Hybrid Initial Value Problems of Fractional Order with -Derivatives

Abstract

In the present article, it has been tried to extend the theory of q-fractional hybrid differential equations that entails Riemann–Liouville q-differential operators. We have also attempted to prove an existence theorem for FHDEs under Lipschitz and Caratheodory conditions. We have also presented some fundamental fractional differential inequalities that can be employed to indicate that some external solutions exist for such problems. The required tools are taken into consideration and the principle for comparison is offered which can be used to conduct further research work regarding qualitative behavior of solutions.

1. Introduction

A long history is often attributed to Fractional calculus that can trace back to the emergence of the classical calculus. Even though a number of research have been accomplished during the last few decades, only recently, this novel trend in calculus plus dynamic equation have received more attention. There are several categories of noninteger derivative, yet the most commonly used definitions include Caputo derivatives and Riemann–Liouville derivatives. While the first one has an abstract mathematical nature, the second one is most extensively employed among engineers.

Investigations into differential equations featuring fractional derivative has appreciably improved within current years, the fact that signifies the crucial importance of the calculus featuring fractional derivative and fractional integral in engineering, sciences, and technologies. There are some more books related to fractional calculus for interested readers [1, 2].

During the last decade, having the solution of liner initial FDEs regarding special functions was explored in such sorts of problems, which often investigated solutions (or positive solution) applying Leray–Schoder theory and FPT [3, 4].

To put it in other words, there have been the similar necessary conditions for boundary conditions and initial conditions [5, 6].

In the last few years, researchers’ attention has often been drawn towards perturbations of various types on nonlinear differential equation in particular hybrid differential equation.

Dhage and Lakshmikantham [7], for instance, took the next HDE under study comprising linear perturbations of first kind:

While employing a FPT in Banach algebra (BA), Zhao et al. [6] examined the existence of solution based on the noninteger derivative version of the above initial value problem (IVP), i.e.,

The possibility of taking solutions for the following FHDE including supremumin which , , and was investigated by Caballero et al. [8], whose major tool was the approach of measures of noncompactness in the Banach space.

In the present article, we will tackle with the HDE via fractional -derivatives mentioned below:in which is the Riemann–Liouville -derivative which , , , for each is measurable map and continuous for each

Through utilizing a FPT in BA due to Bai and Lü [5], with respect to mixed Lipschitz and Caratheodory condition (LCC) and by supposing the following hypotheses, we have been able to obtain the existence of solution to the FHDEs.

Accordingly, a number of basic -differential inequalities have been determined, which are employed to show the existence of minimal and maximal solutions. Following that, a number of comparison theorems have been proved.

Hypotheses:(i)The function is increasing in almost everywhere for .(ii)There exists a constant such that for all and .(iii)There exists a function such that in which , for all .

2. -Calculus

We now will give preliminaries of -calculus, definitions, and lemmas that will be used in the remainder of our article. The presentation here can be gained in [9].

Let and define

The -analogue of the function featuring can be described byin which .

In more general terms, if and , thus,

It is obvious that, as therefore, The -gamma function can be described viaand satisfies .

The -derivative of a function can be described whereinwhich is the -derivative of higher order due to

The -integral of a function , is resulted fromwhich is defined in interval

The -integral of a function , from to is resulted fromin which and is defined on .

In the same way as to that for derivatives, an operator for is resulted fromand for is

The basic theorem of calculus is applicable in such operators as and , i,e.,where is continuous, hence,

The formulas mentioned below will be employed in the rest of the article, i.e., the integration by parts formula:andin which stands for the derivative.

Definition 1. The fractional -integral of Riemann–Liouville of order and in which is a function on for , is described byand , for .

Definition 2. The fractional -derivative of Riemann–Liouville of order is described via and

Lemma 1 (see [9]). Assume that and consequently, .

Lemma 2 (see [9]). Assume that be a function which is defined on Following that two properties can be stipulated:(1),(2),in which .

Lemma 3 (see [9]). Assume that and . Accordingly, the next equality holds:

Lemma 4 (see [10]). Let and . If ∈ , then, and .

