On fuzzy fractional Schrödinger equations under Caputo’s H -differentiability 1

Abstract

In this paper, we initiate to investigate the existence and uniqueness of solutions to initial value problems for fuzzy fractional Schrödinger equations involving the Caputo’s H-derivative. Continuous dependence on initial values of the solution is also considered. Our results are based on a successive approximation method and the Banach contraction mapping principle. Two examples are presented for our new results.

Keywords

Fuzzy fractional differential equation Schrödinger equation initial value problem strongly generalized Hukuhara differentiability fixed point theorem

1 Introduction

Schrödinger equation, also known as Schrödinger wave equation, is a basic equation in quantum mechanics(cf. [1]). It combines the concept of matter wave and wave equation to establish a second order partial differential equation describing the motion of microscopic particles. By solving this kind of equations, a specific form of a wave function and a corresponding energy can be obtained. Schrödinger equation shows that particles appear in a probabilistic manner with uncertainty in quantum mechanics but not in macro scale. One of the most classical examples is the non-relativistic Schrödinger equation describing motion of a single electron in an electric field: $i ℏ \frac{\partial}{\partial t} Ψ (r, t) = [\frac{- ℏ^{2}}{2 μ} \nabla^{2} + V (r, t)] Ψ (r, t),$ where Ψ is the wave function of the quantum system, i is the imaginary unit, ℏ is the reduced Planck constant(Planck constant divided by 2π), μ is the particle’s reduced mass, V is its potential energy and ∇² is the Laplacian operator. The other one is a nonlinear Schrödinger equation which is mainly applied in the propagation of light in nonlinear optical fibers and Bose-Einstein condensates confined to highly anisotropic traps(cf. [2]): $\begin{matrix} i ℏ \frac{\partial}{\partial t} Ψ (r, t) & = & [\frac{- ℏ^{2}}{2 μ} \nabla^{2} + V (r, t)] Ψ (r, t) \\ + κ | Ψ (r, t) |^{2} Ψ (r, t), \end{matrix}$ where κ is a real number.

In recent years, fractional Schrödinger equations, discovered by Laskin(cf. [3]), have gradually become a hot topic in the fields of physics and mathematics. In 2007, Rida et al. found the solution to a generalized fractional nonlinear Schrödinger equation(cf. [4]) $i \frac{\partial^{α} u}{\partial t^{α}} + β \frac{\partial^{2} u}{\partial x^{2}} + V (x) u + γ | u |^{2} u = 0, t > 0,$ where 0 < α < 1 and β and γ are two real constants. In 2015, Li et al. studied the existence of positive solutions to the boundary value problem of a class of one-dimensional time-independent fractional q-difference Schrödinger equations(cf. [5]) $^{RL} D_{q}^{α} u (x) + \frac{m}{h} (E - V (x)) u (x) = 0,$ where $^{RL} D_{q}^{α}$ is the Riemann-Liouville derivative of q-difference, q ∈(0, 1), α ∈(2, 3), m is the mass of a particle, h is the Planck constant and E is the energy of a particle.

When modeling real-world phenomena, we have to consider some uncertainties because information about the behavior of a dynamical system is usually not known precisely enough. Fortunately, fuzzy differential equations are natural tools to model these uncertainties. In 2014, Golmankhaneh et al. studied boundary value problems for fuzzy Schrödinger equations involving Kandel-Fridman Ming derivatives and derived some analytical results(cf. [6]). In fact, as early as 2009, Allahviranloo et al. proposed the definition of higher order strongly generalized Hukuhara differentiability(GH-differentiability for short) and investigated the following initial value problem for a second-order nonlinear fuzzy differential equation(cf. [7]): ${\begin{matrix} u_{GH}^{″} (t) = f (t, u (t), u_{GH}^{'} (t)), \\ u (t_{0}) = k_{1}, u_{GH}^{'} (t_{0}) = k_{2}, \end{matrix}$ where $f : [t_{0}, T] \times E^{1} \times E^{1} \to E^{1}$ is a continuous fuzzy-valued function and $k_{1}, k_{2} \in E^{1}$ . In 2015, Hoa(cf. [8]) and Hv et al.(cf. [9]) studied initial value problems for second-order interval-valued differential equations and second-order random fuzzy differential equations as above form, respectively. In 2017, Tapaswini et al. proposed a new method for solving nth-order fuzzy differential equations(cf. [10]).

On the other hand, fuzzy fractional differential equations, which not only possess the advantages of fractional derivatives’ non-locality and memory but also reflect subjectivity and uncertainty in some determining rules, are an important and new research direction of fuzzy differential equations. In 2012, Salahshour et al. proposed the concept of Caputo’s H-derivative and studied the existence and uniqueness of solutions to the following fuzzy fractional differential equation under this derivative(cf. [11]): $_{GH}^{C} D_{*}^{q} u (t) = f (t, u (t)), u (t_{0}) = u_{0} \in E^{1},$ where $_{GH}^{C} D_{*}^{q}$ is the Caputo’s H-derivative, q ∈(0, 1] is a real number, $f : [t_{0}, T] \times E^{1} \to E^{1}$ is a continuous fuzzy-valued function and t₀ ≥ 0. Approximate solutions to this kind of equations are also discussed in [11]. Some other recent results to fuzzy fractional differential equations can be seen in [12 –17] and references contained therein.

As far as we know, however, most of the study of fuzzy fractional differential equations is only limited to low order until now. As a special kind of fuzzy fractional differential equations, fuzzy fractional Schrödinger equations(FFSEs) may describe the movement of particles more “accurately” because they can take the uncertainty in the process into consideration very well. Therefore, it is foreseeable that there is a very wide range of applications in quantum mechanics and atomic physics for this kind of equations. Next, the following kind of one-dimensional time-independent FFSEs will be regarded as our main research object: $_{GH}^{C} D_{*}^{q} u (x) = \frac{2 m}{ℏ^{2}} (E - V (x)) u (x) \oplus κ | u (x) |^{2} u (x),$ where q ∈(1, 2] and $κ \in ℝ$ . Motivated by these works mentioned above, we initiate to consider the existence and uniqueness of solutions to this kind of equations. More generally, we will study a class of nonlinear fuzzy fractional equations $_{GH}^{C} D_{*}^{q} u (x) = λ f (x, u (x)), x \in (0, + \infty)$ (1) subject to initial conditions $u (0) = k_{1}, u_{GH}^{'} (0) = k_{2},$ (2) where q ∈(1, 2] is a real number, λ is a positive real number and $f : [0, + \infty) \times E^{1} \to E^{1}$ is a continuous fuzzy-valued function. We will prove the local and global existence and uniqueness of solutions to this kind of initial value problems by utilizing a successive approximation method and an appropriate fixed point theorem. The continuous dependence on initial values of the solution is showed by inequality technique.

The remainder of this paper is organized as follows. In Section 2, some basic definitions, properties and lemmas about fuzzy set theory and fractional calculus theory are collected or derived. In Section 3, the existence, uniqueness and continuous dependence on initial values of solutions to initial value problem(1)–(2) are investigated. In Section 4, two mathematical models in the fuzzy sense are present to illustrate our main results. In Section 5, conclusion and some prospect for future work are drawn.

2 Preliminaries

In this section, we present some basic knowledge which is used throughout this paper. More detailed information can be found in monographs [18 –20] and references contained therein.

