖−1) ] . (27)

Case 2.WhenV_2𝑖= 2𝑖,𝑖 = 1, 2, . . . , 𝑁, then

Δ𝑥_𝑚(𝑡_2𝑖) =𝐷₁(𝑖)

𝐷₂(𝑖), (28)

where

𝐷₁(𝑖) = 𝐹_𝑚−1(𝑡_2𝑖) + 0.5𝑘₁(𝑡_2𝑖, 𝑦₀(𝑡_2𝑖))

× [Δ𝑥_𝑚(𝑡_2𝑖−2) 𝑥₀(𝑦₀(𝑡_2𝑖))

(𝑡_2𝑖− 𝑦₀(𝑡_2𝑖)) (𝑦₀(𝑡_2𝑖) − 𝑡_2𝑖−2) 𝑡_2𝑖− 𝑡_2𝑖−2 ] , 𝐷₂(𝑖) = [1 − 0.5 (𝑡_2𝑖− 𝑦₀(𝑡₂))𝑘₁(𝑡_2𝑖, 𝑡_2𝑖)

𝑥₀(𝑡_2𝑖)

− 0.5𝑘₁(𝑡_2𝑖, 𝑦₀(𝑡_2𝑖)) (𝑡_2𝑖− 𝑦₀(𝑡_2𝑖))² 𝑥₀(𝑦₀(𝑡_2𝑖)) (𝑡_2𝑖− 𝑡_2𝑖−2)] .

(29)

Also, to computeΔ𝑦_𝑚(𝑡) on the grid (𝜔), (18) can be re- presented in the form

Δ𝑦_𝑚(𝑡_2𝑖) = 1 𝐻 (𝑡_2𝑖)

× [∫^𝑡2𝑖

𝑦₀(𝑡_2𝑖)ℎ (𝑡_2𝑖, 𝜏)Δ𝑥_𝑚(𝜏) 𝑥₀(𝜏) 𝑑𝜏 + ∫^𝑡2𝑖

𝑦𝑚−1(𝑡2𝑖)ℎ (𝑡_2𝑖, 𝜏)log󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏

− Δ𝑥_𝑚(𝑡_2𝑖) − 𝑥_𝑚−1(𝑡_2𝑖) + 𝑔 (𝑡_2𝑖) ] . (30)

Let us setV_2𝑖= 𝑦₀(𝑡_2𝑖)and𝑢_2𝑖= 𝑦_𝑚−1(𝑡_2𝑖)and

𝑡_V2𝑖 ={ {{

𝑡_2𝑖, 𝑡_2𝑖−2 ≤ 𝑦₀(𝑡_2𝑖) < 𝑡_2𝑖, 𝑡_V2𝑖, 𝑡₀≤ 𝑦₀(𝑡_2𝑖) < 𝑡_2𝑖−2,

𝑡_𝑢2𝑖={ {{

𝑡_2𝑖, 𝑡_2𝑖−2≤ 𝑦_𝑚−1(𝑡_2𝑖) < 𝑡_2𝑖, 𝑡_𝑢2𝑖, 𝑡₀≤ 𝑦_𝑚−1(𝑡_2𝑖) < 𝑡_2𝑖−2.

(31)

Then(30)can be written as

Δ𝑦_𝑚(𝑡_2𝑖) = 1 𝐻 (𝑡_2𝑖)

× [∫^𝑡V2𝑖

𝑦0(𝑡2𝑖)ℎ (𝑡_2𝑖, 𝜏)Δ𝑥_𝑚(𝜏) 𝑥₀(𝜏) 𝑑𝜏 + ∑^𝑖−1

𝑗=V2𝑖

∫^𝑡2𝑗+2

𝑡2𝑗

ℎ (𝑡_2𝑖, 𝜏)Δ𝑥_𝑚(𝜏) 𝑥₀(𝜏) 𝑑𝜏 + ∫^{𝑡𝑢2𝑖}

𝑦_𝑚−1(𝑡_2𝑖)ℎ (𝑡_2𝑖, 𝜏)log󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏 + ^𝑖−1∑

𝑗=𝑢_2𝑖∫^𝑡2𝑗+2

𝑡2𝑗

ℎ (𝑡_2𝑖, 𝜏)log󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏

− Δ𝑥_𝑚(𝑡_2𝑖) − 𝑥_𝑚−1(𝑡_2𝑖) + 𝑔 (𝑡_2𝑖) ] , (32)

and by applying mixed formula for(32)we obtain the follow- ing four cases.

