A second-order finite difference scheme for the multi-dimensional nonlinear time-fractional Schrödinger equation

Abstract

This paper is concerned with a linearized second-order finite difference scheme for solving the nonlinear time-fractional Schrödinger equation in d (d = 1,2,3) dimensions. Under a weak assumption on the nonlinearity, the optimal error estimate of the numerical solution is established without any restriction on the grid ratio. Besides the standard energy method, the key tools for analysis include the mathematical induction method, several inverse Sobolev inequalities, and a discrete fractional Gronwall-type inequality. The convergence rate of the proposed scheme is of O(τ² + h²) with time step τ and mesh size h. Numerical results are carried out to confirm the theoretical analysis.

Appendix. Proof of the time-fractional Gronwall inequality given in Lemma 3.1

In this appendix, we present two useful lemmas which are main tools used for proving Lemma 3.1.

Lemma A.1

Let {p_n} be a sequence defined by

$$ p_{0} = \frac{1}{C_{0}^{\sigma}}, \quad p_{n} = \frac{1}{C_{0}^{\sigma}} \sum\limits_{j=1}^{n} (C_{j-1}^{\sigma} - C_{j}^{\sigma}) p_{n-j}, \quad n \ge 1. $$

(A.1)

Then it holds that

$$ \begin{array}{@{}rcl@{}} &&\text{(i)}\quad 0\le p_{n} \le \frac{1}{C_{0}^{\sigma}}, \quad \sum\limits_{j=k}^{n+1} p_{n-j+1} C_{j-k}^{\sigma} = 1, \quad 1 \le k \le n; \end{array} $$

(A.2)

$$ \begin{array}{@{}rcl@{}} &&\text{(ii)}\quad{\Gamma}(2-\alpha) \sum\limits_{j=0}^{n} p_{n-j} \le \frac{(n+1)^{\alpha}}{\Gamma(1+\alpha)}; \end{array} $$

(A.3)

$$ \begin{array}{@{}rcl@{}} &&\text{(iii)}\quad\frac{\Gamma(2-\alpha)}{\Gamma(1+(m-1)\alpha)} \sum\limits_{j=1}^{n} p_{n-j+1} j^{(m-1)\alpha}\\ &&\qquad \leq \frac{(n+1)^{m\alpha}}{\Gamma(1+m\alpha)}, \quad m=1,2,\cdots. \end{array} $$

(A.4)

Proof

(i) Since $C_{0}^{\sigma } \geq C_{1}^{\sigma } \geq {\cdots } \geq C_{j}^{\sigma } \ge 0$ for j ≥ 0, it is easy to verify inductively from (A.1) that $0 \leq p_{n} \leq 1/C_{0}^{\sigma } \ (n \geq 1)$ by mathematical induction. Moreover, we have

$$ {\Phi}_{n} = \sum\limits_{j=1}^{n+1} p_{n-j+1} C_{j-1}^{\sigma} = \sum\limits_{j=0}^{n} p_{n-j} C_{j}^{\sigma} = \sum\limits_{j=1}^{n} p_{n-j} C_{j-1}^{\sigma} = {\Phi}_{n-1}, \qquad n \ge 1. $$

(A.5)

This implies ${\Phi }_{n} = {\Phi }_{0} = p_{0} C_{0}^{\sigma } = 1$ for n ≥ 1. Substituting j = l + k − 1, we further find

$$ \sum\limits_{j=k}^{n+1} p_{n-j+1} C_{j-k}^{\sigma} = \sum\limits_{l=1}^{n-k+2} p_{n-k+2-l} C_{l-1}^{\sigma} = {\Phi}_{n-k+1} = {\Phi}_{n} = 1, \qquad 1 \leq k \leq n. $$

(A.6)

The equality (A.2) is proved.

(ii) To prove (A.3) and (A.4), we introduce an auxiliary function q(t) = t^mα/Γ(1 + mα) for m ≥ 1. Then for j ≥ 1, we have

$$ {\int}_{0}^{j+\sigma} \frac{(j+\sigma-s)^{-\alpha} q^{\prime}(s)}{\Gamma(1-\alpha)} ds = \frac{B(1-\alpha, m\alpha) (j+\sigma)^{(m-1)\alpha}}{\Gamma(1-\alpha) {\Gamma}(m\alpha)} = \frac{(j+\sigma)^{(m-1)\alpha}}{\Gamma(1+(m-1)\alpha)}. $$

(A.7)

