On a general class of $n$th order sequential hybrid fractional differential equations with boundary conditions

1.   Introduction

Recently fractional calculus has gotten much attention from researchers like traditional calculus. The mentioned area has the ability to describe real-world phenomena in more excellent ways. Also, such derivatives have numerous applications in the description of those systems with memory effects. Due to various applications, the said calculus has been used to investigate various infectious disease models like in ^[1,2,3]. Also, in the field of physics, engineering, and cosmology, fractional calculus has very well been used. For instance, we refer ^[4,5,6]. It has been shown that in many applications, the use of fractional calculus provides more realistic models than those obtained via classical ordinary derivatives. Due to this reason, the study of fractional models has received great attention from many researchers in the last few years. As the fractional order derivatives have important characteristics known as the memory effect which ordinary derivatives do not involve. Also, fractional differential operators are nonlocal as compared to the local behavior of integer derivatives. Recently researchers have published some very important results in this regard like ^[7,8]. Also, authors ^[9] have published some new results on the numerical scheme for fractional order SEIR epidemic of measles. Here it is remarked that some authors have discussed the geometry of fractional order derivatives. For instance, authors ^[10] have suggested a geometric interpretation of the fractional derivatives which is based on modern differential geometry and the geometry of jet bundles. In fact, the fractional differential operators are definite integrals that create the complete accumulation or spectrum of the function on whose these applications include the corresponding integer-order counterpart of a special case. In the same way, authors ^[11] have given the definition of the geometric interpretation of gradient of order $(0, 1]$. In this way, they have suggested some geometric interpretations of the differentiability of real order.

Different problems under the concept of fractional calculus have been studied for the existing theory and stability analysis. One of the important areas is devoted to investigating hybrid problems under the aforesaid concepts. In this regard, different problems under boundary and initial conditions have been studied by using nonlinear analysis and tools of advanced functional analysis. In the present time class of fractional differential equations devoted to the quadratic perturbation of nonlinear differential equations (called hybrid differential equations) has attracted much attention from researchers. This is due to the fact that they include several dynamical systems as special cases. For instance, Dhage and Lakshmikantham ^[12] discussed the existence theory of the following problem of hybrid differential equations

where

They used hybrid fixed point theory to establish the existence of solutions to the proposed problems. In the same way, Dhage and Jadhav^[13] studied the existence and uniqueness results for the ordinary first-order hybrid differential equation with perturbation of second type given by

where $f, g \in C([t_0, t_0+a]\times \mathbb{R}, \mathbb{R}).$ Motivated from the mentioned mentioned work, Lu et al.^[14] studied the following class of FHDEs by replacing the ordinary derivative by Caputo fractional type with $0 < \sigma\leq 1$ as

They used the hybrid fixed point theory of Dhage to study the existence and uniqueness of the solution to the mentioned problem.

In the last few years, hybrid fixed point theory has been used to study different problems of hybrid nature for the existence theory in ^{[15,16,17,18]}. Also, using the aforementioned tools, authors have established various results devoted to existence theory for different boundary value problems of HFDEs in ^{[19,20,21,22,23]}. Also, some authors have studied integral-type FHDEs in ^[24,25]. Authors ^[26] have studied a system of FHDEs of thermostats type by using a fixed point approach. Furthermore, authors ^[27] have investigated a class of FHDEs under mixed-type hybrid integral boundary conditions. In the same way authors in ^{[28,29,30,31,32]} have used the tools of nonlinear functional analysis for studying various problems and systems of HFDEs. Here, we remark that authors ^[33] established a review of the analytical solutions for some generalized classes. In the same line, a class of HFDEs has also been considered in ^[34]. Authors in ^[35,36,37] have studied different boundary value problems of fractional order using topological degree theory for the existence theory. Here it should be kept in mind that a hybrid system is a dynamic system that interacts with continuous and discrete dynamics. For, example, the novel multiplex engineering systems involve numerous kinds of process and abstract decision-making units that present the image of various systems simultaneously exhibiting continuous and discrete time dynamics (see details in ^[38]). Further, the applications of hybrid systems can be found in embedded control systems also (see ^[39]).