3. Existence Results

In the present part of the article, it has been attempted to show the existence outcomes for the -FHIVP equations (4) and (5) both on the bounded and closed on with LCCs on the nonlinearities included in it. We have posited the -FHIVP equations (4) and (5) in the space of continuous functions with real-value considered on In a supremum norm is defined based onand a multiplied in based onfor Considering the above norm and multiplication in it, it is evident that is a BA. Based on denotes the space of Lebesgue which is an integrable function with real-value in interval related to the defined by

What follows is considered a FPT in BA according to Dhage [11], which is a fundamental theorem that confirms our major result.

Theorem 1 (see [11]). Assume that be a closed, convex, nonempty, and bounded subcategory of the BA on and assume and be two operators in a manner wherein(i) and .(ii) is Lipschitz and is a Lipschitz constant.(iii) is complete and continuous.(iv),(v), in which . Meanwhile, for the operator equation has a solution in

The lemma given below is useful for the following sections.

Lemma 5. Presume that hypothesis holds. Thus, for and any , the function is a solution of the following IVP:if and only if fulfills the following hybrid integral equation:in which .

Proof. see [6].

Theorem 2. Assume that hypotheses hold. Furthermore, if , afterward for -FHIVP equations (4) and (5) has a solution defined on .

Proof. Set and consider a subset:where and is .
It can certainly be claimed that, is a convex, closed, and bounded subcategory of the Banach space . With Lemma 5, equations (4) and (5) is equivalent to the equation below:Assume two operators , by and . Then, equation (34) can be converted to the operator equation as ,
In [6], we showed which of the two operators and fulfill all the requirements of Theorem 1 and consequently, for -FHIVP equations (4) and (5), we can get to a solution defined on .

4. An Illustrative Example

Now, assume the FHIVP given below:Here, and We have

Since . Also,

Then, hypothesis and hold also for , we have

Therefore, the -FHIVP equation (35) has a solution.

5. Fractional Hybrid -Differential Inequalities

In the next section, we dissert a fundamental relevance to strict inequalities for the hybrid differential equation with fractional -derivative.

Lemma 6. Presume that be a locally Holder continuous in a way which for any , the result for is

Then, it follows that

Proof. We know thatwhere Let .
For , we haveSince for and by hypothesis, we have For by,In as much as is a locally Holder continuous and such that, for , there exists a constant :where is such that and As a result, we get toHence, for small enough .
Counting we have .
Then, for small enough . Now, it is simple to show that and we have the full proof.

Theorem 3. Presume that hypothesis hold. Also, assume that there exists the function that is a locally Holder continuous in a manner thatandin case of one of the inequalities being strict. Thence,implies for all

Proof. Assume that inequality equation (46) is strict. Take the claim is not true. Thus there exists a , in a way that and for
Define and .
Then, we have and by virtue of hypothesis we take for all For putting for we can gain and Thence, by Lemma 6,This repugn via . Accordingly, equation (48) is a valid conclusion and we have the full proof.
The result which comes next is related to strict fractional -differential inequalities that is regarded a sort of one way Lipschitz condition.

Theorem 4. Assume that the requirement of Theorem 3 hold via q-inequalities equations (45) and (46). Presume that there exists in such a way that for all via .

Therefore, implies,

for all provided .

Proof. For small we putThence, it follows thatLet .
For , so thatSincefor all and , one hasLikewise, we have Consequently, applying Theorem 3 with results in for all By assuming , considering the limits as we have for all which gives us the full proof.

Remark 1. Let and . We can facilely determine that and fulfill the condition equation (50).

6. Existence of Minimal and Maximal Solutions

In the present part, the existence of minimal and maximal solutions is confirmed in order to the FHDE equation (30) in interval Accordingly, the definition below is required for the rest of the article.

Definition 3. Let . A solution of the -FHDE equation (20) be maximal, if , where on , is any solution of the -FHDE equation (30).

Definition 4. Let . A solution of the q-FHDE equation (20) be minimal, if , where on , is any solution of the -FHDE equation (30).
We only tussle the condition of minimal solution, whereas the condition of maximal solution is equivalent and may be achieved via some arguments with some suitable variations, presuming a small number
Now assume the following IVP of -FHDE:in which , and .
Now, we give an existence theorem for the -FHDE equation (71) given below:

Theorem 5. Assume that hypotheses hold. Presume that inequality in Theorem 2 holds. Therefore, for per small number , the FHDE equation (57) has a solution defined on .

