Multiple solutions for a class of boundary value problems of fractional differential equations with generalized Caputo derivatives

1.   Introduction

The fractional calculus is a branch of mathematics, which studies the integration and differentiation of any order in real or complex field. In 1832, the fractional derivative was first formally proposed by Liouville. See ^[1,2] for more knowledge on fractional calculus. The fractional order differential equation (FDE, for short) is a generalization of classical integer order differential equation as well, which can describe complex with simple modeling, clear physical meaning of parameters, accurate selection and so on. Hence, it becomes an important tool for mathematical modeling of complex machines, physical processes, fluid dynamics, finance and other areas of applications (see ^[3,4] and references therein). In recent decades, more and more researchers pay much attention to the fractional differential equations and have obtained substantial achievements. For example, S. Salahshour and A. Ahmadian et al. researched the heat transfer problem with a approach of fractional modeling ^[5], successive approximation method for Caputo q-fractional IVPs ^[6] and M-fractional derivative under interval uncertainty ^[7]. N. Sene investigated chaotic system involving Caputo fractional derivatives in ^[8,9]. ^[10,11,12] were concerned with fractional diffusion equation. ^[13,14] studied infinitely many solutions for impulsive fractional boundary value problem with p-Laplacian and for fractional schrodinger-maxwell equations, respectively. ^[15,16,17] analyzed the Ulam stability of nonlinear FDEs. ^[18,19] investigated the controllability for two classes of semilinear fractional evolution systems.

In the last few years, boundary value problems of fractional differential equations (FBVPs, for short) have been extensively studied. Most of them have been considered in the frame of standard fractional derivatives such as Rieman-Liouvile and Caputo derivatives. For instance, ^[20] is concerned with positive solutions of a two-point boundary value problem for singular fractional differential equations in Banach space. ^[21,22,23] developed bifurcation techniques for FBVPs. ^[24,25,26] investigated positive solutions for FBVPs. ^[27,28] dealt with coupled fractional differential systems with nonlocal boundary conditions. ^[29,30,31] studied FBVPs via critical point theory. ^[32,33] were concerned with the solvability for multi-order nonlinear fractional systems and periodic boundary value problems of nonlinear fractional hybrid differential equations. ^[34] investigated positive solutions for nonlinear discrete FBVPs with a p-laplacian operator.

More generally, A. Babakhani in ^[35] considered

where $^{C}D^{\alpha}_{0+}$ is the Caputo fractional derivative of order $\alpha$, $f : [0, 1]\times \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a given continuous function and $g:[0, 1] \rightarrow [0, +\infty)$ is nondecreasing function. By constructing a special cone, the existence of at least one positive solution was obtained under some suitable assumptions. In ^[36], Y. Li studied the following fractional q-difference equations involving q-integral boundary conditions:

where $2 < \alpha < 3$, $1 < \nu < 2$, $D_{q}^{\nu}$ is $\alpha$-order Riemann-Liouville's fractional $q$-derivative. The existence of nontrivial solutions is obtained by using topological degree theory.

With the development of investigation on fractional derivatives, new concepts are constantly being put forward. For instance, F. Jarad et al. proposed a new kind of generalized fractional derivatives and studied their properties in ^[37,38]. N. Sene investigates fractional advection-dispersion equation described by the Caputo left generalized fractional derivative in ^[39]. ^[40] studied Ulam stabilities of fractional differential equations including generalized Caputo fractional derivative. However, to our best knowledge, there are few studies on the existence of multiple solutions and Ulam-Hyers stability for integral boundary value problems of FDES with generalized Caputo derivatives. The purpose of present paper is to fill this gap.