Let Q(t) be a quadratic interpolation of q(t) using the points (s − 1,q(s − 1)), (s,q(s)), (s + 1,q(s + 1)) for 1 ≤ s ≤ j, and a linear interpolation of q(t) using the points (j,q(j)), (j + 1,q(j + 1)). We define the approximation error by

$$ \begin{array}{@{}rcl@{}} &&{\int}_{0}^{j+\sigma} \frac{q^{\prime}(s) - Q^{\prime}(s)}{\Gamma(1-\alpha)(j+\sigma-s)^{\alpha}} ds \\ &=& \sum\limits_{k=1}^{j} {\int}_{k-1}^{k} \frac{q^{\prime}(s) - Q^{\prime}(s)}{\Gamma(1-\alpha)(j+\sigma-s)^{\alpha}} ds + {\int}_{j}^{j+\sigma} \frac{q^{\prime}(s) - Q^{\prime}(s)}{\Gamma(1-\alpha)(j+\sigma-s)^{\alpha}} ds \\ &:=& \sum\limits_{k=1}^{j} {R_{k}^{j}} + R_{\sigma}^{j}, \end{array} $$

(A.8)

where

$$ \begin{array}{@{}rcl@{}} {R_{k}^{j}} &=& {\int}_{k-1}^{k} \frac{q^{\prime}(s) - Q^{\prime}(s)}{\Gamma(1-\alpha)(j+\sigma-s)^{\alpha}} ds \\ &=& -\frac{\alpha}{\Gamma(1-\alpha)} {\int}_{k-1}^{k} \frac{q(s) - Q(s)}{(j+\sigma-s)^{\alpha+1}} ds \\ &=& -\frac{\alpha}{\Gamma(1-\alpha)} {\int}_{k-1}^{k} \frac{ \frac{1}{6} q^{\prime\prime\prime}(\xi_{s}) (s-(k-1))(s-k)(s-(k+1)) }{(j+\sigma-s)^{\alpha+1}} ds, \quad\\ && 1 \leq k \leq j, \end{array} $$

(A.9)

$$ \begin{array}{@{}rcl@{}} R_{\sigma}^{j} &=& {\int}_{j}^{j+\sigma} \frac{q^{\prime}(s) - Q^{\prime}(s)}{\Gamma(1-\alpha)(j+\sigma-s)^{\alpha}} ds \\ &=& {\int}_{j}^{j+\sigma} \frac{q^{\prime}(s) - q(j+\frac{1}{2}) + q(j+\frac{1}{2}) - Q^{\prime}(s)}{\Gamma(1-\alpha)(j+\sigma-s)^{\alpha}} ds \\ &=& {\int}_{j}^{j+\sigma} \frac{2q^{\prime\prime}(j+\frac{1}{2})(s-(j+\frac{1}{2})) + q^{\prime\prime\prime}(\xi_{s})(s-(j+\frac{1}{2}))^{2}}{2{\Gamma}(1-\alpha)(j+\sigma-s)^{\alpha}} ds \\ &&- {\int}_{j}^{j+\sigma} \frac{q^{\prime\prime\prime}(\xi_{s})}{24{\Gamma}(1-\alpha)(j+\sigma-s)^{\alpha}} ds \\ &=& {\int}_{j}^{j+\sigma} \frac{q^{\prime\prime\prime}(\xi_{s})(s-(j+\frac{1}{2}))^{2}}{2{\Gamma}(1-\alpha)(j+\sigma-s)^{\alpha}} ds\\ &&- {\int}_{j}^{j+\sigma} \frac{q^{\prime\prime\prime}(\xi_{s})}{24{\Gamma}(1-\alpha)(j+\sigma-s)^{\alpha}} ds. \end{array} $$

(A.10)

Combining (A.7) and (A.8) yields

$$ \frac{(j+\sigma)^{(m-1)\alpha}}{\Gamma(1+(m-1)\alpha)} = \frac{1}{\Gamma(2-\alpha)} \sum\limits_{k=0}^{j} C_{j-k}^{\sigma} (q(k+1) - q(k)) + \sum\limits_{k=1}^{j} {R_{k}^{j}} + R_{\sigma}^{j}. $$

(A.11)