The existence theory of solutions to nonlinear problems is an important area of research in the current scenario. Because the said theory predicts the existence of a solution to a dynamic problem whether it has a solution or not. Usually, for the said theory fixed point theory has been used very well. But fixed point theory needs strong compact conditions which restricted its use in some situations. Also, to deal HFDEs, Dhage established some hybrid-type fixed point theorems to study the existence and uniqueness of the solution to the mentioned problems. However, it also needs the same strong compact conditions. To relax, the criteria and replace strong compact conditions with some weaker compactness, the degree theory has been introduced. It has a sophisticated tool to be used to investigate numerous nonlinear problems of integral, differential, and difference equations for their corresponding solution. The mentioned theory has been used very well for usual problems of fractional order equations. However, in the case of HFDEs, it has not been used properly. For some important contributions by using degree theory to study various problems in fractional calculus, we refer few papers as ^{[40,41,42,43]}.

In this work, we study a more general class of nonlinear boundary value problems (BVPs) consisting of a more general class of $n$th order S-HFDEs) together subject to nonlinear boundary conditions. Also, we choose the general case in which the nonlinear functions involved depend on the non-integer order derivatives. Further, necessary conditions required for the uniqueness of a solution to the proposed problem, we implement Caratheódory conditions along with techniques of measure of non-compactness and degree theory. Some new and interesting results are developed. Also, a result devoted to U-H stability is derived for the considered problem. Our proposed problem is stated as

where $0 < \vartheta\leq 1, \, n-1 < {\omega}, \beta_i, \ \gamma\leq n, \, n-2 < \rho \leq n-1$ with $n\geq 2$, $0 < \eta < 1$, the functions $f:\mathrm{I}\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}-\{0\}, \, h_i:\mathrm{I}\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}\, (i = 1, 2, ...., m) \text{ and } g:\mathrm{I}\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ satisfy the Caratheódory conditions. Moreover, $\psi_1, \, \psi_2:\mathbb{R}\rightarrow \mathbb{R}$ are nonlinear functions. Also, the notation $^{c}{\bf D}$ denotes the Caputo fractional order derivative and ${\bf I}$ represents fractional integral. Here, we use the tools mentioned in ^[44,45,46] to establish a detailed analysis of the considered problem. Also, stability is an important aspect of qualitative theory. In this regard, U-H stability analysis has also considered for some problems of HFDEs. For instance, see ^[47,48].

The present article is organized as: Section 1 is devoted to the introduction. Section 2 is related to basic results from fractional calculus and degree theory. Section 3 is devoted to the first part of our main results. Section 4 is related to the second part of our main results. The section is consisted of applications to verify our results. Section 6 is devoted to a brief conclusion.

2.   Fundamental results

Here it should be kept in mind that we have used the following basic definitions from ^[1,2] of fractional order derivative and integration.

Definition 2.1. If $\vartheta > 0,$ then the fractional order integration of a function $u\in L^1([0, 1])$ is given by

Definition 2.2. The fractional derivative in Caputo sense of a function $u$ over the interval $[0, 1]$ is defined as

where $n-1 = [\vartheta].$

Theorem 2.3. The solution of

is derived as

for arbitrary $\mathscr{C}_i\in \mathbb{R}$, $i = 0, 1, 2, \dots, n-1 = [\vartheta]$.

Let $\mathbb{E} = \{{u}\in C(\mathrm{I}):^{c}{\bf D}^{{\omega}-1} {u}\in C(\mathrm{I})\}$ is Banach space under the norm $\| {u}\|_{\rho} = \max \limits_{{0}\leq t\leq {1}} | {u}(t)|+\max \limits_{{0}\leq t\leq {1}} |^{c}{\bf D}^{\rho}{u}|$.

