Existence and uniqueness of positive solution of a nonlinear differential equation with higher order Erdélyi-Kober operators

1.   Introduction

In this paper, the initial value problem of the nonlinear differential equation

is studied, where ${_*D_\beta^{\alpha, \gamma}}$ is given in Definition 2.3 and called a Caputo type modification of the Erdélyi-Kober fractional derivative (EKFD) with $\gamma$-th order, $\gamma\in (n-1, n]$, $n\in \Bbb N^+$, $\beta\in\Bbb R^+$ and $\alpha \in\Bbb R^+$.

Caputo type fractional derivative is an important subject with numerous applications to several fields outside mathematics, such as engineering, biology, economics, physics, etc., because it describes the property of memory and heredity of many materials. In general, every type of fractional derivative has its own merits and also the shortages. The Riemann-Liouville fractional derivative (RLFD) is a natural generalization of the integer order derivative and usually employed in mathematical texts and not so frequently in applications, and the Caputo fractional derivative (CFD) is often used to formulate mathematical models of some applied problems by means of fractional differential equations since its physical interpretation is clear. The EKFD as a generalization of RLFD and CFD is often applied both in the mathematical texts and applications. The so-called Caputo type modification of the EKFD was first introduced by Gorenflo, Luchko and Mainardi in ^[1] and applied to describe of the scale-invariant solutions of the diffusion-wave equation. In fact, fractional differential equations with Caputo type initial data are natural and useful in modeling the reality. Finding simple and effective methods to solve these kinds of models is fundamental and has become more and more active ^[2,3]. Integral transform method is suitable for solving Cauchy problems of fractional differential equations with constant coefficients ^[4] in half space. For fractional differential equations with variable coefficients, Kiryakova and her collaborators established the explicit solution by use of the transmutation method ^[5,6]. In ^[7], the series method works only for relatively special domain, although it allows for finding solutions of fractional differential equations with arbitrary order. For more analytical approaches and numerical algorithm, one can refer to ^[8,9] and the references therein. As the deepening of research on calculus theory and its applications, the so-called Caputo type modification of the EKFD has attracted more and more attention ^[10,11,12]. In this paper, we develop some new related Gronwall integral inequality to enrich the analytical methods and overcome the difficulties that are caused by nonlinearity of equations and the low regularity of solutions. Recently, this idea has been generalized for studying the qualitative theory of fractional differential equations ^{[13,14,15,16,17]}.

In order to establish the solution of problem (1), we focus on a weakly singular integral whose kernel involves Mittag-Leffler functions. Some new related Gronwall type integral inequalities are established, then based on these inequalities and fixed point theorems, we show the existence and uniqueness of the positive solution of nonlinear fractional differential equations with higher order Caputo type modification of the EKFDs. This method is not confined with the order of the equation.

The main contributions of this work are that:

(ⅰ) The well posedness of the initial value problem of linear fractional differential equation with higher order Caputo type modification of the EKFDs is established.

(ⅱ) The established new Gronwall-type integral inequalities extends the results in ^[18].

(ⅲ) The equivalence between the nonlinear differential equation with higher order Caputo type modification of the EKFDs and a nonlinear Volterra-type integral equation with a complicated singular kernel is established.

(ⅳ) The existence and uniqueness of the positive solution of the nonlinear fractional differential equation with higher order Caputo type modification of the EKFDs are obtained.

This paper is organized as follows: In Section 2, the basic knowledge of some fractional calculus and Mittag-Leffler functions are recalled. In Section 3, we show the well posedness of the initial value problem of fractional differential equation. In Section 4, we analyze an integral whose kernel involves Mittag-Leffler functions and singular power functions, and then derive some new useful Gronwall-type inequalities. In Section 5, we study nonlinear fractional differential equations by use of Gronwall-type inequalities and fixed point theorems. In final, we establish the existence and uniqueness of the positive solution of the nonlinear fractional differential equation with higher order Caputo type modification of the EKFD operators.

2.   Preliminary

In this section, we start by writing down definitions of the Erdélyi-Kober fractional integral (EKFI) given in ^[5,11,19], and then give some recalls of fractional calculus that will be used later.