Case 1.WhenV_2𝑖 ̸= 2𝑖and𝑢_2𝑖 ̸= 2𝑖, we have Δ𝑦_𝑚(𝑡_2𝑖)

= 1

𝐻 (𝑡_2𝑖)

× [ [

0.5 (𝑡_V2𝑖− 𝑦₀(𝑡_2𝑖))

× (ℎ (𝑡_2𝑖, 𝑡_V2𝑖)Δ𝑥_𝑚(𝑡_V2𝑖) 𝑥₀(𝑡_V2𝑖)

+ ℎ (𝑡_2𝑖, 𝑦₀(𝑡_2𝑖))Δ𝑥_𝑚(𝑦₀(𝑡_2𝑖)) 𝑥₀(𝑦₀(𝑡_2𝑖)) )

+ ^𝑖−1∑

𝑗=V2𝑖

(𝑡_2𝑗+2− 𝑡_2𝑗) 6

× (ℎ (𝑡_2𝑖, 𝑡_2𝑗)Δ𝑥_𝑚(𝑡_2𝑗) 𝑥₀(𝑡_2𝑗)

+ 4ℎ (𝑡_2𝑖, 𝑡_2𝑗+1)Δ𝑥_𝑚(𝑡_2𝑗+1) 𝑥₀(𝑡_2𝑗+1)

+ ℎ (𝑡_2𝑖, 𝑡_2𝑗+2)Δ𝑥_𝑚(𝑡_2𝑗+2) 𝑥₀(𝑡_2𝑗+2) ) + 0.5 (𝑡_𝑢2𝑖− 𝑦_𝑚−1(𝑡_2𝑖))

× (ℎ (𝑡_2𝑖, 𝑡_𝑢2𝑖)log󵄨󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝑡_𝑢2𝑖)󵄨󵄨󵄨󵄨󵄨

+ ℎ (𝑡_2𝑖, 𝑦_𝑚−1(𝑡_2𝑖))log󵄨󵄨󵄨󵄨(𝑥^𝑚−1(𝑦_𝑚−1(𝑡_2𝑖)))󵄨󵄨󵄨󵄨 ) + ^𝑖−1∑

𝑗=𝑢2𝑖

(𝑡_2𝑗+2− 𝑡_2𝑗) 6

× (ℎ (𝑡_2𝑖, 𝑡_2𝑗)log(𝑥_𝑚−1(𝑡_2𝑗)) + 4ℎ (𝑡_2𝑖, 𝑡_2𝑗+1)log󵄨󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝑡_2𝑗+1)󵄨󵄨󵄨󵄨󵄨

+ ℎ (𝑡_2𝑖, 𝑡_2𝑗+2)log󵄨󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝑡_2𝑗+2)󵄨󵄨󵄨󵄨󵄨)

− Δ𝑥_𝑚(𝑡_2𝑖) − 𝑥_𝑚−1(𝑡_2𝑖) + 𝑔 (𝑡_2𝑖) ] ] .

(33)

Case 2.IfV_2𝑖= 2𝑖and𝑢_2𝑖 ̸= 2𝑖, then Δ𝑦_𝑚(𝑡_2𝑖)

= 1

𝐻 (𝑡_2𝑖)

× [ [

0.5 (𝑡_2𝑖− 𝑦₀(𝑡_2𝑖))

× (ℎ (𝑡_2𝑖, 𝑡_2𝑖)Δ𝑥_𝑚(𝑡_V2𝑖) 𝑥₀(𝑡_2𝑖)

+ ℎ (𝑡_2𝑖, 𝑦₀(𝑡_2𝑖))Δ𝑥_𝑚(𝑦₀(𝑡_2𝑖)) 𝑥₀(𝑦₀(𝑡_2𝑖)) ) + 0.5 (𝑡_𝑢2𝑖− 𝑦_𝑚−1(𝑡_2𝑖))

× (ℎ (𝑡_2𝑖, 𝑡_𝑢2𝑖)log󵄨󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝑡_𝑢2𝑖)󵄨󵄨󵄨󵄨󵄨

+ ℎ (𝑡_2𝑖, 𝑦_𝑚−1(𝑡_2𝑖))log󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝑦_𝑚−1(𝑡_2𝑖))󵄨󵄨󵄨󵄨 ) + ∑^𝑖−1

𝑗=𝑢2𝑖

(𝑡_2𝑗+2− 𝑡_2𝑗) 6

× (ℎ (𝑡_2𝑖, 𝑡_2𝑗)log󵄨󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝑡_2𝑗)󵄨󵄨󵄨󵄨󵄨

+ 4ℎ (𝑡_2𝑖, 𝑡_2𝑗+1)log󵄨󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝑡_2𝑗+1)󵄨󵄨󵄨󵄨󵄨

+ ℎ (𝑡_2𝑖, 𝑡_2𝑗+2)log󵄨󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝑡_2𝑗+2)󵄨󵄨󵄨󵄨󵄨)

− Δ𝑥_𝑚(𝑡_2𝑖) − 𝑥_𝑚−1(𝑡_2𝑖) + 𝑔 (𝑡_2𝑖) ] ] .