Let $E^{1}$ be the space of all fuzzy sets in $ℝ$ , that is, $E^{1}$ is the space of all functions $u : ℝ \to [0, 1]$ satisfying the following conditions:

u is normal, i. e., there exists $x_{0} \in ℝ$ , such that u(x₀) =1;

$[u]^{0} = \bar{{x \in ℝ | u (x) > 0}}$ is compact;

u is a convex fuzzy-valued function, i.e., for any $x_{1}, x_{2} \in ℝ$ and λ ∈(0, 1), we have $u (λ x_{1} + (1 - λ) x_{2}) \geq min {u (x_{1}), u (x_{2})};$

u is an upper semi-continuous function on $ℝ$ .

Usually, $E^{1}$ is also called the space of fuzzy numbers. The set $[u]^{α} = {x \in ℝ | u (x) \geq α}$ is called the α-level set of the fuzzy set u, where α ∈(0, 1].

For $u, v \in E^{1}$ and $λ \in ℝ$ , the following operations, which are based on a generalized Zadeh’s extension principle, define a semi-linear structure on $E^{1}$ :

$(u \oplus v) (x) = sup_{x_{1} + x_{2} = x} min {u (x_{1}), v (x_{2})}$ ;

$(λ u) (x) = {\begin{matrix} u (x / λ), & λ \neq 0, \\ χ_{{0}}, & λ = 0 . \end{matrix}$

Lemma 2.1. [21]

If we denote $\hat{0} = χ_{{0}}$ then $\hat{0} \in E^{1}$ is neutral element with respect to ⊕, i. e., $u \oplus \hat{0} = \hat{0} \oplus u = u$ , for all $u \in E^{1}$ ;

With respect to $\hat{0}$ , none of $u \in E^{1} \ ℝ$ has inverse in $E^{1}$ (with respect to ⊕);

For any $a, b \in ℝ$ , with a, b ≥ 0 or a, b ≤ 0 and any $u \in E^{1}$ , we have(a + b) u = au ⊕ bu. For general $a, b \in ℝ$ , the above property does not hold;

For any $λ \in ℝ$ and any $u, v \in E^{1}$ , we have λ(u ⊕ v) = λu ⊕ λv;

For any $λ, μ \in ℝ$ and any $u, \in E^{1}$ , we have λ(μu) =(λμ) u.

As a generalization of Hausdorff-Pompeiu metric on compact and convex set, a metric d on $E^{1}$ can be defined by $d (u, v) = sup_{α \in [0, 1]} d_{H} ([u]^{α}, [v]^{α}), u, v \in E^{1} .$

Definition 2.1. [20, 22] Hukuhara difference P23 Definition 2.8; P556 Let $u, v \in E^{1}$ . If there exists a fuzzy number $w \in E^{1}$ such that v ⊕ w = u, then w is called the Hukuhara difference(H-difference for short) of u and v, which is denoted by u ⊖ v.

Lemma 2.2. [18, 19] If $u, v, x, y \in E^{1}$ , then

d(u ⊕ x, v ⊕ x) = d(u, v);

d(λu, λv) = |λ|d(u, v), $λ \in ℝ$ ;

d(u ⊕ x, v ⊕ y) ≤ d(u, v) + d(x, y);

$d (λ u, μ u) = | λ - μ | d (u, \hat{0})$ , λ, μ ≥ 0;

d(u ⊖ x, v ⊖ y) ≤ d(u, v) + d(x, y), provided the differences u ⊖ x and v ⊖ y exist;

$(E^{1}, d)$ is a complete metric space.

Also, we introduce some notations that are used throughout the paper:

$C^{E} [a, b]$ denotes the set of all continuous fuzzy-valued functions on [a, b];

$L^{E} [a, b]$ denotes the set of all measurable and integrable fuzzy-valued functions on [a, b].

As we know, $(C^{E} [a, b], D)$ is a complete metric space, where $D (u, v) : = sup_{t \in [a, b]} d (u (t), v (t)) .$

For a positive constant N we denote $D_{N} (u, v) = sup_{t \in [0, + \infty)} d (u (t), v (t)) e^{- Nt}$ as the metric in space $C^{E} ([0, + \infty), E^{1})$ . Obviously, metric space $(C^{E} ([0, + \infty), E^{1}), D_{N})$ is a complete space according to the proof of Lemma 4.1 in [23].

Definition 2.2. [20, 24] The Aumann integral of a fuzzy-valued function $F : I \to E^{1}$ over I is defined levelwise $\begin{matrix} {[\int_{I} F (t) dt]}^{α} & = & \int_{I} F_{α} (t) dt \\ = & {\int_{I} g (t) dt : g \in S (F_{α})} \end{matrix}$ for all α ∈(0, 1], where S(F_α) is the subset of all integrable selections of set-valued mapping F_α.

Lemma 2.3. [25] Let $f, g : I \to E^{1}$ be integrable and $λ \in ℝ$ . Then

∫_I(f ⊕ g) = ∫_If ⊕ ∫_Ig;

∫_I(λf) = λ∫_If;

d(f, g) is integrable on I;

d(∫_If, ∫_Ig) ≤ ∫_Id(f, g).

Definition 2.3. [26, Definition 2.10] Let function $F \in C^{E} (a, b] ⋂ L^{E} (a, b]$ . The fuzzy fractional integral of order q > 0 of F is defined as $I^{q} F (t) = \frac{1}{Γ (q)} \int_{a}^{t} (t - s)^{q - 1} F (s) ds, t \in (a, b),$ provided that the integral in the right-hand side is pointwise well-defined. For q = 1 we obtain $I^{1} F (t) = \int_{a}^{t} F (s) ds$ , that is, the classical fuzzy integral operator.

As a generalization for Hukuhara differentiability of fuzzy-valued functions, the conception of GH-differentiability is presented by B. Bede and S. Gal as early as 2004(cf. [27, Definition 9]). This definition contains four types of situations, which can be referred as(i),(ii),(iii) or(iv)-differentiable, respectively.

Denote $f_{GH}^{'}$ is the GH-derivative of a fuzzy-valued function f. According to a fact that $f_{GH}^{'} \in ℝ$ if the function f is(iii) or(iv)-differentiable(cf. [28, Theorem 7]), we only consider(i) and(ii)-differentiability in this paper. Next, we list an important result:

Lemma 2.4. [28, Lemma 20] For $t_{0} \in ℝ$ , the initial value problem for fuzzy differential equation ${\begin{matrix} u_{GH}^{'} (t) = f (t, u (t)), \\ u (t_{0}) = u_{0} \in E^{1}, \end{matrix}$ where $f : ℝ \times E^{1} \to E^{1}$ is continuous, is equivalent to one of the integral equations: $\begin{matrix} u (t) = u_{0} \oplus \int_{t_{0}}^{t} f (s, u (s)) ds, t \in [t_{0}, t_{1}], \\ u (t) = u_{0} ⊖ (- 1) \int_{t_{0}}^{t} f (s, u (s)) ds, t \in [t_{0}, t_{1}] \end{matrix}$ on some interval $(t_{0}, t_{1}) \subset ℝ$ , depending on the strongly differentiability considered,(i) or(ii), respectively.

Caputo’s H-derivative whose order is in(0, 1], is presented by [11, Definition 4.1]. Fuzzy gH-fractional Caputo derivative is proposed in [29]. Similarly, we give the definition of Caputo’s H-derivative with arbitrary order based on the concept of GH-differentiability of the mth-order(cf. [7, Definition 3.1]).