Noting that $q^{\prime \prime \prime }(t) \geq 0$ for m = 1, we have ${R_{k}^{j}} \leq 0$ and

$$ \begin{array}{@{}rcl@{}} R_{\sigma}^{j} &\leq& \frac{q^{\prime\prime\prime}(\xi)}{24{\Gamma}(1-\alpha)} {\int}_{j}^{j+\sigma} \frac{12(s-(j+\frac{1}{2}))^{2} - 1}{(j+\sigma-s)^{\alpha}} ds \\ &=&\frac{q^{\prime\prime\prime}(\xi)}{24{\Gamma}(1-\alpha)} \frac{4(\sigma-1)}{(2\sigma+1)\sigma(2\sigma-1)} \leq 0, \end{array} $$

(A.12)

so we have

$$ \begin{array}{@{}rcl@{}} \frac{1}{\Gamma(2-\alpha)} \sum\limits_{k=0}^{j} C_{j-k}^{\sigma} (q(k+1) - q(k)) \geq 1. \end{array} $$

(A.13)

Multiplying (A.13) by Γ(2 − α)p_n−j and summing it over for j from 0 to n, we have

$$ \begin{array}{@{}rcl@{}} {\Gamma}(2-\alpha) \sum\limits_{j=0}^{n} p_{n-j} &\leq& \sum\limits_{j=0}^{n} p_{n-j} \sum\limits_{k=1}^{j+1} C_{j-k+1}^{\sigma} (q(k) - q(k-1))\\ &= & \sum\limits_{k=1}^{n+1} (q(k) - q(k-1)) \sum\limits_{j=k-1}^{n} p_{n-j} C_{j-k+1}^{\sigma} \\ &= & \sum\limits_{k=1}^{n+1} (q(k) - q(k-1)) \sum\limits_{j=k}^{n+1} p_{n-j+1} C_{j-k}^{\sigma} \\ &= & \sum\limits_{k=1}^{n+1} (q(k) - q(k-1))\\ &= & \frac{(n+1)^{\alpha}}{\Gamma(1+\alpha)}, \end{array} $$

(A.14)

where we the equality (A.2) was used.

(iii) Consider of (A.11), we have

$$ \begin{array}{@{}rcl@{}} \frac{j^{(m-1)\alpha}}{\Gamma(1+(m-1)\alpha)} &\leq &\frac{(j+\sigma)^{(m-1)\alpha}}{\Gamma(1+(m-1)\alpha)} \\ &=&\frac{1}{\Gamma(2-\alpha)} \sum\limits_{k=0}^{j} C_{j-k}^{\sigma} (q(k+1) - q(k)) \\ &&+ \sum\limits_{k=1}^{j} {R_{k}^{j}} + R_{\sigma}^{j}. \end{array} $$

(A.15)

We multiply (A.15) by Γ(2 − α)p_n−j+ 1 and sum the resulting inequality for j from 1 to n to obtain

$$ \begin{array}{@{}rcl@{}} &&\frac{\Gamma(2-\alpha)}{\Gamma(1+(m-1)\alpha)} \sum\limits_{j=1}^{n} p_{n-j+1} j^{(m-1)\alpha}\\ &&\leq \sum\limits_{j=1}^{n} p_{n-j+1} \sum\limits_{k=0}^{j} C_{j-k}^{\sigma}\left( q(k+1) - q(k)\right) \\ &&\quad + {\Gamma}(2-\alpha) \sum\limits_{j=1}^{n} p_{n-j+1} \left( \sum\limits_{k=1}^{j} {R_{k}^{j}} + R_{\sigma}^{j}\right). \end{array} $$

(A.16)

If 1 ≤ m ≤ 1/α, $q^{\prime \prime \prime }(t) \geq 0$, then ${R_{k}^{j}} \leq 0$ and $R_{\sigma }^{j} \le 0$, so (A.4) follows immediately from the above estimate. If m ≥ 1/α, by (A.8), we have