Definition 2.1. The EKFI of $f(t)\in \Bbb C_\mu$ is defined by

with arbitrary parameters $\mu\in \Bbb R$, $\gamma\in \Bbb R^+$, $\alpha\in \Bbb R$ and $\beta\in \Bbb R^+$, where weighted space of continuous functions $\Bbb C^{(n)}_\mu$ is defined by

The special case for $\gamma = 0$, the EKFI is defined as the identity operator. For $\gamma < 0$, the interpretation is derived by

For $\alpha = 0$, $\beta = 1$, the EKFI is reduced to the well-known Riemann-Liouville fractional integral with a power weight.

Definition 2.2. The Riemann-Liouille type modification of the EKFD of order $\gamma$ of a function $f(t)\in \Bbb C_\mu^{(n)}$, $\mu > -\beta(\alpha+1)$ is defined by

where $D_n = \Pi_{i = 1}^{n}(\alpha+i+\frac{1}{\beta}t\frac{d}{dt})$, $\gamma\in(n-1, n]$, $n\in \Bbb N$.

Definition 2.3. The Caputo type modification of the EKFD of order $\gamma$ of a function $f(t)\in \Bbb C_\mu^{(n)}$, $\mu > -\beta(\alpha+1)$ is defined by

where $\gamma\in(n-1, n]$, $n\in \Bbb N$ and $D_n = \Pi_{i = 1}^{n}(\alpha+i+\frac{1}{\beta}t\frac{d}{dt})$.

Lemma 2.4. For $f(t)\in C_\mu$, $\mu\geq-\beta(\alpha+1)$, the right-hand sided EKFD ${_*D_\beta^{\alpha, \gamma}}$ and $D_\beta^{\alpha, \gamma}$ satisfy

Lemma 2.5. For $f(t)\in C_\mu^{(n)}$, $\mu\geq-\beta(\alpha+\gamma+1)$ and $\gamma\in(n-1, n]$, $n\in\Bbb N^+$, the right-hand sided EKFD ${_*D_\beta^{\alpha, \gamma}}$ and integral $I_\beta^{\alpha, \gamma}$ satisfy

where constants $p_k = \lim_{t\to 0}t^{\beta(1+\alpha+k)}\Pi_{i = k+1}^{n-1}(1+\alpha+i+\frac{1}{\beta}t\frac{d}{dt})f(t)$, $k = 0, 1, 2, \cdots, n-1$, and the right-hand sided EKFD ${D_\beta^{\alpha, \gamma}}$ and integral $I_\beta^{\alpha, \gamma}$ satisfy

where constants $c_k = \frac{\Gamma(n-k)}{\Gamma(\alpha-k)}\lim_{t\to 0}t^{\beta(1+\alpha+k)}\Pi_{i = k+1}^{n-1}(1+\alpha+i+\frac{1}{\beta}t\frac{d}{dt})(I_\beta^{\alpha+\gamma, n-\gamma}f)(t)$, $k = 0, 1, 2, \cdots, n-1$, and $\Gamma(\cdot)$ is a Gamma function.

The Mittag-Leffler function is defined by the convergent series

where $\alpha\in \Bbb C$, $\beta\in\Bbb C$, $\Re(\cdot)$ denotes the real part of a complex number. The Mittag-Leffler function is an entire function that is a natural extension of the exponential, trigonometric and incomplete gamma functions. Particularly, it is easy to verify that $E_{\alpha, \beta}(t)$ is positive for $t > 0$, $\alpha\in \Bbb R^+$, $\beta\in\Bbb R^+$. The following asymptotic expansions of the Mittag-Leffler function are given in ^[20].

Lemma 2.6. For $0 < \alpha < 2$, $\beta\in\mathbb R$ and $\frac{\pi\alpha}{2} < \mu < \min\{\pi, \pi\alpha\}$, we have

for large $|t|$ and $|\arg t|\leq\mu$.

For $\alpha\ge 2$, $\beta\in\mathbb R$, we have

for large $|t|$, $|\arg t|\leq\frac{\pi\alpha}{2}$ and where the first sum is taken over all integers $n$ such that $|\arg t+2n\pi|\leq \frac{\pi\alpha}{2}$.

