Using Variational Iteration Method for Solving Linear Fuzzy Random Ordinary Differential Equations

In this section, the basic concepts related to this work will be presented, which are including essentially two topics, namely random process and then fuzzy logic.

2.1 Random variables concept

A random variable is a real-valued function X(ω), $\omega \in \Omega$, that is measurable in terms of the probability measure P, where Ω is the sample space [28, 29]. The family of random variables x_t(ω) (or just x_t or x(ω) or x(t, ω)) for two variables is called stochastic process. Assuming $\mathrm{t} \in\left[t_0, \mathrm{~T}\right] \subset[0, \infty), \omega \in \Omega$ has a common probability [30], then a stochastic Process W_t, for every $\mathrm{t} \in[0, \infty)$, is a Brownian motion or Wiener process if [31]:

1. $P\left(\left\{\omega \in \Omega \mid w_0(\omega)=0\right\}\right)=1$.

2. For $0<t_0<t_1<\ldots<, t_3$ the increments $W_{t_1}-W_{t_0}, W_{t_2}-$ $W_{t_1}, \ldots, W_{t_n}-W_{t_{n-1}}$   are independent.

3. For an arbitrary t and h>0, then W_t+h-W_t has a normal distribution with mean 0 and variance h.

It is possible to study the convergence of the sequence of random variables denoted by $\left\{x_n(\omega)\right\}$, $\mathrm{n} \in \mathbb{N}$, using a variety of methods. One of these methods is known as converges with probability one. (Denoted by P-w.p.1 or w.p.1), which is also referred to as almost sure convergence (for short a.s.) if:

$P\left(\left\{\omega \in \Omega: \lim _{n \rightarrow \infty} x_n(\omega)=x(\omega)\right\}\right)=1$

Also, another type of convergence is the converges in probability to x(ω), which satisfies for all $\varepsilon>0$ [29]:

$\lim _{n \rightarrow \infty} P\left(\left\{\omega \in \Omega:\left|x_n(\omega)-x(\omega)\right| \geq \varepsilon\right\}\right)=0$

2.2 Fuzzy set theory

Fuzzy set theory is a generalization of the classical or crisp sets. Suppose that X is the universal set, which is nonempty and $\tilde{A}$ be a fuzzy subset of X, the function $\mu_{\tilde{A}}$ is used to denote the membership function or the grades of an element x which belongs to $\tilde{A}$, which is a real valued function from X onto the unit interval [0,1]. The fuzzy set $\tilde{A}$ can be represented as a set of ordered pairs in the form [32]:

$\tilde{A}=\left\{\left(x, \mu_{\tilde{A}}(x)\right): x \in X, \mu_{\tilde{A}}(x) \in[0,1]\right\}$.

Fuzzy numbers (FN) are special cases of a fuzzy sets, which is defined over the field of real numbers $\mathbb{R}$. In addition, among the main notations in fuzzy set theory, which are of importance in this work, are the a-cut or a-level of a fuzzy set $\tilde{A}$ is the set of all $\mathrm{x} \in \mathrm{X}$, such that $\mu_{\tilde{A}}(x) \geq \alpha$. Convex fuzzy set if:

$\mu_{\tilde{A}}(\lambda x+(1-\lambda) y) \geq \min \left\{\mu_{\tilde{A}}(x), \mu_{\tilde{A}}(y)\right\}$,

for all $\lambda \in[0,1]$, while normal fuzzy sets are those sets with maximum membership value equals 1, otherwise it may be normalized. Thus, a fuzzy number assumed to be convex normalized fuzzy subset $\tilde{A}$ of $\mathbb{R}$ with a piecewise continuous membership function [32].

Again, an arbitrary FN may be represented in parametric form using a-level sets as an ordered pair of functions $(\underline{u}(\alpha), \bar{u}(\alpha)), \quad 0 \leq \alpha \leq 1$; which satisfy the following requirements [33]:

(i) $\underline{u}(\alpha)$ is a bounded right continuous non-decreasing function over [0,1].

(ii) $\bar{u}(\alpha)$ is a bounded left continuous non-increasing function over [0,1].