$$ \begin{array}{@{}rcl@{}} {R_{k}^{j}} &=& {\int}_{k-1}^{k} \frac{q^{\prime}(s) - Q^{\prime}(s)}{\Gamma(1-\alpha)(j+\sigma-s)^{\alpha}} ds \\ &=& {\int}_{k-1}^{k} \frac{q^{\prime}(s) - [q(k)-q(k-1) + (q(k-1) - 2q(k) + q(k+1))(s-(k-\frac{1}{2}))]}{\Gamma(1-\alpha)(j+\sigma-s)^{\alpha}} ds \\ &=& {\int}_{k-1}^{k} \frac{q^{\prime}(s) - (q(k)-q(k-1))}{\Gamma(1-\alpha)(j+\sigma-s)^{\alpha}} ds \\ && + {\int}_{k-1}^{k} \frac{(s-(k-\frac{1}{2}))[(q(k)-q(k-1)) - (q(k+1) - q(k))]}{\Gamma(1-\alpha)(j+\sigma-s)^{\alpha}} ds \\ &=& {\int}_{k-1}^{k} \frac{(j+\sigma-s)^{-\alpha}}{\Gamma(1-\alpha)} {\int}_{k-1}^{k} {\int}_{\mu}^{s} q^{\prime\prime}(\eta) d\eta d\mu ds \\ &&+ {\int}_{k-1}^{k} \frac{s-(k-\frac{1}{2})}{\Gamma(1-\alpha)(j+\sigma-s)^{\alpha}} {\int}_{k-1}^{k} q^{\prime}(\mu) d\mu ds \\ && - {\int}_{k-1}^{k} \frac{s-(k-\frac{1}{2})}{\Gamma(1-\alpha)(j+\sigma-s)^{\alpha}} {\int}_{k}^{k+1} q^{\prime}(\mu) d\mu ds \\ &\leq& {\int}_{k-1}^{k} \frac{(j+\sigma-s)^{-\alpha}}{\Gamma(1-\alpha)} {\int}_{k-1}^{k} {\int}_{\mu}^{k} q^{\prime\prime}(\eta) d\eta d\mu ds\\ && + {\int}_{k-1}^{k} \frac{s-(k-\frac{1}{2})}{\Gamma(1-\alpha)(j+\sigma-s)^{\alpha}} {\int}_{k-1}^{k} q^{\prime}(\mu) d\mu ds \\ && - {\int}_{k-1}^{k} \frac{s-(k-\frac{1}{2})}{\Gamma(1-\alpha)(j+\sigma-s)^{\alpha}} {\int}_{k}^{k+1} q^{\prime}(\mu) d\mu ds \\ &=& \frac{a_{j-k+1}^{\sigma}}{\Gamma(2-\alpha)} {\int}_{k-1}^{k} {\int}_{\mu}^{k} q^{\prime\prime}(\eta) d\eta d\mu + \frac{b_{j-k+1}^{\sigma}}{\Gamma(2-\alpha)} {\int}_{k-1}^{k} q^{\prime}(\mu) d\mu \\ && - \frac{b_{j-k+1}^{\sigma}}{\Gamma(2-\alpha)} {\int}_{k}^{k+1} q^{\prime}(\mu) d\mu \end{array} $$

$$ \begin{array}{@{}rcl@{}} &=& \frac{a_{j-k+1}^{\sigma}}{\Gamma(2-\alpha)} {\int}_{k-1}^{k} \frac{k^{m\alpha-1} - \mu^{m\alpha-1}}{\Gamma(m\alpha)} d\mu + \frac{b_{j-k+1}^{\sigma}}{\Gamma(2-\alpha)} {\int}_{k-1}^{k} \frac{\mu^{m\alpha-1}}{\Gamma(m\alpha)} d\mu \\ && - \frac{b_{j-k+1}^{\sigma}}{\Gamma(2-\alpha)} {\int}_{k}^{k+1} \frac{\mu^{m\alpha-1}}{\Gamma(m\alpha)} d\mu \\ &=& \frac{a_{j-k+1}^{\sigma}}{\Gamma(2-\alpha)} {\int}_{k-1}^{k} \frac{k^{m\alpha-1} - \mu^{m\alpha-1}}{\Gamma(m\alpha)} d\mu - \frac{b_{j-k+1}^{\sigma}}{\Gamma(2-\alpha)} {\int}_{k-1}^{k} \frac{k^{m\alpha-1} - \mu^{m\alpha-1}}{\Gamma(m\alpha)} d\mu \\ &&+ \frac{b_{j-k+1}^{\sigma}}{\Gamma(2-\alpha)} {\int}_{k}^{k+1} \frac{k^{m\alpha-1} - \mu^{m\alpha-1}}{\Gamma(m\alpha)} d\mu \\ &\leq& \frac{a_{j-k+1}^{\sigma} - b_{j-k+1}^{\sigma}}{\Gamma(2-\alpha)} {\int}_{k-1}^{k} \frac{k^{m\alpha-1} - \mu^{m\alpha-1}}{\Gamma(m\alpha)} d\mu \\ && + \frac{b_{j-k+1}^{\sigma}}{\Gamma(2-\alpha)} {\int}_{k}^{k+1} \frac{(k+1)^{m\alpha-1} - \mu^{m\alpha-1}}{\Gamma(m\alpha)} d\mu \end{array} $$