Fixed point theory is one of the most powerful and fruitful tools of various theoretical and applied fields, such as linear inequalities, variational inequalities, the approximation theory, integral and differential equations and inclusions, nonlinear analysis, the dynamic systems theory, mathematics of fractals, mathematical physics, economics and mathematical modeling. In particular, the fixed point theory is always considered a core subject of nonlinear analysis. The following results can be found in ^[21,22].

Theorem 2.7. Assume $S_0$ is a Banach space, $S_1$ is a closed, convex subset of $S_0$, $S_2$ is an open subset of $S_1$, and $p\in S_2$. Suppose that $M : \overline{ S_2} \to S_1$ is a continuous, compact map, then either

(a) F has a fixed point in $\overline{ S_2}$; or

(b) there are $s\in \partial S_2$ (the boundary of $S_2$ in $S_1$) and $\nu \in (0, 1)$ with $s = \nu M(s) + (1-\nu)p$.

Theorem 2.8. Assume $S_0$ is a Hausdorff locally convex linear topological space, $S_1$ is a convex subset of $S_0$, $S_2$ is an open subset of $S_1$, and $p\in S_2$. Suppose that $M: \overline{ S_2}\to S_1$ is a continuous, compact map, then either

(a) F has a fixed point in $\overline{ S_2}$; or

(b) there are $s\in \partial S_2$ (the boundary of $S_2$ in $S_1$) and $\nu \in (0, 1)$ with $s = \nu M(s) + (1-\nu)p$.

3.   Well posedness of the initial value problem of linear fractional differential equation

In this section, we consider the initial value problem of the linear fractional differential equation

where $u_k$ $(k = 0, 1, 2, \cdots, n-1)$ are positive numbers.

We refer to ^[23] for a study of the initial value problem involving Riemann-Liouille type modification of EKFD

where $\alpha\in\Bbb R$, $\beta\in\Bbb R^+$, $\gamma\in (n-1, n]$ and $n\in \Bbb N^+$. The explicit solution was established by the transmutation method.

Lemma 3.1. Let $f\in C_{\beta\mu}$, $\mu\ge \max\{0, -\alpha-\gamma\}-1$, then there exists a solution $u\in C_{\beta\mu}^{(n)}$ of problem (3.2) with the form

The parameter that appears in initial conditions of the initial value problem and is defined by the Riemann-Liouille type modification of the EKFDs whose physical interpretation is not clear. For this reason, the Caputo type modification of the EKFDs is introduced and the following problem is considered:

where $\alpha\in\Bbb R$, $\beta\in\Bbb R^+$, $\gamma\in (n-1, n]$ and $n\in \Bbb N^+$.

In order to use of the result given in Lemma 3.1, we recall the relation between ${_*D_\beta^{\alpha, \gamma}}$ and ${D_\beta^{\alpha, \gamma}}$, which was established in ^[12] by use of operational methods.

Lemma 3.2. The Caputo type modification of the EKFD ${_*D_\beta^{\alpha, \gamma}}$ coincides with Riemann-Liouille type modification of the EKFD ${D_\beta^{\alpha, \gamma}}$ for a function $u\in C_\nu^{n}$, $\gamma\in(n-1, n]$, $n\in \Bbb N$, $\nu\ge-\beta(\alpha+1)$, if and only if for conditions $\tilde u_k = \bar u_k, k = 0, 1, 2, \cdots, n-1$ are fulfilled.

Since ${_*D_\beta^{\alpha, \gamma}}$ and $D_\beta^{\alpha, \gamma}$ are on the right-hand sided EKFDs of order $\gamma$, then applying Lemmas 2.4 and 2.5, we have

then, there exists

which is equivalent to

in terms of Lemma 2.5. According to formulas (45) and (67) in ^[11], there exists

Then, these yield $\bar{u_k} = 0$. Hence, the problem (3.3) is equivalent to the following problem

where $\alpha\in \Bbb R$, $\beta\in\Bbb R^+$, $\gamma\in (n-1, n]$ and $n\in \Bbb N^+$. Based on the above analysis and Lemmas 3.1 and 3.2, we confirm the well-posedness of the initial value problem of inhomogeneous linear differential equation involving Caputo type modification of the EKFDs.