(iii) $\underline{u}(\alpha) \leq \bar{u}(\alpha) ; 0 $\tilde{v}=(\underline{v}(\alpha), \bar{v}(\alpha))$\leq \alpha \leq 1:$

A crisp number a is simply represented by $\underline{u}(\alpha)=\bar{u}(\alpha)=\mathrm{a}$; $0 \leq \alpha \leq 1$ and for an arbitrary two fuzzy numbers $\tilde{u}=$ $(u(\alpha) ; \bar{u}(\alpha))$ and $\tilde{v}=(\underline{v}(\alpha), \bar{v}(\alpha))$, the following algebraic operations may be carried on:

(a) $\tilde{u}=\tilde{v}$ if and only if $\underline{u}(\alpha)=\underline{v}(\alpha)$; and $\bar{u}(\alpha)=\bar{v}(\alpha)$.

(b) $\tilde{u} \oplus \tilde{v}=(\underline{u \oplus v} ; \overline{u \oplus v})=(\underline{u}(\alpha)+(\underline{v}(\alpha) ; \bar{u}(\alpha)+\bar{v}(\alpha))$.

(c) $k \odot \tilde{u}=\left\{\begin{array}{l}(\underline{k \odot u}, \overline{k \odot u})=(k \underline{u}(\alpha), k \bar{u}(\alpha)), k \geq 0 \\ \left(\underline{k \odot u}, \overline{k \odot u}\right)=(k \bar{u}(\alpha), k \underline{u}(\alpha)), k<0\end{array}\right.$

Now, the general form of the n-th order FRODE with constant coefficients related to Eq. (1) is given [34, 35]:

$\begin{aligned} \tilde{x}^{(n)}(t, \omega)+c_{n-1} \, \tilde{x}^{(n-1)}(t, \omega) & +\cdots+c_1 \tilde{x}^{\prime}(t, \omega) +c_0 \tilde{x}(t, \omega)=\widetilde{R}(t, \omega), t \geq 0\end{aligned}$               (3)

with initial conditions:

$\tilde{x}(0, \omega)=\tilde{b}_0, \tilde{x}^{\prime}(0, \omega)=\tilde{b}_1, \ldots, \tilde{x}^{(n-1)}(0, \omega)=\tilde{b}_{n-1}$

where, all $0 \leq i \leq n-1$ have real constants called ci's and $\tilde{x}$ is the solution that will be described as a fuzzy function, $\tilde{b}_i$, for all $i$ $=0,1, \ldots, n-1$ are triangular fuzzy numbers and $\widetilde{R}$ is any given function plays the role of the nonhomgeneuous term.

Also, we can write Eq. (3) in the terms of lower and upper solutions, which is related to the a-levels of the fuzzy solution, and is given as:

$\begin{gathered}(\left.\underline{x}^{(n)}(t ; \omega, \alpha), \bar{x}^{(n)}(t ; \omega, \alpha)\right.)+ \\ c_{n-1}\left(\underline{x}^{(n-1)}(t ; \omega, \alpha), \bar{x}^{(n-1)}(t ; \omega, \alpha)\right) \\ +\cdots+c_1\left(\underline{x}^{\prime}(t ; \omega, \alpha), \overline{x^{\prime}}(t ; \omega, \alpha)\right) \\ +c_0(\underline{x}(t ; \omega, \alpha), \bar{x}(t ; \omega, \alpha)) \\ =(\underline{R}(t ; \omega, \alpha), \bar{R}(t ; \omega, \alpha))\end{gathered}$         (4)

Thus, three cases related to Eq. (3) are considered depending on the linear coefficients of the differential equation [34, 36, 37]:

Case (i): When all of the coefficients in Eq. (4), from $c_{n-1}, c_{n-2}, \ldots, c_1, c_0$, are positive. Next, the new equations are rewritten in terms of the lower and upper bounds as:

$\begin{gathered}\underline{x}^{(n)}(t ; \omega, \alpha)+c_{n-1} \underline{x}^{(n-1)}(t ; \omega, \alpha)+\cdots+ \\ c_1 \underline{x}^{\prime}(t ; \omega, \alpha)+c_0 \underline{x}(t ; \omega, \alpha)=\underline{R}(t ; \omega, \alpha), t \geq 0, \alpha \in {[0,1]}\end{gathered}$            (5)

with initial conditions:

$\underline{x}(0, \omega, \alpha)=\underline{b}_0, \underline{x}^{\prime}(0, \omega, \alpha)=\underline{b}_1, \ldots, \underline{x}^{(n-1)}(0, \omega, \alpha)=\underline{b}_{n-1}$

and:

$\begin{aligned} \bar{x}^{(n)}(t ; \omega, \alpha)+c_{n-1}\, \bar{x}^{(n-1)}(t ; \omega, \alpha)+\cdots +c_1 \overline{x^{\prime}}(t ; \omega, \alpha)+c_0 \bar{x}(t ; \omega, \alpha) =\bar{R}(t ; \omega, \alpha)\end{aligned}$                (6)

with initial conditions:

$\bar{x}(0 ; \omega, \alpha)=\bar{b}_0, \bar{x}^{\prime}(0 ; \omega, \alpha)=\bar{b}_1, \ldots, \bar{x}^{(n-1)}(0 ; \omega, \alpha)=\bar{b}_{n-1}$  

Case (ii): When some of the coefficients in Eq. (4), say $c_{m-1}\, , \ldots, c_{m-n}$ are positive and $c_{m-n-1}\,\, , c_{m-n-2}\,\, , \ldots, c_1, c_0$ are negative, then one may obtain the lower- and upper-bound equations as:

If some of the coefficients in Eq. (4), such as $c_{m-1}\, , \ldots, c_{m-n}$, have positive values, but other coefficients in the equation, such as $c_{m-n-1}\,\, , c_{m-n-2}\,\, , \ldots, c_1, c_0$ have negative values and so the equations for the lower and upper bounds may be written as follows:

$\begin{gathered}\underline{x}^{(n)}(t ; \omega, \alpha)+c_{n-1} \underline{x}^{(n-1)}(t ; \omega, \alpha)+\cdots+ \\ c_{n-m} \underline{x}^{(n-m)}(t ; \omega, \alpha)+c_{n-m-1}\,\, \bar{x}^{(n-m-1)}\, (t ; \omega, \alpha)+ \\ \cdots+c_0 \bar{x}(t ; \omega, \alpha)=\underline{R}(t ; \omega, \alpha), t \geq 0, \alpha \in[0,1]\end{gathered}$         (7)

with initial conditions:

$\begin{gathered}\underline{x}^{(n-1)}\, (0 ; \omega, \alpha)=\underline{b}_{n-1}, \ldots, \underline{x}^{(n-m)}(0 ; \omega, \alpha)=\underline{b}_{n-m} \,  , \\ \bar{x}^{(n-m-1)}\, (0 ; \omega, \alpha)=\bar{b}_{n-m-1}\, , \ldots, \bar{x}^{\prime}(0 ; \omega, \alpha)=\bar{b}_1, \bar{x}(0 ; \omega, \alpha) =\bar{b}_0\end{gathered}$

and the upper solution:

$\begin{gathered}\bar{x}^{(n)}(t ; \omega, \alpha)+c_{n-1} \bar{x}^{(n-1)}\, (t ; \omega, \alpha)+\cdots+ \\ c_{n-m} \bar{x}^{(n-m)}(t ; \omega, \alpha)+c_{n-m-1}\, \underline{x}^{(n-m-1)}\, (t ; \omega, \alpha)+ \\ \cdots+c_0 \underline{x}(t ; \omega, \alpha)=\bar{R}(t ; \omega, \alpha)\end{gathered}$             (8)

with initial conditions:

$\begin{gathered}\bar{x}^{(n-1)}(0 ; \omega, \alpha)=\bar{b}_{n-1}, \ldots, \bar{x}^{(n-m)}(0 ; \omega, \alpha)= \\ \bar{b}_{n-m} \underline{x}^{(n-m-1)}(0 ; \omega, \alpha)=\underline{b}_{n-m-1}, \ldots, \underline{x}^{\prime}(0 ; \omega, \alpha)= \\ \underline{b}_1, \underline{x}(0 ; \omega, \alpha)=\underline{b}_0\end{gathered}$

Case (iii): When all of the coefficients in the Eq. (4), $c_{n-1}, c_{n-2}, \ldots, c_1, c_0$, are negative. Then after, we will be able to obtain the lower- and upper-bound equations that are as follows:

$\begin{gathered}\underline{x}^{(n)}(t ; \omega, \alpha)+c_{n-1} \bar{x}^{(n-1)}(t ; \omega, \alpha)+\cdots+ \\ c_1 \bar{x}^{\prime(t ; \omega, \alpha)}+c_0 \bar{x}(t ; \omega, \alpha)=\underline{R}(t ; \omega, \alpha), t \geq 0, \alpha \in[0,1]\end{gathered}$                      (9)

with initial conditions:

$\bar{x}(0, \omega, \alpha)=\bar{b}_0, \bar{x}^{\prime}(0, \omega, \alpha)=\bar{b}_1, \ldots, \bar{x}^{(n-1)}(0, \omega, \alpha)=\bar{b}_{m-1}$

and

$\begin{gathered}\bar{x}^{(n)}(t ; \omega, \alpha)+c_{n-1} \underline{x}^{(n-1)}(t ; \omega, \alpha)+\cdots+ \\ c_1 \underline{x}^{\prime}(t ; \omega, \alpha)+c_0 \underline{x}(t ; \omega, \alpha)=\bar{R}(t ; \omega, \alpha)\end{gathered}$      (10)

and the initial conditions:

$\underline{x}(0 ; \omega, \alpha)=\underline{b}_0, \underline{x}^{\prime}(0 ; \omega, \alpha)=\underline{b}_1, \ldots, \underline{x}^{(n-1)}(0 ; \omega, \alpha)=\underline{a}_{n-1}$

It is notable that there are no certain scenario applications for the above three cases, except that the application of these cases depends on the signs of the coefficients $c_{n-1}, c_{n-2}, \ldots, c_1, c_0$ in Eq. (3).

In the previous section, we had presented an alternative approach of the VIM [40] for solving FRODEs, which can be used, in an effective and reliable way to handle the FRODEs in operator form given by Eq. (11). Thus, in this section, and according to the above three cases, we will consider only the convergence of the first case of the approximation using VIM (15) to the exact solution of the FRODEs (1), however the other two cases may be verified similarly.

When all the coefficients $c_0, c_1, \ldots, c_{n-1}$ are positive. Then the correction functional to be determined as an approximate solution to the problem under consideration using the VIM for all m=0, 1, … consists of the following lower and upper solutions:

$\begin{gathered}\underline{x}_{m+1}(t ; \omega, \alpha)=\underline{x}_m(t ; \omega, \alpha)+\frac{(-1)^n}{(n-1) !} \int_{t_0}^T(s- \\ t)^{n-1}\left\{\underline{x}_m^{(n)}(s ; \omega, \alpha)+c_{n-1} \underline{x}_m^{(n-1)}(s ; \omega, \alpha)+\cdots+\right. \\ \left.c_1 \underline{x}_m^{\prime}(s ; \omega, \alpha)+c_0 \underline{x}_m(s ; \omega, \alpha)-\underline{R}(s ; \omega, \alpha)\right\} d s \\ \bar{x}_{m+1}(t ; \omega, \alpha)=\bar{x}_m(t ; \omega, \alpha)+\frac{(-1)^n}{(n-1) !} \int_{t_0}^T(s- \\ t)^{n-1}\left\{\bar{x}_m^{(n)}(s ; \omega, \alpha)+c_{n-1} \bar{x}_m^{(n-1)}(s ; \omega, \alpha)+\cdots+\right. \\ \left.c_1 \bar{x}_m^{\prime}(s ; \omega, \alpha)+c_0 \bar{x}_m(s ; \omega, \alpha)-\bar{R}(s ; \omega, \alpha)\right\} d s\end{gathered}$

The above FRODEs is given subjected to the following initial conditions:

$\bar{x}^k(0 ; \omega, \alpha)=\bar{x}_0^k, k=0,1, \ldots, n-1$               (21)

$\underline{x}^k(0 ; \omega, \alpha)=\underline{x}_0^k, k=0,1, \ldots, n-1$              (22)

where, $\bar{x}_0^k$ and $\underline{x}_0^k$ are given real numbers for all k = 0, 1, …, n - 1