(34)

Case 3.WhenV_2𝑖 ̸= 2𝑖and𝑢_2𝑖= 2𝑖, we get Δ𝑦_𝑚(𝑡_2𝑖)

= 1

𝐻 (𝑡_2𝑖)

× [ [

0.5 (𝑡_V2𝑖− 𝑦₀(𝑡_2𝑖))

× (ℎ (𝑡_2𝑖, 𝑡_V2𝑖)Δ𝑥_𝑚(𝑡_V2𝑖) 𝑥₀(𝑡_V2𝑖)

+ ℎ (𝑡_2𝑖, 𝑦₀(𝑡_2𝑖))Δ𝑥_𝑚(𝑦₀(𝑡_2𝑖)) 𝑥₀(𝑦₀(𝑡_2𝑖)) )

+ ^𝑖−1∑

𝑗=V2𝑖

(𝑡_2𝑗+2− 𝑡_2𝑗) 6

× (ℎ (𝑡_2𝑖, 𝑡_2𝑗)Δ𝑥_𝑚(𝑡_2𝑗) 𝑥₀(𝑡_2𝑗) + 4ℎ (𝑡_2𝑖, 𝑡_2𝑗+1)Δ𝑥_𝑚(𝑡_2𝑗+1)

𝑥₀(𝑡_2𝑗+1) + ℎ (𝑡_2𝑖, 𝑡_2𝑗+2)Δ𝑥_𝑚(𝑡_2𝑗+2) 𝑥₀(𝑡_2𝑗+2) ) + 0.5 (𝑡_2𝑖− 𝑦_𝑚−1(𝑡_2𝑖))

× (ℎ (𝑡_2𝑖, 𝑡_2𝑖)log󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝑡_2𝑖)󵄨󵄨󵄨󵄨

+ ℎ (𝑡_2𝑖, 𝑦_𝑚−1(𝑡_2𝑖))log󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝑦_𝑚−1(𝑡_2𝑖))󵄨󵄨󵄨󵄨 )

− Δ𝑥_𝑚(𝑡_2𝑖) − 𝑥_𝑚−1(𝑡_2𝑖) + 𝑔 (𝑡_2𝑖) ] ] .

(35) Case 4.IfV_2𝑖= 2𝑖and𝑢_2𝑖= 2𝑖, then

Δ𝑦_𝑚(𝑡_2𝑖)

= 1

𝐻 (𝑡_2𝑖)

× [ [

0.5 (𝑡_2𝑖− 𝑦₀(𝑡_2𝑖))

× (ℎ (𝑡_2𝑖, 𝑡_2𝑖)Δ𝑥_𝑚(𝑡_2𝑖) 𝑥₀(𝑡_2𝑖)

+ ℎ (𝑡_2𝑖, 𝑦₀(𝑡_2𝑖))Δ𝑥_𝑚(𝑦₀(𝑡_2𝑖)) 𝑥₀(𝑦₀(𝑡_2𝑖)) ) + 0.5 (𝑡_2𝑖− 𝑦_𝑚−1(𝑡_2𝑖))

× (ℎ (𝑡_2𝑖, 𝑡_2𝑖)log󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝑡_2𝑖)󵄨󵄨󵄨󵄨

+ ℎ (𝑡_2𝑖, 𝑦_𝑚−1(𝑡_2𝑖))log󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝑦_𝑚−1(𝑡_2𝑖))󵄨󵄨󵄨󵄨 )

− Δ𝑥_𝑚(𝑡_2𝑖) − 𝑥_𝑚−1(𝑡_2𝑖) + 𝑔 (𝑡_2𝑖) ] ] .

(36)

Thus,(32)can be computed by one of(33)–(36)according to the cases.

4. The Convergence Analysis of the Method

On the basis of general theorems of Newton-Kantorovich method [21, Chapter XVIII] for the convergence, we state the following theorem regarding the successive approximations described by(18)–(20).

