1 Introduction

Partial differential equations have played an important role in various scientific areas, such as physics and engineering [1–8]. There are many interesting studies on uniqueness and existence of solutions, based on the theory of fixed points, for fractional nonlinear PDEs and corresponding initial or boundary value problems, as well as for integral equations [9, 10]. Ouyang and Zhu et al. [11–13] studied the time fractional PDEs given below:

$$\begin{aligned} \textstyle\begin{cases} \frac{ _{c} \partial ^{\alpha}}{\partial t^{\alpha}} u(t, x) - a(t) \frac{\partial ^{2}}{\partial x^{2}} u(t, x) = v (t, u( \tau _{1}(t), x), \dots, u(\tau _{l}(t), x) ),\quad t \in [0, T_{0}], \\ u(t, x) = 0, \quad (t, x) \in [0, T_{0}] \times \partial \Omega, \\ u(0, x) = \psi (x), \quad x \in \Omega, \end{cases}\displaystyle \end{aligned}$$

where \(0 < \alpha \leq 1\), the function \(a(t)\) is a diffusion coefficient, l is a positive integer, \(\Omega \subset R^{l}\) is a bounded domain with a smooth boundary ∂Ω, \(\psi \in L^{2}(\Omega )\), and the function \(v: [0, T_{0}] \times R^{l} \rightarrow R\) satisfies certain conditions. Ouyang [11] investigated the existence of the local solutions using Leray–Schauder’s fixed point theorem. Additionally, Zhu et al. [12, 13] converted the above time fractional partial differential equations into a form of the time fractional differential equations in the Banach space \(L^{2}(\Omega )\), and, using Banach’s fixed point theorem and strict contraction principle, derived results on the existence and uniqueness.

Let \(a(x) \in C[0, T]\), \(g: [0, T] \times R \rightarrow R\), and \(f: C[0, T] \rightarrow R\). Very recently, Li [14] studied the uniqueness of solutions for the following nonlinear integro-differential equation with a nonlocal boundary condition and variable coefficients for \(l < \alpha \leq l + 1\):

$$\begin{aligned} \textstyle\begin{cases} _{C} D^{\alpha }u(x) + a(x) I^{\beta} u(x) = g(x, u(x)),\quad x \in [0, T], \\ u(0) = - f(u), \qquad u''(0) = \cdots = u^{(l )}(0) = 0, \\ \int _{0}^{T} u(x) \,d x = \lambda, \end{cases}\displaystyle \end{aligned}$$
(1.1)

where λ is a constant. In particular for \(l = 1\), equation (1.1) turns out to be

$$\begin{aligned} \textstyle\begin{cases} _{C} D^{\alpha }u(x) + a(x) I^{\beta} u(x) = g(x, u(x)), \quad x \in [0, T], \\ u(0) = - f(u), \qquad\int _{0}^{T} u(x) \,d x = \lambda. \end{cases}\displaystyle \end{aligned}$$

This paper aims to study the uniqueness of solutions for the following new equation with \(0 < \alpha \leq 1\) and \(m = 1, 2, \dots \), in the space \(S([0, 1]^{2} )\):

$$\begin{aligned} \textstyle\begin{cases} \frac{ _{c} \partial ^{\alpha}}{\partial t^{\alpha}} u(t, x) + \sum_{i = 1}^{m} \lambda _{i} I_{t}^{\gamma _{i}} I_{x}^{\beta _{i}} u(t, x) = v (t, x, u(t, x)), \quad \gamma _{i} \geq 0, \beta _{i} \geq 0, \\ u(0, x) + u(1, x) - \psi (x) = 0, \quad (t, x) \in [0, 1]^{2}, \end{cases}\displaystyle \end{aligned}$$
(1.2)

where all \(\lambda _{i}\) are constants, \(\psi (x)\) is a continuous function on \([0, 1]\), and \(v: [0, 1]^{2} \times R \rightarrow R\) is a function which satisfies conditions to be given. Equation (1.2) with its initial condition is new and, to the best of our knowledge, has never been investigated earlier.

The remainder of the paper is organized in the following manor. Section 3 studies the uniqueness of solutions for equation (1.2) by the newly introduced generalized multivariate Mittag-Leffler function and Banach’s fixed point theory. Section 4 presents a demonstrative example which illuminates applications of the key results based on the value of a generalized multivariate Mittag-Leffler function calculated by our Python code. Finally, in Sect. 5, we provide a summary of the work.

2 Preliminaries

We define \(I_{t}^{\alpha}\) as the partial Riemann–Liouville fractional integral of order \(\alpha > 0\) [15, 16] given by

$$\begin{aligned} \bigl(I_{t}^{\alpha }u \bigr) (t, x) = \frac{1}{\Gamma (\alpha )} \int _{0}^{t} (t - s)^{\alpha - 1} u(s, x) \,d s, \end{aligned}$$

and \(\frac{ _{c} \partial ^{\alpha}}{\partial t^{\alpha}}\) as the partial Liouville–Caputo fractional derivative of order \(\alpha > 0\) [15] by

$$\begin{aligned} \biggl(\frac{ _{c} \partial ^{\alpha}}{\partial t^{\alpha}} u \biggr) (t, x) = \frac{1}{\Gamma (1 - \alpha )} \int _{0}^{t} (t - s)^{ - \alpha} u_{s}^{\prime }(s, x) \,d s,\quad 0 < \alpha \leq 1. \end{aligned}$$