Simply, we will consider the convergence analysis for the upper case and the following operator is defined to be used in the proof, which depends on the Banach fixed point theorem:

$\begin{gathered}A(\bar{x}(t ; \omega, \alpha))=\frac{(-1)^n}{(n-1) !} \int_{t_0}^T(s-t)^{n-1}\left\{\bar{x}_m^{(n)}(s ; \omega, \alpha)+\right. \\ c_{n-1} \bar{x}_m^{(n-1)}(s ; \omega, \alpha)+\cdots+c_1 \bar{x}_m^{\prime}(s ; \omega, \alpha)+ \\ \left.c_0 \bar{x}_m(s ; \omega, \alpha)-\bar{R}(s ; \omega, \alpha)\right\} d s\end{gathered}$        (23)

Also, define the following new components $\bar{v}_k$ $(t ; \omega, \alpha)$, k = 0, 1, …; as follows:

$\begin{gathered}\bar{v}_0(t ; \omega, \alpha)=\bar{x}_0(t ; \omega, \alpha)=\bar{x}_0 \\ \bar{v}_1(t ; \omega, \alpha)=A\left[\bar{v}_0(t ; \omega, \alpha)\right] \\ \bar{v}_2(t ; \omega, \alpha)=A\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)\right] \\ \vdots \\ \bar{v}_{k+1}(t ; \omega, \alpha)=\mathrm{A}\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\ldots+\right. \left.\bar{v}_k(t ; \omega, \alpha)\right]\end{gathered}$     (24)

Hence, for the convergence of the VIM, one must have:

$\begin{gathered}\bar{x}(t ; \omega, \alpha)=\lim _{k \rightarrow \infty} \bar{x}_k(t ; \omega, \alpha) \\ =\sum_{k=0}^{\infty} \bar{v}_k(t ; \omega, \alpha)\end{gathered}$

Thus, the solution of problem (11) may be obtained using the following series:

$\bar{x}(t ; \omega, \alpha)=\sum_{k=0}^{\infty} \bar{v}_k(t ; \omega, \alpha)$     (25)

The zeroth (initial guess) approximation $\bar{v}_0(t ; \omega, \alpha)=\bar{x}_0(t ; \omega, \alpha)=\bar{x}_0$ can be freely chosen if it satisfies the initial requirements, as well as, the boundary conditions of the problem that is being considered. The success of the fast convergence of the approximate solution to the exact solution depends on the proper selection of the initial approximation $\bar{v}_0$  (t; w, $\alpha$). However, using the initial values $\bar{x}^k(0 ; \omega, \alpha)=\bar{x}_0^k, k=0,1, \ldots, n-1$ are preferably used for the selective zeroth approximation $\bar{v}_0(t ; \omega, \alpha)$ as will be seen later. In an alternative approach, and for simplicity, one can select the initial approximation $\bar{v}_0(t ; \omega, \alpha)$ as:

$\bar{v}_0(t ; \omega, \alpha)=\sum_{k=0}^{\infty} \frac{\bar{x}_0^k}{k!}(t ; \omega, \alpha)$            (26)

As a result, the shortened series up to the m-terms can approximate the exact solution as:

$\bar{x}(t ; \omega, \alpha)=\sum_{k=0}^{\infty} \bar{v}_k(t ; \omega, \alpha)$

For the convergence of the VIM when the alternative method described above is used to solve Eq. (11), in which the sufficient condition of the method and the error estimate are presented. The following theorem is a special case of the Banach fixed point theorem with random variables and fuzzy initial conditions.

Theorem (1): Consider the operator (23) given by:

$\begin{gathered}A(\bar{x}(t ; \omega, \alpha))=\frac{(-1)^n}{(n-1) !} \int_{t_0}^T(s-t)^{n-1}\left\{\bar{x}_m^{(n)}(s ; \omega, \alpha)+\right. \\ c_{n-1} \bar{x}_m^{(n-1)}(s ; \omega, \alpha)+\cdots+c_1 \bar{x}_m^{\prime}(s ; \omega, \alpha)+c_0 \bar{x}_m(s ; \omega, \alpha)- \bar{R}(s ; \omega, \alpha)\} d s\end{gathered}$

to be defined by transforming a Hilbert space H into itself. The series solution is then $\bar{x}(t ; \omega, \alpha)=\sum_{k=0}^{\infty} \bar{v}_k(t ; \omega, \alpha)$ converges if there exists $\gamma \in(0,1)$, such that:

$\begin{gathered}|| A\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\ldots+\bar{v}_{k+1}(t ; \omega, \alpha)\right] \| \leq \\ \gamma\left\|A\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\ldots+\bar{v}_k(t ; \omega, \alpha)\right]\right\|\end{gathered}$

that is:

$\left\|\bar{v}_{k+1}(t ; \omega, \alpha)\right\| \leq \gamma\left\|\bar{v}_k(t ; \omega, \alpha)\right\|, \forall k=0,1, \ldots$

Proof: Define the sequence $\left\{\bar{s}_m(t ; \omega, \alpha)\right\}_{m=0}^{\infty}$ as:

$\left.\begin{array}{rl}\bar{s}_0= & \bar{v}_0(t ; \omega, \alpha) \\ \bar{s}_1= & \bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha) \\ \bar{s}_2= & \bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\bar{v}_2(t ; \omega, \alpha) \\ & \vdots \\ \bar{s}_m= & \bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\cdots+  \bar{v}_m(t ; \omega, \alpha)\end{array}\right\}$         (27)

and hence to show that $\left\{\bar{s}_m(t ; \omega, \alpha)\right\}_{m=0}^{\infty}$ is a Cauchy sequence in the Hilbert space H. For this purpose, consider $m \in \mathbb{N}$, then:

$\begin{aligned}\left\|\bar{s}_{m+1}-\bar{s}_m\right\|= & \left\|\bar{v}_{m+1}(t ; \omega, \alpha)\right\| \\ & \leq \gamma\left\|\bar{v}_m(t ; \omega, \alpha)\right\| \\ & \leq \gamma^2\left\|\bar{v}_{m-1}(t ; \omega, \alpha)\right\| \\ & \vdots \\ \leq & \gamma^{m+1}\left\|\bar{v}_0(t ; \omega, \alpha)\right\|\end{aligned}$                   (28)

Now, for every $m, j \in \mathbb{N}, m \geq j$, we have:

$\begin{aligned} & \left\|\bar{s}_m-\bar{s}_j\right\|=\|\left(\bar{s}_m-\bar{s}_{m-1}\right)+\left(\bar{s}_{m-1}-\bar{s}_{m-2}\right)+\ldots+  \quad\left(\bar{s}_{j+1}-\bar{s}_j\right) \| \\ & \leq\left\|\bar{s}_m-\bar{s}_{m-1}\right\|+\left\|\bar{s}_{m-1}-\bar{s}_{m-2}\right\|+\ldots+\left\|\bar{s}_{j+1}-\bar{s}_j\right\| \\ & \leq \gamma^m\left\|\bar{v}_0(t ; \omega, \alpha)\right\|+\gamma^{m-1}\left\|\bar{v}_0(t ; \omega, \alpha)\right\|+\ldots+ \\ & \gamma^{j+1}\left\|\bar{v}_0(t ; \omega, \alpha)\right\| \leq \frac{1-\gamma^{m-j}}{1-\gamma} \gamma^{j+1}\left\|\bar{v}_0(t ; \omega, \alpha)\right\|\end{aligned}$                        (29)

and since 0<g<1, then:

$\lim _{j m \rightarrow \infty}\left\|\bar{s}_m(t ; \omega, \alpha)-\bar{s}_j(t ; \omega, \alpha)\right\|=0$                  (30)

Therefore, $\left\{\bar{s}_m(t ; \omega, \alpha)\right\}_{m=0}^{\infty}$ is a Cauchy sequence in the Hilbert space $H$, which implies that the series solution $\bar{x}(t ; \omega, \alpha)=\sum_{k=0}^{\infty} \bar{v}_k(t ; \omega, \alpha)$ defined in Eq. (25) will be converge to the exact solution.

