Existence, uniqueness and stability of solutions for generalized proportional fractional hybrid integro-differential equations with Dirichlet boundary conditions

1.   Introduction

Fractional calculus has received a lot of attention in the last decades ^[1]. This type of calculus has become an important area of research due to its intensive development and diverse applications with several complex problems of everyday life have been modeled by differential equations and integro-differential equations of fractional order ^[2,3,4,5]. Such applications can be found in variety of fields like economics, physics, chemistry, astronomy, thermal acoustic engineering, biology, probability theory, control theory, viscous fluid dynamics, signal processing, electromagnetism, robotics, anomalous diffusion, potential theory and electrical statistics, etc. We refer the reader to the papers ^[6,7,8] and the references cited therein for more details.

Many authors have taken into consideration the numerical solutions of boundary value problems or initial value problems of ordinary/partial value problems (see ^[9,10] and the references mentioned). However, studying the qualitative properties of differential equations has still had priority.

In mathematical analysis, boundary conditions are constraints on the solutions of ordinary or partial differential equations in a given domain. There is a large number of possible boundary conditions depending on the formulation of the number of variables involved and thus depending on the nature of the differential equation. Boundary value problems of differential equations of fractional order have been extensively studied. In many works, the authors presented some excellent results about the existence of solutions and utilized the fixed point theorems for this purpose; for example, see ^{[11,12,13,14,15,16]}. Concerning the fractional integro-differential equations, please refer to ^{[17,18,19,20,21]}. On the other hand, some researchers were interested in studying the stability of solutions to fractional differential equations. One of the most important types of stability that played a noticeable role in the development of fractional differential equations is the so-called Ulam-Hyers stability. Ulam-Hyers stability analysis was recognized as a simple method of investigation of the solutions of a differential equation. We refer the readers to ^{[22,23,24,25,26,27]} and the references contained therein.

Some basic theory for the hybrid differential (or hybrid integro-differential) equations was discussed in some articles; for instance, in ^[28], Dhage and Lakshmikantham discussed the existence and the uniqueness of solutions for the following classical Cauchy problem of hybrid differential equations

where $g\in C([0, T]\times \mathbb{R}, \mathbb{R})$ and$\ f\in C([0, T]\times \mathbb{R}, \mathbb{R} \backslash \{0\}).$

Herzallah and Baleanu ^[29] discussed the existence of solutions and some fundamental differential inequalities for the following Cauchy problems of hybrid differential equations involving Caputo-Liouville derivative.

and

where $t\in [0, T],$ $0 < \beta \leq 1,$ $g\in C([0, T]\times \mathbb{R}, \mathbb{R})$ and$\ f, h\in C([0, T]\times \mathbb{R}, \mathbb{R} \backslash \{0\}).$

Bashir et al. ^[30] investigated the existence of solutions for a nonlocal boundary value problem of hybrid fractional integro-differential inclusions given by

where where $^{C}D_{0^{+}}^{\beta }$ denotes the Caputo-Liouville fractional derivative of order $\beta \in (1, 2]$, $I_{0^{+}}^{\alpha _{i}}$ denotes the Reimann-Liouville fractional integral of order$\ \alpha _{i}$ and $h_{i}\in C([0, T]\times \mathbb{R}, \mathbb{R}),$ $i = 1, ..., m.$

In ^[31], Dhage proved the existence and approximations of the solutions for the following initial value problems of nonlinear hybrid fractional integro-differential equations

where $0 < \beta \leq 1,$ $\alpha > 0$ and $h, k:[0, T]\times \mathbb{R} \longrightarrow \mathbb{R},$ $f$ $:[0, T]\times \mathbb{R} \longrightarrow \mathbb{R} \backslash \{0\},$ $g:[0, T]\times \mathbb{R} \times \mathbb{R} \longrightarrow \mathbb{R}$ are given functions.