Proof. By a hypothesis as long asthere exists an in a way thatfor all Now, the remaining part of the proof just like Theorem 4.

Theorem 6. Assume that hypotheses hold. Moreover, if the condition of Theorem 2 is valid, then, on for the -FHDE equations (4) and (5), we can find a maximal defined (definite) solution.

Proof. Assume be a declining sequence of real numbers and positive in such a manner that , in which is a real number and positive fulfilling the next inequality:The number exists in view of inequality . With the help of Theorem 5, therefore, there exists a solution defined on of the FHDETherefore, achieving (obtaining) (attaining) any solution of the q-FHDE equations (4) and (5) fulfillsand any solution of supportive problem equation (73) fulfillsin which With the help of Theorem 4 we deduce thatfor all and
Since by Theorem 5, we deduce that Hence, is a declining sequence of real numbers and positive, the limitexists. We can indicate that the convergence in equation (65) is uniform on To put it to an end, what we need is to demonstrate that the sequence is equicontinuous in Assume with be arbitrary. Thus,where . Because is continuous on the compact set it is uniformly continuous there. Hereupon, for all ,as uniformly.
Accordingly, regarding the inequality mentioned above, it can be concluded that as uniformly for all Afterwards, for all as .
In later section, we indicate that the function is a solution of q-FHDE equations (4) and (5) defined on . In as much as is a solution of the -FHDE equation (61) we will get tofor all Imagining the limit as in the equation (48) leads tofor all . Therefore, on , the function z is a solution of q-FHDE equations (4) and (5). Ultimately, from inequality equation (64) it can be concluded that for all Thus, for the -FHDE equations (4) and (5), we can get to a maximal solution on , which results in the full proof

7. Comparison Theorems

It can be argued that the major problem related to differential inequalities is to appraise a bound in order to attain the solution set for the differential inequality attributed to the q-FHDE equations (4) and (5).

In the following part, we will try to confirm that the minimal and maximal solution function as bounds for the solution of the corresponding differential inequality to the q-FHDE equations (4) and (5) in interval

Theorem 7. Assume that hypotheses and condition are valid. Presume that a real number exists in such a manner that for ,for all with where . Moreover, in case there exists a function, in such a way thatin which . Therefore, for all ,in which is a maximal solution of the q-FHDE equations (4) and (5) on

Proof. Presume be arbitrarily small. By Theorem 6, there is a maximal solution of the q-FHDE equation (57), and that the limitis uniform on and the function is a maximal solution of the FHDE equation (20) on As a result, for we will get toFrom the above inequality for , it follows thatAccordingly, Theorem 4 can be applied to inequalities equations (71) and (75), and one can get to the conclusion that for all Hence, this in view of limit equation (73) suggests that inequality equation (72) holds true on which in consequence leads to make the proof full.

Theorem 8. Assume that hypotheses and condition hold. Suppose that a real number exists in such a manner that for ,for all with where . Furthermore, if exists in a way that

Therefore, which is a minimal solution of the -FHDE equations (4) and (5) on .

Theorem 9. Assume that hypothesis and condition hold. Presuming that a real number exists so that in order to , and for all ,with where . If an identical zero function for is the only solution of the differential equation,

Thus, the -FHDE equations (4) and (5) has a unique solution on .

Proof. With the assumption that the Theorem 2, equations (4) and (5) has a solution defined on . Assume that there are two solutions and of the -FHDE equations (4) and (5) on with Define a function , so,With regards to hypothesis , it can be concluded that Accordingly, we get tofor almost everywhere and that .
Following that, we execute Theorem 7 via to gain that for where an identical zero function is the only solution of differential equation (79). , which is contradictory to . Hence, we can gain . This completes the proof.

8. Conclusion

In this paper, we have developed the theory of the HDE via fractional Riemann–Liouville q-derivatives of order . Through utilizing a FPT in BA due to Bai and Lü [5], with respect to mixed Lipschitz and Caratheodory condition (LCC) and by supposing the above hypotheses, we have been able to obtain the existence of a solution to the FHDEs. We proved Lemma 6, which is the main tools to raise new existence results in -calculus. Some fundamental fractional differential inequalities have been established which are used to prove the existence of extremal solutions. The necessary tools have been considered and the comparison principle has been proved, which will be useful for further study of qualitative behavior of solutions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.