Motivated by the above discussions, this paper studies multiple solutions and Ulam-Hyers stability for the following FBVPs with generalized Caputo derivatives

where $2 < \alpha < 3$, $1 < \nu < 2$, $\alpha-\nu-1 > 0$, $f\in C([0, 1]\times \mathbb{R}^{+}, \mathbb{R}^{+})$, $g' > 0$, $h \in C([0, 1], \mathbb{R}^{+})$, $\mathbb{R}^{+} = [0, +\infty)$. The main features of this paper are as follows. Firstly, compared with the above mentioned references, BVP (1.1) is studied in the sense of generalized Caputo fractional derivatives, which is also different from fractional $q$-difference equations in ^[36]. Secondly, the fractional boundary value condition we consider here is of integral form, and that makes BVP (1.1) more widely applicable in solving practical problems. Thirdly, the used approach in this paper has certain advantages over some references listed as above. In detail, the distinctive tool used here is the first eigenvalue of corresponding linear operator. At the same time, a suitable cone is established by researching properties of Green's function deeply. So the positive solutions can be obtained by means of the cone expansion and compression fixed point theorem and Leggett-Williams theorem. Finally, the Ulam-Hyers stability and generalized Ulam-Hyers stability for BVP (1.1) are also studied under some suitable assumptions.

The remainder of this paper is organized as follows: Some basic knowledge of fractional calculous and some preliminary results are given in Section 2. The existence results will be given and proved in Section 3. And in Section 4, the Ulam-Hyers stability and generalized Ulam-Hyers stability will be established. Three examples are worked out to illustrate the main results in Section 5. Finally, the conclusion and some future works are given in Section 6.

2.   Preliminaries

Definition 2.1. ^[38] (1) Let $g \in C^{n}[a, b]$ such that $g'(t) > 0$ on $[a, b]$. Define

where $f^{[n-1]} = ({ }\frac{1}{g'(t)}\frac{d}{dt})^{n-1}f$.

(2)

where $C_{0, g} = C[a, b].$

(3)

where $C^{n}_{0, g} = C[a, b].$

Definition 2.2. ^[38] (1) The Riemann-Liouville fractional integral of order $\alpha\in \mathbb{R}^{+}$ of a function $f$ on a finite or infinite interval $(a, b)$ is defined as follows:

(2) The Riemann-Liouville fractional derivative of order $\alpha\in \mathbb{R}^{+}$ of a function $f$ on a finite or infinite interval $(a, b)$ is defined as follows:

where $n = [\alpha] + 1$, $g^{(i)} \neq 0, i = 1, 2, ..., n$.

Definition 2.3. ^[38] The Caputo fractional derivative of order $\alpha\in \mathbb{R}^{+}$ of a function $f$ on a finite or infinite interval $(a, b)$ is defined as follows:

Definition 2.4. ^[38] (1) The generalized Riemann-Liouville fractional integral of order $\alpha\in \mathbb{R}^{+}$ of a function $f$ with respect to the function $g$ on a finite or infinite interval $(a, b)$ is defined as follows:

(2) The generalized Riemann-Liouville fractional derivative of order $\alpha\in \mathbb{R}^{+}$ of a function $f$ with respect to the function $g$ on a finite or infinite interval $(a, b)$ is defined as follows:

where $n = [\alpha] + 1$, $g^{(i)} \neq 0, i = 2, ..., n$.

Definition 2.5. ^[38] The generalized Caputo fractional derivative of order $\alpha\in \mathbb{R}^{+}$ of a function $f$ with respect to another function $g$ on a finite or infinite interval $(a, b)$ is defined as follows:

where $y^{[n]} = (\frac{1}{g'(t)}\frac{d}{dt})^{n}$, $n = [\alpha] + 1$, $g^{(i)} \neq 0, i = 2, ..., n$.

Remark 2.6. From the Definition 2.4 (1) and Definition 2.5 above, we can see that

Lemma 2.7. ^[38] Let $g \in C^{n}[a, b]$ such that $g'(t) > 0$ on $[a, b]$. Then $y \in AC^{n}_{g}$ if and only if it can be written as that

Lemma 2.7. ^[38] Let $\alpha > 0$, $n = [\alpha]+1$ and $y \in AC^{n}_{g}[a, b]$. Then the fractional derivative of $y$ with respect to $g$ exists almost everywhere and

Remark 2.9. ^[38] Equation (2.2) can be written as that

and thus, one can define the Caputo fractional derivative of a function with respect to another function as

Similar to Caputo fractional derivative, we can easily obtain the following properties.