Motivated by the above mentioned works, in this article, we handle the existence of solutions for the following hybrid proportional fractional integro-differential equations with Dirichlet boundary conditions

where $\sigma \in (0, 1],$ $1 < \beta \leq 2,$ $\alpha _{i} > 0, (i = 1, ..., m), m \in \mathbb{N}^{\ast},$ $u_{0}, u_{T} \in \mathbb{R}$ and $\ _{p}^{\mathcal{ C}}\mathfrak{D}_{0^{+}}^{\beta, \sigma }$ represents the Caputo-Liouville proportional fractional derivative of order $\beta$ and $J_{0^{+}}^{\alpha _{i}, \sigma }$ denotes the Reimann-Liouville proportional fractional integral of order $\alpha _{i}$, and $h_{i}:[0, T]\times \mathbb{R} \longrightarrow \mathbb{R},$ $f$ $:[0, T]\times \mathbb{R} \longrightarrow \mathbb{R} \backslash \{0\}, g:[0, T]\times \mathbb{R} \longrightarrow \mathbb{ R}$ are given functions. We discuss existence of solutions by applying the Dhage's fixed point and benefiting from the incomplete Gamma function and its properties that were presented by Laadjal et al. ^[32]. On the top of this, we discuss the uniqueness of these solutions and their Ulam-Hyers-Rassias stability for the case $f(t, u(t)) = 1$ for all $t\in [0, T]$ of the hybrid problem (1.6).

We note that, we deduce the following main cases from problem (1.6).

Case 1. If $\sigma = 1.$ The problem (1.6) reduces to a hybrid integro-differential equations involving the usual Caputo-Liouville fractional derivative with the usual Reimann-Liouville fractional integral.

Case 2. If $f\left(t, u\left(t\right) \right) = 1$ and $h_{i}\left(t, u\left(t\right) \right) = 0,$ $(i = 1, ..., m)$ for all $\left(t, u\left(t\right) \right) \in \lbrack 0, T]\times \mathbb{R},$ we get the following Caputo-Liouville proportional fractional boundary value problem

Case 3. If $\sigma = 1, f\left(t, u\left(t\right) \right) = 1,$ and$\ h_{i}\left(t, u\left(t\right) \right) = 0,$ $(i = 1, ..., m)$ for all $\left(t, u\left(t\right) \right) \in \lbrack 0, T]\times \mathbb{R},$ we get the following Caputo-Liouville fractional boundary value problem

The paper is organized as follows: Next section presents some definitions and some properties needed in next sections. Section 3 proves the existence of solutions for the given problem. Section 4 investigates the uniqueness result of solutions for the given problem (with $f(t, u(t)) = 1$). Section 5 discusses the Ulam-Hyers-Rassias stability results. Lastly, section 6 provides an example to clarify the results obtained.

2.   Preliminaries

In this section, we present some definitions and properties associated with the fractional calculus ^[33,34] and the incomplete gamma function ^[32] that are helpful for our discussion.

Definition 4 (^[33]). Let $\beta \geq 0.$ The left fractional integral of Reimann-Liouville type of the function $\Theta \in L^{1}([0, T], \mathbb{R})$ is defined by $I_{a^{+}}^{0}\Theta\left(t\right) = \Theta\left(t\right)$ and

where $t\in [a, b]$.

Definition 5 (^[33]). Let $\beta \geq 0.$ The left fractional derivative of Caputo-Liouville type of the function $\Theta \in C^{n}([0, T], \mathbb{R})$ is defined by $^{C}D_{a^{+}}^{0}\Theta\left(t\right) = \Theta\left(t\right)$ and

where $n-1 < \beta \leq n$, $n\in \mathbb{N}.$

Definition 6 (^[34]). Let $\sigma \in (0, 1],$ and $\beta \geq 0.$ The left generalized proportional fractional integral of Reimann-Liouville type of the function $\Theta \in L^{1}([0, T], \mathbb{R})$ is defined by $J_{a^{+}}^{0, \sigma }\Theta\left(t\right) = \Theta\left(t\right)$ and

where $t\in [a, b]$.

