Existence theorems for $\Psi$-fractional hybrid systems with periodic boundary conditions

1.   Introduction

Fractional calculus ^[1,2] is a field of mathematics that deals with integrals and derivatives of fractional orders. The most used fractional operators are the Riemann-Liouville and Caputo types. There are other types of fractional derivatives as well, we allude to ^{[3,4,5,6,7,8,9,10]} and references therein. More recently, Almeida ^[11] and Sousa et al. ^[12] have provided fractional operarors to generalize Caputo and Hilfer types respectively with respect to another function, which have become known as $\psi$-Caputo and $\psi$-Hilfer, also Jarad and Abdeljawad in ^[13] have introduced interesting properties of this generalized operator in the frame of a $\psi$ function including the generalized Laplace transform.

Due to the rapid and intense growth in fractional calculus and its applications, fractional differential equations (FDEs) have been of extraordinary interest, so, several authors have applied some generalized fractional operators to investigate the qualitative analysis of FDEs, see ^{[14,15,16,17,18,19,20,21,22,23]}.

Hybrid differential equations include the fractional derivatives of an unknown function hybrid with the nonlinearity relying upon it. This class of equations emerges from a wide range of spaces of applied and physical sciences, e.g., in the redirection of a bent pillar having a consistent or changing cross-area, a three-layer shaft, electromagnetic waves, or gravity-driven streams, etc. Hybrid FDEs have been investigated using various types of fractional derivatives in literature (see, e.g. ^{[24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39]}. For instance, Dhage and Lakshmikanathm ^[25] investigated the existence and uniqueness results for a hybrid differential equation:

where $\mathsf{g}_{1}:[0, 1]\times \mathbb{R}\rightarrow \mathbb{R}-\{0\}$ and $\mathsf{g}_{2}:[0, 1]\times \mathbb{R}\rightarrow \mathbb{R}$ are continuous.

Zhao et al. ^[24] studied the existence and uniqueness results for a hybrid FDE in the frame of Riemann-Liouville operators:

where $\mathsf{g}_{1}:[0, T]\times \mathbb{R}\rightarrow \mathbb{R}-\{0\}$ and $\mathsf{g}_{2}:[0, T]\times \mathbb{R}\rightarrow \mathbb{R}$ are continuous.

Motivated by the above investigations, we discuss two nonlinear fractional differential hybrid systems subjected to periodic boundary conditions. The first fractional nonlinear system is given by

and the second system has the following form

where $\vartheta \in \mho : = [a, b], \; ^{C}\mathcal{D}_{a^{+}}^{\varrho, \Psi }$ is the $\Psi$-Caputo fractional derivative, $\mathsf{g}_{1}:\mho \times \mathbb{R}\rightarrow \mathbb{R}-\{0\}$ and $\mathsf{g}_{2}:\mho \times \mathbb{R}\rightarrow \mathbb{R}$ are continuous with $\mathsf{g}_{1}$ and $\mathsf{g}_{2}$ are identically zero at the origin and $\mathsf{g} _{2}(\vartheta, 0) = 0.$

In this respect, we study the existence of solutions to two types of hybrid FDEs involving generalized Caputo fractional derivatives rather than the classical Caputo one. The hybrid problems have been discussed in the literature under classical FDEs, while we investigate the generalized FDEs under similar boundary conditions which is the novel contribution of this research paper. Furthermore, the special cases generated from various values of the positive increasing function $\Psi$ are covered in our examination.

The paper is coordinated as follows. In Section 2, we present a few documentations, definitions and lemmas. In Section 3, we demonstrate existence results for problems (1.1) and (1.2) by utilizing Dhage's fixed point theorem. In Section 4, we illustrate the acquired outcomes by examples. At last, we close our paper with a conclusion.

2.   Preliminaries

Let $\Psi \in C^{1}(\mho, \mathbb{R})$ be an increasing differentiable function such that $\Psi ^{\prime }(\vartheta)\neq 0$ for all $\vartheta \in \mho$. Now, we start by defining $\Psi$-fractional integral and derivative:

Definition 2.1. ^[1] The $\Psi$–Riemann-Liouville fractional integral of order $\varrho > 0$ for an integrable function $\upsilon :\mho \longrightarrow \mathbb{R}$ is given by

One can deduce that

where $D_{\vartheta } = \frac{d}{dt}.$

Definition 2.2. ^[11] For $n-1 < \varrho < n \; (n\in \mathbb{N)}$ and $\upsilon, \Psi \in C^{n}(\mho, \mathbb{R})$, the $\Psi$-Caputo fractional derivative of a function $\upsilon$ of order $\varrho$ is given by

where $n = [\varrho ]+1$ for $\varrho \notin \mathbb{N}$, $n = \varrho$ for $\varrho \in \mathbb{N}$.