Lemma 2.10. Let $\alpha > 0$, $^{C}_{a}D_{g}^{\alpha}$ be a generalized Caputo fractional derivative of $\alpha$, $y(t)\in C[0, 1]$. Then,

where $g^{(k)}\neq 0$, \ $k = 0, \ 1, \ 2, \ \cdots, \ n$, \ $n = [\alpha]+1.$

Proof. The proof is done by using Remark 2.6 and Lemma 2.7.

where $g^{(k)}\neq 0$, $k = 0, \ 1, \ 2, \ \cdots, \ n$, $n = [\alpha]+1$.

The proof is completed.

Lemma 2.11. Let $A = 1-\int_{0}^{1}h(t)(\frac{g(t)-g(0)}{g(1)-g(0)})^{\alpha-v-1}g'(t)dt\neq 0$. Then the following boundary value problem

has a unique solution

where

Proof. Using Definition 2.4 and Definition 2.5, one can obtain that

Noticing that $x(0) = (_{0}^{C}D_{g}^{1}x)(0) = 0$, one can deduce that $c_{0} = c_{1} = 0$. Hence,

By (2.4), one can easily get that

Obviously,

Applying

and (2.5), it is immediate to see that

Hence, we deduce that

Moreover, by (2.4), one can get that

The proof is completed.

Suppose that $(2-\nu)(g(t)-g(s))\leq (\alpha-\nu-1)(g(t)-g(0))$, for $0\leq s\leq t\leq1$ in the rest of the paper. $

Lemma 2.12. The functions $G_{i}(i = 0, 1)$ has the following properties:

(1) $G_{0}(t, s)\geq 0\ for\ s, \ t\in [0, 1]$;

(2) $G_{1}(t, s)\geq 0\ for\ s, \ t\in[0, 1]$;

(3) $G_{0}(t, s)\leq G_{0}(1, s)\ for\ s, \ t\in [0, 1];$

(4) $G_{0}(t, s)\geq ({ }\frac{g(t)-g(0)}{g(1)-g(0)})^{2}G_{0}(1, s)\ for\ s, \ t\in[0, 1].$

Proof. (1) On the one hand, for $0 \leq s \leq t \leq 1$, we know

By careful calculation, we can see

With the property of $g(t)$, it implies that

Similarly,

Thus, it is easy to see that

On the other hand, for $0 \leq t \leq s \leq 1$, it is easy to see that from Lemma 2.10, $G_{0}(t, s) \geq 0\ $ for $s, \ t \in [0, 1]$.

(2) For $0\leq s\leq t\leq1$, from $(2-\nu)(g(t)-g(s))\leq (\alpha-\nu-1)(g(t)-g(0))$, ${ }\frac{(g(t)-g(s))^{\alpha-\nu-1}}{(g(t)-g(0))^{2-\nu}}$ is a nondecreasing function for $t\ $\ on\ $[0, 1]$.

Then,

On the other hand, for $0 \leq t \leq s \leq 1$, it is easy to see that from Lemma 2.10 the conclusion is obviously established. Therefore, $G_{1}(t, s) \geq 0$ for $s, t \in [0, 1]$.

(3) For $s, t \in [0, 1]$, from (1) and Lemma 2.10, one can easily obtain that $G_{0}(t, s)$ is an increasing function with respect to $t$. Then, $G_{0}(t, s)\leq G_{0}(1, s).$

(4) For $0\leq s\leq t\leq1$, from $(2-\nu)(g(t)-g(s))\leq (\alpha-\nu-1)(g(t)-g(0))$, ${ }\frac{(g(t)-g(s))^{\alpha-1}}{(g(t)-g(0))^{2}}$ is a nondecreasing function with respect to $t$.