Theorem 3.3. Let $f\in C_{\beta\mu}$, $\mu\ge \max\{0, -\alpha-\gamma\}-1$, then there exists a solution $u\in C_{\beta\mu}^{(n)}$ of problem (3.3) with the form

According to formula (67) in ^[11], there exists

then, we obtain

Hence, set

and we arrive at

where $\mathrm B(\cdot, \cdot)$ denotes a Beta function.

Finally, applying the above results and Theorem 3.3 with $u_k = \tilde u_{n-1-k}$, we obtain Theorem 3.4.

Theorem 3.4. Set $f\in C_{\beta\mu}$, $\mu\ge \max\{0, -\alpha-\gamma\}-1$, then there exists an explicit solution $u\in C_{\beta\mu}^{(n)}$ of problem (3.1), which is given in the form

Remark 3.5. If the continuous function $f(t)$ is nonnegative and constants $\lambda$, $u_k(k = 0, 1, 2, \cdots, n-1)$ are positive, then the explicit solution of problem (5) presented in Theorem 3.4 is positive.

4.   Gronwall type integral inequalities involving Erdélyi-Kober fractional integral

Gronwall type inequalities play an important role in solving solutions of numerous differential and integral equations. The classical form of this type of inequality ^[24] is described as follows.

Theorem 4.1. Assume $u(t)$, $v_i(t)$ $(i = 1, 2)$ and $g(t)$ are continuous, nonnegative functions on $[0, +\infty)$ with $p\geq 1$ such that

then,

where $w(t) = \exp(-\int_0^tg(\tau)v_2^p(\tau)d\tau)$.

In order to establish a new useful Gronwall inequalities for studying the problem (1.1), we recall Kalla's results with the form

that one can refer to see (^{[25,26,27,28,29]}). Some extended theories of this integral are developed and have been applied to different topics, like as Gronwall inequalities and other operational calculus, integral transforms, and classes of integral and differential equations.

Consider integral

with parameters $\alpha\in\Bbb R$, $\beta\in\Bbb R^+$, $\gamma\in (n-1, n]$ and $n\in \Bbb N^+$, the kernel is expressed by

where $E_{\gamma, \gamma}(\cdot)$ denotes the Mittag-Leffler function, function $f(\cdot)$ is given. It is easy to find that integral (4.1) is a direct generalization of Kalla's and closely connected with EKFD formally. To establish some new related Gronwall inequalities based on (4.1)–(4.2), one should consider the more complicated kernel with singularity compared with the form ^[3,14,17,18].

In the following, we show our results.

Theorem 4.2. Let $\lambda\in \Bbb R^+$, $\alpha\in\Bbb R$, $\beta\in\Bbb R^+$, $\gamma\in (n-1, n]$ and $n\in \Bbb N^+$, assume $v_i(t)$ $(i = 1, 2)$, $g(t)$ and $u(t)$ are continuous, nonnegative functions on $(0, +\infty)$ with

then,

where $w(t) = \exp (-\int_0^t (gV_2)^{r}(\tau)d\tau)$, $V_1(t) = v_1(t)$, $V_2(t) = Cv_2(t)h(t)t^{\frac{1-\beta}{p}+\frac{1}{q}}$, constant $C = C(\alpha, \beta, \gamma, p)$ only depends on $\alpha, \beta, \gamma, p$, and

for $(1-\beta)q+p > 0$, $p(\alpha+\gamma+1-\frac{1}{\beta})+1 > 0$, $\frac{1}{p}+\frac{1}{q}+\frac{1}{r} = 1$, $p, q, r\in \Bbb R^+$.

Proof: By a direct computation, it is easy to verify that

In terms of the series expansion (2.2) and the asymptotic behavior of the entire function $E_{\alpha, \beta}(t)$ in Lemma 2.6, we have

for $C_1 = C_1(\alpha, \beta, \gamma, p)$, $p(\alpha+\gamma+1-\frac{1}{\beta})+1 > 0$ and any $t\in[0, +\infty)$.