From [17, 18], we have for \(0 < \alpha \leq 1\),

$$\begin{aligned} & \bigl(I_{t}^{0} u \bigr) (t, x) = u (t, x), \\ & I_{t}^{\alpha } \biggl( \frac{ _{c} \partial ^{\alpha}}{\partial t^{\alpha}} u \biggr) (t, x) = u (t, x) - u(0, x). \end{aligned}$$

The set \(S([0, 1]^{2} )\) is a Banach space equipped with the following norm:

$$\begin{aligned} \lVert u \rVert = \sup_{t \in [0, 1], x \in [0, 1]} \bigl\vert u(t, x) \bigr\vert \quad\text{for } u \in S \bigl([0, 1]^{2} \bigr), \end{aligned}$$

where u is continuous on \([0, 1]^{2}\).

Definition 1

A generalized multivariate Mittag-Leffler function is defined by the following series:

$$\begin{aligned} & E_{(\alpha _{1}, \dots, \alpha _{m}), \epsilon}^{(\beta _{1}, \dots, \beta _{m}), \delta}(\zeta _{1}, \dots, \zeta _{m}) \\ &\quad =\sum_{\mathit{l} = 0}^{\infty }\sum _{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m} = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m} \geq 0 }} \binom{\mathit{l}}{\mathit{l}_{1}, \dots, \mathit{l}_{m}} \frac{\zeta _{1}^{\mathit{l}_{1}} \cdots \zeta _{m}^{\mathit{l}_{m}} }{\Gamma (\alpha _{1} \mathit{l}_{1} + \cdots + \alpha _{m} \mathit{l}_{m} + \epsilon ) \Gamma (\beta _{1} \mathit{l}_{1} + \cdots + \beta _{m} \mathit{l}_{m} + \delta )}, \end{aligned}$$

where \(\alpha _{i}, \epsilon, \delta > 0, \beta _{i} \geq 0, \zeta _{j} \in {\mathcal {C}} \) for \(1 \leq j \leq m\) and

$$\begin{aligned} \binom{\mathit{l}}{\mathit{l}_{1}, \dots, \mathit{l}_{m}} = \frac{\mathit{l}!}{\mathit{l}_{1} ! \cdots \mathit{l}_{m} !}. \end{aligned}$$

In particular,

$$\begin{aligned} E_{(\alpha _{1}, \dots, \alpha _{m}), \epsilon}^{(0, \dots, 0), 1}(\zeta _{1}, \dots, \zeta _{m}) &= E _{(\alpha _{1}, \dots, \alpha _{m}), \epsilon}(\zeta _{1}, \dots, \zeta _{m}) \\ & = \sum_{\mathit{l} = 0}^{\infty }\sum _{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m} = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m} \geq 0 }} \binom{\mathit{l}}{\mathit{l}_{1}, \dots, \mathit{l}_{m}} \frac{\zeta _{1}^{\mathit{l}_{1}} \cdots \zeta _{m}^{\mathit{l}_{m}} }{\Gamma (\alpha _{1} \mathit{l}_{1} + \cdots + \alpha _{m} \mathit{l}_{m} + \epsilon ) }, \end{aligned}$$

which is the multivariate Mittag-Leffler function given in [19] since \(\Gamma (1) = 1\). Moreover,

$$\begin{aligned} E_{\alpha, \epsilon}^{0, 1} (\zeta ) = E_{\alpha, \epsilon}(\zeta ) = \sum _{\mathit{l} = 0}^{\infty } \frac{\zeta ^{l}}{\Gamma (\alpha \mathit{l} + \epsilon )}, \quad\zeta \in { \mathcal {C}}, \end{aligned}$$

which is the well-known two-parameter Mittag-Leffler function.

Babenko’s approach [20] is a highly effective method that can be employed to solve various integral and differential equations [9, 17] by treating a bounded integral operator as a “normal” variable and using the inverse operator to deduce solutions. The method itself is similar to the Laplace transform while working on differential and integral equations with constant coefficients, but it can be applied to equations with continuous and bounded variable coefficients. To show this approach, we will consider the following equation in the space \(C[0, 1]\) (the space of all continuous functions on \([0, 1]\)) for constants a and b:

$$\begin{aligned} \textstyle\begin{cases} _{c} D_{0}^{\beta} u(t) + a {_{c} D_{0}^{\beta _{1}}} u(t) + b I_{0}^{\alpha }u(t) = t^{2}, \quad 0 < \beta _{1} < \beta \leq 1, \alpha > 0, \\ u(0) = 0, \end{cases}\displaystyle \end{aligned}$$
(2.1)

where

$$\begin{aligned} _{c} D_{0}^{\beta} u(t) = \frac{1}{\Gamma (1 - \beta )} \int _{0}^{t} (t - s)^{- \beta} u'(s) \,ds \end{aligned}$$

and

$$\begin{aligned} I_{0}^{\alpha }u(t) = \frac{1}{\Gamma (\alpha )} \int _{0}^{t} (t - s)^{ \alpha - 1} u(s) \,ds. \end{aligned}$$