Let ${\bf P}$ represents family of all bounded sub sets of $\mathbb{E}$, then we define the following measure of non-compactness.

Definition 2.4. ^[44] The measure due to non-compactness $\mu: {\bf P}\rightarrow \mathcal{R}^+$ is Kuratowski measure which is defined as

Definition 2.5. ^[25] If $\mathrm{T_1}, \mathrm{T_2}:\mathbb{E} \rightarrow \mathbb{E}$ are $\mu$-Lipschtiz with constants $\mathrm{C}$ and $\mathrm{C}^{'}$ respectively, then $\mathrm{T_1}+\mathrm{T_2}:\mathbb{E}\rightarrow \mathbb{E}$ is $\mu$-Lipschitz with constant $\mathrm{\mathrm{C}+\mathrm{C}}^{'}$.

Definition 2.6. ^[25] If $\mathrm{T_1}:\mathrm{S}\rightarrow \mathbb{E}$ is compact, then $\mathrm{T_1}$ is $\mu$-Lipschitz with constant $\mathrm {\mathrm{C}} = 0$.

Definition 2.7. ^[25] If $\mathrm{T_1}:\mathrm{S} \rightarrow \mathbb{E}$ is Lipschitz with constant $\mathrm{C}$, then $\mathrm{T_1}$ is $\mu$-Lipschitz with the same constant $\mathrm{C}$.

We need the given theorem.

Theorem 2.8. ^[25] Let $\mathrm{T}:\mathbb{E}\rightarrow \mathbb{E}$ be $\mu$-condensing and

If $\mathrm{S}$ is a bounded set in $\mathbb{E}$, so we can find $r > 0,$ such that $\mathrm{S}\subset \mathrm{D}_r(0)$, then the degree

Thus $\mathrm{T}$ has at least one fixed point and the set of the fixed points of $\mathrm{T}$ lies in $\mathrm{D}_r(0)$.

3.   Existence theory

Here, we derive first part of our main results.

Lemma 3.1. The solution of (1.1) can be described as

such that

represent value of integral

at $t = 1$ and ${\bf I}^{{\omega}+\beta_i} h_i(1, {u}(1), ^{c}{\bf D}^{\rho}{u}(1))$ denotes the value of the integral ${\bf I}^{{\omega}+\beta_i}h_i(t, {u}(t), ^{c}{\bf D}^{\rho}{u}(t))$ at $t = 1,$ for $i = 1, 2, 3, ...m.$

Proof. On using ${\bf I}^\vartheta$ at both sides of (1.1), one has

Hence, we obtain

Using ${\bf I}^{\omega}$ and the semi group property of integrals, one has

Taking $j$th order ordinary derivative, one has

Also, $u'(0) = 0, \, u''(0) = 0, ..., u^{n-1}(0) = 0$ yield $\mathscr{C}_2 = 0, \, \mathscr{C}_3 = 0, ...., \mathscr{C}_n = 0$. Thus

Further $u(0) = \psi_1 (u(\eta))$ yields $\mathscr{C}_1 = \psi_1 ({u}(\eta))$ and using ${u}(1) = \psi_2({u}(\eta))$, one has

which implies

Hence, we get the given result

Hence, (3.2) becomes

which can be rewritten as

From (3.3), it follows that

Let define $\mathbb{A}, \, \mathbb{B}, \mathbb{C}: \mathbb{E}\rightarrow \mathbb{E}$ by $\mathbb{A} = \mathbb{\bar{A}}+\psi_1(u(\eta)), \, \mathbb{C} = (\psi_2(u(\eta))-\psi_1(u(\eta)))-\mathbb{\bar{C}},$ where

then (3.3) takes the form of the operator equation

and fixed points of the operator Eq (3.6) are solutions of the BVP (1.1). Further, from (3.4), it follows that

Using (3.5), and (3.7), we obtain

The following data depended results need to be hold to establish our main results.

$(H_1)$ $f, h_i, g$ fulfill the criteria of Caratheódory conditions.