Definition 2.4. The Caputo’s H-derivative with arbitrary order of fuzzy-valued function F is defined as following: $\begin{matrix} (_{GH}^{C} D_{*}^{q} F) (t) \\ = (I_{a}^{m - q} F_{GH}^{(m)}) (t) \\ = \frac{1}{Γ (m - q)} \int_{a}^{t} (t - s)^{m - q - 1} F_{GH}^{(m)} (s) ds, \end{matrix}$ provided that the integral in the right-hand side is pointwise well-defined, where m - 1 < q ≤ m, $m \in ℕ_{+}$ and $F_{GH}^{(m)}$ is GH-derivative of the mth-order.

Definition 2.4 is [11, Definition 4.1] if we take m = 1. And according to Definition 2.4, it is straightforward to get the following property of Caputo’s H-derivative.

Lemma 2.5. Let F have qth-order Caputo’s H-derivative and q > 1. Then $(_{GH}^{C} D_{*}^{q} F) (t) = (_{GH}^{C} D_{*}^{q - 1} F_{GH}^{'}) (t) .$

Remark 2.1. From [11], we say that a function F is ^C[(i)-GH]-differentiable if it is Caputo’s H-differentiable and(i)-differentiable; or F is ^C[(ii)-GH]-differentiable if it is Caputo’s H-differentiable and(ii)-differentiable.

Lemma 2.6. [11, Lemma 5.3] Let q ∈(0, 1] and $t_{0} \in ℝ$ . The initial value problem for fuzzy fractional differential equation ${\begin{matrix} {}_{G H}^{C}D_{*}^{q} u (t) = f (t, u (t)), \\ u (t_{0}) = u_{0} \in E^{1}, \end{matrix}$ where $f : ℝ \times E^{1} \to E^{1}$ is continuous, is equivalent to one of the following integral equations: $u (t) = u_{0} \oplus \frac{1}{Γ (q)} \int_{t_{0}}^{t} (t - s)^{q - 1} f (s, u (s)) ds,$ if u is ^C[(i)-GH]-differentiable, and $u (t) = u_{0} ⊖ \frac{- 1}{Γ (q)} \int_{t_{0}}^{t} (t - s)^{q - 1} f (s, u (s)) ds,$ if u is ^C[(ii)-GH]-differentiable, provided that the Hukuhara difference exists.

In the sequel, a similar result can be found in [7, Theorem 3.1].

Lemma 2.7. Assume that $f : [0, + \infty) \times E^{1} \to E^{1}$ is continuous. A fuzzy-valued function $u : [0, + \infty) \to E^{1}$ is a solution to problem(1)–(2) if and only if u and $u_{GH}^{'}$ are continuous and satisfy one of the following fuzzy integral equations on [0, + ∞): $u (x) = k_{1} \oplus k_{2} x \oplus λ \int_{0}^{x} I^{q - 1} f (t, u (t)) dt,$ (3) where u is(i)-differentiable and $u_{GH}^{'}$ is ^C[(i)-GH]-differentiable, or $u (x) = k_{1} \oplus k_{2} x ⊖ (- λ) \int_{0}^{x} I^{q - 1} f (t, u (t)) dt,$ where u is(i)-differentiable and $u_{GH}^{'}$ is ^C[(ii)-GH]-differentiable, or $u (x) = k_{1} ⊖ (- 1) [k_{2} x \oplus λ \int_{0}^{x} I^{q - 1} f (t, u (t)) dt],$ where u is(ii)-differentiable and $u_{GH}^{'}$ is ^C[(i)-GH]-differentiable, or $u (x) = k_{1} ⊖ (- 1) [k_{2} x ⊖ (- λ) \int_{0}^{x} I^{q - 1} f (t, u (t)) dt],$ where u is(ii)-differentiable and $u_{GH}^{'}$ is ^C[(ii)-GH]-differentiable, provided that the above Hukuhara differences exist.

Proof. According to Lemma 2.5, Equation(1) can be rewritten as $(_{GH}^{C} D_{*}^{q - 1} u_{GH}^{'}) (x) = λ f (x, u (x)) .$ (4)

Without loss of generality, we only discuss the case that u is(i)-differentiable and $u_{GH}^{'}$ is ^C[(i)-GH]-differentiable. The proofs of other cases are similar.

Notice continuity of $f : [0, + \infty) \times E^{1} \to E^{1}$ and $u : [0, + \infty) \to E^{1}$ . According to Lemma 2.6, Equation(4) with the condition $u_{GH}^{'} (0) = k_{2}$ is equivalent to $u_{GH}^{'} (x) = k_{2} \oplus \int_{0}^{x} \frac{λ (x - t)^{q - 2}}{Γ (q - 1)} f (t, u (t)) dt$ (5) on [0, + ∞). From Lemma 2.4 and initial condition u(0) = k₁, Equation(5) is equivalent to $u (x) = k_{1} \oplus k_{2} x \oplus \int_{0}^{x} \int_{0}^{t} \frac{λ (t - s)^{q - 2}}{Γ (q - 1)} f (s, u (s)) dsdt$ on [0, + ∞), that is, Equation(3). The proof is completed.

3 Basic results

At the beginning of this section, we transform initial value problem(1)–(2) to a fuzzy fractional coupled system with order 0 < q ≤ 1. Let y(x) = u(x) and $z (x) = y_{GH}^{'} (x)$ . Then we consider the following system of two fuzzy differential equations: $y_{GH}^{'} (x) = z (x),$ (6) $_{GH}^{C} D_{*}^{q - 1} z (x) = λ f (x, y (x)),$ (7) together with the initial values $y (0) = k_{1},$ (8) $z (0) = k_{2} .$ (9)

For convenience, we introduce the vector notation U(x) = [y(x), z(x)] ^T and U⁰ = [k₁, k₂] ^T.

Definition 3.1. A function $U \in C^{E} (J, E^{2})$ is said to be a ^C[(m-n)-GH]-differentiable solution(m and n can be taken as i or ii) to initial value problem(6)–(9) on interval J ⊆ [0, + ∞), if y is(m)-differentiable and z is ^C[(n)-GH]-differentiable on the entire interval J as well as y and z satisfy(6)–(9).

Similarly to Lemma 2.7, we can obtain formulations of equivalence between system(6)–(9) and the following system of fuzzy fractional integral equations. A similar result can be found in [8, Lemma 3.3] and [9, Lemma 3.1].

Lemma 3.1. Assume that $f : [0, + \infty) \times E^{1} \to E^{1}$ is continuous. A fuzzy-valued function $U : [0, + \infty) \to E^{2}$ is a solution to problem(6)–(9) if and only if U is continuous and satisfies one of the following fuzzy integral systems: $U (x) = [\begin{matrix} k_{1} \oplus \int_{0}^{x} z (t) dt \\ k_{2} \oplus \int_{0}^{x} \frac{λ (x - t)^{q - 2}}{Γ (q - 1)} f (t, y (t)) dt \end{matrix}]$ (10) if U is ^C[(i-i)-GH]-differentiable on [0, + ∞), or $U (x) = [\begin{matrix} k_{1} \oplus \int_{0}^{x} z (t) dt \\ k_{2} ⊖ \int_{0}^{x} \frac{- λ (x - t)^{q - 2}}{Γ (q - 1)} f (t, y (t)) dt \end{matrix}]$ if U is ^C[(i-ii)-GH]-differentiable on [0, + ∞), or $U (x) = [\begin{matrix} k_{1} ⊖ (- 1) \int_{0}^{x} z (t) dt \\ k_{2} \oplus \int_{0}^{x} \frac{λ (x - t)^{q - 2}}{Γ (q - 1)} f (t, y (t)) dt \end{matrix}]$ if U is ^C[(ii-i)-GH]-differentiable on [0, + ∞), or $U (x) = [\begin{matrix} k_{1} ⊖ (- 1) \int_{0}^{x} z (t) dt \\ k_{2} ⊖ \int_{0}^{x} \frac{- λ (x - t)^{q - 2}}{Γ (q - 1)} f (t, y (t)) dt \end{matrix}]$ if U is ^C[(ii-ii)-GH]-differentiable on [0, + ∞), provided that the above Hukuhara differences exist.