(A.17)

and

$$ \begin{array}{@{}rcl@{}} R_{\sigma}^{j} &=& {\int}_{j}^{j+\sigma} \frac{q^{\prime}(s) - Q^{\prime}(s)}{\Gamma(1-\alpha)(j+\sigma-s)^{\alpha}} ds \\ &=&{\int}_{j}^{j+\sigma} \frac{(j+\sigma-s)^{-\alpha}}{\Gamma(1-\alpha)} {\int}_{j}^{j+1} q^{\prime}(s) - q^{\prime}(\mu) d\mu ds \\ &\leq& \frac{a_{0}^{\sigma}}{\Gamma(2-\alpha)} {\int}_{j}^{j+1} \frac{(j+1)^{m\alpha-1} - \mu^{m\alpha-1}}{\Gamma(m\alpha)} d\mu, \end{array} $$

(A.18)

so that

$$ \sum\limits_{k=1}^{j}{R_{k}^{j}} + R_{\sigma}^{j} \leq \sum\limits_{k=0}^{j} C_{j-k}^{\sigma} {\int}_{k}^{k+1} \frac{(k+1)^{m\alpha-1} - \mu^{m\alpha-1}}{\Gamma(m\alpha)} d\mu. $$

(A.19)

Because of

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{j=1}^{n} p_{n-j+1} \sum\limits_{k=0}^{j} C_{j-k}^{\sigma} (q(k+1) - q(k)) \\ &=& \sum\limits_{k=0}^{n}(q(k+1)-q(k)) \sum\limits_{j=k}^{n} p_{n-j+1} C_{j-k}^{\sigma} - p_{n+1} c_{0}^{\sigma} (q(1)-q(0)) \\ &\leq&\sum\limits_{k=0}^{n} (q(k+1) - q(k)) \sum\limits_{j=k}^{n+1} p_{n-j+1} C_{j-k}^{\sigma} \\ &=& \frac{(n+1)^{m\alpha}}{\Gamma(1+m\alpha)}, \end{array} $$

(A.20)

and

$$ \begin{array}{@{}rcl@{}} &&{\Gamma}(2-\alpha) \sum\limits_{j=1}^{n} p_{n-j+1} \left( \sum\limits_{k=1}^{j} {R_{k}^{j}} + R_{\sigma}^{j}\right) \\ &\leq&\sum\limits_{j=1}^{n} p_{n-j+1} \sum\limits_{k=0}^{j} C_{j-k}^{\sigma} {\int}_{k}^{k+1} \frac{(k+1)^{m\alpha-1} - \mu^{m\alpha-1}}{\Gamma(m\alpha)} d\mu \\ &=& \sum\limits_{k=0}^{n} {\int}_{k}^{k+1} \frac{(k+1)^{m\alpha-1} - \mu^{m\alpha-1}}{\Gamma(m\alpha)} d\mu \sum\limits_{j=k}^{n} p_{n-j+1} C_{j-k}^{\sigma} - p_{n+1} C_{0}^{\sigma} {{\int}_{0}^{1}} \frac{1-\mu^{m\alpha-1}}{\Gamma(m\alpha)} d\mu \\ &\leq& \sum\limits_{k=0}^{n} {\int}_{k}^{k+1} \frac{(k+1)^{m\alpha-1} - \mu^{m\alpha-1}}{\Gamma(m\alpha)} d\mu \sum\limits_{j=k}^{n+1} p_{n-j+1} C_{j-k}^{\sigma} \\ &=& \sum\limits_{k=0}^{n} \frac{(k+1)^{m\alpha-1}}{\Gamma(m\alpha)} - \frac{(n+1)^{m\alpha}}{\Gamma(1+m\alpha)} \\ &\leq& \frac{(n+1)^{m\alpha}}{\Gamma(1+m\alpha)} - \frac{(n+1)^{m\alpha}}{\Gamma(1+m\alpha)} \\ &=& 0, \end{array} $$

(A.21)

one can immediately get (A.4), and the proof of Lemma A.1 is completed. □

Lemma A.2

Let $\vec {e} = (1,1,\cdots ,1)^{T} \in R^{n+1}$ and

$$ J = 2 {\Gamma}(2-\alpha) \lambda \tau^{\alpha} \begin{bmatrix} 0 & p_{1} & {\cdots} & p_{n-1} & p_{n} \\ 0 & 0 & {\cdots} & p_{n-2} & p_{n-1} \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} & {\vdots} \\ 0 & 0 & {\cdots} & 0 & p_{1} \\ 0 & 0 & {\cdots} & 0 & 0 \end{bmatrix}_{(n+1) \times (n+1)} $$