Definition 7 (^[34]). Let $\sigma \in (0, 1],$ and $\beta \geq 0.$ The left generalized proportional fractional derivative of Caputo-Liouville type of the function $\Theta \in C^{n}([0, T], \mathbb{R})$ is defined by $_{p}^{\mathcal{C}} \mathfrak{D} _{a^{+}}^{0, \sigma }\Theta\left(t\right) = \Theta\left(t\right)$ and

where $n-1 < \beta \leq n$, $n\in \mathbb{N}.$ and ($D^{1, \sigma }\Theta)(t) = (D^{\sigma }\Theta)(t) = (1-\sigma)\Theta (t)+\sigma \Theta ^{\prime }(t),$ and

Remark 8. Note that, for $\sigma = 1,$ Definitions 6 and 7 reduce to the usual definitions of Riemann-Liouville fractional integral and Caputo-Liouville fractional derivative, respetively.

Proposition 9 (^[34]). Let $\sigma \in (0, 1],$ $\alpha > 0$ and $\beta > 0$ with $n-1 < \beta \leq n$, and $\Theta \in L^{1}([0, T], \mathbb{R}),$ we have the following properties.

where $c_{k} = \frac{\left(D^{k, \sigma }\Theta\right) \left(a\right) }{ \sigma ^{k}k!}.$ In particular, when $1 < \beta \leq 2$ and $a = 0,$ we have

Definition 10 (^[32,35,36]). Let $\beta \in$ $(\Re (\beta) > 0).$

The upper incomplete gamma function is defined by

The lower incomplete gamma function is defined by

The upper regularized incomplete gamma function is defined by

The lower regularized incomplete gamma function is defined by

Remark 11 (^[32]). Let $\varsigma \in \mathbb{R}^{+}$ and $\beta \in \mathbb{C}$, where $(\Re (\beta) > 0),$ It is clear that $P(\beta, \varsigma (t-a))$ is a non-decreasing function with respect to$\ t\in [a, b].$ Moreover

Lemma 12 (^[32,37]). Let $\sigma \in (0, 1],$ and $\omega > 0.$ Then

where $t\in [a, b]$ and the function $P$ is defined by Eq (2.13). Moreover

and

Lemma 13 (^[32,37]). Let $\sigma \in (0, 1),$ $t_{1}, t_{2}\in [a, b]$ $(t_{1}\leq t_{2})$ and $\omega > 0.$ Then

where the function $P$ is defined by Eq (2.13).

Lemma 14 (^[32,37]). Let $\sigma \in (0, 1],$ $\beta > 0$ and$\ a\leq \rho \leq t_{1} < t_{2}\leq b.$ Then

Theorem 15 (Dhage ^[38]). Let $\mho$ be a non-empty, closed convex and bounded subset of a Banach algebra $\Sigma$ and let $A, C:\Sigma \rightarrow \Sigma$ and $B:\mho\rightarrow \Sigma$ be three operators satisfying the following conditions.

(1) $A$ and $C$ are Lipschitz with Lipschitz constants $\widetilde{A}$ and $\widetilde{C}$, respectively

(2) $B$ is completely continuous

(3) $\ \widetilde{A}M_{B}+\widetilde{C} < 1$, where $M_{B} = \left\Vert B(\mho)\right\Vert = \sup \left\{ \parallel Bx\parallel :x\in \mho\right\}$

(4) $AxBy+Cx = x\Rightarrow x\in \mho$ for all $y\in \mho$.

Then, the operator equation

has a solution in $\mho.$

3.   Existence results

In this section, we provide our essential findings concerning the problem defined by (1.6) and then derive the existence results for the special cases 1, 2 and 3.

Lemma 16. Let $\sigma \in (0, 1],$ $1 < \beta \leq 2,$ $\alpha _{i} > 0, (i = 1, ..., m).$ For $\Theta \in C([0, T], \mathbb{R}),$ the solution of

is equivalent to the integral equation

Proof. Applying the operator $J_{0^{+}}^{\beta, \sigma }$ on both sides of the equation in Eq (3.1), we get

From the boundary conditions $u(0) = u_{0}$ and $u(T) = u_{T}$ we get

and

Substituting the values of $c_{0}\ $and $c_{1}$ in Eq (3.3), we obtain

Inversely, it is obvious that if $u(t)$ satisfies Eq (3.2), then Eq (3.1) holds. The proof is complete.