Proof. It is obtained immediately by Lemmas 2.4 and 2.6. In the sequel we only prove this for the case that U is ^C[(i-i)-GH]-differentiable. The proofs of other cases are similar.

In fact, assume that $U : [0, + \infty) \to E^{2}$ is ^C[(i-i)-GH]-differentiable and a solution to problem(6)–(9). Then y is(i)-differentiable and z is ^C[(i)-GH]-differentiable on [0, + ∞); $y_{GH}^{'} (x)$ and $_{GH}^{C} D_{*}^{q - 1} z (x)$ are integrable as continuous fuzzy-valued functions. Applying Lemmas 2.4 and 2.6 we obtain $[\begin{matrix} y (x) \\ z (x) \end{matrix}] = [\begin{matrix} y (0) \oplus \int_{0}^{x} z (t) dt \\ z (0) \oplus \int_{0}^{x} \frac{λ (x - t)^{q - 2}}{Γ (q - 1)} f (t, y (t)) dt \end{matrix}] .$

Noticing initial conditions(8)–(9), we immediately have (10).

To show that the opposite implication is true, suppose that $y, z : [0, + \infty) \to E^{1}$ are continuous fuzzy-valued functions and they satisfy Equation(10), that is, $\begin{matrix} y (x) = k_{1} \oplus \int_{0}^{x} z (t) dt, \\ z (x) = k_{2} \oplus \int_{0}^{x} \frac{λ (x - t)^{q - 2}}{Γ (q - 1)} f (t, y (t)) dt . \end{matrix}$

Notice initial conditions(8) and (9) again. According to Lemma 2.4, we know y(x) is(i)-differentiable and equivalent to Equation(6) with condition(7). Meanwhile, from Lemma 2.6, z(x) is ^C[(i)-GH]-differentiable and equivalent to Equation(7) with condition(9). The proof is completed.

In vector form, for $U, V \in C^{E} ([a, b], E^{2})$ , we define $D (U, V) = sup_{t \in [a, b]} d (U (t), V (t)),$ where d(U, V) = max {d(u₁, v₁), d(u₂, v₂)}, U = [u₁, u₂] ^T and V = [v₁, v₂] ^T. Then metric space $(C^{E} ([a, b], E^{2}), D)$ is a complete space.

In the following, we start to establish some existence and uniqueness results to initial value problem(6)–(9).

Theorem 3.1. For h, ρ > 0, assume that f(x, u) is a continuous fuzzy-valued function on $R : 0 \leq x \leq h, d (u, k_{1}) \leq ρ$ and satisfies (H1) there exists a constant M_f ≥ 0 such that $d (f (x, u), \hat{0}) \leq M_{f}$ for(x, u) ∈ R;(H2) there exists a constant L ≥ 0 such that $d (f (x, u), f (x, v)) \leq Ld (u, v)$ for(x, u),(x, v) ∈ R. Set U₀(x) ≡ U₀ = [y₀, z₀] ^T = [y(0), z(0)] ^T = [k₁, k₂] ^T and $m \in ℕ$ . Then the following successive approximations given by $U_{m + 1} (x) = [\begin{matrix} k_{1} \oplus \int_{0}^{x} z_{m} (t) dt \\ k_{2} \oplus \int_{0}^{x} \frac{λ (x - t)^{2 - q}}{Γ (q - 1)} f (t, y_{m} (t)) dt \end{matrix}]$ for case ^C[(i-i)-GH]-differentiability, and $U_{m + 1} (x) = [\begin{matrix} k_{1} \oplus \int_{0}^{x} z_{m} (t) dt \\ k_{2} ⊖ \int_{0}^{x} \frac{- λ (x - t)^{2 - q}}{Γ (q - 1)} f (t, y_{m} (t)) dt \end{matrix}]$ for case ^C[(i-ii)-GH]-differentiability, and $U_{m + 1} (x) = [\begin{matrix} k_{1} ⊖ (- 1) \int_{0}^{x} z_{m} (t) dt \\ k_{2} \oplus \int_{0}^{x} \frac{λ (x - t)^{2 - q}}{Γ (q - 1)} f (t, y_{m} (t)) dt \end{matrix}]$ for case ^C[(ii-i)-GH]-differentiability, and $U_{m + 1} (x) = [\begin{matrix} k_{1} ⊖ (- 1) \int_{0}^{x} z_{m} (t) dt \\ k_{2} ⊖ \int_{0}^{x} \frac{- λ (x - t)^{2 - q}}{Γ (q - 1)} f (t, y_{m} (t)) dt \end{matrix}]$ for case ^C[(ii-ii)-GH]-differentiability, converge uniformly to the corresponding unique solution U(x) to problem(6)–(9) on [0, T], provided that the above Hukuhara differences exist, where $T = min {\frac{ρ}{M}, 1}$ , $M = M_{0} + \frac{λ M_{f}}{Γ (q + 1)}$ and $M_{0} = d (k_{2}, \hat{0})$ .

Proof. Without loss of generality, we only need consider the case of ^C[(ii-ii)-GH]-differentiability. The proofs of other cases are similar. According to Lemmas 3.1, initial value problem(6)–(9) are equivalent to $U (x) = [\begin{matrix} k_{1} ⊖ (- 1) \int_{0}^{x} z (t) dt \\ k_{2} ⊖ \int_{0}^{x} \frac{- λ (x - t)^{q - 2}}{Γ (q - 1)} f (t, y (t)) dt \end{matrix}] .$ (11)

In order to apply successive approximation method, the proof can subsequently be separated into the following four steps.

Step 1. To construct the iterative sequence $U_{m} : [0, T] \to E^{2}$ , $m \in ℕ$ . We define the sequence as $\begin{matrix} U_{m + 1} (x) & = & [\begin{matrix} y_{m + 1} (x) \\ z_{m + 1} (x) \end{matrix}] \\ = & [\begin{matrix} k_{1} ⊖ (- 1) \int_{0}^{x} z_{m} (t) dt \\ k_{2} ⊖ \int_{0}^{x} \frac{- λ (x - t)^{2 - q}}{Γ (q - 1)} f (t, y_{m} (t)) dt \end{matrix}] \end{matrix}$ with U₀(x) = [y₀, z₀] ^T = [k₁, k₂] ^T. Obviously, U₀ is already well-defined. Now, we should check U_m is well-defined on [0, T] for all $m \in ℕ_{+}$ . In fact, it is enough to check y_m for $m \in ℕ_{+}$ . According to Lemmas 2.2–2.3 and(H1), for x ∈ [0, T], we have $\begin{matrix} d (y_{1} (x), k_{1}) \\ = d (k_{1} ⊖ (- 1) \int_{0}^{x} z_{0} dt, k_{1}) \\ \leq \int_{0}^{x} d (k_{2}, \hat{0}) dt \leq M_{0} T < (\frac{λ M_{f}}{Γ (q + 1)} + M_{0}) T, \\ d (y_{2} (x), k_{1}) \\ = d (k_{1} ⊖ (- 1) \int_{0}^{x} z_{1} dt, k_{1}) \leq \int_{0}^{x} d (z_{1}, \hat{0}) dt \\ = \int_{0}^{x} d (k_{2} ⊖ \int_{0}^{t} \frac{- λ (t - s)^{q - 2}}{Γ (q - 1)} f (s, y_{0}) ds, \hat{0}) dt \\ \leq \int_{0}^{x} [d (\int_{0}^{t} \frac{- λ (t - s)^{q - 2}}{Γ (q - 1)} f (s, y_{0}) ds, \hat{0}) \\ + d (k_{2}, \hat{0})] dt \\ \leq \int_{0}^{x} [\int_{0}^{t} \frac{λ (t - s)^{q - 2}}{Γ (q - 1)} d (f (s, y_{0}), \hat{0}) ds + M_{0}] dt \\ \leq \frac{λ M_{f}}{Γ (q + 1)} x^{q} + M_{0} x \leq (\frac{λ M_{f}}{Γ (q + 1)} + M_{0}) T . \end{matrix}$