From Definition 2.2, we can express $\Psi$-Caputo fractional derivative by formula

Lemma 2.3. ^[1] For $\varrho _{1}, \varrho _{2} > 0,$ and $\upsilon \in C(\mho, \mathbb{R})$, we have

Lemma 2.4. ^[11] Let $\varrho > 0$. If $\upsilon \in C(\mho, \mathbb{R}),$ then

and if $\upsilon \in C^{n-1}(\mho, \mathbb{R}),$ then

Lemma 2.5. ^[1,11] For $\vartheta > a, \ \varrho \geq 0, \; \beta > 0.$ If $\varkappa _{\beta }(\vartheta) = (\Psi (\vartheta)-\Psi (a))^{\beta -1}$, then

● $\mathcal{I}_{a^{+}}^{\varrho; \Psi }\varkappa _{\beta }(\vartheta) = \frac{\Gamma (\beta)}{\Gamma (\beta +\varrho)}\varkappa _{\beta +\varrho }(\vartheta)$;

● $^{C}\mathcal{D}_{a^{+}}^{\varrho; \Psi }\varkappa _{\beta }(\vartheta) = \frac{\Gamma (\beta)}{\Gamma (\beta -\varrho)}\varkappa _{\beta -\varrho }(\vartheta)$;

● $^{C}\mathcal{D}_{a^{+}}^{\varrho; \Psi }\varkappa _{k+1}(\vartheta) = 0, \;$for all$\; k\in \{0, \dots, n-1\}, n\in \mathbb{N}.$

Now, we mention the key outcomes to the forthcoming analysis.

Theorem 2.6. ^[40,41] Let $\mathbb{X}$ be a closed convex and bounded subset of the Banach algebra $\mathcal{\aleph }$ and let $\mathcal{A}:\aleph \rightarrow \aleph$ and $\mathcal{B}:\mathbb{X}\rightarrow \aleph$ be two operators such that

(a) $\mathcal{A}$ is Lipschitzian with Lipschitz constant $L_{\mathcal{A}}$,

(b) $\mathcal{B}$ is compact and continuous,

(c) $\upsilon = \mathcal{A}\upsilon \mathcal{B}\upsilon ^{\star }\Rightarrow \upsilon \in \mathbb{X}$ for all $\upsilon ^{\star }\in \mathbb{X},$ and

(d) $L_{\mathcal{A}}M_{\mathcal{B}} < 1,$ where $M_{\mathcal{B}} = \Vert \mathcal{B}(\mathbb{X})\Vert = \sup \{\Vert \mathcal{B}\upsilon \Vert :\upsilon \in \mathbb{X}\}$

Then the operator equation $\upsilon = \mathcal{A}\upsilon \mathcal{B} \upsilon$ has a solution in $\mathbb{X}$.

2.1.   Solutions representation

Lemma 2.7. A function $\upsilon$ is a solution of the fractional integral equation

if and only if $\upsilon$ is a solution of the periodic hybrid system

Proof. Applying the operator $\mathcal{I}_{a^{+}}^{\varrho, \Psi }$ on both sides for the first equation of (2.2) and using Lemma 2.4, we have

Then, at $\vartheta = a$ and $\vartheta = b,$ we get

The periodic condition $\left(\upsilon (a) = \upsilon (b)\right)$ implies that

Hence

Substituting the value of $c_{0}$ into (2.3), we get the solution (2.1).

Conversely, it is clear that if $\upsilon$ satisfies Eq (2.1), then system (2.2) is satisfied by $\upsilon,$ due to Lemma 2.4 and Lemma 2.5. The proof is completed.