Now, consider the Banach algebra space defined by $\Sigma = C([0, T], \mathbb{R}),$ with the norm $\left\Vert u\right\Vert = \underset{0\leq t\leq T}{\sup } \left\vert u(t)\right\vert.$ Then, we define the multiplication in $\Sigma$ by $(uv)(t) = u(t)v(t)$ for all $u, v\in \Sigma$ and the operator $\Phi :\Sigma \rightarrow \Sigma$ $\ $by

Remark 17. The problem described by Eq (1.6) has a solution $u\in \Sigma$ if only if$\ u$ is fixed point of the operator $\Phi$.

To prove the existence, we need the following assumption:

$(H_{1})$ Assume that $g:[0, T]\times \mathbb{R} \rightarrow \mathbb{R}$ is continuous function, there exists a continuous function $\widetilde{g}$ from $[0, T]$ to $\mathbb{R}_{+}$ and a continuous nondecreasing function $\psi$ from $\mathbb{R}_{+}$ to $\mathbb{R}_{+}\backslash \{0\}$ such that $\left\vert g(t, u(t))\right\vert \leq \widetilde{g}(t)\psi (\left\Vert u\right\Vert),$ for all $t\in [0, T]$ with $\underset{0\leq t\leq T}{\sup } \widetilde{g}(t) = \left\Vert \widetilde{g} \right\Vert.$

$(H_{2})$ Assume that $h_i:[0, T]\times \mathbb{R}\rightarrow \mathbb{R}$, $i = 1, ..., 4,$ are continuous functions, there exist $L_{h_{i}} > 0,$ $i = 1, ..., m$ such that $\left\vert h_{i}(t, \lambda_1)-h_{i}(t, \lambda_2)\right\vert \leq L_{h_{i}}\left\vert \lambda_1-\lambda_2\right\vert,$ for all $(t, \lambda_1, \lambda_2)\in [0, T]\times \mathbb{R}\times \mathbb{R}$ and $\left\vert h_{i}(t, 0)\right\vert \leq \theta _{i}(t),$ where $\theta _{i}(t)$ are positive and continuous functions on $[0, T]$ for $i = 1, ..., m$ with $\underset{ 0\leq t\leq T}{\sup }\theta _{i}(t) = \left\Vert \theta _{i} \right\Vert.$

$(H_{3})$ Assume that there exists $L_{f} > 0$ such that $\left\vert f(t, \lambda _{1})-f(t, \lambda _{2})\right\vert \leq L_{f}\left\vert \lambda _{1}-\lambda _{2}\right\vert,$ for all $(t, \lambda _{1}, \lambda _{2})\in [0, T]\times \mathbb{R} \times \mathbb{R},$ and $\left\vert f(t, 0)\right\vert \leq \zeta (t),$ where $\zeta (t)$ is positive and continuous function on $[0, T]$ with $\underset{0\leq t\leq T}{\sup }\zeta (t) = \left\Vert \zeta \right\Vert.$

Now, we present the following existence theorem for the problem described in Eq (1.6) using the Dhage's fixed point theorem.

Theorem 18. Let $\sigma \in (0, 1)$ and the hypotheses $(H_{1}), (H_{2})$ and $(H_{3})$ hold. If there exists a real number $R > 0$ such that

where $P$ denotes the lower regularized incomplete gamma function with

and

Then, the problem described in Eq (1.6) has at least one solution on $[0, T].$

Proof. Consider $\mho = \{u\in \Sigma,$ s.t. $\left\Vert u\right\Vert \leq R\}$ and let

where $A, C:\Sigma \rightarrow \Sigma$ and $B:\mho \rightarrow \Sigma$ are three operators defined by

and

Claim (1). $A$ and $C$ are Lipschitz with Lipschitz constants $\widetilde{A}$ and $\widetilde{C}$, respectively.