(A.22)

Then, it holds that

$$ \begin{array}{@{}rcl@{}} &&\text{(i)} \quad J^{l} = 0, \quad l \geq n+1; \end{array} $$

(A.23)

$$ \begin{array}{@{}rcl@{}} &&\text{(ii)} \quad J^{m} \vec{e} \le \frac{1}{\Gamma(1+m\alpha)} \left( (2\lambda t_{n+1}^{\alpha})^{m}, (2\lambda t_{n}^{\alpha})^{m}, \cdots, (2\lambda t_{1}^{\alpha})^{m} \right)^{T}, \quad \\ &&\qquad m = 0,1,2,\cdots; \end{array} $$

(A.24)

$$ \begin{array}{@{}rcl@{}} &&\text{(iii)} \quad \sum\limits_{j=0}^{l} J^{j} \vec{e} = \sum\limits_{j=0}^{n} J^{j} \vec{e} \leq \left( E_{\alpha}(2\lambda t_{n+1}^{\alpha}), E_{\alpha}(2\lambda t_{n}^{\alpha}), \cdots, E_{\alpha}(2\lambda t_{1}^{\alpha}) \right)^{T}, \quad \\ &&\qquad l \geq n+1. \end{array} $$

(A.25)

Proof

The proof is similar to that of Lemma 3.3 in [24], and we here omit it for brevity. □

We now turn back to the proof of Lemma 3.1

Proof Proof of Lemma 3.1

By the definition of $D_{\sigma }^{\alpha }$, we get

$$ \sum\limits_{k=0}^{j} C_{j-k}^{\sigma} (\omega^{k+1} - \omega^{k}) \le {\Gamma}(2-\alpha)\tau^{\alpha} (\lambda_{1} \omega^{j+1} + \lambda_{2} \omega^{j} + \lambda_{3} \omega^{j-1}) + {\Gamma}(2-\alpha)\tau^{\alpha} g^{j}, \quad j \geq 1. $$

(A.26)

Multiplying inequality (A.26) by p_n−j and summing over for j from 1 to n, we have

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{j=1}^{n} p_{n-j} \sum\limits_{k=1}^{j+1} C_{j-k+1}^{\sigma} (\omega^{k} - \omega^{k-1}) \\ &&\leq {\Gamma}(2-\alpha)\tau^{\alpha} \sum\limits_{j=1}^{n} p_{n-j} (\lambda_{1} \omega^{j+1} + \lambda_{2} \omega^{j} + \lambda_{3} \omega^{j-1}) \\ && \quad + {\Gamma}(2-\alpha)\tau^{\alpha} \sum\limits_{j=1}^{n} p_{n-j} g^{j}. \end{array} $$

(A.27)

By using the result (A.2) in Lemma A.1, we obtain

$$ \begin{array}{@{}rcl@{}} && \sum\limits_{j=1}^{n} p_{n-j} \sum\limits_{k=1}^{j+1} C_{j-k+1}^{\sigma} (\omega^{k} - \omega^{k-1}) \\ &=&\sum\limits_{k=1}^{n+1} (\omega^{k} - \omega^{k-1}) {\sum}_{j=k+1}^{n} p_{n-j} C_{j-k+1}^{\sigma} - p_{n} C_{0}^{\sigma} (\omega^{1} - \omega^{0}) \\ &=&\sum\limits_{k=1}^{n+1} (\omega^{k} - \omega^{k-1}) - p_{n} C_{0}^{\sigma} (\omega^{1} - \omega^{0}) \\ &=&\omega^{n+1} - \omega^{0} - p_{n} C_{0}^{\sigma} (\omega^{1} - \omega^{0}), \quad n \geq 1. \end{array} $$

(A.28)

It follows that

$$ \begin{array}{@{}rcl@{}} \omega^{n+1} &\leq&\omega^{0} + p_{n} C_{0}^{\sigma} (\omega^{1} - \omega^{0}) + {\Gamma}(2-\alpha)\tau^{\alpha} \sum\limits_{j=1}^{n} p_{n-j} g^{j} \\ &&+ {\Gamma}(2-\alpha)\tau^{\alpha} \sum\limits_{j=1}^{n} p_{n-j} \left( \lambda_{1} \omega^{j+1} + \lambda_{2} \omega^{j} + \lambda_{3} \omega^{j-1}\right). \end{array} $$

