Existence criteria for fractional differential equations using the topological degree method

1.   Introduction

Fractional calculus is a field of mathematics that expands the notion of differentiation and integration beyond integer orders. These operations are applicable to all real numbers, including non-integer values. Fractional calculus has found practical applications in a variety of physical systems, as can be seen in ^{[2,16,17,20,30,40,41]} and some classic books ^{[22,27,29,39]}.

Based on the literature, fixed point theory has been applied for many years to establish that differential equations have a solution ^{[26,31,35,46,48,49]}. Mahwin ^[32] in their paper made use of the topological degree theory (TDT) to solve integral equations for the first time. Isaia ^[24] theoretically applied TDT to analyze some integral equations. Use of TDT can also be observed in ^{[14,42,43,47]}.

To date, a lot of good work with integro-differential equations has been conducted, including the studies described in ^[7,34]. Zuo et al. ^[33] derived the following fractional integro-differential equations with impulsive and antiperiodic boundary conditions:

where $J^\prime = J \setminus \{q_1, q_2, \ldots, q_m\}, 0 < \gamma \leq 1, \lambda > 0$ and $D^{\gamma}$ is denoted as CFD, $1 < \gamma \leq 2$. Here $f \in C(J \times \mathscr{R} \times \mathscr{R} \times \mathscr{R}, \mathscr{R}), J = [0, 1]$ is the integro-differential function and $P$ and $S$ are linear operators:

where $q \in J, \; k \in C(D, \mathscr{R}), \; D = \{(q, s) \in J \times J : q \geq s\}$, $h \in C(J \times J, \mathscr{R})$.

It is observed from the literature that for last many years, fixed point theory has been used to prove the existence of a solution to the differential equations. However, the use of fixed point theory requires strong conditions, which severely limits its applicability. Also the uniqueness is proved via the Banach contraction principle, which is applied to find a unique solution for the defined problem. It is noticed that most of the work on the topic of fractional differential equations (FDEs) involves either the RL or CFD. While these derivatives are common place in the study of FDEs, the Hadamard fractional derivative (HFD) is another kind of fractional derivatives. This kind of derivative was introduced by Hadamard ^[21]. This fractional derivative differs from the other ones in the sense that the kernel of the integral (in the definition of Hadamard derivative) contains the logarithmic function of an arbitrary exponent. In ^[25], we see the modification of the HFD into a more suitable one called the Caputo-Hadamard fractional derivative (CHFD). Applications of where Hadamard derivative and the Hadamard derivative integral can be found in papers by Butzer et al. ^[11,12,13]. Other important results dealing with studies on fractional calculus using Hadamard derivatives can be seen in ^{[4,6,8,9,18,19,23,36,37,38,45]}.

A paper by Jarad et al. ^[25], deals with the CHFD by modifying the to be of caputo type. This is familiar with different kinds of boundary conditions like the Neumann boundary condition and the Dirichlet boundary condition. A weighed combination of these boundary conditions is called the Robin boundary condition. It finds its applications in fields such as physics. The Caputo-Hadamard (CH) derivative type of FDEs with boundary value problems are described in ^[1,3,5,10].

TDT-based existence results for FIDEs with CH-derivatives have the following form:

where $^C_HD^{\nu}$, $^C_HD^{\gamma}$ are CH-derivatives of order $\nu, \gamma$, respectively with $1 < \nu\leq 2, 0 < \gamma \leq 1$, HD integral of order $^H\mathcal{I}^{\nu_i}$, $\nu_i, \; i\in[1,2]$ and $f \in C(J \times \mathscr{R} \times \mathscr{R} \times \mathscr{R}, \mathscr{R})$ is the continuous function; $P$ and $S$ are linear operators;

Let $a, b, c, d \in \mathscr{R}$ such that

In this paper, we determine the existence results via TDT in Section 2. Additionally, we discuss the FIDEs existence results under boundary conditions. An appropriate illustration and conclusion are provided in Sections 4 and 5.