First, consider the following classes of functions:

(i)𝐶_[𝑡0,𝑇]the set of all continuous functions𝑓(𝑡)defined on the interval[𝑡₀, 𝑇],

(ii)𝐶_{[𝑡0,𝑇]×[𝑡0,𝑇]}the set of all continuous functions𝜓(𝑡, 𝜏) defined on the region[𝑡₀, 𝑇] × [𝑡₀, 𝑇],

(iii)𝐶 = {𝑋 : 𝑋 = (𝑥(𝑡), 𝑦(𝑡)) : 𝑥(𝑡), 𝑦(𝑡) ∈ 𝐶_[𝑡0,𝑇]}, (iv)𝐶^<_[𝑡0,𝑇]= {𝑦(𝑡) ∈ 𝐶¹_[𝑡0,𝑇]: 𝑦(𝑡) < 𝑡}.

And define the following norms

‖𝑥‖ = max

𝑡∈[𝑡₀,𝑇]|𝑥 (𝑡)| ,

‖Δ𝑋‖_𝐶=max{‖Δ𝑥‖_{𝐶[𝑡0,𝑇]}, 󵄩󵄩󵄩󵄩Δ𝑦󵄩󵄩󵄩󵄩𝐶_[𝑡0,𝑇]} ,

‖𝑋‖_𝐶¹ =max{‖𝑥‖𝐶_[𝑡0,𝑇], 󵄩󵄩󵄩󵄩󵄩𝑥^󸀠󵄩󵄩󵄩󵄩󵄩𝐶_[𝑡0,𝑇]} ,

󵄩󵄩󵄩󵄩󵄩𝑋󵄩󵄩󵄩󵄩󵄩𝐶=max{‖𝑥‖_{𝐶[𝑡0,𝑇]}, 󵄩󵄩󵄩󵄩𝑦󵄩󵄩󵄩󵄩𝐶_[𝑡0,𝑇]}

‖ℎ (𝑡, 𝜏)‖ = 𝐻₁, 󵄩󵄩󵄩󵄩󵄩ℎ^󸀠𝜏(𝑡, 𝜏)󵄩󵄩󵄩󵄩󵄩 = 𝐻^1󸀠,

‖𝑘 (𝑡, 𝜏)‖ = 𝐻₂, 󵄩󵄩󵄩󵄩󵄩𝑘^󸀠𝜏(𝑡, 𝜏)󵄩󵄩󵄩󵄩󵄩 = 𝐻^2󸀠,

󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩1

𝑥₀󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 = max

𝑡∈[𝑡0,𝑇]󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 1 𝑥₀(𝑡)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 = 𝑐¹,

󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩

󵄩 1

𝑥²₀󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩= max

𝑡∈[𝑡0,𝑇]󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 1 𝑥²₀(𝑡)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨= 𝑐₂,

󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 1

𝐺 (𝑡)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 = max

𝑡∈[𝑡0,𝑇]󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 1 𝐺 (𝑡)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 = 𝑐³,

󵄩󵄩󵄩󵄩𝑥⁰󵄩󵄩󵄩󵄩 = max

𝑡∈[𝑡0,𝑇]󵄨󵄨󵄨󵄨𝑥⁰(𝑡)󵄨󵄨󵄨󵄨 = 𝐻³,

󵄩󵄩󵄩󵄩󵄩𝑥^󸀠0󵄩󵄩󵄩󵄩󵄩 = max

𝑡∈[𝑡₀,𝑇]󵄨󵄨󵄨󵄨󵄨𝑥^󸀠0(𝑡)󵄨󵄨󵄨󵄨󵄨 = 𝐻^3󸀠,

𝑡∈[𝑡min0,𝑇]󵄨󵄨󵄨󵄨𝑦⁰(𝑡)󵄨󵄨󵄨󵄨 = 𝐻⁴,

󵄩󵄩󵄩󵄩log󵄩󵄩󵄩󵄩 = max

𝑡∈[𝑡0,𝑇]󵄨󵄨󵄨󵄨log(𝑥 (𝑡))󵄨󵄨󵄨󵄨 = 𝐻⁵,

󵄩󵄩󵄩󵄩𝑔󵄩󵄩󵄩󵄩 = max

𝑡∈[𝑡₀,𝑇]󵄨󵄨󵄨󵄨𝑔(𝑡)󵄨󵄨󵄨󵄨 = 𝐻⁶,

󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩 = max

𝑡∈[𝑡0,𝑇]󵄨󵄨󵄨󵄨𝑓(𝑡)󵄨󵄨󵄨󵄨 = 𝐻⁷.