Therefore, U₁(x) and U₂(x) are well-defined. Assume U_m(x) be well-defined on [0, T]. Noticing(H1) once again, for x ∈ [0, T], we have $\begin{matrix} d (y_{m + 1} (x), y_{0}) \\ \leq \int_{0}^{x} d (k_{2} ⊖ \frac{- λ}{Γ (q - 1)} \int_{0}^{t} \frac{f (s, y_{m - 1} (s))}{(t - s)^{2 - q}} ds, \hat{0}) dt \\ \leq \int_{0}^{x} [d (\frac{- λ}{Γ (q - 1)} \int_{0}^{t} \frac{f (s, y_{m - 1} (s))}{(t - s)^{2 - q}} ds, \hat{0}) \\ + d (k_{2}, \hat{0})] dt \\ \leq \frac{λ M_{f}}{Γ (q + 1)} x^{q} + M_{0} x \leq (\frac{λ M_{f}}{Γ (q + 1)} + M_{0}) T . \end{matrix}$

In addition, for 0 ≤ x₁ ≤ x₂ ≤ T and $m \in ℕ$ , we have $\begin{matrix} d (y_{m + 1} (x_{2}), y_{m + 1} (x_{1})) \\ \leq \int_{x_{1}}^{x_{2}} d (k_{2} ⊖ \int_{0}^{t} \frac{- λ (t - s)^{q - 2}}{Γ (q - 1)} f (s, y_{m - 1} (s)) ds, \hat{0}) dt \\ \leq \int_{x_{1}}^{x_{2}} [\int_{0}^{t} \frac{λ (t - s)^{q - 2}}{Γ (q - 1)} d (f (s, y_{m - 1} (s)), \hat{0}) ds + M_{0}] dt \\ \leq \frac{λ M_{f}}{Γ (q + 1)} (x_{2}^{q} - x_{1}^{q}) + M_{0} (x_{2} - x_{1}), \\ d (z_{m + 1} (x_{2}), z_{m + 1} (x_{1})) \\ \leq \int_{0}^{x_{1}} \frac{λ | (x_{2} - t)^{q - 2} - (x_{1} - t)^{q - 2} |}{Γ (q - 1)} d (f (t, y_{m} (t)), \hat{0}) dt \\ + \int_{x_{1}}^{x_{2}} \frac{λ (x_{2} - t)^{q - 2}}{Γ (q - 1)} d (f (t, y_{m} (t)), \hat{0}) dt \\ \leq \frac{λ M_{f}}{Γ (q)} (x_{2}^{q - 1} - x_{1}^{q - 1}) . \end{matrix}$ }It follows that U_m+1(x) is well-defined and continuous on [0, T]. By the mathematical induction, we know that ${U_{m} (x)}_{m \in ℕ}$ is well-defined and continuous on [0, T].Step 2. To show that the iterative sequence ${U_{m}}_{m \in ℕ}$ is a Cauchy sequence in $C^{E} ([0, T], E^{2})$ .On the one hand, for x ∈ [0, T], we have $\begin{matrix} d (y_{1} (x), y_{0}) \leq M_{0} x, \\ d (z_{1} (x), z_{0}) \\ \leq \frac{λ}{Γ (q - 1)} \int_{0}^{t} (x - t)^{q - 2} d (f (t, y_{0}), \hat{0}) dt \\ \leq \frac{λ M_{f} x^{q - 1}}{Γ (q)} . \end{matrix}$ Further, since f satisfies Lipschitz condition(H2), for x ∈ [0, T], we get $\begin{matrix} d (y_{2} (x), y_{1} (x)) \leq \int_{0}^{x} d (z_{1} (t), z_{0}) dt \leq \frac{λ M_{f} x^{q}}{Γ (q + 1)}, \\ d (z_{2} (x), z_{1} (x)) \\ \leq \frac{λ}{Γ (q - 1)} \int_{0}^{x} (x - t)^{q - 2} d (f (t, y_{1} (t)), f (t, y_{0})) dt \\ \leq \frac{λ L}{Γ (q - 1)} \int_{0}^{x} (x - t)^{q - 2} d (y_{1} (t), y_{0}) dt \\ \leq \frac{λ L M_{0} x^{q}}{Γ (q + 1)} . \end{matrix}$ For x ∈ [0, T] and $k \in ℕ_{+}$ , we assume $\begin{matrix} d (y_{m} (x), y_{m - 1} (x)) \\ \leq {\begin{matrix} \frac{(λ L)^{k} M_{0} x^{kq + 1}}{Γ (kq + 1)}, & m = 2 k - 1, \\ \frac{(λ L)^{k - 1} λ M_{f} x^{kq}}{Γ (kq + 2)}, & m = 2 k \end{matrix} \end{matrix}$ (12) and $\begin{matrix} d (z_{m} (x), z_{m - 1} (x)) \\ \leq {\begin{matrix} \frac{(λ L)^{k} λ M_{f} x^{(k + 1) q - 1}}{Γ ((k + 1) q)}, & m = 2 k - 1, \\ \frac{(λ L)^{k} M_{0} x^{kq}}{Γ (kq + 1)}, & m = 2 k . \end{matrix} \end{matrix}$ (13) By the mathematical induction, it is easy to see that inequalities(12) and (13) hold for any $m \in ℕ_{+}$ .On the other hand, we show that D(U_m+p, U_m) →0 for any $m \in ℕ$ and $p \in ℕ_{+}$ when m→ + ∞. Without loss of generality, just take m = 2k₁ - 1 and p = 2k₂, $k_{1}, k_{2} \in ℕ_{+}$ . Then for any $k_{2} \in ℕ_{+}$ and x ∈ [0, T], one has $\begin{matrix} d (y_{m + p} (x), y_{m} (x)) \\ \leq d (y_{2 (k_{1} + k_{2}) - 1} (x), y_{2 (k_{1} + k_{2}) - 2} (x)) \\ + d (y_{2 (k_{1} + k_{2}) - 2} (x), y_{2 (k_{1} + k_{2}) - 3} (x)) \\ + \cdot \cdot \cdot + d (y_{2 k_{1}} (x), y_{2 k_{1} - 1} (x)) \\ \leq \frac{(λ L)^{k_{1} + k_{2}} M_{0} x^{(k_{1} + k_{2}) q + 1}}{Γ ((k_{1} + k_{2}) q + 1)} \\ + \frac{(λ L)^{k_{1} + k_{2} - 2} λ M_{f} x^{(k_{1} + k_{2} - 1) q}}{Γ ((k_{1} + k_{2} - 1) q + 2)} \\ + \cdot \cdot \cdot + \frac{(λ L)^{k_{1} - 1} λ M_{f} x^{k_{1} q}}{Γ (k_{1} q + 1)} \\ \leq \sum_{n = k_{1}}^{k_{1} + k_{2}} \frac{(λ L)^{n} x^{nq} (M_{0} + M_{f} / L)}{Γ (nq + 1)} \\ \leq \frac{M_{0} L + M_{f}}{L} \sum_{n = k_{1}}^{k_{1} + k_{2}} \frac{(λ L T^{q})^{n}}{n!}, \end{matrix}$ which means D(y_m+p, y_m) →0 as m→ + ∞(k₁→ + ∞). Similarly, we have D(z_m+p, z_m) →0 for any $p \in ℕ_{+}$ as m→ + ∞. Therefore, it indicates that $D (U_{m + p}, U_{m}) \to 0$ for any $p \in ℕ_{+}$ when m→ + ∞. So, ${U_{m}}_{m \in ℕ}$ is a Cauchy sequence in $C^{E} ([0, T], E^{2})$ and there exists $U \in C^{E} ([0, T], E^{2})$ such that $lim_{m \to + \infty} U_{m} = U$ .Step 3. Denote $U (x) = lim_{m \to + \infty} U_{m} (x) for any x \in [0, T] .$ We show U(x) is the continuous solution to system(11) on [0, T]. According to(H2), we deduce that $d (f (x, y_{m} (x)), f (x, y (x))) \to 0$ uniformly on [0, T] as m→ + ∞. From $\begin{matrix} d (\frac{λ}{Γ (q - 1)} \int_{0}^{x} \frac{f (t, y_{m} (t))}{(x - t)^{2 - q}} dt, \frac{λ}{Γ (q - 1)} \int_{0}^{x} \frac{f (t, y (t))}{(x - t)^{2 - q}} dt) \\ \leq \frac{λ}{Γ (q - 1)} \int_{0}^{x} (x - t)^{q - 2} d (f (t, y_{m} (t)), f (t, y (t))) dt \end{matrix}$ }for x ∈ [0, T], one gets $\begin{matrix} z (x) & = & lim_{m \to + \infty} z_{m} (x) \\ = & k_{2} ⊖ \frac{- λ}{Γ (q - 1)} \int_{0}^{x} (x - t)^{q - 2} f (t, y (t)) dt . \end{matrix}$ Then we have $\begin{matrix} y (x) & = & lim_{m \to + \infty} y_{m} (x) \\ = & k_{1} ⊖ (- 1) \int_{0}^{x} z (t) dt, x \in [0, T] . \end{matrix}$ These illustrate U(x) is a continuous solution to system(11) on [0, T]. According to Lemma 3.1, U(x) is also a ^C[(ii-ii)-GH]-differentiable solution to problem(6)–(9) on [0, T].Step 4. To discuss the uniqueness of the solution. Let V(x) = [v(x), w(x)] ^T be another ^C[(ii-ii)-GH]-differentiable solution to problem(6)–(9). Then for each x ∈ [0, T], we get $\begin{matrix} d (y (x), v (x)) \\ \leq \int_{0}^{x} d (z (t), w (t)) dt \leq \int_{0}^{x} d (U (t), V (t)) dt \\ \leq \int_{0}^{x} (x - t)^{q - 2} d (U (t), V (t)) dt, \\ d (z (x), w (x)) \\ \leq \frac{λ}{Γ (q - 1)} \int_{0}^{x} (x - t)^{q - 2} d (f (t, y (t)), f (t, v (t))) dt \\ \leq \frac{λ L}{Γ (q - 1)} \int_{0}^{x} (x - t)^{q - 2} d (y (t), v (t)) dt \\ \leq \frac{λ L}{Γ (q - 1)} \int_{0}^{x} (x - t)^{q - 2} d (U (t), V (t)) dt . \end{matrix}$ Consequently, $d (U (x), V (x)) \leq Λ_{1} \int_{0}^{x} (x - t)^{q - 2} d (U (t), V (t)) dt$ for x ∈ [0, T], where $Λ_{1} : = (\frac{λ L}{Γ (q - 1)} + 1)$ . From the generalized Gronwall’s inequality(cf. [30, Corollary 2]), we obtain that d(U(x), V(x)) =0, that is, U(x) = V(x) on [0, T]. The proof is completed.The following theorem shows the continuous dependence of the solution to problem(6)–(9) on the initial function.