Obviously,

$$\begin{aligned} I_{0}^{\beta } \bigl( _{c} D_{0}^{\beta} u(t) \bigr) = u(t) - u(0) = u(t). \end{aligned}$$

Applying \(I_{0}^{\alpha}\) to equation (2.1), we have

$$\begin{aligned} \bigl(1 + a I_{0}^{\beta - \beta _{1}} + b I_{0}^{\beta + \alpha} \bigr) u(t) = I_{0}^{\beta }t^{2} = \frac{2}{\Gamma (\beta + 3)} t^{ \beta + 2}. \end{aligned}$$

Considering the inverse operator of \((1 + a I_{0}^{\beta - \beta _{1}} + b I_{0}^{\beta + \alpha} )\), we informally get by Babenko’s technique

$$\begin{aligned} u(t) & = \frac{2}{\Gamma (\beta + 3)} \bigl(1 + a I_{0}^{\beta - \beta _{1}} + b I_{0}^{\beta + \alpha} \bigr)^{-1} t^{\beta + 2} \\ & = \frac{2}{\Gamma (\beta + 3)} \sum_{n = 0}^{\infty }(-1)^{n} \bigl( a I_{0}^{\beta - \beta _{1}} + b I_{0}^{\beta + \alpha} \bigr)^{n} t^{\beta + 2} \\ & = \frac{2}{\Gamma (\beta + 3)} \sum_{n = 0}^{\infty }(-1)^{n} \sum_{n_{1} + n_{2} = n} \binom{n}{n_{1}, n_{2}} a^{n_{1}} I_{0}^{( \beta - \beta _{1}) n_{1}} b^{n_{2}} I_{0}^{(\beta + \alpha ) n_{2}} t^{\beta + 2} \\ & = 2 \sum_{n = 0}^{\infty }(-1)^{n} \sum_{n_{1} + n_{2} = n} \binom{n}{n_{1}, n_{2}} a^{n_{1}} b^{n_{2}} \frac{t^{n_{1} (\beta - \beta _{1}) + n_{2} (\beta + \alpha ) + \beta + 2}}{\Gamma ((\beta - \beta _{1}) n_{1} + (\beta + \alpha )n_{2} + \beta + 3)} \\ & = 2 t^{\beta + 2} \sum_{n = 0}^{\infty }(-1)^{n} \sum_{n_{1} + n_{2} = n} \binom{n}{n_{1}, n_{2}} a^{n_{1}} b^{n_{2}} \frac{t^{(\beta - \beta _{1}) n_{1} + (\beta + \alpha )n_{2}}}{\Gamma ((\beta - \beta _{1}) n_{1} + (\beta + \alpha )n_{2} + \beta + 3)}, \end{aligned}$$

using

$$\begin{aligned} I_{0}^{(\beta - \beta _{1})n_{1} + (\beta + \alpha )n_{2}} t^{\beta + 2} = \frac{\Gamma (\beta + 3) t^{(\beta - \beta _{1}) n_{1} + (\beta + \alpha )n_{2} + \beta + 2}}{\Gamma ((\beta - \beta _{1}) n_{1} + (\beta + \alpha )n_{2} + \beta + 3)}. \end{aligned}$$

This implies that

$$\begin{aligned} \lVert u \rVert & \leq 2 \sum_{n = 0}^{\infty } \sum_{n_{1} + n_{2} = n} \binom{n}{n_{1}, n_{2}} \frac{ \vert a \vert ^{n_{1}} \vert b \vert ^{n_{2}}}{\Gamma ((\beta - \beta _{1}) n_{1} + (\beta + \alpha )n_{2} + \beta + 3)} \\ & = 2 E_{(\beta - \beta _{1}, \beta + \alpha ), \beta + 3} \bigl( \vert a \vert , \vert b \vert \bigr)< + \infty, \end{aligned}$$

which gives that the series solution

$$\begin{aligned} u(t) = 2 t^{\beta + 2} \sum_{n = 0}^{\infty }(-1)^{n} \sum_{n_{1} + n_{2} = n} \binom{n}{n_{1}, n_{2}} a^{n_{1}} b^{n_{2}} \frac{t^{(\beta - \beta _{1}) n_{1} + (\beta + \alpha )n_{2}}}{\Gamma ((\beta - \beta _{1}) n_{1} + (\beta + \alpha )n_{2} + \beta + 3)} \end{aligned}$$

is an element in \(C[0, 1]\).

3 Uniqueness of solutions

Theorem 2

Let \(\psi \in C[0, 1]\), \(\lambda _{i}\) be real constants, \(\beta _{i}, \gamma _{i} \geq 0\) for all \(i = 1, 2, \dots, m\), and \(v: [0, 1]^{2} \times R \rightarrow R\) be a continuous and bounded function. In addition, we assume that \(0 < \alpha \leq 1\) and

$$\begin{aligned} q = 1 - \frac{1}{2} \sum_{i = 1}^{m} \frac{ \vert \lambda _{i} \vert }{\Gamma (\gamma _{i} + \alpha + 1) \Gamma (\beta _{i} + 1)} E_{(\gamma _{1} + \alpha, \dots, \gamma _{m} + \alpha ), 1}^{( \beta _{1}, \dots, \beta _{m}), 1} \bigl( \vert \lambda _{1} \vert , \dots, \vert \lambda _{m} \vert \bigr) > 0. \end{aligned}$$