(37)

Let

𝜂₁=max{𝐻₁𝑐₂(𝑇 − 𝐻₄) , 𝐻₁𝑐₁, 𝐻^󸀠₁𝐻₅ + 𝐻₁𝐻₃^󸀠𝑐₁,

𝐻₂𝑐₂(𝑇 − 𝐻₄) , 𝐻₂𝑐₁, 𝐻₂^󸀠𝐻₅+ 𝐻₂𝐻₃^󸀠𝑐₁} . (38)

Let us consider real valued function

𝜓 (𝑡) = 𝐾 (𝑡 − 𝑡₀)²− (1 + 𝐾𝜂) (𝑡 − 𝑡₀) + 𝜂, (39) where𝐾 > 0and𝜂are nonnegative real coefficients.

Theorem 1. Assume that the operator𝑃(𝑋) = 0in(5)is de- fined inΩ = {𝑋 ∈ 𝐶([𝑡₀, 𝑇]) : ‖𝑋 − 𝑋₀‖ ≤ 𝑅}and has continuous second derivative in closed ballΩ₀ = {𝑋 ∈ 𝐶([𝑡₀, 𝑇]) : ‖𝑋 − 𝑋₀‖ ≤ 𝑟}where𝑇 = 𝑡₀+ 𝑟 ≤ 𝑡₀+ 𝑅. Suppose the following conditions are satisfied:

(1)‖Γ₀𝑃(𝑋₀)‖ ≤ 𝜂/(1 + 𝐾𝜂),

(2)‖Γ₀𝑃^󸀠󸀠(𝑋)‖ ≤ 2𝐾/(1+𝐾𝜂), when‖𝑋−𝑋₀‖ ≤ 𝑡−𝑡₀≤ 𝑟, where𝐾and𝜂as in(39). Then the function𝜓(𝑡)defined by (39)majorizes the operator𝑃(𝑋).

Proof. Let us rewrite(5)and(39)in the form

𝑡 = 𝜙 (𝑡) , 𝜙 (𝑡) = 𝑡 + 𝑐₀𝜓 (𝑡) , (40) 𝑋 = 𝑆 (𝑋) , 𝑆 (𝑋) = 𝑋 − Γ0𝑃 (𝑋) , (41) where𝑐₀= −1/𝜓^󸀠(𝑡₀) = 1/(1 + 𝐾𝜂)andΓ₀= [𝑃^󸀠(𝑋₀)]⁻¹.

Let us show that (40) and (41) satisfy the majorizing conditions [21, Theorem 1, page 525]. In fact

󵄩󵄩󵄩󵄩𝑆(𝑋⁰) − 𝑋₀󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩−Γ⁰𝑃 (𝑋₀)󵄩󵄩󵄩󵄩 ≤ 𝜂

1 + 𝐾𝜂 = 𝜙 (𝑡₀) − 𝑡₀, (42) and for the‖𝑋 − 𝑋₀‖ ≤ 𝑡 − 𝑡₀with the Remark in [21, Remark 1, page 504] we have

󵄩󵄩󵄩󵄩󵄩𝑆^󸀠(𝑋)󵄩󵄩󵄩󵄩󵄩 =󵄩󵄩󵄩󵄩󵄩𝑆^󸀠(𝑋) − 𝑆^󸀠(𝑋₀)󵄩󵄩󵄩󵄩󵄩

≤ ∫^𝑋

𝑋₀󵄩󵄩󵄩󵄩󵄩𝑆^󸀠󸀠(𝑋)󵄩󵄩󵄩󵄩󵄩 𝑑𝑋 = ∫

𝑋

𝑋₀󵄩󵄩󵄩󵄩󵄩Γ⁰𝑃^󸀠󸀠(𝑋)󵄩󵄩󵄩󵄩󵄩 𝑑𝑋

≤ ∫^𝑡

𝑡0

𝑐₀𝜓^󸀠󸀠(𝜏) 𝑑𝜏 = ∫^𝑡

𝑡0

2𝐾 1 + 𝐾𝜂𝑑𝜏

= 2𝐾

1 + 𝐾𝜂(𝑡 − 𝑡₀) = 𝜙^󸀠(𝑡) .

(43)

Hence𝜓(𝑡) = 0is a majorant function of𝑃(𝑋) = 0.