Let $u, v\in \Sigma$, for all $t\in [0, T]$ we have

which yields

Next, we have

which yields

Hence $A$ and $C$ are Lipschitz with Lipschitz constants $\widetilde{A}$ and $\widetilde{C}$, respectively.

Claim (2). $B$ is completely continuous.

Step 1. We will show that the operator $B:\mho\rightarrow \Sigma$ is uniformly bounded.

For any $u\in \mho$, we have

For $u\in \mho$ and $t\in [0, T]$, using $(H_{3})$ we get

Then, using Lemma 13, we obtain

Therefore, $\Vert Bu\Vert < +\infty$, and consequently $B(\mho)$ is bounded. Hence, $B$ is uniformly bounded.

Step 2.We show that the continuity of $B$ on $\mho.$ Let $\left(u_{n}\right) _{n\in \mathbb{N}}$ be a sequence where $\underset{n\rightarrow +\infty }{\lim }u_{n} = u$ with $u\in \mho$.

For all $t\in [0, T]$, we have

Using the Lebesgue dominated convergence theorem, we get

which yields $\lim_{n\rightarrow +\infty }\left\Vert Bu_{n}-Bu\right\Vert = 0.$ Therefore, $B$ is continuous on $\mho$.

Step 3.We prove that $B(\mho)$ is equicontinuous.

Let $t_{1}, t_{2}\in I = [0, T]$ with $t_{1} > t_{2}.$ Then, for any $u\in \mho$, we have

Using Lemmas 16 and 14, we obtain

Therefore, $Bu(t)$ is equicontinuous on $[0, T]$.

Making use of the Arzela-Ascoli theorem, we have $B(\mho)$ is relatively compact. Thus $B$ is a compact operator and as a consequence $B$ is completely continuous.

Claim (3). $\widetilde{A}M_{B}+\widetilde{C} < 1$, where $M_{B} = \sup \left\{ \parallel Bu\parallel :u\in \mho\right\}.$

We have $M_{B} = \left\Vert B(\mho)\right\Vert = \sup \left\{ \parallel Bu\parallel :u\in \mho\right\} \leq \varrho$ where $\varrho$ is given by(3.9). So,

Claim (4).$AuB\overline{u}+Cu = u\Rightarrow u\in \mho$ for all $\overline{u}\in \mho.$

Let $\overline{u}\in \mho$ we have

which yields

Therefore, $u\in \mho$.

Thus, all the conditions of Dhage fixed point theorem are satisfied; hence, the operator $\Phi$ has a fixed point in $\mho$. As a result, the proportional fractional boundary value problem declaired in Eq (1.6) has at least one solution on $[0, T].$ This completes the proof.

In the following, we present the existence theorem for the given problems in the special cases 1, 2 and 3.

Remark 19. From Lemma 12, in case $\sigma = 1,$ we can replace the formula $\frac{P(\omega, \frac{1-\sigma }{\sigma }T)}{\left(1-\sigma \right) ^{\omega }}$ by the formula $\frac{T^{\omega }}{\Gamma (\omega +1)}$ ($\omega \in \left\{ \beta, \alpha _{1}, ..., \alpha _{m}\right\}$). Then, by using Theorem 18 we can conclude the existence results of the given problem with usual Caputo fractional derivative.

Corollary 20. Let $\sigma = 1$ (i.e., $_{p}^{\mathcal{C}}\mathfrak{D} _{0^{+}}^{\beta, \sigma } =$ $^{C}D_{0^{+}}^{\beta }$) and assume that the hypotheses $(H_{1}), (H_{2})$ and $(H_{3})$ hold. If there exists a real number $R > 0$ such that

where

and

Then, the identified problem in Eq (1.6) has at least one solution on $[0, T].$

Corollary 21. Let $\sigma \in (0, 1)$ and assume that the hypothesis $(H_{1})$ holds. If there exists a real number $R > 0$ such that

where

Then, the problem in (1.7) has at least one solution on $[0, T].$

Corollary 22. Let $\sigma = 1,$ and assume that the hypothesis $(H_{1})$ holds. If there exists a real number $R > 0$ such that

where

Then, the problem (1.8) has at least one solution on $[0, T].$

4.   Uniqueness of the solution

In this section, we discuss the existence and uniqueness of solution for the following problem.