2.   Facts

Here, we shall establish basic results and definitions for our analysis. We shall refer to the notations and results from ^[15]. Let the Banach space (BS) be $X$ and $\mathscr{B} \subset \mathscr{P}(X)$ be bounded subsets.

Definition 2.1. ^[14] Let $\epsilon: \mathscr{B} \to \mathscr{R}_+$,

where, Kuratowski- measure of non compactness is $B \in \mathscr{B}$.

Let $B$ be a compact set and set $B$ of a space $X$ is compact if and only if it is complete and totally bounded. The value of $\epsilon$ is the measured value, and the value of $\epsilon(B)$ is $0$. The set is compact, when the value is $0$. The larger the value of $\epsilon$, the less it is like a compact set.

Proposition 2.2. ^[14] For bounded subsets $\mathscr{D}$, $\mathscr{D}_1$, $\mathscr{D}_2$ on a BS,

(1) $\epsilon(\mathscr{D}) = 0 \Leftrightarrow \overline{\mathscr{D}}$ is compact,

(2) $\epsilon(\lambda \mathscr{D}) = \vert \lambda \vert \epsilon(\mathscr{D}), \quad \lambda\in \mathscr{R}$,

(3) $\zeta(\mathscr{D}_1+\mathscr{D}_2) \leq \epsilon(\mathscr{D}_1)+\epsilon(\mathscr{D}_2), \quad \mathscr{D}_1, \mathscr{D}_2 \in \mathscr{B}$,

(4) $\mathscr{D}_1 \subset \mathscr{D}_2 \Rightarrow \epsilon(\mathscr{D}_1) \leq \epsilon(\mathscr{D}_2)$,

(5) $\epsilon(\mathscr{D}_1 \cup \mathscr{D}_2) = \max \{\epsilon(\mathscr{D}_1), \epsilon(\mathscr{D}_2)\}$,

(6) $\epsilon(\mathit{\text{conv}}\mathscr{D}) = \epsilon(\mathscr{D})$,

(7) $\epsilon(\overline{\mathscr{D}}) = \epsilon(\mathscr{D}).$

Let $\Psi: = \{\zeta: [0, \mathscr{T}] \to \mathscr{R}: \zeta \in C(I^\prime)\}$ and $(\Psi, \vert \vert \cdot \vert \vert)$ be a BS under $\vert \vert \zeta \vert \vert: = \sup \{\vert \zeta(q)\vert : q \in [0, \mathscr{T}]\}$.

Definition 2.3. ^[14] Suppose a continuous bounded map is $\mathscr{F}: \varpi \to \mathscr{Y}$ and $\varpi \subset \mathscr{Y}$ and $\exists$ $k \geq 0$ such that $\epsilon(\mathscr{F}(\mathscr{H})) \leq k \epsilon(\mathscr{H}) \forall \mathscr{H} \subset \varpi$. If $\mathscr{F}$ is an $\epsilon$-contraction that implies $k < 1$.

Definition 2.4. ^[14,15,24] Let a $C_{\epsilon}(\varpi)$ be a class of all $\epsilon$-condensing maps $\mathscr{F}: \varpi \to X$. An $\epsilon$-condensing map $\mathscr{F}: \mathscr{Y} \to \mathscr{F}$ if $\forall A \in \mathscr{B}$, $\epsilon(\mathscr{F}(A)) \leq \epsilon(A)$.

Theorem 2.5. ^[14] Let map $\mathscr{F}:\mathscr{Y} \to \mathscr{Y}$ be $\epsilon$-condensing; then,

such that $\mathscr{H} \subset \mathscr{B}_r(0)$, also $\mathscr{H}$ is a bounded set in $\Psi$, so $\exists\; r > 0$

Therefore, $T$ has a fixed point.

Definition 2.6. ^[29] Let $\nu > 0$ be an Hadamard derivative integral of order $\zeta \in L^1(J)$ is;

where

Let $\delta = q\dfrac{d}{dt}, \quad \nu > 0, n = [\nu]+1$.