Theorem 2. Let the functions𝑓(𝑡), 𝑔(𝑡) ∈ 𝐶_[𝑡0,𝑇],𝑥₀(𝑡) ∈ 𝐶¹[𝑡₀, 𝑇],𝑥₀(𝑦₀(𝑡)) ̸= 0,𝑥²₀(𝑡) ̸= 0, and the kernelsℎ(𝑡, 𝜏), 𝑘(𝑡, 𝜏) ∈ 𝐶¹_{[𝑡0,𝑇]×[𝑡0,𝑇]}and(𝑥₀(𝑡), 𝑦₀(𝑡)) ∈ Ω₀; then

(1)the system(7)has unique solution in the interval[𝑡₀, 𝑇]; that is, there exists Γ₀, and ‖Γ₀‖ ≤ ∑^∞_𝑗=1(𝑐₁𝐻₁ + 𝑐₁𝑐₃𝐻₂)^𝑗((𝑇 − 𝐻₄)^𝑗−1/(𝑗 − 1)!) = 𝜂₂,

(2)‖Δ𝑋‖ ≤ 𝜂/(1 + 𝐾𝜂), (3)‖𝑃^󸀠󸀠(𝑋)‖ ≤ 𝜂₁,

(4)𝜂 > 1/𝐾and𝑟 < 𝜂 + 𝑡₀,

where𝐾 and 𝜂 as in (39). Then the system(4) has unique solution𝑋^∗ in the closed ballΩ₀ and the sequence𝑋_𝑚(𝑡) = (𝑥_𝑚(𝑡), 𝑦_𝑚(𝑡)),𝑚 ≥ 0of successive approximations

Δ𝑦_𝑚(𝑡) = 1 𝐻 (𝑡)[∫^𝑡

𝑦0(𝑡)ℎ (𝑡, 𝜏)Δ𝑥_𝑚(𝜏) 𝑥₀(𝜏) 𝑑𝜏 + ∫^𝑡

𝑦𝑚−1(𝑡)ℎ (𝑡, 𝜏)log󵄨󵄨󵄨󵄨𝑥^𝑚−1(𝜏)󵄨󵄨󵄨󵄨 𝑑𝜏

− Δ𝑥_𝑚(𝑡) − 𝑥_𝑚−1(𝑡) + 𝑔 (𝑡) ] ,

Δ𝑥_𝑚(𝑡) − ∫^𝑡

𝑦0(𝑡)𝑘₁(𝑡, 𝜏)Δ𝑥_𝑚(𝜏)

𝑥₀(𝜏) 𝑑𝜏 = 𝐹_𝑚−1(𝑡) ,

(44)

whereΔ𝑥_𝑚(𝑡) = 𝑥_𝑚(𝑡)−𝑥_𝑚−1(𝑡)andΔ𝑦_𝑚(𝑡) = 𝑦_𝑚(𝑡)−𝑦_𝑚−1(𝑡), 𝑚 = 2, 3, . . ., and𝑋_𝑚converge to the solution𝑋^∗. The rate of convergence is given by

󵄩󵄩󵄩󵄩𝑋^∗− 𝑋_𝑚󵄩󵄩󵄩󵄩 ≤ ( 21 + 𝐾𝜂)^𝑚(1

𝐾) . (45)

Proof. It is shown that(7)is reduced to(17). Since(17)is a linear Volterra integral equation of 2nd kind with respect to Δ𝑥(𝑡)and since𝑘(𝑡, 𝑦₀(𝑡)) ̸= 0, ∀𝑡 ∈ [𝑡₀, 𝑇]which implies that the kernel𝑘₁(𝑡, 𝜏)defined by(18)is continues it follows that (17) has a unique solution which can be obtained by the method of successive approximations. Then the function Δ𝑦(𝑡)is uniquely determined from(16). Hence the existence ofΓ₀is archived.

To verify thatΓ₀is bounded we need to establish the resol- vent kernelΓ₀(𝑡, 𝜏)of(17), so we assume the integral operator 𝑈from𝐶[𝑡₀, 𝑇] → 𝐶[𝑡₀, 𝑇]is given by

𝑍 = 𝑈 (Δ𝑥) , 𝑍 (𝑡) = ∫^𝑡

𝑦0(𝑡)𝑘₂(𝑡, 𝜏) Δ𝑥 (𝜏) 𝑑𝜏, (46) where𝑘₂(𝑡, 𝜏) = 𝑘₁(𝑡, 𝜏)/𝑥₀(𝜏), and𝑘₁(𝑡, 𝜏)is defined in(18).