Note that we can write the equivalent equation for problem described by Eq (4.1) as follows.

where $B_{1} = B$ (with $f(0, u_{0}) = 1 = f(T, u_{T})$) and $B$ and $C$ are given by Eq (3.12) and Eq (3.13), respectively.

The following assumption is essential.

$(H_{4})$ Assume $g\in C([0, T]^{2}\times \mathbb{R}, \mathbb{R})$ and there exists $L_{g} > 0$ such that $\left\vert g(t, \lambda _{1})-g(t, \lambda _{2})\right\vert \leq L_{g}\left\vert \lambda _{1}-\lambda _{2}\right\vert,$ for all $(t, \lambda _{1}, \lambda _{2})\in [0, T]\times \mathbb{R}\times \mathbb{R}$ and $\left\vert g(t, 0)\right\vert \leq \Upsilon (t),$ where $\Upsilon (t)$ is positive and continuous function on $[0, T]$, with $\underset {0\leq t\leq T}{\sup }\Upsilon (t) = \left\Vert \Upsilon \right\Vert.$

Theorem 23. Let $\sigma \in (0, 1)$ and assume that $(H_{2})$ and $(H_{4})$ are satisfied. Then, the problem described by Eq (4.1) has a unique solution on $[0, T]$ if

where

with $P$ defined as in Eq (2.13).

Proof. Let us set $\overline{\mho } = \{u\in \Sigma,$ s.t. $\left\Vert u\right\Vert \leq r\},$ where $r > 0$ satisfying:

where $\Delta$ is given by Eq (4.4) and

We will show that $\Phi _{1}\overline{\mho }\subset \overline{\mho }.$

For $u\in \overline{\mho }$ and $t\in [0, T],$ we have

On the other hand, we have

We procure that

which implies that $\Phi _{1}\overline{\mho }\subset \overline{\mho }.$

Next, we show that the operator $\Phi _{1}$ is a contraction mapping.

For $u, v\in \Sigma$ and for all $t\in [0, T],$ we have

Taking the supermum over all $t\in [0, T]$ and using the inequality (3.14) yield

From the condition (4.3), we conclude that the operator $\Phi _{1}$ is a contraction mapping. Hence, the problem (4.1) has a unique solution on $[0, T].$ The proof is completed.

Now, for $\sigma = 1$, using Remark 19 above, we can easily come by the following result.

Corollary 24. Let $\sigma = 1$ (i.e., $_{p}^{\mathcal{C}}\mathfrak{D} _{0^{+}}^{\beta, \sigma } =$ $^{C}D_{0^{+}}^{\beta }$) and assume that $(H_{2})$ and $(H_{4})$ are satisfied. Then, the problem describe by Eq (4.1) has a unique solution on $[0, T]$ if

where

5.   Ulam-Hyers stability results

We begin introducing the concept of Ulam-type stability for the problem described by (4.1).

Definition 25. The solution of the problem described by (4.1) is said to be Ulam-Hyers stable if there exists a real constant $K_{g} > 0$ such that for given $\varepsilon > 0$ and for each solution $v\in {C([0, T], \mathbb{R})}$ of the inequality

there exists a solution $u\in {C([0, T], \mathbb{R})}$ of problem described by Eq (4.1) with

Definition 26. The solution of the problem described by Eq (4.1) is said to be generalized Ulam-Hyers stable if there exists a function $\phi _{g}\in {C(\mathbb{R}^{+}, \mathbb{R} ^{+})}$ with $\phi _{g}(0) = 0$ such that, for any given $\varepsilon > 0$ and for each solution $v\in {C([0, T], \mathbb{R})}$ of the inequality (5.1), there exists a solution $u\in {C([0, T], \mathbb{R})}$ of problem described by (4.1) with

The stability mentioned in Definition 25 can be generalized if the constant $\varepsilon$ is replaced by a certain type of functions. Such stability is called the Ulam-Hyers-Rassias stability.