(A.29)

Because of

$$ \frac{a_{0}^{\sigma}}{\Gamma(2-\alpha)\tau^{\alpha}} \left( \omega^{1} - \omega^{0}\right) = D_{\sigma}^{\alpha} \omega^{0} \leq \lambda_{1} \omega^{1} + \lambda_{2} \omega^{0} + g^{0} $$

and

$$ \frac{C_{0}^{\sigma}}{a_{0}^{\sigma}} = \frac{a_{0}^{\sigma} + b_{1}^{\sigma}}{a_{0}^{\sigma}} \leq 6, $$

we get

$$ \begin{array}{@{}rcl@{}} \omega^{n+1} &\leq& \omega^{0} + 6{\Gamma}(2-\alpha)\tau^{\alpha} p_{n} (\lambda_{1} \omega^{1} + \lambda_{2} \omega^{0} + g^{0}) + {\Gamma}(2-\alpha)\tau^{\alpha} \sum\limits_{j=1}^{n} p_{n-j} g^{j} \\ && + {\Gamma}(2-\alpha)\tau^{\alpha} \sum\limits_{j=1}^{n} p_{n-j} (\lambda_{1} \omega^{j+1} + \lambda_{2} \omega^{j} + \lambda_{3} \omega^{j-1}) \\ &=& {\Gamma}(2-\alpha) \lambda_{1} \tau^{\alpha} p_{0} \omega^{n+1} + (1 + 6{\Gamma}(2-\alpha) \lambda_{2} \tau^{\alpha} p_{n} + {\Gamma}(2-\alpha)\lambda_{3} \tau^{\alpha} p_{n-1}) \omega^{0} \\ && + 6{\Gamma}(2-\alpha) \tau^{\alpha} p_{n} g^{0} + {\Gamma}(2-\alpha)\tau^{\alpha} {\sum}_{j=1}^{n} p_{n-j} g^{j} \\ && + 6{\Gamma}(2-\alpha) \lambda_{1} \tau^{\alpha} p_{n} \omega^{1} + {\Gamma}(2-\alpha) \lambda_{1} \tau^{\alpha} \sum\limits_{j=1}^{n-1} p_{n-j} \omega^{j+1} \\ && + {\Gamma}(2-\alpha) \lambda_{2} \tau^{\alpha} \sum\limits_{j=1}^{n} p_{n-j} \omega^{j} + {\Gamma}(2-\alpha) \lambda_{3} \tau^{\alpha} \sum\limits_{j=2}^{n} p_{n-j} \omega^{j-1}. \end{array} $$

(A.30)

It follows that

$$ \begin{array}{@{}rcl@{}} \omega^{n+1} & \leq& 6 \omega^{0} + 12 {\Gamma}(2-\alpha) \tau^{\alpha} \sum\limits_{j=0}^{n} p_{n-j} g^{j} + 12 {\Gamma}(2-\alpha) \lambda_{1} \tau^{\alpha} \sum\limits_{j=0}^{n-1} p_{n-j} \omega^{j+1} \\ && + 2 {\Gamma}(2-\alpha) \lambda_{2} \tau^{\alpha} \sum\limits_{j=1}^{n} p_{n-j} \omega^{j} \\ && + 2 {\Gamma}(2-\alpha) \lambda_{3} \tau^{\alpha} \sum\limits_{j=2}^{n} p_{n-j} \omega^{j-1}, \quad n \geq 1, \end{array} $$

(A.31)

when τ ≤ τ^∗. By using the result (A.3) in Lemma A.1, we obtain

$$ \begin{array}{@{}rcl@{}} {\Gamma}(2-\alpha) \tau^{\alpha} \sum\limits_{j=0}^{n} p_{n-j} g^{j} &\leq& {\Gamma}(2-\alpha) \tau^{\alpha} \max_{0\le j\le n} g^{j} \sum\limits_{j=0}^{n} p_{n-j} \\ &\leq& \frac{t_{n+1}^{\alpha}}{\Gamma(1+\alpha)} \max_{0\leq j\le n} g^{j}, \quad n \geq 1. \end{array} $$

(A.32)

It follows that

$$ \omega^{n+1} \leq {\Psi}_{n} + {\Gamma}(2-\alpha) \left( 12 \lambda_{1} \tau^{\alpha} \sum\limits_{j=0}^{n-1} p_{n-j} \omega^{j+1} + 2 \lambda_{2} \tau^{\alpha} \sum\limits_{j=1}^{n} p_{n-j} \omega^{j} + 2 \lambda_{3} \tau^{\alpha} \sum\limits_{j=2}^{n} p_{n-j} \omega^{j-1} \right), $$