Lemma 2.8. A function $\upsilon$ is a solution of the fractional integral equation

if and only if $\upsilon$ is a solution of the periodic hybrid system

where

and

Proof. Applying the operator $\mathcal{I}_{a^{+}}^{\varrho, \Psi }$ on both sides for the first equation of (2.5) and using Lemma 2.4, we have

Differentiating Eq (2.6) with respect to $\vartheta$ and using Leibniz rule yields that

Then, at $\vartheta = a$ and $\vartheta = b,$ we get

and

The boundary conditions imply that

and

Solving (2.7) and (2.8) in terms of $c_{0}$ and $c_{1}$, we obtain

and

Substituting the values of $c_{0}$ and $c_{1}$ into (2.6), we get the solution (2.4).

The converse of the lemma follows by direct computation along with Lemmas 2.4 and 2.5.

This finishes the proof.

2.2.   Existence result of (1.1)

In order to achieve our main results, we list the following hypotheses:

(H1) $\mathsf{g}_{2}:\mho \times \mathbb{R}\rightarrow \mathbb{R}$ and $\mathsf{g}_{1}: \mho \times \mathbb{R}\rightarrow \mathbb{R} -\{0\}$ are continuous.

(H2) $\mathsf{g}_{1}^{-1}:\mho \times \mathbb{R}\rightarrow \mathbb{R}$ is continuous and

i) There exists a positive function $\omega$ with bounds $\Vert \omega \Vert$, such that

for each $(\vartheta, \upsilon _{1}), (\vartheta, \upsilon _{2})\in \mho \times \mathbb{R}$;

ii) The mapping $\upsilon \rightarrow \mathsf{g}_{1}^{-1}(\vartheta, \upsilon)$ is increasing in $< /p > < p > \mathbb{R} < /p > < p >$ a.e. for each $\vartheta \in \mathrm{\mho }$.

(H3) There exists constant $\mathrm{M}_{\mathsf{g}_{2}}$ such that

To simplify, we will use the following notations

Theorem 2.9. Suppose (H1)–(H3) hold. If

then hybrid problem (1.1) has a solution on $\mho$.

Proof. Define the set $\mathbb{X} = \left\{ \upsilon \in \mathcal{C}:\left\Vert \upsilon \right\Vert \leq R\right\}.$ Clearly, $\mathbb{X}$ is a convex, closed, bounded subset of $\mathcal{C}$. Choose

where $\mathsf{g}_{10} = \sup_{\vartheta \in \mho }|\mathsf{g} _{1}^{-1}(\vartheta, 0)|.$ From Lemma 2.7, the nonlinear hybrid problem (1.1) is equivalent to the nonlinear fractional integral equation

Define two operators $\mathcal{A}:\mathcal{C}\rightarrow \mathcal{C}$ and $\mathcal{B}:\mathbb{X}\rightarrow \mathcal{C}$ by

Then, (2.13) can be express in the operator form as

To achieve Theorem 2.6, we will summarize the proof in the following steps:

$\textsf{Step1:}\; \mathcal{A}$ is Lipschitzian on $\mathcal{C}$.

Let $\upsilon, \upsilon ^{\star }\in \mathcal{C}$. Then by (H2), for $\vartheta \in \mho$

which leads to

So. $\mathcal{A}$ is Lipschitzian on $\mathcal{C}$ with Lipschitz constant $\Vert \omega \Vert$.

$\textsf{Step 2:}\; \mathcal{B}$ is completely continuous on $\mathbb{X}$.

Firstly, $\mathcal{B}$ is continuous on $\mathcal{C}$, due to the continuity of $\mathsf{g}_{2}, \mathsf{g}_{1}$ implies that $\mathcal{B}$ is continuous too. Next, we shall prove that $\mathcal{B}(\mathbb{X})$ is uniformly bounded in $\mathbb{X}$. For any $\upsilon \in \mathbb{X}$, we have

which implies

This shows that $\left\{ \mathcal{B\upsilon }:\upsilon \in \mathbb{X} \right\}$ is uniformly bounded set.

To prove that $\mathcal{B(}\mathbb{X)}$ is an equicontinuous set in $\mathbb{ X},$ let $\vartheta _{1}, \vartheta _{2}\in \mho \; (\vartheta _{1} < \vartheta _{2})$. Then for any $\upsilon \in \mathbb{X}$ and by (H3), we get

Distinctly, the right-hand side of the a bove inequality tends to zero independently of $\upsilon \in \mathbb{X}$ as $\vartheta _{2}\rightarrow \vartheta _{1}$. As a result of the Ascoli-Arzela theorem, $\mathcal{B}$ is a completely continuous operator on $\mathbb{X}$.