Then \(u(t, x)\) is a solution to equation (1.2) if and only if it satisfies the following integral equation in the space \(S([0, 1]^{2} )\):

$$\begin{aligned} u(t, x) ={}& \sum_{l = 0}^{\infty }(-1)^{l} \sum_{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m}} \lambda _{1}^{\mathit{l}_{1}} \cdots \lambda _{m}^{\mathit{l}_{m}} \\ &{} \times I_{t}^{l_{1} (\alpha + \gamma _{1}) + \cdots + l_{m} (\alpha + \gamma _{m}) + \alpha} I_{x}^{ \beta _{1} l_{1} + \cdots + \beta _{m} l_{m}} v(t, x, u) \\ &{} - \frac{1}{2}\sum_{l = 0}^{\infty }(-1)^{l} \sum_{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m}} \lambda _{1}^{\mathit{l}_{1}} \cdots \lambda _{m}^{\mathit{l}_{m}} \\ & {}\times I_{t}^{l_{1} ( \alpha + \gamma _{1}) + \cdots + l_{m}(\alpha + \gamma _{m}) } I_{t = 1}^{ \alpha }I_{x}^{\beta _{1} l_{1} + \cdots + \beta _{m} l_{m}} v(t, x, u) \\ & {}+ \frac{1}{2} \sum_{l = 0}^{\infty }(-1)^{l} \sum_{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m }} \lambda _{1}^{\mathit{l}_{1}} \cdots \lambda _{m}^{\mathit{l}_{m}} \\ &{} \times I_{t}^{l_{1} ( \alpha + \gamma _{1}) + \cdots + l_{m}(\alpha + \gamma _{m}) } I_{x}^{ \beta _{1} l_{1} + \cdots + \beta _{m} l_{m}} \psi (x) \\ &{} + \frac{1}{2} \sum_{i = 1}^{m} \lambda _{i} \sum_{l = 0}^{ \infty }(-1)^{l} \sum_{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m }}\lambda _{1}^{l_{1}} \cdots \lambda _{m}^{l_{m}} \\ &{} \times I_{t}^{l_{1} ( \alpha + \gamma _{1}) + \cdots + l_{m}(\alpha + \gamma _{m}) } I_{x}^{ \beta _{1} l_{1} + \cdots + \beta _{m} l_{m}} I_{t = 1}^{\gamma _{i} + \alpha} I_{x}^{\beta _{i}} u. \end{aligned}$$
(3.1)

Furthermore,

$$\begin{aligned} \lVert u \rVert \leq{}& \frac{1}{q} E_{(\gamma _{1} + \alpha, \dots, \gamma _{m} + \alpha ), \alpha + 1}^{(\beta _{1}, \dots, \beta _{m}), 1} \bigl( \vert \lambda _{1} \vert , \dots, \vert \lambda _{m} \vert \bigr) \sup_{(t, x) \in [0, 1] \times [0, 1] u \in R} \bigl\vert v (t, x, u) \bigr\vert \\ &{} + \frac{1}{2 q} \biggl(\frac{1}{ \Gamma (\alpha + 1)} + 1 \biggr)E_{(\gamma _{1} + \alpha, \dots, \gamma _{m} + \alpha ), 1}^{( \beta _{1}, \dots, \beta _{m}), 1} \bigl( \vert \lambda _{1} \vert , \dots, \vert \lambda _{m} \vert \bigr) \\ & {}\times \Bigl(\sup_{(t, x) \in [0, 1] \times [0, 1] u \in R} \bigl\vert v (t, x, u) \bigr\vert + \max_{x \in [0, 1]} \bigl\vert \psi (x) \bigr\vert \Bigr) < + \infty. \end{aligned}$$

Proof

Applying \(I_{t}^{\alpha}\) to equation (1.2), we get

$$\begin{aligned} & I_{t}^{\alpha} \frac{ _{c} \partial ^{\alpha}}{\partial t^{\alpha}} u(t, x) + \sum _{i = 1}^{m} \lambda _{i} I_{t}^{\alpha + \gamma _{i}} I_{x}^{ \beta _{i}} u(t, x) = I_{t}^{\alpha} v \bigl(t, x, u(t, x) \bigr). \end{aligned}$$

This implies that

$$\begin{aligned} & u(t, x) - u(0, x) + \sum_{i = 1}^{m} \lambda _{i} I_{t}^{\alpha + \gamma _{i}} I_{x}^{\beta _{i}} u(t, x) = I_{t}^{\alpha} v \bigl(t, x, u(t, x) \bigr), \quad \text{and} \\ & u(1, x) - u(0, x) + \sum_{i = 1}^{m} \lambda _{i} I_{t = 1}^{ \alpha + \gamma _{i}} I_{x}^{\beta _{i}} u(t, x) = I_{t = 1}^{\alpha} v \bigl(t, x, u(t, x) \bigr). \end{aligned}$$

Using

$$\begin{aligned} - u(0, x) - u(1, x) = - \psi (x), \end{aligned}$$

we get

$$\begin{aligned} u(0, x) = \frac{1}{2} \psi (x) + \frac{1}{2} \sum _{i = 1}^{m} \lambda _{i} I_{t = 1}^{\alpha + \gamma _{i}} I_{x}^{\beta _{i}} u(t, x) - \frac{1}{2} I_{t = 1}^{\alpha} v \bigl(t, x, u(t, x) \bigr). \end{aligned}$$

