Lie symmetry analysis of conformable differential equations

1.   Introduction

In this paper, we study the fractional iterative functional differential equation with a convection term and nonlocal boundary condition

where $^CD_{0^+}^\alpha$ denotes the Caputo derivative of order $\alpha, \ 1 < \alpha < 2, \ \lambda\in \mathbb R, \ u^{[0]}(t) = t, \ u^{[1]}(t) = u(t), \cdots, u^{[N]}(t) = u^{[N-1]}(u(t)).\ \varphi(u) = \int_0^1 u(s)dA(s)$ is a Stieltjes integral with a signed measure, that is, $A$ is a function of bounded variation.

During the recent few decades, a vast literature on fractional differential equations has emerged, see ^{[1,2,3,4,5,6]} and the references therein. On the excellent survey of these related documents it is pointed out that the applicability of the theoretical results to fractional differential equations arising in various fields, for instance, chaotic synchronization ^[3], signal propagation ^[4], viscoelasticity ^[5], dynamical networks with multiple weights ^[6], and so on. Recently, we notice that the study of Caputo fractional differential equations with a convection term has become a heat topic (see ^[7,8,9,10]).

In ^[7], Meng and Stynes considered the Green function and maximum principle for the following Caputo fractional boundary value problem (BVP)

where $1 < \alpha < 2, \ b, \; \beta_0, \; \beta_1, \; \gamma_0, \; \gamma_1\in \mathbb R$ and $f\in C[0, 1]$ are given. Bai et al. ^[8] studied the Green function of the above problem, and the results obtained improve some conclusions of ^[7] to some degree.

Wang et al. ^[9] used operator theory to establish the Green function for the following problem

where $1 < \alpha < 2$, the constants $\lambda, \beta_0, \beta_1, \gamma_0, \gamma_1$ and the function $h\in C[a, b]$ are given. The methods are entirely different from those used in ^[7,8], and the results generalize corresponding ones in ^[7,8].

In ^[10], Wei and Bai investigated the following fractional order BVP

where $1 < \alpha\leq2$ and $b, \; \beta_0, \; \beta_1\in \mathbb R$ are constants. By employing the Guo-Krasnoselskii fixed point theorem and Leggett-Williams fixed point theorem, the existence and multiplicity results of positive solutions are presented.

Now, not only fractional differential equations have been studied constantly (see ^{[11,12,13,14,15,16,17]}), but also iterative functional differential equations have been discussed extensively as valuable tools in the modeling of many phenomena in various fields of scientific and engineering disciplines, for example, see ^{[18,19,20,21,22,23,24,25,26]} and the references cited therein. In ^[22], Zhao and Liu used the Krasnoselskii fixed point theorem to discuss the existence of periodic solutions of an iterative functional differential equation

For the general iterative functional differential equation

the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions was considered in ^[23].

In ^[24], the authors studied the following BVP

where $h:[-b, b]\times \mathbb{R}^N \rightarrow \mathbb{R}$ is continuous. By using the fixed point theorems, the authors established the existence, uniqueness and continuous dependence of a bounded solution.

To the best of our knowledge, there are few researches on fractional iterative functional differential equations integral boundary value problem with a convection term. Motivated by the above works and for the purpose to contribute to filling these gaps in the literature, this paper mainly focuses on handing with the existence, uniqueness, continuous dependence and multiplicity of positive solutions for the fractional iterative functional differential equation nonlocal BVP (1.1).

By a positive solution $u$ of (1.1), we mean $u(t) > 0$ for $t\in [0, 1]$ and satisfies (1.1).

2.   Preliminaries and lemmas

Definition 2.1.(^[1], ^[2]) The Riemann-Liouville fractional integral of order $\alpha > 0$ of a function $f:(0, +\infty)\rightarrow \mathbb R$ is given by

provided the right-hand side is pointwise defined on $(0, +\infty)$, where $\Gamma(x) = \int_0^{+\infty} t^{x-1}e^{-t}dt\ (x > 0)$ is the gamma function.

Definition 2.2.(^[1], ^[2]) The Caputo fractional derivative of order $\alpha > 0$ of a function $f:(0, \infty)\rightarrow \mathbb{R}$ is given by

provided the right-hand side is pointwise defined on $(0, +\infty)$, where $m = [\alpha]+1$.