Due to(46),(17)can be written as

Δ𝑥 − 𝑈 (Δ𝑥) = 𝐹₀. (47)

The solutionΔ𝑥^∗of(47)is expressed in terms of𝐹₀by means of the formula

Δ𝑥^∗ = 𝐹₀+ 𝐵 (𝐹₀) , (48) where 𝐵is an integral operator and can be expanded as a series in powers of𝑈[21, Theorem 1, page 378]:

𝐵 (𝐹₀) = 𝑈 (𝐹₀) + 𝑈²(𝐹₀) + ⋅ ⋅ ⋅ + 𝑈^𝑛(𝐹₀) + ⋅ ⋅ ⋅ , (49) and it is known that the powers of𝑈are also integral opera- tors. In fact

𝑍_𝑛 = 𝑈^𝑛, 𝑍_𝑛(𝑡) = ∫^𝑡

𝑦0(𝑡)𝑘^(𝑛)₂ (𝑡, 𝜏) Δ𝑥 (𝜏) 𝑑𝜏, (𝑛 = 1, 2, . . .) ,

(50)

where𝑘^(𝑛)₂ is the iterated kernel.

Substituting(50)into(48)we obtain an expression for the solution of(47):

Δ𝑥^∗= 𝐹₀(𝑡) +∑^∞

𝑗=1

∫^𝑡

𝑦0(𝑡)𝑘^(𝑗)₂ (𝑡, 𝜏) 𝐹₀(𝜏) 𝑑𝜏. (51) Next, we show that the series in(51)is convergent uniformly for all𝑡 ∈ [𝑡₀, 𝑇]. Since

󵄨󵄨󵄨󵄨𝑘²(𝑡, 𝜏)󵄨󵄨󵄨󵄨 =󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑘₁(𝑡, 𝜏) 𝑥₀(𝜏) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

≤󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨ℎ (𝑡, 𝜏)

𝑥₀(𝜏)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 𝑘 (𝑡, 𝜏)

𝑥₀(𝜏) 𝐺 (𝑡)󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 𝑐¹𝐻₁+ 𝑐₁𝑐₃𝐻₂. (52)

Let𝑀 = 𝑐₁𝐻₁+ 𝑐₁𝑐₃𝐻₂; then by mathematical induction we get

󵄨󵄨󵄨󵄨󵄨𝑘⁽²⁾² (𝑡, 𝜏)󵄨󵄨󵄨󵄨󵄨 ≤ ∫

𝑡

𝑦0(𝑡)󵄨󵄨󵄨󵄨𝑘²(𝑡, 𝑢) 𝑘₂(𝑢, 𝜏)󵄨󵄨󵄨󵄨 𝑑𝑢 ≤ 𝑀²(𝑡 − 𝐻₄) (1)! ,

󵄨󵄨󵄨󵄨󵄨𝑘⁽³⁾² (𝑡, 𝜏)󵄨󵄨󵄨󵄨󵄨 ≤ ∫

𝑡

𝑦0(𝑡)󵄨󵄨󵄨󵄨󵄨𝑘²(𝑡, 𝑢) 𝑘⁽²⁾₂ (𝑢, 𝜏)󵄨󵄨󵄨󵄨󵄨 𝑑𝑢 ≤ 𝑀³(𝑡 − 𝐻₄)² (2)! , ...

󵄨󵄨󵄨󵄨󵄨𝑘^(𝑛)2 (𝑡, 𝜏)󵄨󵄨󵄨󵄨󵄨 ≤ ∫

𝑡

𝑦0(𝑡)󵄨󵄨󵄨󵄨󵄨𝑘²(𝑡, 𝑢) 𝑘^(𝑛−1)₂ (𝑢, 𝜏)󵄨󵄨󵄨󵄨󵄨 𝑑𝑢

≤ 𝑀^𝑛(𝑡 − 𝐻₄)^𝑛−1 (𝑛 − 1)! ,

(𝑛 = 1, 2, . . .) ; (53) then

󵄩󵄩󵄩󵄩𝑈^𝑛󵄩󵄩󵄩󵄩 = max

𝑡∈[𝑡0,𝑇]∫^𝑡

𝑦0(𝑡)󵄨󵄨󵄨󵄨󵄨𝑘^(𝑛)2 (𝑡, 𝜏)󵄨󵄨󵄨󵄨󵄨 𝑑𝜏 ≤𝑀^𝑛(𝑇 − 𝐻₄)^(𝑛−1) (𝑛 − 1)! .