This further implies that

$$\begin{aligned} & \Biggl(1 + \sum_{i = 1}^{m} \lambda _{i} I_{t}^{\alpha + \gamma _{i}} I_{x}^{\beta _{i}} \Biggr) u(t, x) \\ &\quad = I_{t}^{\alpha} v \bigl(t, x, u(t, x) \bigr) + \frac{1}{2} \psi (x) + \frac{1}{2} \sum _{i = 1}^{m} \lambda _{i} I_{t = 1}^{\alpha + \gamma _{i}} I_{x}^{\beta _{i}} u(t, x) - \frac{1}{2} I_{t = 1}^{ \alpha} v \bigl(t, x, u(t, x) \bigr). \end{aligned}$$

Using Babenko’s method, we deduce that

$$\begin{aligned} & u(t, x) \\ &\quad = \Biggl(1 + \sum_{i = 1}^{m} \lambda _{i} I_{t}^{\alpha + \gamma _{i}} I_{x}^{\beta _{i}} \Biggr)^{-1} \\ & \qquad{}\times \Biggl(I_{t}^{\alpha} v \bigl(t, x, u(t, x) \bigr) + \frac{1}{2} \psi (x) + \frac{1}{2} \sum _{i = 1}^{m} \lambda _{i} I_{t = 1}^{\alpha + \gamma _{i}} I_{x}^{\beta _{i}} u(t, x) - \frac{1}{2} I_{t = 1}^{ \alpha} v \bigl(t, x, u(t, x) \bigr) \Biggr) \\ & \quad= \sum_{l = 0}^{\infty }(-1)^{l} \Biggl( \sum_{i = 1}^{m} \lambda _{i} I_{t}^{\alpha + \gamma _{i}} I_{x}^{\beta _{i}} \Biggr)^{l} \\ &\qquad{} \times \Biggl(I_{t}^{\alpha} v \bigl(t, x, u(t, x) \bigr) + \frac{1}{2} \psi (x) + \frac{1}{2} \sum _{i = 1}^{m} \lambda _{i} I_{t = 1}^{\alpha + \gamma _{i}} I_{x}^{\beta _{i}} u(t, x) - \frac{1}{2} I_{t = 1}^{ \alpha} v \bigl(t, x, u(t, x) \bigr) \Biggr) \\ & \quad= \sum_{l = 0}^{\infty }(-1)^{l} \sum_{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m }} \lambda _{1}^{l_{1}} \cdots \lambda _{m}^{l_{m}} I_{t}^{(\alpha + \gamma _{1}) l_{1} + \cdots + (\alpha + \gamma _{m})l_{m}} I_{x}^{\beta _{1} l_{1} + \cdots + \beta _{m} l_{m} } \\ &\qquad{} \times \Biggl(I_{t}^{\alpha} v \bigl(t, x, u(t, x) \bigr) + \frac{1}{2} \psi (x) + \frac{1}{2} \sum _{i = 1}^{m} \lambda _{i} I_{t = 1}^{\alpha + \gamma _{i}} I_{x}^{\beta _{i}} u(t, x) - \frac{1}{2} I_{t = 1}^{ \alpha} v \bigl(t, x, u(t, x) \bigr) \Biggr) \\ &\quad = \sum_{l = 0}^{\infty }(-1)^{l} \sum_{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m}} \lambda _{1}^{\mathit{l}_{1}} \cdots \lambda _{m}^{\mathit{l}_{m}} \\ &\qquad{} \times I_{t}^{l_{1} (\alpha + \gamma _{1}) + \cdots + l_{m} (\alpha + \gamma _{m}) + \alpha} I_{x}^{ \beta _{1} l_{1} + \cdots + \beta _{m} l_{m}} v(t, x, u) \\ &\qquad{} - \frac{1}{2}\sum_{l = 0}^{\infty }(-1)^{l} \sum_{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m}}\lambda _{1}^{\mathit{l}_{1}} \cdots \lambda _{m}^{\mathit{l}_{m}} \\ &\qquad{} \times I_{t}^{l_{1} ( \alpha + \gamma _{1}) + \cdots + l_{m}(\alpha + \gamma _{m}) } I_{t = 1}^{ \alpha }I_{x}^{\beta _{1} l_{1} + \cdots + \beta _{m} l_{m}} v(t, x, u) \\ &\qquad{} + \frac{1}{2} \sum_{l = 0}^{\infty }(-1)^{l} \sum_{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m }} \lambda _{1}^{\mathit{l}_{1}} \cdots \lambda _{m}^{\mathit{l}_{m}} \\ &\qquad{} \times I_{t}^{l_{1} ( \alpha + \gamma _{1}) + \cdots + l_{m}(\alpha + \gamma _{m}) } I_{x}^{ \beta _{1} l_{1} + \cdots + \beta _{m} l_{m}} \psi (x) \\ & \qquad{}+ \frac{1}{2} \sum_{i = 1}^{m} \lambda _{i} \sum_{l = 0}^{ \infty }(-1)^{l} \sum_{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m }}\lambda _{1}^{l_{1}} \cdots \lambda _{m}^{l_{m}} \\ &\qquad{} \times I_{t}^{l_{1} (\alpha + \gamma _{1}) + \cdots + l_{m}(\alpha + \gamma _{m}) } I_{x}^{ \beta _{1} l_{1} + \cdots + \beta _{m} l_{m}} I_{t = 1}^{\gamma _{i} + \alpha} I_{x}^{\beta _{i}} u, \end{aligned}$$