On the other hand, for $0 \leq t \leq s \leq 1$, it is easy to see that

Therefore, $({ }\frac{g(t)-g(0)}{g(1)-g(0)})^{2}G_{0}(1, s)\leq G_{0}(t, s)$ for $s, \ t \in [0, 1].$

The proof is completed.

By Lemma 2.12, the following conclusion is established obviously.

Lemma 2.13. Under the assumption in Lemma 2.12, the function $G$ has the following properties:

where $\varphi(s) = G_{0}(1, s)+{ }\frac{(g(1)-g(0))^{2}}{BL\Gamma(\alpha-\nu)}\int_{0}^{1}h(t)G_{1}(t, s)g'(t)dt.$

The following lemmas will be used in the proof of the main results.

Lemma 2.14. ^[41] Let $\Omega\subset E$ be a bounded open set and $0\in\Omega$. $T:P\cap \overline{\Omega} \rightarrow P$ be a completely continuous operator. If $T$ satisfies

then $i(T, P\cap\Omega, p) = 1$.

Lemma 2.15. ^[41] Let $\Omega\subset E$ be a bounded open set and $0\in\Omega$. $T:P\cap\overline{\Omega}\rightarrow P$ be a completely continuous operator. If there is $\varphi\in P, \varphi\neq0$ such that $T$ satisfies

then $i(T, P\cap\Omega, p) = 0$.

Lemma 2.16. ^[42] (Leggett-Williams theorem) Let $P$ be a cone in a real Banach space $E$, $P_{c} = \{x \in P|\ \vert\vert x\vert\vert < c\}$, $\theta$ be a nonnegative continuous concave functional on $P$ such that $\theta(x) \leq \vert\vert x\vert\vert$, for $\forall x \in P_{c}$; and $P(\theta, b, d) = \{x \in P |\ b \leq \theta(x), \ \vert\vert x\vert\vert \leq d\}$. Suppose that $T : P_{c} \rightarrow P_{c}$ is completely continuous and there exist constants $0 < a < b < d \leq c$ such that

(A1) $\{x \in P(\theta, b, d)|\ \theta(x) > b\} \neq \emptyset$ and $\theta(Tx) > b$ for $x \in P(\theta, b, d)$;

(A2) $\vert\vert Tx\vert\vert < a$ for $\|x\| \leq a$;

(A3) $\theta(Tx) > b$ for $x \in P(\theta, b, c)$ with $\vert\vert Tx\vert\vert > d$.

Then $T$ has at least three fixed points $x_{1}$, $x_{2}$, $x_{3}$ with $\vert\vert x_{1}\vert\vert < a$; $b < \theta(x_{2})$; $a < \vert\vert x_{3}\vert\vert$ and $\theta(x_{3}) < b$.

3.   Existence results

In this section, we establish the existence and multiplicity results for BVP (1.1). Let $E = C[0, 1]$, $\vert\vert x\vert\vert: = max_{t\in [0, 1]}|x(t)|$ and $P: = \{x\in E: x(t)\geq 0, \ \forall t\in[0, 1]\}.$ Then $(E, \ \vert\vert \cdot \vert\vert)$ is a real Banach space and $P$ is a cone on $E$. Hence $E$ is an ordered Banach space and the cone $P$ is normal. Obviously, the normal constant is $N = 1$. Define operator $T:P\rightarrow P$ as follows:

For any $x\in P$, by the continuity of $G, f$ and $g'$, $Tx$ is well defined. Since $f$ is bounded, It is easy to see that $T$ is also bounded. By Lemma (2.11), one can easily see that the existence of solutions for BVP (1.1) is equivalent to the existence of positive fixed point of $T$. Therefore, we need only to find the positive fixed point of $T$ in the following work.