$(H_2)$ For constants $\tau_1, \, \tau_2, \, d_1, \, d_2, \, c_1, \, c_2,$ one has

$(H_3)$ Let we have continuous mappings $\theta_i, \, \mu, \, \delta :\mathrm{I}\rightarrow \mathbb{R}$, with $0 < \xi, \, \lambda,$ such that for $u\in \mathbb{E}$ that

$(H_4)$ There exists $\tau_i > 0,$ such that for $u_1, u_2\in \mathbb{E},$

where

Under the hypothesis $(H3)$, we have the following relation

Using (3.8), and (3.9) together with the hypothesis $H_2, H_3$, we obtain the following relations

where $\Lambda = |{\bf I}^{\omega}\Psi(1, {u}(1), ^{c}{\bf D}^{\rho}{u}(1)).$ Thus, under the hypothesis $H_4$, we have the following relation

Further, we have

Thus, one has

which in view of (3.9) implies that

Similarly, in view of (3.9), we obtain

Hence, it follows that

which in view (3.11), and $H_4$ implies that

In addition, we have

which in view (3.11), and $H_4$ implies that

From (3.23), and (3.19), it follows that

where $\kappa_{1} = \frac{(\delta\mu+\rho_{0}(\|u\|_{\rho}+\lambda)}{\Gamma(\vartheta+1)}(\frac{1}{\Gamma(\omega+1)}+\frac{1}{\Gamma(\omega-\rho+1)}) +\sum_1^{m}(\frac{|\theta_i|}{\Gamma(\omega+\beta_i+1)}+\frac{|\theta_i|}{\Gamma(\omega+\beta_i-\rho +1)})+\tau_1$. Now

using $H_4$ yields

Hence, it follows that

where$\kappa_2 = \frac{\mu}{\Lambda}(\frac{1}{\Gamma(\omega+1)}+\frac{1}{\Gamma(\omega-\rho+1)})$.

Using(3.11), and $H_4$, we obtain

where $\kappa_3 = \frac{(\delta\mu+\rho_{0}(\|u \|_{\rho}+\lambda)}{\Gamma(\omega+1)\Gamma(\vartheta +1)}+\frac{\mu_0}{\Gamma(w+1)}+\sum_1^{m}\frac{|\theta_i|}{\Gamma(\omega+\beta_i +1)}+\tau_1+\tau_2.$

Theorem 3.2. Under the hypothesis $H_1$–$H_3$, the operator $\mathbb{\bar{A}}$ is compact and satisfies the following growth condition $\|\mathbb{\bar{A}} u\|_{\rho}\leq \Delta_{1}\|u\|_{\rho}+\Delta_{2}$, where

and

Proof. Here $(\mathbb{\bar{A}}u)(t) = {\bf I}^\omega\Psi(t, {u}(t), ^{c}{\bf D}^{\rho}{u}(t))+\sum_{1}^{m}{\bf I}^{\omega+\beta_i}h_i(t, {u}(t), ^{c}{\bf D}^{\rho}{u}(t))$, clearly, $\mathbb{\bar{A}}$ is continuous on $\mathbb{E}$. Now, for $u\in \mathbb{E}$, using (3.10), we have

Hence

(3.26) yields that $\mathbb{\bar{A}}$ is uniformly bounded for bounded set on $\mathbb{E}$. Let, $t_1 < t_2 \in I$, and consider

In view of (3.12), and (3.14), we obtain

in view of (3.13), and (3.15), we obtain

From (3.27) and (3.28), it follows that

Thus $\mathbb{\bar{A}}$ is equi continuous, and by Arzelá- Ascoli theorem $\mathbb{\bar{A}}$ is relatively compact. Hence, $\mathbb{\bar{A}}$ is $\mu$- Lipschtiz with constant $0$.