by the multinomial theorem. We will now show that \(u \in S([0, 1]^{2} )\). Indeed,

$$\begin{aligned} \lVert u \rVert \leq{}& \sum_{l = 0}^{\infty }\sum _{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m} \geq 0 }} \binom{l}{l_{1}, \dots, l_{m }} \frac{ \vert \lambda _{1} \vert ^{l_{1}} \cdots \vert \lambda _{m} \vert ^{l_{m}} }{\Gamma (l_{1} (\alpha + \gamma _{1}) + \cdots + l_{m }(\alpha + \gamma _{m}) + \alpha + 1)} \\ & {}\times \frac{1}{\Gamma (l_{1} \beta _{1} + \cdots + l_{m} \beta _{m} + 1)} \sup_{(t, x) \in [0, 1] \times [0, 1], u \in R} \bigl\vert v(t, x, u) \bigr\vert \\ &{} + \frac{1}{2 \Gamma (\alpha + 1)} \sum_{l = 0}^{\infty } \sum_{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m }} \frac{ \vert \lambda _{1} \vert ^{l_{1}} \cdots \vert \lambda _{m} \vert ^{l_{m}} }{\Gamma (l_{1} (\alpha + \gamma _{1}) + \cdots + l_{m}(\alpha + \gamma _{m}) + 1)} \\ &{} \times \frac{1}{\Gamma (l_{1} \beta _{1} + \cdots + l_{m} \beta _{m} + 1)} \sup_{(t, x) \in [0, 1] \times [0, 1], u \in R} \bigl\vert v(t, x, u) \bigr\vert \\ &{} + \frac{1}{2} \sum_{l = 0}^{\infty } \sum_{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m }} \\ & {}\times \frac{ \vert \lambda _{1} \vert ^{l_{1}} \cdots \vert \lambda _{m} \vert ^{l_{m}} }{\Gamma (l_{1} (\alpha + \gamma _{1}) + \cdots + l_{m }(\alpha + \gamma _{m}) + 1) \Gamma (l_{1} \beta _{1} + \cdots + l_{m} \beta _{m} + 1)} \max_{x \in [0, 1]} \bigl\vert \psi (x) \bigr\vert \\ &{} + \frac{1}{2} \sum_{i = 1}^{m} \frac{ \vert \lambda _{i} \vert }{\Gamma (\gamma _{i} + \alpha + 1) \Gamma (\beta _{i} + 1)} \sum_{l = 0}^{\infty }\sum _{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m }} \\ &{} \times \frac{ \vert \lambda _{1} \vert ^{l_{1}} \cdots \vert \lambda _{m} \vert ^{l_{m}} }{\Gamma (l_{1} (\alpha + \gamma _{1}) + \cdots + l_{m }(\alpha + \gamma _{m}) + 1) \Gamma (l_{1} \beta _{1} + \cdots + l_{m} \beta _{m} + 1)} \lVert u \rVert. \end{aligned}$$

Since

$$\begin{aligned} q = 1 - \frac{1}{2} \sum_{i = 1}^{m} \frac{ \vert \lambda _{i} \vert }{\Gamma (\gamma _{i} + \alpha + 1) \Gamma (\beta _{i} + 1)} E_{(\gamma _{1} + \alpha, \dots, \gamma _{m} + \alpha ), 1}^{( \beta _{1}, \dots, \beta _{m}), 1} \bigl( \vert \lambda _{1} \vert , \dots, \vert \lambda _{m} \vert \bigr) > 0, \end{aligned}$$

we come to

$$\begin{aligned} \lVert u \rVert \leq{}& \frac{1}{q} E_{(\gamma _{1} + \alpha, \dots, \gamma _{m} + \alpha ), \alpha + 1}^{(\beta _{1}, \dots, \beta _{m}), 1} \bigl( \vert \lambda _{1} \vert , \dots, \vert \lambda _{m} \vert \bigr) \sup_{(t, x) \in [0, 1] \times [0, 1] u \in R} \bigl\vert v (t, x, u) \bigr\vert \\ &{} + \frac{1}{2 q} \biggl(\frac{1}{ \Gamma (\alpha + 1)} + 1 \biggr)E_{(\gamma _{1} + \alpha, \dots, \gamma _{m} + \alpha ), 1}^{( \beta _{1}, \dots, \beta _{m}), 1} \bigl( \vert \lambda _{1} \vert , \dots, \vert \lambda _{m} \vert \bigr) \\ &{} \times \Bigl(\sup_{(t, x) \in [0, 1] \times [0, 1] u \in R} \bigl\vert v (t, x, u) \bigr\vert + \max_{x \in [0, 1]} \bigl\vert \psi (x) \bigr\vert \Bigr) < + \infty, \end{aligned}$$

since v is bounded. Hence \(u \in S([0, 1]^{2} )\). This marks the completion of the proof. □