Definition 27. The solution of the problem described by (4.1) is said to be Ulam-Hyers-Rassias stable with respect to $\phi _{g}\in {C}([0, T], \mathbb{R}^{+})$ if there exists $K_{g, \phi } > 0$ such that, for any given $\varepsilon > 0$ and for each solution $v\in {C}([0, T], \mathbb{R})$ of the inequality

there exists a solution $u\in {C([0, T], \mathbb{R})}$ of the problem described by (4.1) with

Definition 28. The solution of the problem described by (4.1) is a generalized Ulam-Hyers-Rassias stable with respect to $\phi _{g}\in {C}([0, T], \mathbb{R}^{+})$ if there exists $K_{g, \phi } > 0$ such that, for any given $\varepsilon > 0$ and for each solution $v\in {C([0, T], \mathbb{R})}$ of inequality

there exists a solution $u\in {C([0, T], \mathbb{R})}$ of problem (4.1) with

Remark 29. A function $v\in {C([0, T], \mathbb{R})}$ is a solution of the inequality (5.2) if and only if there exists a small perturbation $\Xi \in C([0, T], \mathbb{R})$ (dependent on $v$) such that

i) $\left\vert \Xi (t)\right\vert \leq \varepsilon \phi _{\Xi }(t),$ $t\in { [0, T], }$

ii) $_{p}^{\mathcal{C}}\mathfrak{D}_{0^{+}}^{\beta, \sigma }\bigg(u\left(t\right) -\sum_{i = 1}^{i = m}J_{0^{+}}^{\alpha _{i}, \sigma }h_{i}(t, u(t)) \bigg) = g\left(t, u\left(t\right) \right) +\Xi (t),$ $t\in {[0, T]}.$

Lemma 30. If $v\in {C}([0, T], \mathbb{R})$ represents a solution of inequality (5.2), then $v$ is a solution of the following integral inequality

where

Proof. From Remark 29, we get

this yields that

which leads to the inequality in (5.4).

In the following, we present the theorem related the Ulam-Hyers-Rassias stability of the solution of the problem described by Eq (4.1).

Theorem 31. Let $\sigma \in (0, 1).$ Assume that $(H_{1}),$ $(H_{2})$ and$\ (H_{4})$ are satisfied with

Then, the solution to the problem described by Eq (4.1) is both Ulam-Hyers-Rassias stable and generalized Ulam-Hyers Rassias stable on $[0, T].$

Proof. From Lemma 30 we get, for $t\in [0, T]$

This yields that

By setting $K_{g, \phi } = \frac{1}{1-\Delta }$, we end up with

Hence, the solution of problem described by Eq (4.1) is Ulam-Hyers-Rassias stable with respect to $\Pi _{\phi _{\Xi }}.$ Moreover, if we set $\phi _{g}(t) = \varepsilon \Pi _{\phi _{\Xi }}(t)$ with $\phi _{g}(0) = 0,$ then the same solution is generalized Ulam-Hyers-Rassias stable. The proof is completed.

Corollary 32. Let $\sigma = 1$ (i.e., $_{p}^{\mathcal{C}}\mathfrak{D} _{0^{+}}^{\beta, \sigma } =$ $^{C}D_{0^{+}}^{\beta }$) and assume that $(H_{2})$ and $(H_{4})$ are satisfied with $\Delta _{1} < 1$ where $\Delta _{1}$ is given by (4.7). Then, the solution of the problem described by Eq (4.1) is both Ulam-Hyers-Rassias stable and generalized Ulam-Hyers Rassias stable on $[0, T].$

Remark 33. Let $\sigma \in (0, 1].$ By using Theorem 31 and Corollary 32, we can conclude that:

1) If $\Pi _{\phi _{\Xi }}(t) = 1,$ then the solution of the problem described by (4.1) is Ulam-Hyers stable.