Definition 2.7. ^[29] Let $\nu > 0$ be the Hadamard derivative and $\zeta \in \Psi$ is;

Definition 2.8. ^[25,29] Let $\nu > 0$ be the CH-derivative and $\zeta \in \Psi$ is;

Lemma 2.9. ^[25,29] Let $\nu > 0, r > 0$ and $n = [\nu]+1$.

(1) $^H\mathscr{I}^{\nu}(\log\frac{q}{a})^{r-1} = \dfrac{\Gamma(r)}{\Gamma(\nu+r)}(\log\frac{q}{a})^{\nu+r+1}.$

Let $a = 1$ and $r = 1$; we get that $^H\mathscr{I}^{\nu}(1)(\nu) = \dfrac{1}{\Gamma(\nu+1)}(\log (q))^{\nu+1}$.

(2) $^H_CD^{\nu}(\log\frac{q}{a})^{r-1} = \begin{cases} \dfrac{\Gamma(r)}{\Gamma(r-\nu)}(\log\frac{q}{a})^{r-\nu-1}, \quad r > n\\ 0, \quad r \in \{0, 1, \ldots, n-1\} \end{cases}.$

Lemma 2.10. ^[25,29] Let $nu_1, \nu_2 > 0$ and $\zeta \in \Psi$; then

(1) $^H\mathscr{I}^{\nu_1} (^H\mathscr{I}^{\nu_2}\zeta(q)) = (^H\mathscr{I}^{\nu_1+\nu_2}\zeta(q));$

(2) $^H_CD^{\nu_1} (^H\mathscr{I}^{\nu_2}\zeta(q)) = (^H\mathscr{I}^{\nu_2-\nu_1}\zeta(q));$

(3) $^H_CD^{\nu_1} (^H\mathscr{I}^{\nu_1}\zeta(q)) = \zeta(q).$

Lemma 2.11. ^[25,29] Let $\nu > 0$ and $n = [\nu]+1$. Let $\zeta \in \Psi$ be the CH derivative of the FDEs

has a solution as

3.   Main results

We shall define some hypotheses:

$({\rm{A1}})$ $\exists$ constants $A_1, A_2 > 0$ and $p \in [0, 1)$ such that

$({\rm{A2}})$ $\exists$ constants $\alpha, \beta, \chi$ such that

We define $k_{\max} = \sup\limits_{q \in J}\int_0^q \vert k(q, s) \vert ds$ and $h_{\max} = \sup\limits_{q \in J}\int_0^1 \vert h(q, s) \vert ds$.

Thus using (A2),

Now we prove the existence result:

Lemma 3.1. ^[14] Let $h$ be a continuous function on $J$; then, we have the following FIDE:

have a unique solution given by

where,

● $K_1(q) = c_1(\chi_1-\chi_2q), \quad K_2(q) = c_1\chi_3+\chi_4q$

● $\chi_1 = \dfrac{1}{\chi}\Big(c\log L + \dfrac{d (\log L)^{1-\gamma}}{\Gamma(2-\gamma)} - \dfrac{c_2 (\log \zeta_2)^{\nu_2+1}}{\Gamma(\nu_2+2)}\Big)$

● $\chi_2 = \dfrac{1}{\chi}\Big(c\log L - \dfrac{c_2 (\log\zeta_2)^{\nu_2}}{\Gamma(\nu_2+1)}\Big)$

● $\chi_3 = \dfrac{1}{\chi}\Big(\dfrac{c_1 (\log\zeta_1)^{\nu_1+1}}{\Gamma(\nu_1+2)}\Big)$

● $\chi_4 = \dfrac{1}{\chi}\Big(a - \dfrac{c_1 (\log\zeta_1)^{\nu_1}}{\Gamma(\nu_1+1)}\Big)$

and $\chi$ is given by (1.2).