Definition 2.3.^[1] The two-parameter Mittag-Leffler function is defined by

Lemma 2.1. ^[7] Let $F_\beta(x) = x^{\beta - 1}E_{\alpha-1}, _\beta(\lambda x^{\alpha-1})$. Then $F_\beta$ has the following properties:

$(P_1): [F_{\beta+1}(x)]' = F_\beta(x)$ for $\beta \geq 0$ and $x \geq 0$;

$(P_2): F_1(0) = 1, \ F_\beta(0) = 0$ for $\beta > 1$;

$(P_3): F_1(x) > 0$ for $x > 0, \ F_2(x)$ is increasing for $x \geq 0$;

$(P_4): F_{\alpha-1}(x) > 0$ for $x > 0, \ F_\alpha(x)$ is increasing for $x > 0$;

$(P_5): F_1(x) = \lambda F_\alpha(x) + 1$ for $0\leq x\leq1$.

For $\beta > 0$ and $\nu > 0$ one has by ^[1]

That is to say,

Lemma 2.2. Suppose that $h \in A C[0, 1]$ and $\varphi(\mathbf{1})\neq 1$. Then $u \in AC^2[0, 1]$ is the solution of

if and only if the function $u$ satisfies $u(t) = \int_0^1 H(t, s)h(s)ds$, where

Proof. Applying $I_{0+}^\alpha$ to the both sides of the Eq (2.2), we know by simple calculation that the general solution of (2.2) is given by

Then

In view of $u'(0) = 0$ and $u(1) = \varphi(u)$, we deduce by (2.3) that

Therefore,

Direct computations yield

It follows that

Substituting it to (2.4), we have $u(t) = \int_0^1 H(t, s)h(s)ds, \ t\in [0, 1]$.

Conversely, due to

we obtain

and

Then, $u(1) = \varphi(u)$ and $u'(0) = 0.$

Let

Then, for almost all $t\in[0, 1]$,

and

Then using (2.1), we calculate

and

Combining (2.5) with (2.6), we obtain

where we utilize $h \in A C[0, 1]$ and $(P_5).$ Consequently, we obtain $-^CD_{0^+}^\alpha u(t) + \lambda u'(t) = h(t).$ The proof is finished.

As a direct consequence of the previous results, we deduce the following properties that, as we will see, will be fundamental for subsequent studies.

Lemma 2.3. Assume that $0 \leq \varphi(\mathbf{1}) < 1$ and $\mathcal{G}_A(s) \geq 0$ for $s\in[0, 1]$. Then for $t, s\in[0, 1]$,

1). $G(t, s)$ and $H(t, s)$ are continuous;

2). $0 \leq G(t, s) \leq F_\alpha(1-s)$, and $G(t, s)$ is decreasing with respect to $t$;

3). $0 \leq H(t, s) \leq \omega(s)$, where $\omega(s) = \frac{1}{1-\varphi(\mathbf{1})}\mathcal{G}_A(s) + F_\alpha(1-s)$, and $H(t, s)$ is decreasing with respect to $t$.

Let $E = C[0, 1]$. Then $E$ is a Banach space with the usual maximum norm $||u|| = \max_{t\in[0, 1]}|u(t)|$. For $0 \leq P \leq 1$ and $L > 0$, define

It is easy to show that $\Omega(P, L)$ is a convex and compact set.

Define an operator $T_\lambda:E\rightarrow E$ as follows:

By Lemma 2.2, we can easily know that $u$ is a solution of BVP (1.1) iff $u$ is the fixed point of the operator $T$.

Suppose that

$(H_1)$ for the function $f:[0, 1]^{N+1} \rightarrow [0, +\infty),$ there exist constants $0 < k_0, k_1, \cdots, k_N < +\infty$ such that

$(H_2)\ 0 \leq \varphi(\mathbf{1}) < 1$, and $A$ is a function of bounded variation such that $\mathcal{G}_A(s) \geq 0$ for $s \in [0, 1]$.

Clearly, using $(H_1)$, we obtain

where $\beta = |f(0, 0, \cdots, 0)|$.