Subsequently, for simplicity and convenience, set

Let $r(S)$ be the spectral radius of the linear bounded operator $S$ defined by

From the Krein-Rutman theorem, we know that $r (S)$ is positive and $S$ has a positive eigenfunction $\varphi_{1}$ corresponding to $\lambda_{1}$ such that $\lambda_{1}S(\varphi_{1}) = \varphi_{1}$, where $\lambda_{1}$ is the first eigenvalue of $S$ and $\lambda_{1} = (r(S))^{-1}$.

Now let's list the following assumptions satisfied throughout the paper.

(H1) $\lim\limits_{x\rightarrow 0^{+}}\sup \limits_{t\in [0, 1]}{ }\frac {f(t, x)}{x} < \lambda_{1}$.

(H2) $\lim\limits_{x\rightarrow +\infty}\inf \limits_{t\in [0, 1]}{ }\frac {f(t, x)}{x} > \lambda_{1}$.

(H3) $\lim\limits_{x\rightarrow 0^{+}}\inf \limits_{t\in [0, 1]}{ }\frac{f(t, x)}{x} > \lambda_{1}$.

(H4) $\lim\limits_{x\rightarrow +\infty}\sup \limits_{t\in [0, 1]}{ }\frac {f(t, x)}{x} < \lambda_{1}$.

By the Arzela-Ascoli theorem, the following conclusion is established obviously.

Lemma 3.1. The operator $T:\ P \rightarrow P$ is completely continuous.

Now we are in a position to give our main results.

Theorem 3.2. Under the assumptions (H1) and (H2), BVP (1.1) admits at least one positive solution.

Proof. For the sake of obtaining the desired result, we need only to prove $T$ has at least a positive fixed point in $P\cap(B_{\overline{R}_{1}}\backslash{\overline{B}}_{r_{1}})$.

First, the assumption (H1) implies that there exist $r_{1} > 0$ and $\varepsilon_{0}\in(0, \lambda_{1})$ such that

We claim that for $\mu\in (0, 1],$

Suppose on the contrary that there exist $x_{0}\in P\cap \partial B_{r_{1}}$, $\mu_{0} \in (0, 1]$ such that

Then,

By $n$th iteration, we can get that

From the definition of the norm $\|\cdot\|$, one can deduce that

It is easy to see that

Hence we have

This is a contradiction with

which means that (3.2) holds. By Lemma 2.14, we get

Second, the assumption (H2) implies that there exist $\varepsilon_{1} > 0$ and $R_{1} > 0$ such that

Let $\widetilde{M} = max_{t\in[0, 1], x \in[0, {R}_{1}]}[f(t, x)+(\lambda_{1}+\varepsilon_{1})x]$, we can see

Choose $\overline{R}_{1} > max\{r_{1}, R_{1}, { }\frac{ \widetilde{M}\| [(\lambda_{1}+\varepsilon_{1})S-I]^{-1}\parallel}{M}\}$. We claim that for $\mu\in[0, +\infty),$

Suppose on the contrary that there exist $x_{1}\in P\cap \partial B_{\overline{R}_{1}}$ and $\mu_{1}\geq0$ such that

Therefore,

Thus, we can see

Since $\lambda_{1}+\varepsilon_{1} > \lambda_{1}$, $[(\lambda_{1}+\varepsilon_{1})S-I]$ is a positive linear operator. Hence, it has the inverse operator $[(\lambda_{1}+\varepsilon_{1})S-I]^{-1}$. By normality of cone $P$, we know

This is a contradiction, which implies that (3.4) hold. By Lemma 2.15, one can get that

Together with (3.3) and according to the regional additivity of the fixed point index, we have

The proof is completed.

Theorem 3.3. Under the assumptions (H3) and (H4), BVP (1.1) admits at least one positive solution.

Proof. For the sake of obtaining the desired result, we need only to prove that $T$ has a positive fixed point in $P\cap (B_{\overline{R}_{2}}\backslash \overline{B}_{r_{2}})$.