Theorem 3.2. Let ω₁, ω₂ > 0. Assume that $f : [0, T] \times E^{1} \to E^{1}$ is continuous and satisfies(H2). If $U (x, φ) : [0, ω_{1}) \to E^{2}$ and $U (x, ψ) : [0, ω_{2}) \to E^{2}$ are two solutions in the same differentiable case to problem(6)–(9) with U(0, φ) = φ and U(0, ψ) = ψ, where $φ = [φ_{1}, φ_{2}]^{T} \in E^{2}$ and $ψ = [ψ_{1}, ψ_{2}]^{T} \in E^{2}$ , then $\begin{matrix} d (U (x, φ), U (x, ψ)) \\ \leq d (φ, ψ) E_{q - 1} ([λ L + Γ (q - 1)] x^{q - 1}), \\ x \in [0, ω) \end{matrix}$ where ω = min { ω₁, ω₂, 1}, E_q-1 is the one-parametric Mittag-Leffler function and L is a Lipschitz constant.Proof. Without loss of generality, we only show the case of ^C[(ii-ii)-GH]-differentiable solution. According to (11), for x ∈ [0, ω), we have $\begin{matrix} d (y (x, φ_{1}), y (x, ψ_{1})) \\ \leq d (φ_{1}, ψ_{1}) + \int_{0}^{x} d (z (t, φ_{2}), z (t, ψ_{2})) dt \\ \leq d (φ, ψ) + \int_{0}^{x} (x - t)^{q - 2} d (U (t, φ), U (t, ψ)) dt, \\ d (z (x, φ_{2}), z (x, ψ_{2})) \leq d (φ_{2}, ψ_{2}) \\ + \int_{0}^{x} \frac{λ L (x - t)^{q - 2}}{Γ (q - 1)} d (y (t, φ_{1}), y (t, ψ_{1})) dt \\ \leq d (φ, ψ) \\ + \int_{0}^{x} \frac{λ L (x - t)^{q - 2}}{Γ (q - 1)} d (U (t, φ), U (t, ψ)) dt . \end{matrix}$ Consequently, $\begin{matrix} d (U (x, φ), U (x, ψ)) \leq 2 d (φ, ψ) \\ + Λ_{1} \int_{0}^{x} (x - t)^{q - 2} d (U (t, φ), U (t, ψ)) dt \end{matrix}$ for x ∈ [0, ω). Noticing d(φ, ψ) is a constant and from the generalized Gronwall’s inequality, we obtain that $\begin{matrix} d (U (x, φ), U (x, ψ)) \\ \leq 2 d (φ, ψ) E_{q - 1} ([λ L + Γ (q - 1)] x^{q - 1}) \end{matrix}$ for all x ∈ [0, ω). The proof is completed.For $U, V \in C^{E} ([0, + \infty), E^{2})$ , we define $D_{N} (U, V) = sup_{t \in [0, + \infty)} d (U (t), V (t)) e^{- Nt} .$ Then metric space $(C^{E} ([0, + \infty), E^{2}), D_{N})$ is also a complete space.Next, we apply the Banach contraction mapping principle to show the global existence and uniqueness of the solution to problem(6)–(9).