Lemma 2.4.^[22] For any $u, v \in \Omega(P, L)$,

Lemma 2.5.^[27] Let $P$ be a cone in a real Banach space $E, \ P_c = \{u\in\ P: \|u\|\leq c \}, \ P(\theta, a, b) = \{u\in P: a\leq \theta(u), \ \|u\| \leq b\}$. Suppose $A:P_c \rightarrow P_c$ is completely continuous, and suppose there exists a concave positive functional $\theta$ with $\theta(u) \leq \|u\|\ (u\ \in P$) and numbers $a, b$ and $d$ with $0 < d < a < b\leq c$, satisfying the following conditions:

$(C1)\ \{ u\in P(\theta, a, b): \theta(u) > a\} \neq \emptyset$ and $\theta(Au) > a$ if $u \in P(\theta, a, b)$;

$(C2)\ \|Au\| < d$ if $u\in P_d$;

$(C3)\ \theta(Au) > a$ for all $u\ \in\ P(\theta, a, c)$ with $\|Au\|\ > \ b$.

Then $A$ has at least three fixed points $u_1, u_2, u_3\in P_c$ with

Remark. If $b = c$, then $(C1)$ implies $(C3)$.

3.   Existence and uniqueness of positive solution

Theorem 3.1. Suppose that $(H_1)$ and $(H_2)$ hold. If

and

then problem (1.1) has a unique nonnegative solution. If in addition $A$ is an increasing function, and there exists $t_0\in [0, 1]$ such that $f(t_0, 0, \cdots, 0) > 0$, then problem $(1.1)$ has a unique positive solution.

Proof. For any $u \in \Omega(P, L)$, in view of (2.7) and Lemma 2.4, we deduce that

and hence

Therefore, $0 \leq (Tu)(t) \leq P$ for $t\in [0, 1]$.

On the other hand, for any $t_1, t_2 \in [0, 1]$ and $t_1 < t_2,$ by means of Lagrange mean value theorem, there exists $\xi\in (t_1, t_2) \subset (0, 1)$ such that

It follows from (3.2) that

Therefore, $T(\Omega(P, L)) \subset \Omega(P, L)$.

Next, we show that $T$ is a contraction mapping on $\Omega(P, L)$. Indeed, let $u, v \in \Omega(P, L)$. Then

and

It follows from (3.1) that

This shows that $T$ is a contraction mapping on $\Omega(P, L)$. It follows from the contraction mapping theorem that $T$ has a unique fixed point $u$ in $\Omega(P, L)$. In other words, problem (1.1) has a unique nonnegative solution.

Suppose $u$ is the nonnegative solution to problem (1.1). Then

By the monotonicity of $H(t, s)$, we have $u(t)\geq u(1)\geq 0$ for $t\in [0, 1]$. If $A$ is an increasing function, and there exists $t_0\in [0, 1]$ such that $f(t_0, 0, \cdots, 0) > 0$, we must have $u(1) > 0$. Otherwise, $u(1) = 0$ and we have $\int_0^1 u(s)dA(s) = \varphi(u) = u(1) = 0$. Then $u(t)\equiv 0$ for $t\in [0, 1]$. By the equation of (1.1), we conclude $f(t, 0, \cdots, 0) \equiv 0$ for $t\in [0, 1]$, which is a contradiction. We have thus proved $u(1) > 0$ and $u(t)\geq u(1) > 0$ for $t\in [0, 1]$.

4.   Continuous dependence

Theorem 4.1. Assume that the conditions of Theorem 3.1 are satisfied. Then the unique positive solution of problem (1.1) continuously depends on function $f$.

Proof. For two continuous functions $f_1, f_2:[0, 1]^{N+1} \rightarrow [0, +\infty)$, they correspond respectively to unique solutions $u_1$ and $u_2$ in $\Omega(P, L)$ such that

By $(H_1)$, we find that

It follows from Lemma 2.3 that

Then

The proof is complete.

5.   Multiplicity of positive solutions

Define

Obviously, $\theta$ is a continuous concave functional and $\theta(u)\leq\|u\|$ for $u\in\Omega.$

Theorem 5.1.Assume that $f\in AC([0, 1]^{N+1}, [0, +\infty))$ and $(H_2)$ hold. If there exist constants $0 < a < \frac{1}{6}\leq b < c\leq\frac{5}{6}$ such that

$(D1)\ f(t, u_1, u_2, \cdots, u_N) < Ma, \quad (t, u_1, u_2, \cdots, u_N)\in [0, 1] \times [0, a]^N$;

$(D2)\ f(t, u_1, u_2, \cdots, u_N) > mb, \quad (t, u_1, u_2, \cdots, u_N)\in [\frac{1}{6}, \frac{5}{6}] \times [b, c]^N$;