Lemma 3.3. Under the hypothesis $H_1$–$H_3$, the operator $\mathbb{A}$ is $\mu$-Lipschtiz with constant $\tau_1$ and satisfes the following growth condition

Proof. By $H_2$, the operator $\psi_{1}(\eta)$ is $\mu$-Lipschtiz with constant $\tau_1$ and by Lemma (3.2), the operator $\mathbb{\bar{A}}$ is $\mu$-Lipschtiz with constant $0$. Hence, the operator $\mathbb{A} = \mathbb{\bar{A}}+\psi_{1}(\eta)$ is $\mu$-Lipschtiz with constant $\tau_1$. Since, $\|\mathbb{\bar{A}} u\|_{\rho}\leq \Delta_{1}\|u\|_{\rho}+\Delta_{2}$ by Lemma (3.2), it follows that $\|\mathbb{A} u\|_{\rho}\leq (\Delta_{1}+c_1)\|u\|_{\rho}+(\Delta_{2}+d_1)$.

Lemma 3.4. Under the hypothesis $H_1$–$H_3$, the operator $\mathbb{B}$ is continuous and compact.

Proof. Here, $(\mathbb{B}u)(t) = \frac{{\bf I}^{\omega}f(t, {u}(t), ^{c}{\bf D}^{\rho}{u}(t))}{{\bf I}^{\omega}f(1, {u}(1), ^{c}{\bf D}^{\rho}{u}(1))}.$ Clearly, $\mathbb{B}$ is continuous on $E$ and bounded as

For equi-continuity, choose $t_1 < t_2 \in I$, and consider

In view of (3.16), we obtain

in view of (3.17), we obtain

From (3.32) and (3.33), it follows that

Therefore, $\mathbb{B}$ is equi continuous. Therefore, using Arzelá- Ascoli theorem, $\mathbb{B}$ is compact.

Lemma 3.5. Under the hypothesis $H_1$–$H_3$, the operator $\mathbb{\bar{C}}$ is compact and satisfies the following growth condition

where

and

Proof. The continuity of $\mathbb{\bar{C}}$ follows from the definition of $\mathbb{\bar{C}}$. In addition, we have

which is in view of (3.9) implies that

Hence, it follows that

The equi-continuity of $\mathbb{\bar{C}}$ follows from the fact that

Hence, $\mathbb{\bar{C}}$ is compact and it follows that $\mathbb{\bar{C}}$ is $\mu$-Lipschtiz with constant $0$.

Lemma 3.6. Under the hypothesis $H_1$–$H_3$, the operator $\mathbb{C}$ is $\mu$-Lipschtiz with constant $\tau$ and satisfies the following growth condition

Proof. Define $F(u) = \psi_2(u(\eta))-\psi_1(u(\eta))$, for $u_1, \, u_2 \in \mathbb{E}$, consider

which in view of $H_2$ implies that

$\tau = \tau_1 +\tau_2$. Since $\mathbb{C} = (\psi_2(u(\eta))-\psi_1(u(\eta)))-\mathbb{\bar{C}} = F(u)-\mathbb{\bar{C}}$ and by Lemma 3.5 $\mathbb{\bar{C}}$ is $\mu$-Lipschtiz with constant $0$. Hence, $\mathbb{C}$ is $\mu$-Lipschtiz with constant $\tau$. Further, $\|\mathbb{\bar{C}} u\|_{\rho}\leq \Delta_{3}\|u\|_{\rho}+\Delta_{4}$ by Lemma 3.5, it follows that

Choose the parameters such that $\Delta_1+\Delta_3+2c_1+c_2 < 1$. Choose $R\geq \max\{\tau_1+\tau, \, \frac{\Delta_2+\Delta_4+2d_1+d_2}{1-(\Delta_1+\Delta_3+2c_1+c_2)}\}$. Define $S = \{u\in \mathbb{E}:\|u\|_{\rho}\leq R\}$, then $S$ is closed, convex and bounded subset of $\mathbb{E}$.

Theorem 3.7. Under the assumptions $(H_1)$–$(H_3)$, the system (3.6) has at least one solution ${u}\in \mathbb{E}$ provided $\Delta_1+\Delta_3+2c_1+c_2 < 1$.