Theorem 3.3. Assume that $f : [0, + \infty) \times E^{1} \to E^{1}$ is continuous and satisfies (H3) there exists L > 0 such that d(f(x, u), f(x, v)) ≤ Ld(u, v) for $(x, u), (x, v) \in [0, + \infty) \times E^{1}$ . Then there exists a unique solution to problem(6)–(9) on the interval [0, + ∞).Proof. Without loss of generality, we just prove for the ^C[(ii-ii)-GH]-differentiable case. Let $Ω : = C^{E} ([0, + \infty), E^{2})$ . Consider the operator $T : Ω \to E^{2}$ defined by $(T U) (x) = [\begin{matrix} (T_{1} z) (x) \\ (T_{2} y) (x) \end{matrix}],$ where $\begin{matrix} (T_{1} z) (x) = k_{1} ⊖ (- 1) \int_{0}^{x} z (t) dt, \\ (T_{2} y) (x) = k_{2} ⊖ \int_{0}^{x} \frac{- λ (x - t)^{q - 2}}{Γ (q - 1)} f (t, y (t)) dt . \end{matrix}$ According to the continuity of f, it is easy to know that $T U$ is continuous. Hence $T (Ω) \subseteq Ω$ .Let U = [y, z] ^T ∈ Ω and V = [v, w] ^T ∈ Ω. For each x ∈ [0, + ∞), we have $\begin{matrix} d ((T_{1} z) (x), (T_{1} w) (x)) \\ \leq \int_{0}^{x} d (z (t), w (t)) e^{- Nt} e^{Nt} dt \\ \leq D_{N} (z, w) \int_{0}^{x} e^{Nt} dt = \frac{1}{N} (e^{Nx} - 1) D_{N} (z, w), \\ d ((T_{2} y) (x), (T_{2} v) (x)) \\ \leq \frac{λ}{Γ (q - 1)} \int_{0}^{x} (x - t)^{q - 2} d (f (t, y (t), f (t, v (t))) dt \\ \leq D_{N} (y, v) \frac{λ L}{Γ (q - 1)} \int_{0}^{x} (x - t)^{q - 2} e^{Nt} dt, \end{matrix}$ which imply $\begin{matrix} D_{N} (T_{1} z, T_{1} w) \\ = sup_{x \in [0, + \infty)} d ((T_{1} z) (x), (T_{1} w) (x)) e^{- Nx} \\ \leq D_{N} (z, w) sup_{x \in [0, + \infty)} \frac{1}{N} (1 - e^{- Nx}) \\ = \frac{1}{N} D_{N} (z, w), \end{matrix}$ $\begin{matrix} D_{N} (T_{2} y, T_{2} v) \\ = sup_{x \in [0, + \infty)} d ((T_{2} y) (x), (T_{2} v) (x)) e^{- Nx} \\ \leq \frac{λ {LD}_{N} (y, v)}{Γ (q - 1)} sup_{x \in [0, + \infty)} \int_{0}^{x} (x - t)^{q - 2} e^{- N (x - t)} dt \\ \leq \frac{λ {LD}_{N} (y, v)}{Γ (q - 1)} sup_{x \in [0, + \infty)} (\frac{x^{q - 1}}{q - 1} e^{- Nx} + \frac{Γ (q)}{(q - 1) N^{q - 1}}) \\ \leq \frac{(q - 1)^{q - 1} λ {Le}^{1 - q} Γ^{- 1} (q) + 1}{N^{q - 1}} D_{N} (y, v) . \end{matrix}$ Therefore, $D_{N} (T U, T V) \leq Λ_{2} D_{N} (U, V),$ where $Λ_{2} : = \frac{1}{N} + \frac{(q - 1)^{q - 1} {rLe}^{1 - q} Γ^{- 1} (q) + 1}{N^{q - 1}}$ . At this point, we can choose a positive and large enough N such that Λ₂ < 1, which ensure $T$ has a unique fixed point U^* by the Banach contraction mapping principle. Hence problem(6)–(9) has a unique solution. The proof is completed.

4 Fuzzy fractional Schrödinger equations

In this section, we will discuss initial value problems for one-dimensional time-independent FFSEs in linear and nonlinear cases.

Example 4.1. Consider the following initial value problem for linear FFSEs: ${\begin{matrix} {}_{G H}^{C}D_{*}^{1.5} u (x) = \frac{2 μ}{ℏ^{2}} (V (x) - E) u (x), \\ u (0) = k_{1}, u_{G H}^{'} (0) = \hat{0}, \end{matrix}$ (14) where $λ = \frac{2 μ}{ℏ^{2}}$ is a non-negative constant, f(x, u(x)) =(V(x) - E) u(x) is obviously continuous, V(x) is taken as the one-dimensional square well $V (x) = {\begin{matrix} 0, & x < 0, \\ l, & x \geq 0, \end{matrix}$ and k₁ =(1, 2, 3) is a symmetric triangular fuzzy number.

Let L = |l - E|. Then for any $u, v \in E^{1}$ , we have $d (f (x, u), f (x, v)) = | l - E | d (u, v) \leq Ld (u, v),$ which means that condition(H3) holds. Then by Theorem 3.3, initial value problem(14) has a unique solution on [0, + ∞).

In fact, problem(14) is a linear problem and we can directly get its solution $u (x) = k_{1} E_{1.5} (2 μ (l - E) / ℏ^{2} \cdot x^{1.5}),$ (15)x ∈ [0, + ∞). And further, suppose μ = 1.6726231 × 10^-27(alpha particle’s reduced mass), l = 1.6068950 × 10^-40, E = 2.48602310 × 10^-40. Then we get the exact solution(15) and it has the representation in Fig. 1, where ℏ=1.0545717 × 10^-34.

Fig.1

Solution to the problem(14) and its α-level: 1-level or crisp solution(blue solid line), 0.5-level(red dash lines) and 0-level(or boundaries) of the solution(green dot-dash lines).

Example 4.2. Consider the following initial value problem for nonlinear FFSEs: ${\begin{array}{l} _{G H}^{C} D_{*}^{1.5} (x) & = \frac{2 m}{ℏ^{2}} (\frac{1}{2} m ω_{T}^{2} x^{2} - ϱ) u (x) m n \\ \oplus 8 π a N | u (x) |^{2} u (x), \\ u (0) = k_{1,} & u_{G H}^{'} (0) = \hat{0}, \end{array}$ (16) where u(x) is the Bose-Einstein condensate wave function, m is the mass of a single atom, ω_T is the angular frequency of the well, ϱ is the chemical potential of the condensate, N is the number of atoms in the condensate, a is the scattering length and k₁ =(1, 2, 3).

Let h = ρ = 1. Then x ∈ [0, 1] and d(u, k₁) ≤1. According to Definitions 2.3 and 4.1 in [31], some easy manipulation yields $\begin{matrix} d (u, \hat{0}) & \leq & d (u, k_{1}) + d (k_{1}, \hat{0}) \leq 4, \\ d (| u |^{2}, \hat{0}) & = & d (u^{2}, \hat{0}) \leq 16, \\ d (| u |^{2} u, \hat{0}) & = & d (u^{3}, \hat{0}) \leq 64 . \end{matrix}$

Denote $λ : = \frac{2 m}{ℏ^{2}}$ , $f (x, u) : = (\frac{1}{2} m ω_{T}^{2} x^{2} - ϱ) u \oplus \frac{4 π ℏ^{2} aN}{m} | u |^{2} u$ , $M_{f} : = 2 | m ω_{T}^{2} - 2 ϱ | + \frac{256 π ℏ^{2} aN}{m} + 1$ and $L : = | \frac{1}{2} m ω_{T}^{2} - ϱ | + \frac{64 π ℏ^{2} aN}{m} + 1$ . Then for x ∈ [0, 1], one gets $\begin{matrix} d (f (x, u), \hat{0}) \\ \leq | \frac{1}{2} m ω_{T}^{2} - ϱ | d (u, \hat{0}) + \frac{4 π ℏ^{2} aN}{m} d (| u |^{2} u, \hat{0}) \\ \leq 2 | m ω_{T}^{2} - 2 ϱ | + \frac{256 π ℏ^{2} aN}{m} \leq M_{f}, \\ d (f (x, u), f (x, v)) \\ \leq | \frac{1}{2} m ω_{T}^{2} - ϱ | d (u, v) + \frac{4 π ℏ^{2} aN}{m} d (| u |^{2} u, | v |^{2} v) \\ \leq (| \frac{1}{2} m ω_{T}^{2} - ϱ | + \frac{64 π ℏ^{2} aN}{m}) d (u, v) \\ \leq Ld (u, v) . \end{matrix}$

Thus, conditions(H1) and(H2) hold. Noticing $M_{0} = d (k_{2}, \hat{0}) = 0$ , then by Theorem 3.1, initial value problem(14) has a unique solution on [0, T], where $T = min {\frac{Γ (q + 1)}{λ M_{f}}, 1}$ .