$\textsf{Step 3:}$ Assumption (c) of Theorem 2.6 is satisfied.

Let $\upsilon \in \mathcal{C}$ and $\upsilon ^{\star }\in \mathbb{X}$ such that $\upsilon = \mathcal{A}\upsilon \mathcal{B}\upsilon ^{\star }$. Then

which gives

$\textsf{Step 4:}$ Assumption (d) of Theorem 2.6 holds.

To this end, we show that $\left\Vert \omega \right\Vert N < 1$, where $N = \Vert \mathcal{B}(\mathbb{X})\Vert.$ Since

we have

Thus all the assumptions of Theorem 2.6 hold. Hence, $\upsilon = \mathcal{A}\upsilon \mathcal{B}\upsilon$ has a solution in $\mathbb{X}$. So, the hybrid problem 1.1 has a solution on $\mho$.

2.3.   Existence result of (1.2)

In view of Lemma 2.8, we have

where

To simplify, we will use the following notations:

and

Theorem 2.10. Assume that (H1)–(H3) hold. Furthermore, if

then the hybrid problem (1.2) has a least one solution defined on $\mho$.

Proof. Define

In the light of (2.14), $R_{1} > 0.$ Define a subset $\mathbb{X}$ of the Banach algebra $\mathcal{C}$ by

Clearly, $\mathbb{X}$ is a closed, convex and bounded subset of $\mathcal{C}.$ Consider the operators $\mathcal{A}_{1}:\mathcal{C}\rightarrow \mathcal{C }$ and $\mathcal{B}_{1}:\mathbb{X}\rightarrow \mathcal{C}$ defined by

and

where $\upsilon = \mathcal{A}_{1}\upsilon \mathcal{B}_{1}\upsilon, \ \upsilon \in \mathcal{C}$.

Now, we prove that $\mathcal{A}_{1}$ and $\mathcal{B}_{1}$ fulfills assumptions of Theorem 2.6. The proof will be given in forthcoming steps.

$\textsf{Step I:}\; \mathcal{A}_{1}$ is lipschitzian on $\mathcal{C}$ with Lipschitz constants $\Vert \omega \Vert$.

Let $\upsilon, \upsilon ^{\star }\in \mathcal{C}$ and $\vartheta \in \mho$. Then, by using (H2), we have

Thus

That is, $\mathcal{A}_{1}$ is a Lipschitzian with Lipschitz constant $\left\Vert \omega \right\Vert$.

$\textsf{Step II: }$$\mathcal{B}_{1}$ is completely continuous on $\mathbb{X}$. Firstly, $\mathcal{B}_{1}$ is continuous on $\mathcal{C}$, due to the continuity of $\mathsf{g}_{2}, \mathsf{g}_{1}\mathsf{, g}_{1}^{-1}$ implies that $\mu$ and $\nu$ are continuous and hence $\mathcal{B}_{1}$ is continuous too. Next, we shall prove that $\mathcal{B}_{1}(\mathbb{X})$ is uniformly bounded in $\mathbb{X}$. For any $\upsilon \in \mathbb{X}$, we have

which implies

This shows that $\left\{ \mathcal{B}_{1}\mathcal{\upsilon }:\upsilon \in \mathbb{X}\right\}$ is uniformly bounded set. Now, we show that $\mathcal{B} _{1}\mathcal{(}\mathbb{X)}$ is an equicontinuous set in $\mathbb{X},$ let $\vartheta _{1}, \vartheta _{2}\in \mho \; (\vartheta _{1} < \vartheta _{2})$. Then for any $\upsilon \in \mathbb{X}$ and by (H3), we get

As a result of the Ascoli-Arzela theorem, $\mathcal{B}_{1}$ is a completely continuous operator on $\mathbb{X}$.

$\textsf{Step III: }$We prove the third condition (c) of Theorem 2.6 holds. Let $\upsilon \in \mathcal{C}$ and $\upsilon ^{\star }\in \mathbb{X}$ such that $\upsilon = \mathcal{A}_{1}\upsilon \mathcal{B}_{1}\upsilon ^{\star }$. Then

which gives

Thus, $\left\Vert \upsilon \right\Vert \leq R_{1}$ and so the hypothesis (c) of Theorem 2.6 is satisfied.