Meanwhile, by a direct computation, we obtain

for $(1-\beta)q+p > 0$.

Then, the H$\ddot{o}$lder inequality and (4.5)–(4.6) yield

where $V_1(t) = v_1(t)$, $V_2(t) = Cv_2(t)h(t)t^{\frac{1-\beta}{p}+\frac{1}{q}}$ for $\frac{1}{p}+\frac{1}{q}+\frac{1}{r} = 1$, $p, q, r\in \Bbb R^+$.

Hence, in terms of (4.7), Theorem 4.1 yields (4.3).

Thus, we complete the proof.

In solving the fractional differential equations with nonlinear term $f(t, u(t))$, the Gronwall type integral inequality is a powerful tool ^[15,18]. Here and in the following, we define an invertible operator $\Theta$ by

for some positive and nondecreasing function $\theta(t)$.

Theorem 4.3. Let $\lambda\in\Bbb R^+$ $\alpha\in\Bbb R$, $\beta\in\Bbb R^+$, $\gamma\in (n-1, n]$ and $n\in \Bbb N^+$, assume functions $v_i(t)\in C[0, T]$, $i = 1, 2$ are nondecreasing and nonnegative, $g(t)\in C[0, T]$ is nonnegative, $f(t)\in C[0, T]$ is nondecreasing and nonnegative, and $u(t)\in C[0, T]$ is nonnegative, such that

for arbitrary positive number $T$, then

where $V_1(t) = v_1(t)$, $V_2(t) = Cv_2(t)h(t)t^{\frac{1-\beta}{p}+\frac{1}{q}}$, $C = C(\alpha, \beta, \gamma, p)$ is a constant which only depends on $\alpha, \beta, \gamma, p$, and

for $(1-\beta)q+p > 0$, $p(\alpha+\gamma+1-\frac{1}{\beta})+1 > 0$, $\frac{1}{p}+\frac{1}{q}+\frac{1}{r} = 1$, $p, q, r\in \Bbb R^+$.

Proof: Similar to the proof in proving (4.7) of Theorem 4.2, we have

for some $T_0\in (0, T]$. Take $U = u^{r}$, then the last inequality becomes

Next, we show

where $\Theta$ is given by (4.8) with $\epsilon = V(0)$, and

In addition, if we let $\theta(t) = f^{r}(t^{\frac{1}{r}})$, then

holds and (4.10) is derived.

In terms of initial value problem (4.10), we obtain

Since $\Theta$ is inversible, then (4.11) is equivalent to the following equation

Furthermore, applying (4.9), there exists

For the arbitrariness of $t\in [0, T_0]$, we conclude

Furthermore, for the arbitrariness of $T_0\in[0, T]$, we obtain

This completes the proof.

5.   Existence and uniqueness of solution of nonlinear differential equation

In this section, we establish the existence and uniqueness of the positive solution of the initial value problem of the nonlinear differential equation with higher order Caputo type modification of the Erdélyi-Kober operators in terms of Theorems 4.2 and 4.3 with parameters $\gamma$-th order, $\gamma\in (n-1, n]$, $n\in \Bbb N^+$, $\beta\in\Bbb R^+$ and

Theorem 5.1. Set $C_0$ as a positive number, $f$ as a nonnegative and continuous function that satisfies $f(t, u)|_{u = 0} = 0$, $|f(t, u)-f(t, v)|\leq C_0|u-v|$ and initial conditions $u_k > 0$ for all $k = 0, 1, 2, \cdots, n-1$, then there exists a unique positive solution $u(t)\in C(0, +\infty)$ of problem (1.1) and ${_*D_\beta^{\alpha, \gamma}} u(t)\in C(0, +\infty)$.

Proof: Set $S_1 = \{u(t) \in C[0, T]: t^{\beta(\alpha+n)}u(t) \geq u_0\}$, $T\in (0, +\infty)$. Consider the map $M: S_1\to S_1$ defined by

Since $f$ is nonnegative, then Remark 3.5 implies $Mu$ is positive.