First, the assumption (H3) implies that there exist $\varepsilon_{2} > 0$ and $r_{2} > 0$ such that

Now we claim that for $\mu\in[0, +\infty)$,

Hence, suppose on the contrary that there exist $x_{2}\in P\cap \partial B_{r_{2}}$ and $\mu_{2}\geq 0$ such that

Without loss of generality, suppose $\mu_{2} > 0$. Then,

Taking $\mu^{*} = sup\{\mu\ |\ x_{2}\geq\mu\varphi_{1}, \ \mu > 0\}$, we have $0 < \mu_{2}\leq\mu^{*} < +\infty$ and $x_{2}(t)\geq \mu^{*}\varphi_{1}(t)$. By the positivity of operator $S$, we know

This together with (3.7) guarantees that

which is a contradiction with the definition of $\mu^{*}$. Therefore, (3.8) is valid. According to Lemma 2.15, we have

Second, the assumption (H4) implies that there exist $\varepsilon_{3}\in(0, \lambda_{1})$ and $R_{2} > 0$ such that

Let $\widetilde{N} = max_{t\in[0, 1], x\in[0, {R}_{2}]}[f(t, x)+(\lambda_{1}-\varepsilon_{3})x]$, we can see

Choose $\overline{R}_{2} > max\{r_{2}, R_{2}, { }\frac{\widetilde{N}[1-(\lambda_{1}-\varepsilon_{3})\|S\|]^{-1}}{M}\}$. We claim that for $\mu\in(0, 1]$,

If it is not true, there exist $x_{3}\in P\cap \partial B_{\overline{R}_{2}}$ and $\mu_{3} \in (0, 1]$ such that

Then,

which means

Because of $0 < \vert\vert(\lambda_{1}-\varepsilon_{3})S\vert\vert < 1$, $[I-(\lambda_{1}-\varepsilon_{3})S]$ has the bounded and inverse operator $[I-(\lambda_{1}-\varepsilon_{3})S]^{-1}$ and

By normality of cone $P$, we have

This is a contraction, which means that (3.10) is valid. By Lemma 2.14,

It together with (3.9) and the regional additivity of the fixed point index guarantees that

The proof is completed.

Now we are in a position to give the multiple solutions for BVP (1.1).

Theorem 3.4. Assume that (H2) and (H3) hold. In addition, suppose that there exists $R > 0$ such that

Then BVP (1.1) has at least two positive solutions.

Proof. For the sake of obtaining our conclusion, we first claim that for $\mu\in(0, 1]$,

Suppose on the contrary that there exist $x_{4}\in P\cap \partial B_{R}$ and $\mu_{4} \in (0, 1]$ such that

Then,

This is a contradiction, which means that (3.13) is valid. By Lemma 2.14, we have

Next, similar to the process of proving (3.3) and (3.8), there exist ${r} \in(0, R)$ and $\widetilde{R} \geq max\{R, \overline{R}_{1}\}$ such that (3.4) and (3.9) hold.

Together with (3.15), Lemma 2.14 and Lemma 2.15, one can immediately obtain that

Namely, there exist $x_{1}\in P\cap (B_{\widetilde{R}}\backslash \overline{B}_{R})$ and $x_{2}\in P\cap (B_{R}\backslash \overline{B}_{r})$ satisfying $Tx_{i} = x_{i} (i = 1, 2)$.

To sum up, Theorem 3.4 is proved.

Now we are in a position to give at least three solutions for BVP (1.1).

Theorem 3.5. Assume that there exist positive constants $a, \ b, \ c$ with $0 < a < b < c$ such that

(H5) $f(t, x) < Ma, (t, x)\in[0, 1]\times[0, a];$

(H6) $f(t, x) > Nb, (t, x)\in[{ }\frac{1}{2}, 1]\times[b, c];$

(H7) $f(t, x)\leq Mc, (t, x)\in[0, 1]\times[0, c]$.

Then BVP(1.1) has at least three nonnegative solutions $x_{1}, x_{2}, x_{3}$ satisfying $\vert\vert x_{1}\vert\vert < a$; $b < min_{t\in[\frac{1}{2}, 1]}|x_{2}(t)| < \|x_{2}\|\leq c$, $a < \|x_{3}\|\leq c$ and $min_{t\in[\frac{1}{2}, 1]}|x_{3}(t)| \leq b$.