Theorem 3

Let \(\psi \in C[0, 1]\), \(\lambda _{i}\) be real constants, \(\gamma _{i}, \beta _{i} \geq 0\) for all \(i = 1, 2, \dots, m\), and \(v: [0, 1]^{2} \times R \rightarrow R\) be a bounded and continuous function that satisfies the following Lipschitz condition for \({\mathcal {M}}> 0\):

$$\begin{aligned} \bigl\vert v(t, x, u_{1}) - v(t, x, u_{2}) \bigr\vert \leq {\mathcal {M}} \vert u_{1} - u_{2} \vert ,\quad u_{1}, u_{2} \in R. \end{aligned}$$

Furthermore, we suppose that \(0 < \alpha \leq 1\) and

$$\begin{aligned} W = {}&{\mathcal {M}} E_{(\gamma _{1} + \alpha, \dots, \gamma _{m} + \alpha ), \alpha + 1}^{(\beta _{1}, \dots, \beta _{m}), 1} \bigl( \vert \lambda _{1} \vert , \dots, \vert \lambda _{m} \vert \bigr) \\ &{} + \frac{1}{2} \Biggl( \frac{ {\mathcal {M}} }{\Gamma (\alpha + 1)} + \sum _{i = 1}^{m} \frac{ \vert \lambda _{i} \vert }{\Gamma (\gamma _{i} + \alpha + 1)\Gamma (\beta _{i} + 1)} \Biggr) E_{(\gamma _{1} + \alpha, \dots, \gamma _{m} + \alpha ), 1}^{(\beta _{1}, \dots, \beta _{m}), 1} \bigl( \vert \lambda _{1} \vert , \dots, \vert \lambda _{m} \vert \bigr) < 1. \end{aligned}$$

Then there is a unique solution in the space \(S([0, 1]^{2} )\) to equation (1.2).

Proof

Let \({\mathcal {T}}\) be the map** defined on the space \(S([0, 1]^{2} )\) by

$$\begin{aligned} ({\mathcal {T}} u) (t, x) ={}& \sum_{l = 0}^{\infty }(-1)^{l} \sum_{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m}} \lambda _{1}^{\mathit{l}_{1}} \cdots \lambda _{m}^{\mathit{l}_{m}} \\ &{} \times I_{t}^{l_{1} (\alpha + \gamma _{1}) + \cdots + l_{m} (\alpha + \gamma _{m}) + \alpha} I_{x}^{ \beta _{1} l_{1} + \cdots + \beta _{m} l_{m}} v(t, x, u) \\ & {}- \frac{1}{2}\sum_{l = 0}^{\infty }(-1)^{l} \sum_{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m}} \lambda _{1}^{\mathit{l}_{1}} \cdots \lambda _{m}^{\mathit{l}_{m}} \\ &{} \times I_{t}^{l_{1} ( \alpha + \gamma _{1}) + \cdots + l_{m}(\alpha + \gamma _{m}) } I_{t = 1}^{ \alpha }I_{x}^{\beta _{1} l_{1} + \cdots + \beta _{m} l_{m}} v(t, x, u) \\ & {}+ \frac{1}{2} \sum_{l = 0}^{\infty }(-1)^{l} \sum_{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m }} \lambda _{1}^{\mathit{l}_{1}} \cdots \lambda _{m}^{\mathit{l}_{m}} \\ & {}\times I_{t}^{l_{1} ( \alpha + \gamma _{1}) + \cdots + l_{m}(\alpha + \gamma _{m}) } I_{x}^{ \beta _{1} l_{1} + \cdots + \beta _{m} l_{m}} \psi (x) \\ &{} + \frac{1}{2} \sum_{i = 1}^{m} \lambda _{i} \sum_{l = 0}^{ \infty }(-1)^{l} \sum_{ \substack{\mathit{l}_{1} + \cdots + \mathit{l}_{m } = \mathit{l} \\ \mathit{l}_{1} \geq 0, \dots, \mathit{l}_{m } \geq 0 }} \binom{l}{l_{1}, \dots, l_{m }}\lambda _{1}^{l_{1}} \cdots \lambda _{m}^{l_{m}} \\ &{} \times I_{t}^{l_{1} ( \alpha + \gamma _{1}) + \cdots + l_{m}(\alpha + \gamma _{m}) } I_{x}^{ \beta _{1} l_{1} + \cdots + \beta _{m} l_{m}} I_{t = 1}^{\gamma _{i} + \alpha} I_{x}^{\beta _{i}} u. \end{aligned}$$