Based on Theorem 4.2, we obtain that any fixed point of map $M$ is a solution of the Volterra-type integral equation

We know that the fixed point of the Volterra-type integral equation is also a solution of problem (1.1). Set

where

and $h(t)$ is given in Theorem 4.2 for $(1-\beta)q+p > 0$, $p(\alpha+\gamma+1-\frac{1}{\beta})+1 > 0$, $\frac{1}{p}+\frac{1}{q}+\frac{1}{r} = 1$, $p, q, r\in \Bbb R^+$. Take a similar procedure used in ^[18], and we derive that $M: \overline{ S_2}\to S_1$ is continuous and compact.

If $u\in \overline{ S_2}$ is any solution of equation

for $\nu\in(0, 1)$, then it is easy to verify that

Applying (10), we have

Hence, by use of Theorem 2.7, we claim that F has a fixed point in $\overline{ S_2}$, then this fixed point is a positive solution of problem (1.1). Since the arbitrariness of $T\in(0, +\infty)$, then we claim that there exists a positive solution $u\in C(0, +\infty)$ of problem (1.1).

Furthermore, consider the continuity of function $f(\cdot, \cdot)$ and the expression

and we obtain ${_*D_\beta^{\alpha, \gamma}} u(t)\in C(0, +\infty)$.

If there exists two solutions $u$ and $v$ in $\overline{U}$ to problem (15), then it follows

then by use of Theorem 4.2 with $V_1(t) = 0$, we obtain $u = v$. This yields the uniqueness of the solution.

Thus, we complete the proof.

Theorem 5.2. Set $w(t)\in C[0, T]$ and the nondecreasing function $g(t)\in C[0, T]$ as nonnegative, assume $f(t, u(t))\leq w(t)g(u(t))$, which is nonnegative and continuous, and initial conditions $u_k > 0$ for all $k = 0, 1, 2, \cdots, n-1$, then there exists a positive solution $u(t)\in C(0, T]$ of problem (1.1) and ${_*D_\beta^{\alpha, \gamma}} u(t)\in C(0, T]$.

Proof: For any positive number $T > 0$, we form a map $M: S_1\to S_1$, which is defined by

where $S_1 = \{u(t) \in C(0, T]: t^{\beta(\alpha+n)}u(t) \geq u_0, T > 0\}$. It is easy to verify that $Mu$ is positive if $f$ is nonnegative. Set

where

and $h(t)$ is defined in Theorem 4.3 for $(1-\beta)q+p > 0$, $p(\alpha+\gamma+1-\frac{1}{\beta})+1 > 0$, $\frac{1}{p}+\frac{1}{q}+\frac{1}{r} = 1$, $p, q, r\in \Bbb R^+$. By applying the usual techniques used in Theorem 5.1, we obtain that the map $M: \overline{S_2}\to S_1$ is continuous and compact. It follows of Theorem 3.4 that the fixed points of operator $M$ are solutions of problem (1.1).

If $u\in \overline{ S_2}$ is any solution of the Volterra-type integral equation

for $\gamma\in(0, 1)$, then it is easy to verify that

Applying Theorem 4.3 that we obtain

Hence, Theorem 2.8 yields that F has a fixed point in $\overline{ U}$.

Meanwhile, Eq (1.1) and the continuity of function $f(\cdot, \cdot)$ implies ${_*D_\beta^{\alpha, \gamma}} u(t)\in C(0, T]$.

Thus, we confirm the result of Theorem 5.2.

6.   Conclusions

In this paper, the positive solution of a nonlinear differential equation with higher order Caputo type modification of the EKFD was studied. First, the well-posedness of the initial value problem of the higher order linear model was proved and an explicit positive solution was presented based on the transmutation method. Second, some new Gronwall type inequalities involving EKFI with singular kernels were established. At last, by applying the derived results and some fixed point theorems, the existence and uniqueness of the positive solution of this kinds of nonlinear differential equation were obtained. The method is applicable to such kinds of fractional differential equations with any order $\gamma\in (n-1, n]$, $n\in \Bbb N^+$.

Use of AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

I would like to express my sincere gratitude to the anonymous referees for their useful comments and suggestions.

This work was supported by Qinglan Project of Jiangsu Province of China and NNSF of China (No. 11326152).

Conflict of interest

The author declares to have no competing interests.