Proof. By Lemma 2.11, Eq (3.1) becomes,

By using boundary conditions, we have

Solving for $k_0, k_1$ we get the following solutions:

and,

Substituting for $k_0$ and $k_1$ we get (3.2). □

In view of the problem (1.1), by Lemma 3.1, we get,

we denote $g(q \zeta(q), P \zeta(q), S \zeta(q))$ by $g_{\zeta}$; then, we have

and $\chi$ is given by (1.2). The next steps are as follows:

(1) Define $T_1: \Psi \to \Psi$ as $T_1\zeta(q) = ^H\mathscr{I}^{\nu}g_{\zeta}(q)$.

(2) Define $T_2: \Psi \to \Psi$ as $T_2\zeta(q) = K_1(q)^H\mathscr{I}^{\nu_1+\nu}g_{\zeta}(\zeta_1) + K_2(q)c_2^H\mathscr{I}^{\nu_2+\nu}g_{\zeta}(\zeta_2)$.

(3) Define $T_3: \Psi \to \Psi$ as $T_3\zeta(q) = K_2(q)(c^H\mathscr{I}^{\nu}g_{\zeta}(L)+d^H\mathscr{I}^{\nu-\gamma}g_{\zeta}(L)))$.

Let $T: \Psi \to \Psi$ given that $T = T_1+T_2+T_3$. Thus the problem is reduced to finding the fixed points of the operator $T$.

Theorem 3.2. $T_1: \Psi \to \Psi$ is Lipschitz-continuous with the Lipschitz constant $\dfrac{(\log L)^{\nu}}{\Gamma(\nu+1)}(\alpha+\beta k_{\max}+\chi h_{\max})$. It also satisfies the following growth relation:

Proof. Let $\zeta_1, \zeta_2 \in \Psi$; then

This is true for all $q \in J$. Thus when we take the supremum over $q \in J$,

Hence, $T_1$ is a Lipschitz constant provided that

For the growth relation, we have,

Since this is true $\forall \, \, \, \, q \in J$, taking the supremum over all $q$, we have

□

Theorem 3.3. Assume that the operator $T_2$ is continuous and fulfills the following growth relation:

where,

Proof. Let $\zeta_n$ be a sequence in $\Psi$ that converges to $\zeta \in \Psi$. Let $g_{\zeta}$ by continuous; it follows that $g_{\zeta_n} \to g_{\zeta}$. So, $T_{2}$ is continuous according to the Lebesgue dominated convergence theorem (LDCT).

Hence,

□

Theorem 3.4. Assume that the operator $T_3$ is continuous and the following growth relation is satisfied:

where

Proof. Let $\zeta_n$ be a sequence in $\Psi$ that converges to $\zeta \in \Psi$. Since $g_{\zeta}$ is continuous, it follows that $g_{\zeta_n} \to g_{\zeta}$. Thus by the LDCT, it follows that $T_3$ is continuous.

Hence,

□

Theorem 3.5. Suppose that $T_2$ is a compact map implying that $T_2$ is Lipschitz constant zero.

Proof. Let $\zeta \subset B(r)$ be a bounded set. In order to prove that $T_2$ is a compact map. By Theorem 3.3, for $\zeta \in \zeta$,

Hence $T_2(\zeta)$ is uniformly bounded.

Now, for any $\zeta \in \Psi$,

Now, for $q_1, q_2 \in J$,

Thus, because $q_2 \to q_1$, $\vert T_2 \zeta(q_2) - T_2 \zeta(q_1) \vert \to 0$ which implies that $T_2$ is equicontinuous. $T_2$ is compact according to the Arzela-Ascoli theorem. Hence $T_2$ is LC zero. □

Theorem 3.6. If $T_3$ is a compact then $T_3$ is Lipschitz constant zero.

Proof. Assume that $\zeta \subset B(r)$ is a bounded set. In order to prove that $T_3$ is a compact map. From Theorem 3.4, for $\zeta \in \zeta$,

Hence $T_3(\zeta)$ is uniformly bounded.