Theorem (2): The solution $\sum_{k=0}^{\infty} \bar{v}_k(t ; \omega, \alpha)$ defined by $\bar{x}(t ; \omega, \alpha)=\sum_{k=0}^{\infty} \bar{v}_k(t ; \omega, \alpha)$ converges to the exact solution of the nonlinear problem:

$L\{\tilde{x}(t ; \omega)\}+N\{\tilde{x}(t ; \omega)\}=\mathrm{g}(t ; \omega), t \in[0,1]$

Proof: Let us assume that the series solution Eq. (25) converge to a function $\varphi(t ; \omega, \alpha)$, i.e., $\varphi(t ; \omega, \alpha)=\sum_{k=0}^{\infty} \bar{v}_k(t ; \omega, \alpha)$, then:

$\lim _{j \rightarrow \infty} \bar{v}_j(t ; \omega, \alpha)=0$                  (31)

and therefore:

$\begin{gathered}\sum_{j=0}^m\left[\bar{v}_{j+1}(t ; \omega, \alpha)-\bar{v}_j(t ; \omega, \alpha)\right]=\bar{v}_{m+1}(t ; \omega, \alpha)- \bar{v}_0(t ; \omega, \alpha)\end{gathered}$              (32)

and so, as $j \longrightarrow \infty$, implies:

$\begin{aligned} & \sum_{j=0}^{\infty}\left[\bar{v}_{j+1}(t ; \omega, \alpha)-\bar{v}_j(t ; \omega, \alpha)\right]= \lim _{j \rightarrow \infty} \bar{v}_j(t ; \omega, \alpha)-\bar{v}_0(t ; \omega, \alpha) =-\bar{v}_0(t ; \omega, \alpha)\end{aligned}$        (33)

Applying the operator $L=\frac{d^n}{d t^n}, n \in \mathbb{N}$ to the both sides of Eq. (33), then from Eq. (26) yields to:

$\begin{gathered}\sum_{j=0}^{\infty} L\left[\bar{v}_{j+1}(t ; \omega, \alpha)-\bar{v}_j(t ; \omega, \alpha)\right]=-L\left[\bar{v}_0(t ; \omega, \alpha)\right]\end{gathered}$                 (34)

On the other hand, from definition Eq. (24):

$\begin{gathered}L\left[\bar{v}_{j+1}(t ; \omega, \alpha)-\bar{v}_j(t ; \omega, \alpha)\right]= \\ L\left[A\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\ldots+\bar{v}_j(t ; \omega, \alpha)\right]-\right. \\ \left.A\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\ldots+\bar{v}_{j-1}(t ; \omega, \alpha)\right]\right]\end{gathered}$           (35)

when $j \geq 1$, and so using definition Eq. (23):

$\begin{gathered}L\left[\bar{v}_{j+1}(t ; \omega, \alpha)-\bar{v}_j(t ; \omega, \alpha)\right]=L\left\{\frac{(-1)^n}{(\mathrm{n}-1) !} \int_0^t(s-\right. 1)^{n-1}\left(L\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\cdots+\right.\right. \\ \left.\bar{v}_j(t ; \omega, \alpha)\right]-L\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\cdots+\right. \\ \left.\bar{v}_{j-1}(t ; \omega, \alpha)\right]+N\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\cdots+\right. \\ \left.\bar{v}_j(t ; \omega, \alpha)\right]-N\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\cdots+\right. \\ \left.\left.\left.\left.\bar{v}_{j-1}(t ; \omega, \alpha)\right]\right)\right) \mathrm{ds}\right\}, j \geq 1\end{gathered}$           (36)

Now, the operator A defined in Eq. (23) gives the n^th-fold integral of Eq. (11), since the differential operator $L=\frac{d^n}{d t^n}$ of order n is the left inverse to the n^th-fold integral operator, then Eq. (36) becomes as:

$\begin{aligned} & L\left[\bar{v}_{j+1}(t ; \omega, \alpha)-\bar{v}_j(t ; \omega, \alpha)\right]=L\left[\bar{v}_j(t ; \omega, \alpha)\right]+ \\ & N\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\cdots+\bar{v}_j(t ; \omega, \alpha)\right]- \\ & N\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\cdots+\bar{v}_{j-1}(t ; \omega, \alpha)\right], j \geq 1\end{aligned}$          (37)

Consequently, we have:

$\begin{gathered}\sum_{j=0}^m L\left[\bar{v}_{j+1}(t ; \omega, \alpha)-\bar{v}_j(t ; \omega, \alpha)\right]= L\left[\bar{v}_0(t ; \omega, \alpha)\right]+N\left[\bar{v}_0(t ; \omega, \alpha)-\right. \\ g(t ; \omega)+L\left[\bar{v}_1(t ; \omega, \alpha)\right]+N\left[\bar{v}_0(t ; \omega, \alpha)+\right. \\ \left.\bar{v}_1(t ; \omega, \alpha)\right]-N\left[\bar{v}_0(t ; \omega, \alpha)+L\left[\bar{v}_2(t ; \omega, \alpha)\right]+\right. \\ N\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\bar{v}_2(t ; \omega, \alpha)\right]- \\ N\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)\right]+\cdots+ \\ L\left[\bar{v}_m(t ; \omega, \alpha)\right]+N\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\cdots+\right. \\ \left.\bar{v}_m(t ; \omega, \alpha)\right]-N\left[\bar{v}_0(t ; \omega, \alpha)+\bar{v}_1(t ; \omega, \alpha)+\cdots+\right. \\ \left.\bar{v}_{m-1}(t ; \omega, \alpha)\right]\end{gathered}$      (38)

Therefore:

$\begin{gathered}\sum_{j=0}^{\infty} L\left[\bar{v}_{j+1}(t ; \omega, \alpha)-\bar{v}_j(t ; \omega, \alpha)\right]= L\left[\sum_{i=0}^{\infty} \bar{v}_j(t ; \omega, \alpha)\right]+N\left[\sum_{j=0}^{\infty} \bar{v}_j(t ; \omega, \alpha)\right]-g(t ; \omega)\end{gathered}$              (39)

From Eq. (33) and Eq. (38), it can be observed that $\varphi(t ; \omega, \alpha)=\sum_{j=0}^{\infty} \bar{v}_j(t ; \omega, \alpha)$ is an exact solution of Eq. (11). This completes the proof of the theorem.

Theorem (3): Assume that the series solution $\sum_{k=0}^j \bar{v}_k(t ; \omega, \alpha)$ defined by $\bar{x}(t ; \omega, \alpha)=\sum_{k=0}^{\infty} \bar{v}_k(t ; \omega, \alpha)$ is convergent to the solution $\bar{x}(t ; \omega, \alpha)$. If the truncated series $\sum_{k=0}^j \bar{v}_k(t ; \omega, \alpha)$ is used as an approximation to the solution $\bar{x}(t ; \omega, \alpha)$ of Eq. $(22)$ then the maximum error $\bar{E}_j(t ; \omega, \alpha)$ is estimated as follows:

$\bar{E}_j(t ; \omega, \alpha) \leq \frac{1}{1-\gamma} \gamma^{j+1}\left\|\bar{v}_0(t ; \omega, \alpha)\right\|, 0<\gamma<1$                   (40)

Proof: From inequality (29), we have for $m \geq j$:

$\left\|\bar{S}_m-\bar{s}_j\right\| \leq \frac{1-\gamma^{m-j}}{1-\gamma} \gamma^{j+1}\left\|\bar{v}_0(t ; \omega, \alpha)\right\|$

Now, as $m \longrightarrow \infty$, then $\bar{s}_m \longrightarrow \bar{x}(t ; \omega, \alpha)$ and so:

$\begin{gathered}\left\|\bar{x}(t ; \omega, \alpha)-\sum_{k=0}^j \bar{v}_k(t ; \omega, \alpha)\right\| \leq \frac{1-\gamma^{m-j}}{1-\gamma} \gamma^{j+1}\left\|\bar{v}_0(t ; \omega, \alpha)\right\|\end{gathered}$            (41)

Also, since $0<\gamma<1$, we have $1-\gamma^{m-j}<1$. Therefore, inequality (29) becomes:

$\begin{gathered}\left\|\bar{x}(t ; \omega, \alpha)-\sum_{k=0}^j \bar{v}_k(t ; \omega, \alpha)\right\| \leq  \frac{1}{1-\gamma} \gamma^{j+1}\left\|\bar{v}_0(t ; \omega, \alpha)\right\|\end{gathered}$                (42)

which completes the proof of theorem.