And further, suppose m = 3.1061656 × 10^-24(sodium atomic’s mass), ω_T = 1.0686509 × 10^-12, ϱ = 7.4860231 × 10^-45, a = 1.6068950 × 10^-12 and N = 10000. Then we get the numerical solution on [0, 0.0795] by means of Adams-Moulton predictor corrector method, and it has the representation in Fig. 2.

Fig.2

Solution to the problem(14) and its α-level: 1-level or crisp solution(blue solid line), 0.5-level(red dash lines) and 0-level(or boundaries) of the solution(green dot-dash lines).

5 Conclusion

This paper is concerned with the existence and uniqueness of solutions to initial value problems for FFSEs. After transforming problem(1)–(2) into fuzzy fractional coupled system(6)–(9) whose order is no more than one, we show this system has a unique local solution under some appropriate conditions on right-hand side function(see Theorem 3.1 and Example 4.2). Meanwhile, this solution is continuous about initial conditions(see Theorem 3.2). In addition, we also give some conditions that guarantee the global existence and uniqueness of its solution(see Theorems 3.3 and Example 4.1).

It is generally known that solutions to fuzzy fractional differential equations have more plentiful feature than crisp ones. However, present studies are only limited to the differential equations or systems whose order less than one. Our work may be the first attempt to study “higher order” fuzzy fractional differential equations. This makes it possible that fuzzy fractional differential equations are applied to a more extensive modeling process. In this respect, there much more work needs to be taken into account.

References

Schrödinger

, An undulatory theory of the mechanics of atomsand molecules, Physical Review 28(1926), 1049–1070.

Edwards

and K.

Burnett

, Numerical solution of the nonlinear Schrödinger equation for small samples of trapped neutral atoms, Physical Review A 51(1995), 1382–1386.

Laskin

, Fractional Schrödinger equation, Physical Review E 66(2002), 249–264.

Rida

, El-Sherbiny

and A.

Arafa

, On the solution of thefractional nonlinear Schrödinger equation, Physics Letters A 372(2008), 553–558.

, Han

and X.

, Boundary value problems of fractionalqdifference Schrödinger equations, Applied Mathematics Letters 46(2015) 100–105.

Golmankhaneh

and A.

Jafarian

, About fuzzy Schrödingerequation, in: International Conference on Fractional Differ entiation and Its Applications, IEEE, 2014, pp. 1–5. DOI: 526 10.1109/ICFDA.2014.6967457.

Allahviranloo

, Kiani

and M.

Barkhordari

, Toward the existenceand uniqueness of solutions of second-order fuzzy differential equations, Information Sciences 179(2009), 1207–1215.

Hoa

, The initial value problem for interval-valued second-orderdifferential equations under generalized Hdifferentiability, Information Sciences 311(2015), 119–148.

and L.

Dong

, Initial value problem for second-order randomfuzzy differential equations, Advances in Difference Equations 2015(2015), 373.

10.

Tapaswini

, Chakraverty

and T.

Allahviranloo

, A new approach tonth order fuzzy differential equations, Computational Mathematics and Modeling 28(2017), 278–300.

11.

Salahshour

, Allahviranloo

and S.

Abbasbandy

, Existence anduniqueness results for fractional differential equations withuncertainty, Advances in Difference Equations 2012(2012), 112.

12.

Salahshour

, Ahmadian

, Senu

, Baleanu

and P.

Agarwal

, Onanalytical solutions of the fractional differential equation with uncertainty: Application to the Basset problem, Entropy 17(2015), 885–902.

13.

Chehlabi

and T.

Allahviranloo

, Concreted solutions to fuzzylinear fractional differential equations, Applied Soft Computing 44(2016), 108–116.

14.

Armand

, Allahviranloo

, Abbasbandy

and Z.

Gouyandeh

, Fractional relaxation-oscillation differential equations via fuzzyvariational iteration method, Journal of Intelligent and Fuzzy Systems 32(2017), 363–371.

15.

Najafia

and T.

Allahviranloo

, Semi-analytical methods for solvingfuzzy impulsive fractional differential equations, Journal of Intelligent and Fuzzy Systems 33(2017), 3539–3560.

16.

Ahmadian

, Salahshour

, Ali-Akbari

, Ismail

and D.

Baleanu

, A novel approach to approximate fractional derivative with uncertain conditions, Chaos, Solitons and Fractals 104(2017), 68–76.

17.

Alinezhad

and T.

Allahviranloo

, On the solution of fuzzy fractional optimal control problems with the Caputo derivative, Information Sciences 421(2017), 218–236.

18.

Lakshmikantham

, Mohapatra

Theory of Fuzzy Differential Equations and Inclusions, Taylor and Francis, London, 2003.

19.

Lakshmikantham

, Gnana

, Vasundhara

Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, Cambridge, 2006.

20.

Gomes

, Barros

, Bede

Fuzzy Differential Equations in Various Approaches, Springer International Publishing, Berlin, 2015.

21.

Anastassiou

and S.

Gal

, On a fuzzy trigonometric approximationtheorem of Weierstrass-type, The Journal of Fuzzy Mathematics 3(2001), 701–708.

22.

Puri

and D.

Ralescu

, Differentials of fuzzy functions, Journal of Mathematical Analysis and Applications 91(1983), 552–558.

23.

Lupulescu

, On a class of fuzzy functional differential equations, Fuzzy Sets and Systems 160(2009), 1547–1562.

24.

Puri

and D.

Ralescu

, Fuzzy random variables, Journal of Mathematical Analysis and Applications 114(1986), 409–422.

25.

Kaleva

, Fuzzy differential equations, Fuzzy Sets and Systems 24(1987), 301–317.

26.

Khastan

, Nieto

and R.

Rodríguez-López

, Schauder fixedpoint theorem in semilinear spaces and its application to fractional differential equations with uncertainty, Fixed Point Theory and Applications 21(2014), 1–14.

27.

Bede

and S.

Gal

, Almost periodic fuzzy-number-valued functions, Fuzzy Sets and Systems 147(2004), 385–403.

28.

Bede

and S.

Gal

, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems 151(2005), 581–599.

29.

Allahviranloo

, Armand

and Z.

Gouyandeh

, Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, Journal of Intelligent and Fuzzy Systems 26(2014), 1481–1490.

30.

, Gao

and Y.

Ding

, A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications 328(2007), 1075–1081.

31.

Kadak

and F.

Başar

, Power series of fuzzy numbers with realor fuzzy coefficients, Filomat 26(2012), 519–528.