(A.33)

where

$$ {\Psi}_{n} = 6 \omega^{0} + \frac{12t_{n+1}^{\alpha}}{\Gamma(1+\alpha)} \underset{0\leq j\leq n}{\max} g^{j}, $$

(A.34)

and it is easy to get that Ψ_n ≥Ψ_k for n ≥ k ≥ 1. Let V = (ω^n+ 1,ωⁿ,⋯ ,ω¹)^T, then (A.33) can be written in a vector form by

$$ V \leq {\Psi}_{n} \vec{e} + (6\lambda_{1} J_{1} + \lambda_{2} J_{2} + \lambda_{3} J_{3}) V, $$

(A.35)

where

$$ \begin{array}{@{}rcl@{}} J_{1} = 2 {\Gamma}(2-\alpha) \tau^{\alpha} \begin{bmatrix} 0 & p_{1} & {\cdots} & p_{n-1} & p_{n} \\ 0 & 0 & {\cdots} & p_{n-2} & p_{n-1} \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} & {\vdots} \\ 0 & 0 & {\cdots} & 0 & p_{1} \\ 0 & 0 & {\cdots} & 0 & 0 \end{bmatrix}_{(n+1)\times(n+1)}, \end{array} $$

(A.36)

$$ \begin{array}{@{}rcl@{}} J_{2} = 2 {\Gamma}(2-\alpha) \tau^{\alpha} \begin{bmatrix} 0 & p_{0} & {\cdots} & p_{n-2} & p_{n-1} \\ 0 & 0 & {\cdots} & p_{n-3} & p_{n-2} \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} & {\vdots} \\ 0 & 0 & {\cdots} & 0 & p_{0} \\ 0 & 0 & {\cdots} & 0 & 0 \end{bmatrix}_{(n+1)\times(n+1)}, \end{array} $$

(A.37)

$$ \begin{array}{@{}rcl@{}} J_{3} = 2 {\Gamma}(2-\alpha) \tau^{\alpha} \begin{bmatrix} 0 & 0 & {\cdots} & p_{n-3} & p_{n-2} \\ 0 & 0 & {\cdots} & p_{n-4} & p_{n-3} \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} & {\vdots} \\ 0 & 0 & {\cdots} & 0 & 0 \\ 0 & 0 & {\cdots} & 0 & 0 \end{bmatrix}_{(n+1)\times(n+1)}. \end{array} $$

(A.38)

By (A.1), we have

$$ \begin{array}{@{}rcl@{}} p_{i} \leq \frac{C_{0}^{\sigma}}{C_{0}^{\sigma} - C_{1}^{\sigma}} p_{i+1}, \qquad p_{1} \leq \frac{C_{0}^{\sigma}}{C_{1}^{\sigma} - C_{2}^{\sigma}} p_{i+2}, \end{array} $$

(A.39)

therefore,

$$ J_{2} V \leq \frac{C_{0}^{\sigma}}{C_{0}^{\sigma} - C_{1}^{\sigma}} \frac{1}{\lambda} J V, \qquad J_{2} V \leq \frac{C_{0}^{\sigma}}{C_{1}^{\sigma} - C_{2}^{\sigma}} \frac{1}{\lambda} JV, $$

(A.40)

which shows that

$$ V \leq J V + {\Psi}_{n} \vec{e}, $$

(A.41)

where J is defined in (A.22) with $\lambda = 6 \lambda _{1} + \frac {C_{0}^{\sigma } \lambda _{2}}{C_{0}^{\sigma } - C_{1}^{\sigma }} + \frac {C_{0}^{\sigma } \lambda _{3}}{C_{1}^{\sigma } - C_{2}^{\sigma }}$. As a result, we see that

$$ \begin{array}{@{}rcl@{}} V &\leq& JV + {\Psi}_{n} \vec{e} \leq J(JV + {\Psi}_{n} \vec{e}) + {\Psi}_{n} \vec{e} = J^{2}{V} + {\Psi}_{n} \sum\limits_{j=0}^{1} J^{j} \vec{e} \\ &\leq& {\cdots} \leq J^{n+1} V + {\Psi} \sum\limits_{j=0}^{n} J^{j} \vec{e}. \end{array} $$

(A.42)

This together with Lemma A.2 completes the proof. □