Now, for any $\zeta \in \Psi$,

Now, for $q_1, q_2 \in J$, we have,

Thus, because $q_2 \to q_1$, $\vert T_3 \zeta(q_2) - T_3 \zeta(q_1) \vert \to 0 \Rightarrow T_3$ is equicontinuous. $T_3$ is compact according to the Arzelà-Ascoli theorem. Hence $T_3$ is LC zero. □

Since $T = T_1+T_2+T_3$ and $T_1$ is LC $\dfrac{(\log L)^{\nu}}{\Gamma(\nu+1)}(\alpha+\beta k_{\max}+\chi h_{\max})$ and $T_2, T_3$ are LC $0$, it follows that $T$ is LC $\dfrac{(\log L)^{\nu}}{\Gamma(\nu+1)}(\alpha+\beta k_{\max}+\chi h_{\max})$.

If we assume that $\dfrac{(\log L)^{\nu}}{\Gamma(\nu+1)}(\alpha+\beta k_{\max}+\chi h_{\max}) < 1$, then by Definition 2.4, $T$ is $\epsilon$-condensing.

Theorem 3.7 (Existence). Let FODE (1.1) have at least one solution if

Proof. Consider the set

Let $\zeta \in \mathscr{H}$ such that $\lambda T \zeta = \zeta$; then,

Taking $\vert \vert \cdot \vert \vert$ on both sides,

Letting $\vert \vert \zeta \vert \vert \to \infty$, and by using $p \in [0, 1)$ by Theorem 2.5, (1.1) has a solution. □

Theorem 3.8 (Uniqueness). The FODE has a unique solution if

Proof. Let $\zeta_1, \zeta_2 \in \Psi$ be arbitrary and $q \in J$; then

Taking the supremum over all $q \in J$, we have

From Theorems 3.1–3.3,

We have the following from the definition of $T_2$:

Thus,

We have the following from the definition of $T_3$:

Thus,

Hence, we have,

Since $\Big(\dfrac{(\log L)^{\nu}}{\Gamma(\nu+1)}+C_{T_2}+C_{T_3}\Big)(\alpha+\beta k_{\max}+\chi h_{\max}) < 1$, the FODE (1.1), has a unique solution. □

4.   Example

The FIDEs with boundary value conditions are as follows:

where $g(\zeta) = \frac{1}{8}\int_{0}^{1} e^{2s-2}\zeta(s)ds$. Here $\nu = \frac{3}{2}, \gamma = {1}{3}, a, c = 1, b = 3, d = 2, c_1 = 3, \nu_1 = \frac{1}{4},$ $\nu_2 = \frac{3}{4}, \zeta_1 = \frac{1}{4}, \zeta_2 = \frac{3}{4}, L = e$ and

By the above parameters in $C_{T_2}$ and $C_{T_3}$, we get

To prove Theorem 3.6, we take

in (1.1) and then

Hence the condition (A1) holds with $P = 0.087$, where $P = \alpha+\beta k_{\max}+\chi h_{\max}$. We use the following equation to calculate $q_f$ from the given data is;

Let $q \in J, \zeta \in \mathscr{R}$ and

Hence the condition (A2) holds with $\mathcal{W}_1 = 3, \mathcal{W}_2 = \frac{1}{30}, \mathcal{W}_3 = 0.054$ and $\mathcal{W}_4 = 0$.

By Theorem 3.6,

has the following solution set;

Thus,

Hence, $\Big[\dfrac{(\log L)^{\nu}}{\Gamma(\nu+1)}+C_{T_{2}}+C_{T_{3}}\Big]P = 0.2827 < 1.$

5.   Conclusions

This work was performed to investigate the existence and uniqueness of FIDEs with CH derivatives of fractional order by using the Robin boundary condition, TDT and fixed point theorem have been used to accomplish the analysis. The fundamental idea is shown with an efficient example.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R157), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This study was also supported by funding from Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444).

Conflict of interest

The authors declare no conflict of interest.