Proof. We shall prove assumptions of Lemma 2.16 are valid.

Let $\theta(x) = min_{t\in[\frac{1}{2}, 1]}|x(t)|$. Hence, $\theta(x)$ is a nonnegative continuous concave functional on $P$.

First, we prove $T:P_{c}\rightarrow P_{c}$ is completely continuous. In fact, for $x\in \overline{P}_{c}$, from (H7) and Lemma 2.16, one can deduce that

Thus, $T: \overline{P}_{c} \rightarrow \overline{P}_{c}.$ In addition, by the continuity of $G$, $f$ and $g'$, we can conclude that $T:P_{c}\rightarrow P_{c}$ is completely continuous.

Let $x(t) = { }\frac{b+c}{2}$ for $t \in[0, 1]$, it is not difficult to see

This means $\{x\in P(\theta, b, c)|\ \theta(x) > b\}\neq \emptyset$. By condition (H6), for $x\in P(\theta, b, c)$, we have

which means that (A1) in Lemma (2.16) is valid.

By similar analysis, by (H5), one can see that

That is, (A2) in Lemma (2.16) holds. Taking $c = d$, (A3) is valid obviously.

To sum up, all assumptions of Lemma 2.16 are valid. Therefore, BVP (1.1) has at least three nontrivial solutions $x_{1}, \ x_{2}, \ x_{3}$ satisfying $\vert\vert x_{1}\vert\vert < a$; $b < min_{t\in[\frac{1}{2}, 1]}|x(t)| < \|x_{2}\|\leq c$, $a < \|x_{3}\|\leq c$ and $min_{t\in[\frac{1}{2}, 1]}|x(t)| \leq b$.

4.   The Ulam stability analysis

In this section, we shall give the criteria of Ulam stability for BVP (1.1). First, let us list the following assumption.

(H8) For all $x, y\in \mathbb{R}^{+}$, there exists a positive constant $0 < L < M$ such that

Next, for some $\epsilon > 0$, consider the following differential inequalities

Definition 4.1. ^[17] BVP (1.1) is Ulam-Hyers stable if there exists a real number $C_{f} > 0$ such that for each $\epsilon > 0$ and for each solution $x \in E$ of the inequality 4.1, there exists a solution $\overline{x}\in E$ of BVP (1.1) with

Definition 4.2 ^[17] BVP (1.1) is generalized Ulam-Hyers stable if there exist $\Phi_{f} \in C(\mathbb{R}^{+}, (0, +\infty))$, $\Phi_{f} (0) = 0$ such that for each solution $x \in E$ of the inequality 4.1, there exists a solution $\overline{x} \in E$ of BVP (1.1) with

Now, we are in a position to prove the main stable theorem of this section.

Theorem 4.3. Under the assumptions (H1), (H4) and (H8), BVP (1.1) is Ulam-Hyers stable.

Proof. Under the assumptions (H1) and (H4), by process similar to proving Theorems 3.2 and 3.3, BVP (1.1) has at least one positive solution.

Let $\overline{x} \in E$ be the solution of BVP (1.1) and $x\in E$ be a solution of

Then, by Lemma 2.10,

and

where $\mathcal{E}(t) = \ ^{C}_{0}D^{\alpha}_{g}x(t)-f(t, x)$. By (4.2), it is easy to see $|\mathcal{E}(t)| < \epsilon$.

Then,

Hence,

Therefore, BVP (1.1) is Ulam-Hyers stable.

In addition, set $\Phi_{f}(z) = Lz$, then $\Phi_{f}(0) = 0$. By Definition 4.2, BVP (1.1) is generalized Ulam-Hyers stable.

The proof is completed.

5.   Examples

In this section, three illustrative examples are worked out to show the effectiveness of the obtained results.