$\textsf{Step IV}$: Assumption (d) of Theorem 2.6 holds.

To this end, we show that $\left\Vert \omega \right\Vert N < 1$, where $N = \Vert \mathcal{B}_{1}(\mathbb{X})\Vert.$ Since

we have

Thus all the assumptions of Theorem 2.6 hold. Hence, $\upsilon = \mathcal{A}_{1}\upsilon \mathcal{B}_{1}\upsilon$ has a solution in $\mathbb{ X}$. So, the hybrid problem 1.2 has a solution on $\mho$.

3.   Examples

In this section, in order to illustrate our results, we consider two examples.

Example 3.1 Consider the following nonlocal hybrid boundary value problem:

From the system (3.1), and we choose $\Psi (\vartheta) = \vartheta$, $a = 0$, $b = 1$, $\varrho = 4/5$, $\mathsf{g} _{1}^{-1}(\vartheta, \upsilon \left(\vartheta \right)) = \left(\frac{ \vartheta ^{2}}{10}\left(\frac{1}{2}\left(\upsilon (\vartheta)\right) +\vartheta \right) \right) ^{-1}, \; \mathsf{g}_{2}(\vartheta, \upsilon \left(\vartheta \right)) = \frac{e^{(-2\vartheta)}}{\sqrt{9+\vartheta }} \left(\sin \upsilon (\vartheta)\right).$ Clearly, $\mathsf{g}_{2}\mathsf{, g}_{1}^{-1}$ are continuous. Moreover

and

with $\omega = \frac{1}{10}$, $\mathrm{M}_{\mathsf{g}_{2}} = \frac{1}{\sqrt{10}}$ and $g_{0} = \sup_{\vartheta \in \mho }\left\vert g^{-1}(\vartheta, 0)\right\vert = \frac{1}{10}$. Using these values, we get $M\left\Vert \omega \right\Vert \simeq 0.35 < 1$. As all the conditions of Theorem 2.9 are satisfied, problem 3.1 has at least one solution on $\mho$.

Example 3.2. Consider the following nonlocal hybrid boundary value problem:

From the system (3.2), and we choose $\Psi (\vartheta) = \vartheta$, $a = 0$, $b = 1$, $\varrho = 8/5$, $g(\vartheta, \upsilon \left(\vartheta \right)) = \left(\frac{\vartheta }{5}\left(\frac{1}{3}\left(\upsilon (\vartheta)\right) +\vartheta \right) \right) ^{-1}, \; \mathsf{g} _{2}(\vartheta, \upsilon \left(\vartheta \right)) = \frac{\cos ^{2}\left(2\pi \vartheta \right) }{5-\vartheta }\upsilon (\vartheta).$ Clearly, $\mathsf{g}_{2}\mathsf{, g}_{1}^{-1}$ are continuous. Moreover

and

with $\omega = \frac{1}{5}$, $\mathrm{M}_{\mathsf{g}_{2}} = \frac{1}{4}$ and $g_{0} = \sup_{\vartheta \in \mho }\left\vert g^{-1}(\vartheta, 0)\right\vert = \frac{1}{5}$. Using these values, we get $\Omega \left\Vert \omega \right\Vert < 1$. As all the conditions of Theorem 2.10 are satisfied, problem 3.2 has at least one solution on $\mho$.

4.   Conclusions

It is important that we examine the fractional systems of the hybrid with generalized derivatives since these derivatives cover many systems in the literature and they contain a kernel with different values that generate many special cases.

In this research work, we have investigated the sufficient conditions to the existence of solutions to two new types of boundary value problems of nonlinear hybrid fractional differential equations involving generalized fractional derivatives known as $\Psi$-Caputo operators. In order to achieve the objectives, we applied Dhage's fixed point theorem for the sum of three operators. Two examples are provided to confirm the feasibility of the obtained results.

Moreover, we have formulated illustrative examples for this type of hybrid fractional systems to support our main results from a numerical point of view.

Acknowledgments

T. Abdeljawad would like to thank Prince Sultan University for the support through the research lab TAS.

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.