From the proof of Theorem 2, we claim that \({\mathcal {T}} u \in S([0, 1]^{2} )\). We will prove that \({\mathcal {T}}\) is contractive. In fact, for \(u_{1}, u_{2} \in S([0, 1]^{2} )\), we get from Theorem 2 that

$$\begin{aligned} \lVert{\mathcal {T}} u_{1} - {\mathcal {T}} u_{2} \rVert \leq{}& {\mathcal {M}} E_{(\gamma _{1} + \alpha, \dots, \gamma _{m} + \alpha ), \alpha + 1}^{(\beta _{1}, \dots, \beta _{m}), 1} \bigl( \vert \lambda _{1} \vert , \dots, \vert \lambda _{m} \vert \bigr) \lVert u_{1} - u_{2} \rVert \\ &{} + \frac{{\mathcal {M}}}{2 \Gamma (\alpha + 1)} E_{(\gamma _{1} + \alpha, \dots, \gamma _{m} + \alpha ), 1}^{(\beta _{1}, \dots, \beta _{m}), 1} \bigl( \vert \lambda _{1} \vert , \dots, \vert \lambda _{m} \vert \bigr) \lVert u_{1} - u_{2} \rVert \\ &{} + \frac{1}{2} \sum_{i = 1}^{m} \frac{ \vert \lambda _{i} \vert }{\Gamma (\gamma _{i} + \alpha + 1)\Gamma (\beta _{i} + 1)} E_{(\gamma _{1} + \alpha, \dots, \gamma _{m} + \alpha ), 1}^{( \beta _{1}, \dots, \beta _{m}), 1} \bigl( \vert \lambda _{1} \vert , \dots, \vert \lambda _{m} \vert \bigr) \lVert u_{1} - u_{2} \rVert \\ = {}& W \lVert u_{1} - u_{2} \rVert, \end{aligned}$$

by noting that

$$\begin{aligned} \bigl\vert v(t, x, u_{1}) - v(t, x, u_{2}) \bigr\vert \leq {\mathcal {M}} \vert u_{1} - u_{2} \vert . \end{aligned}$$

Since \(W < 1\), there is a unique solution to equation (1.2) in the space \(S([0, 1]^{2})\) by Banach’s fixed point theorem. Hence Theorem 3 follows. □

4 Example

Example 4

Consider the following equation with a boundary condition:

$$\begin{aligned} \textstyle\begin{cases} \frac{ _{c} \partial ^{0.5}}{\partial t^{0.5}} u(t, x) + \frac{1}{15} I_{t}^{1.5} I_{x}^{0.5} u(t, x) + \frac{1}{21} I_{t}^{2.5} I_{x}^{1.1} u(t, x) \\ \quad = \frac{1}{18}\cos (t x u) + t^{2} + \sin x, \\ u(0, x) + u(1, x) = x^{2} + 1, \quad (t, x) \in [0, 1] \times [0, 1]. \end{cases}\displaystyle \end{aligned}$$
(4.1)

Then there is a unique solution in the space \(S([0, 1]^{2} )\) to equation (4.1).

Proof

Let

$$\begin{aligned} v(t, x, u) = \frac{1}{18}\cos (t x u) + t^{2} + \sin x. \end{aligned}$$

Obviously,

$$\begin{aligned} \bigl\vert v(t, x, u_{1}) - v(t, x, u_{2}) \bigr\vert \leq \frac{1}{18} \bigl\vert \cos (t x u_{1}) - \cos (t x u_{2}) \bigr\vert \leq \frac{1}{18} \vert u_{1} - u_{2} \vert , \end{aligned}$$

for all \(u_{1}, u_{2} \in R\), by noting that \((t, x) \in [0,1] \times [0, 1]\). Therefore \({\mathcal {M}} = 1/18\), and

$$\begin{aligned} & \beta _{1} = 0.5,\qquad \beta _{2} = 1.1, \\ & \alpha = 0.5, \qquad\gamma _{1} = 1.5,\qquad \gamma _{2} = 2.5, \\ & \lambda _{1} = \frac{1}{15}, \qquad\lambda _{2} = \frac{1}{21}, \end{aligned}$$

from equation (4.1). We evaluate the following W given in Theorem 3 via Python language to get

$$\begin{aligned} W ={}& \frac{1}{18} E_{(1.5 + 0.5, 2.5 + 0.5), 0.5 + 1}^{(0.5, 1.1), 1} \biggl( \frac{1}{15}, \frac{1}{21} \biggr) \\ &{} + \frac{1}{2} \biggl(\frac{1}{18 \Gamma (0.5 + 1)} + \frac{1}{15 \Gamma (1.5 + 0.5 + 1) \Gamma (0.5 + 1)} \\ &{}+ \frac{1}{21 \Gamma (2.5 + 0.5 + 1) \Gamma (1.1 + 1)} \biggr) \\ &{} \times E_{(1.5 + 0.5, 2.5 + 0.5), 1}^{(0.5, 1.1), 1} \biggl(\frac{1}{15}, \frac{1}{21} \biggr) \approx 0.120560441333871 < 1. \end{aligned}$$

By Theorem 3, the result follows. □

Remark 5

The Python language is quite useful when computing the values of the multivariate Mittag-Leffler function or the newly introduced generalized multivariate Mittag-Leffler function. These functions appear often in many fields and play an important role in studying integral or differential equations with various conditions, as well as in finding approximate solutions, such as for equation (2.1) as an example.

5 Conclusion

We have obtained a sufficient condition for uniqueness of solution to the new boundary value problem (1.2) involving double integral operators by using the new generalized multivariate Mittag-Leffler function, Babenko’s approach, as well as by applying Banach’s fixed point theorem. Moreover, we made use of the Python language to aid in finding the approximate value of a generalized Mittag-Leffler function, which currently seems unfeasible to do so by any existing integral representations of the Mittag-Leffler function. Finally, we presented an example that applies the results of the key theorems derived. The technique used certainly works for different types of PDE and corresponding initial or boundary value problems.