Example 5.1. Consider the following BVP

where $g(t) = e^{\frac{t}{2}}, \ h(t) = 1$ and

Conclusion: BVP (5.1) has at least two positive solutions.

Proof. BVP (5.1) can be regarded as a BVP of the form (1.1).

By careful calculation and Lemma 2.11, one can obtain that

By calculation, we get that

In addition, notice that $M = (\int_{0}^{1}\varphi(s)g'(s)ds)^{-1}\thickapprox11.42$ and choose $R = 5.$

Thus,

Consequently, all conditions in Theorem 3.5 hold, which means that our conclusion follows.

Example 5.2. Consider the following BVP

where $g(t) = e^{\frac{t}{2}}, \ h(t) = 1$ and

Conclusion: BVP (5.2) has at least three nonnegative solutions.

Proof. BVP (5.2) can be regarded as a BVP of the form (1.1). The function $G$, $G_{0}$, $G_{1}$ for BVP (5.2) is the same as that of BVP (5.1) in Example 5.1.

In addition, notice that

Choosing $a = \frac{1}{10}$, $b = 1$, $c = 30$, we have

By Theorem 3.5, BVP (5.3) has at least three nonnegative solutions $x_{1}, \ x_{2}, \ x_{3}$ with $\vert\vert x_{1}\vert\vert < \frac{1}{10}$; $1 < min_{t\in[\frac{1}{2}, 1]}|x(t)| < \|x_{2}\|\leq 30$; $\frac{1}{10} < \|x_{3}\|\leq 30$ and $min_{t\in[\frac{1}{2}, 1]}|x(t)|\leq 1$.

Example 5.3. Consider the following BVP

where $g(t) = e^{\frac{t}{2}}, \ h(t) = 1$ and $f(t, x) = e^{t}ln(1+x^{2})$.

Conclusion: BVP (5.3) has at least one positive solutions and the solution of BVP (5.3) is Ulam-Hyers stable and generalized Ulam-Hyers stable.

Proof. BVP (5.3) can be regarded as a BVP of the form (1.1). The function $G$, $G_{0}$, $G_{1}$ for BVP (5.3) is the same as that of BVP (5.1) in Example 5.1. By calculation, we get that

which implies that (H1) and (H4) hold.

In addition, notice that for all $x, y\in \mathbb{R}^{+},$

This means that (H8) are satisfied if we set $L = e$.

Consequently, by Theorem 4.3, BVP (5.3) has at least one positive solutions and the solution of BVP (5.3) is Ulam-Hyers stable and generalized Ulam-Hyers stable.

The proof is completed.

6.   Conclusions

The existence of solutions is of the fundamental problems for FDEs. This work studies the existence of positive solutions and multiple positive solutions for a class of FBVPs with generalized Caputo derivatives. Taking full advantage of the properties of Green's function, a suitable cone is established. The positive solutions and multiple positive solutions are obtained by means of the first eigenvalue of corresponding linear operator and the cone expansion and compression fixed point theorem. At the same time, by using Leggett-Williams theorem, we obtain that BVP (1.1) has at least three nonnegative solutions. Moreover, Ulam-Hyers stability and generalized Ulam-Hyers stability are also studied under some suitable assumptions.

For our subsequent work, the following issues will continue to be focused on:

(i) The systems studied on this topic will be more and more extensive and complicated. Therefore, it is valuable to investigate impulsive FDEs with generalized derivatives or hybrid FDEs with delay.

(ii) As an important component of technology and mathematical control theory, controllability has already gained considerable attention. Hence, the controllability for fractional differential system with generalized derivatives may be an interesting issue.

(iii) With the development of the theoretical study on FDEs, application area of FDEs with generalized derivatives in reality needs to be investigated in depth.

Acknowledgments

The authors are thankful to the editor and anonymous referees for their valuable comments and suggestions. This research was funded by NNSF of China (62073202), Natural Science Foundation of Shandong Province (ZR2020MA007).

Conflict of interest

The authors declare that there are no conflicts of interest.