2) If we set $\phi _{g}(\varepsilon) = K_{g, \phi }\varepsilon$ with $\phi _{g}(0) = 0$, then the solution of the problem described by (4.1) is generalized Ulam-Hyers stable.

6.   Example

Consider the following hybrid proportional fractional integro-differential equation$:$

where the real constant $\epsilon$ and the function $f(t, u(t))$ will be fixed later.

Here $T = 1, \beta = \frac{3}{2}, \sigma = \frac{3}{4}, \alpha _{i} = \frac{2i-1}{4}, h_{i}(t, u(t)) = \frac{\cos \left\vert u\left(t\right) \right\vert }{t+100i}, (i = 1, ..., 4), g(t, u(t)) = \frac{ \epsilon (1-t)\sin \left\vert u(t)\right\vert }{1+\left\vert u(t)\right\vert }, u_{0} = \frac{1}{10}$ and $u_{1} = \frac{1}{20}.$

We can show that

Also, for all $(t, \lambda)\in [0, 1]\times \mathbb{R},$ we have

It follows that $L_{h_{i}} = \frac{1}{10i} = \left\Vert \theta _{i}\right\Vert, (i = 1, ..., 4), L_{g} = 2\left\vert \epsilon \right\vert, \left\Vert \widetilde{g} \right\Vert = \left\vert \epsilon \right\vert$ and $\psi (\left\Vert u\right\Vert) = \frac{\left\Vert u\right\Vert }{1+\left\Vert u\right\Vert }.$

Using Matlab program, we find

For illustrating Theorem 18, we take $f(t, u(t)) = \frac{1}{10}\sqrt{ 1+\left\vert u(t)\right\vert ^{2}}$ and $\epsilon = 1.$

We can show that

i.e., $L_{f} = \frac{1}{10}$ and $\left\Vert \zeta \right\Vert = \frac{1}{10}.$

We see that the condition (3.7) and (3.8) is followed with a real number $R\in$ $[0.929854125359180, 5.322899777680700].$

Therefore, all conditions in Theorem 18 are satisfied. We conclude that the problem (6.1) has at least one solution on $[0, 1].$

Remark 34. If $R = 0.929854125359180,$ we obtain

and

Also, if $R = 5.322899777680700,$ we obtain

and

Note that all conditions in Theorem 18 are satisfied. We conclude that the problem (6.1) has at least one solution on $[0, 1].$

For illustrating Theorems 23 and 31, we take $f(t, \lambda) = 1$ for all $(t, \lambda)\in$ $[0, 1]\times < /p > < p > \mathbb{R} < /p > < p >$ and $\epsilon = \frac{1}{11\pi }.$

Using the above data, we find:

In virtue of Theorem 23, problem (6.1) has a unique solution. Furthermore, we can compute

Thus, by use of Theorem 31, problem (6.1) is Ulam-Hyers-Rassias stable, and consequently generalized Ulam-Hyers-Rassias stable.

7.   Conclusions

In this work, we inspected the existence of solutions to a certain class of hybrid fractional intego-differential equations in the casement of generalized proportional fractional hybrid integro-differential equations supplemented with Dirichlet boundary conditions. The existence of at least one solution was shown with the assistance of the hybrid fixed point theorem for a product of three operators. In addition, for some special cases, the uniqueness of the solutions for the considered class of equations, and their stability in the sense of Ulam, were discussed.

We believe that the results obtained will be very important for the researchers working on the qualitative aspects of the boundary value problems in the frame work of the generalized fractional derivatives mentioned in the article. This is due to the fact that as far as we know this is the first work in which the lower regularized incomplete gamma function is used to discuss some qualitative aspects of solutions to fractional differential equations.

Conflict of interest

The authors declare there is no conflict of interest.