$(D3)\ f(t, u_1, u_2, \cdots, u_N)\leq Mc, \quad (t, u_1, u_2, \cdots, u_N)\in [0, 1] \times [0, c]^N$,

then problem (1.1) has three non-negative solutions $u_1, u_2, u_3\in \overline{\Omega_c}$ with

Proof. We first prove $T: \overline{\Omega_c}\rightarrow \overline{\Omega_c}$ is completely continuous. For $u \in \overline{\Omega_c}$, we have $\|u\| \leq c < 1$. Then $0\leq u^{[j]}(s)\leq c$ for $j = 1, 2, \cdots, N$ and $0\leq s\leq1$. It follows from $(D3)$ that

Therefore, $T(\overline{\Omega_c}) \subset \overline{\Omega_c}$ and $TD$ is uniformly bounded for any bounded set $D\subset\overline{\Omega_c}$. We denote $\overline{M}$ as the maximum of $f$ on $[0, 1]^{N+1}$. Since $H(t, s)$ is uniformly continuous on $[0, 1]\times[0, 1]$, for any $\varepsilon > 0,$ there exists $\delta > 0$ such that for any $u\in D, t_1, t_2\in[0, 1]$ and $|t_2-t_1| < \delta$, we have $|H(t_2, s)-H(t_1, s)| < \frac{\varepsilon}{\overline{M}}.$ Then,

which implies that $TD$ is equicontinuous. Clearly, the fact that $f$ is continuous implies that $T$ is continuous. Hence, $T: \overline{\Omega_c}\rightarrow \overline{\Omega_c}$ is completely continuous.

For any $u\in\overline{\Omega_a}$, due to $(D1)$, we conclude

Then, $T(\overline{\Omega_a}) \subset \overline{\Omega_a}$, which implies that $(C_2)$ in Lemma 2.5 holds.

Choose $v(t) = \frac{c+b}{2}, \ 0 \leq t \leq 1$. Obviously, $v\in \{u \in \Omega(\theta, b, c): \theta(u) > b \}$. Then $\{u \in \Omega(\theta, b, c): \theta(u) > b \}\neq \emptyset$. For $u \in \Omega(\theta, b, c)$, we have $\frac{1}{6}\leq b \leq u(t) \leq c \leq \frac{5}{6}$ for $\frac{1}{6} \leq t \leq \frac{5}{6}$. Then $b\leq u^{[j]}(s)\leq c$ for $j = 1, 2, \cdots, N$ and $\frac{1}{6} \leq s \leq \frac{5}{6}$. It follows from $(D2)$ that

which implies that $(C_1)$ in Lemma 2.5 holds. By remark under Lemma 2.5, we know that $(C_3)$ in Lemma 2.5 holds. Then according to Lemma 2.5, problem (1.1) has three nonnegative solutions $u_1, u_2, u_3\in \overline{\Omega_c}$ with

6.   An example

We consider the following BVP

where $f(t, u(t), u^{[2]}(t), u^{[3]}(t)) = \frac{1}{200}t+\frac{1}{100}sin(u(t))+\frac{1}{100}sin(u^{[2]}(t))+\frac{1}{100}sin(u^{[3]}(t)), \ A(s) = \frac{1}{2}s$. It follows that $\varphi(\mathbf{1}) = \int_0^1d(\frac{1}{2}s) = \frac{1}{2}$ and $\mathcal{G}_A(s) = \int_0^1G(t, s)d(\frac{1}{2}t)\geq0,$ which implies $(H_2)$ holds. Since

we obtain $k_0 = \frac{1}{200}, \ k_1 = \frac{1}{100}, \ k_2 = \frac{1}{100}, \ k_3 = \frac{1}{100}$ and $\beta = 0,$ which implies $(H_1)$ holds.

Direct computation shows that $\int_0^1\omega(s)ds = \int_0^1[\frac{1}{1-\varphi(\mathbf{1})}\mathcal{G}_A(s) + F_{\frac{3}{2}}(1-s)]ds < \frac{159}{50}.$ Choose $P = \frac{3}{4}$ and $L = 1$, we have

and

Taking $t_0 = \frac{1}{2}\in[0, 1],$ we find that $f(t_0, 0, 0, 0) = \frac{1}{400} > 0$. By Theorem 3.1 we conclude that BVP (6.1) has a unique positive solution.

Acknowledgments

This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2022MA049) and the National Natural Science Foundation of China (11871302, 11501318).

Conflict of interest

The authors declare there is no conflicts of interest.