(54) Therefore the𝑛th root test of the sequence yields

√‖𝑈𝑛 ^𝑛‖ ≤ 𝑀 (𝑇 − 𝐻₄)^1−1/𝑛

√(𝑛 − 1)!𝑛 󳨀󳨀󳨀→_{𝑛 → ∞}0. (55) Hence𝜌 = 1/lim_{𝑛 → ∞}√‖𝑈^{𝑛 𝑛}‖ = ∞and a Volterra integral equations(17)has no characteristic values. Since the series in(51)converges uniformly(48)can be written in terms of resolvent kernel of(17):

Δ𝑥^∗ = 𝐹₀+ ∫^𝑡

𝑦0(𝑡)Γ₀(𝑡, 𝜏) 𝐹₀(𝜏) 𝑑𝜏, (56) where

Γ₀(𝑡, 𝜏) =∑^∞

𝑗=1𝑘^(𝑗)₂ (𝑡, 𝜏) . (57)

Since the series in(57)is convergent we obtain

󵄩󵄩󵄩󵄩Γ⁰󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩𝐵(𝐹⁰)󵄩󵄩󵄩󵄩 ≤∑^∞

𝑗=1󵄩󵄩󵄩󵄩󵄩𝑈^𝑗󵄩󵄩󵄩󵄩󵄩 ≤

∑∞

𝑗=1𝑀^𝑗(𝑇 − 𝐻)^𝑗−1

(𝑗 − 1)! ≤ 𝜂₂. (58)

To establish the validity of second condition, let us represent operator equation

𝑃 (𝑋) = 0, (59)

as in(41)and its the successive approximations is

𝑋_𝑛+1= 𝑆 (𝑋_𝑛) , (𝑛 = 0, 1, 2, . . .) . (60)

For initial guess𝑋₀we have

𝑆 (𝑋₀) = 𝑋₀− Γ₀𝑃 (𝑋₀) . (61)

From second condition of (Theorem 1) we have

󵄩󵄩󵄩󵄩Γ⁰𝑃 (𝑋₀)󵄩󵄩󵄩󵄩 =󵄩󵄩󵄩󵄩𝑆(𝑋⁰) − 𝑋₀󵄩󵄩󵄩󵄩

= 󵄩󵄩󵄩󵄩𝑋¹− 𝑋₀󵄩󵄩󵄩󵄩 = ‖Δ𝑋‖ ≤ 𝜂1 + 𝐾𝜂. (62) In addition, we need to show that‖𝑃^󸀠󸀠(𝑋)‖ ≤ 𝜂₁for all 𝑋 ∈ Ω₀ where 𝜂₁ is defined in (38). It is known that the second derivative 𝑃^󸀠󸀠(𝑋₀)(𝑋, 𝑋) of the nonlinear operator 𝑃(𝑋) is described by 3-dimensional array 𝑃^󸀠󸀠(𝑋₀)𝑋𝑋 = (𝐷₁, 𝐷₂)(𝑋, 𝑋), which is called bilinear operator; that is, 𝑃^󸀠󸀠(𝑋₀)(𝑋𝑋) = 𝐵(𝑋₀, 𝑋, 𝑋)where

𝑃^󸀠󸀠(𝑋₀) (𝑋, 𝑋)

=lim

𝑠 → 0

1

𝑠[𝑃^󸀠(𝑥₀+ 𝑠𝑋) − 𝑃^󸀠(𝑋₀)]

= {lim

𝑠 → 0

1 𝑠[(𝜕𝑃₁

𝜕𝑥 (𝑥₀+ 𝑠𝑥, 𝑦₀+ 𝑠𝑦) − 𝜕𝑃₁

𝜕𝑥 (𝑥₀, 𝑦₀)) 𝑥 + (𝜕𝑃₁

𝜕𝑦 (𝑥₀+ 𝑠𝑥, 𝑦₀+ 𝑠𝑦) − 𝜕𝑃₁

𝜕𝑦 (𝑥₀, 𝑦₀)) 𝑦] ,

𝑠 → 0lim 1 𝑠[(𝜕𝑃₂

𝜕𝑥 (𝑥₀+ 𝑠𝑥, 𝑦₀+ 𝑠𝑦) − 𝜕𝑃₂

𝜕𝑥 (𝑥₀, 𝑦₀)) 𝑥 + (𝜕𝑃₂

𝜕𝑦 (𝑥₀+ 𝑠𝑥, 𝑦₀+ 𝑠𝑦) −𝜕𝑃₂

𝜕𝑦 (𝑥₀, 𝑦₀)) 𝑦]}