Proof. By Lemma 3.3, the operator $\mathbb{A}$ is $\mu$-Lipschitz with constant $\tau_1.$ Using Lemma 3.6, the operator $\mathbb{C}$ is $\mu$-Lipschitz with constant $\tau$. By Lemma 3.4, the operator $\mathbb{B}$ is compact. Now for $v\in S$ and $u\in \mathbb{E}$, consider the equation $u = \mathbb{A}u+\mathbb{B}v \mathbb{C}u$, which implies that

Using (3.30), (3.31) and (3.37), we obtain

That implies

Hence, it follows that

which implies that $u\in S$. Further we have $M = \|\mathbb{B}u\|_\rho = 1$ and $R\geq \tau_1+\tau$. Finally from above, we conclude that (3.6) has at least one solution ${u}\in \mathbb{E}$.

Theorem 3.8. Under the assumptions $(H_1$)–($H_4)$, the system (3.6) has a unique solution in $\mathrm{S}$ provided that

Proof. For $u_1, u_2\in \mathrm{S}$, Consider

By (3.20), (3.21), and (3.23), we have

Using (3.41) in (3.40), we obtain

which in view of (3.31), and (3.38) implies that

Further, the above relation implies that

and uniqueness follows by the Banach contraction principle.

4.   Ulam Hyers stability

U-H stability result is developed for (1.1). For detail introduction and results of U-H stability, we refer ^[47,48].

Definition 4.1. The problem (3.6) is said to be U-H stable, if there exists a constant $\zeta > 0$, such that for a given $\varphi > 0,$ and for each solution ${u}$ of the inequality

there exists a solution $\bar{{u}}(t)$ of (3.6). Then one has

such that

Theorem 4.2. Under the assumptions $(H_2)$ and $(H_4)$, the problem (1.1) is U-H stable provided $k+k_1 < 1$.

Proof. Let ${u}\in \mathbb{E}$ satisfies the inequality (4.1), and $\bar{{u}}\in \mathbb{E}$ be a solution of (1.1) which satisfies the Eq (3.6). Consider

Now using (3.42) and (4.1), we obtain

where $K = \kappa_1+\kappa_2((\Delta_3+c_1)R+(\Delta_4+d_1))+k_3$. Hence, it follows that

5.   Illustrative application

Here, we present an application to demonstrate our results.

Example 5.1. Consider the following problem by taking $n = 2$ as

Consider

It is easy to show that the conditions of Theorem 3.2 and 3.8 are satisfied. Therefore, the given problem (5.1) has at least one solution. Further, the solution uniqueness condition also holds. Also, one can obviously verified the condition of U-H stability given in Theorem 4.2.

6.   Conclusions

In this manuscript, a nonlinear problem of S-HFDEs has been investigated by using a sophisticated tool known as topological degree theory. We have used a degree of non-compactness along with Caratheódory condition to establish appropriate results for the qualitative theory. Usually, fixed point theory involves strong compact conditions which require more restrictions on the nonlinear operators. Therefore, to replace the strong compact condition with some weaker compact condition, the proposed degree theory is a powerful tool. The concerned tool has the ability to relax the criteria and hence can be applied to large numbers of nonlinear problems of differential and integral equations. On the other hand, stability is an important consequence of nonlinear analysis. Therefore, a result based on U-H concepts for stability has been established. Finally, to verify our obtained results, we have given an illustrative problem. In the future, the degree theory will be applied in hybrid problems of fractal-fractional differential equations which have the ability to describe complex and irregular geometry in more diligent ways. Also, the mentioned degree theory has not yet been used in dealing with non-singular type hybrid fractional differential equations. Therefore, the aforesaid area will be our next target.

Acknowledgments

All the authors are thankful to Prince Sultan University for paying the APC and support through TAS research lab.

Conflict of interest

The authors declare no conflict of interest.