Noninstantaneous impulsive inequalities via conformable fractional calculus.

We establish some new noninstantaneous impulsive inequalities using the conformable fractional calculus.

Introduction and preliminaries

The subject of fractional differential equations has evolved as an interesting and important field of research in view of numerous applications in physics, mechanics, chemistry, engineering (like traffic, transportation, logistic, etc.), and so forth [1-3]. The tools of fractional calculus play a key role in improving the mathematical modeling of many real-world processes based on classical calculus. For some recent development on the topic, see [4-12] and the references therein.

Various types of fractional derivatives were introduced: Riemann–Liouville, Caputo, Hadamard, Erdélyi–Kober, Grünwald–Letnikov, Marchaud, and Riesz, to just name a few. Commonly, all they are defined as integrals with different singular kernels, that is, they have a nonlocal structure. Due to this fact, there are many inconsistencies of the existing fractional derivatives with classical derivative. Thus they do not obey the familiar product rule, the quotient rule for two functions, and the chain rule. Also, the fractional derivatives do not have a corresponding Rolle’s theorem or a corresponding mean value theorem.

On the other hand, a recently introduced definition of the so-called conformable fractional derivative involves a limit instead of an integral; see [13, 14]. This local definition enables us to prove many properties analogous to those of integer-order derivatives. The authors in [14] showed that the conformable fractional derivative obeys the product and quotient rules and has results similar to the Rolle theorem and the mean value theorem in classical calculus.

For recent works on conformable derivatives, we refer to [15-19] and the references therein.

Let us recall the definition of the conformable fractional derivative and integral.

Definition 1.1

Let . The conformable fractional derivative starting from a point ϕ of a function is defined by and .

Note that if f is differentiable, then

Definition 1.2

Let . The conformable fractional integral of a function from a point ϕ is defined by

The impulsive differential equations have been used to describe processes that have sudden changes in their states at certain moments. Many mathematical models in physical phenomena that have short-term perturbations at fixed impulse points ,  , caused by external interventions during their evolution appeared in population dynamics, biotechnology processes, chemistry, physics, engineering, and medicine; see [20-22]. In [23, 24], the authors introduced a new class of noninstantaneous impulsive differential equations with initial conditions to describe some certain dynamic changes of evolution processes in the pharmacotherapy. This kind of impulsive differential equations can be distinguished from the usual one as the changing processes containing no ordinary or fractional derivatives of their states work over intervals , whereas the usual does at points ,  . There are some papers on existence and stability theory of this kind of impulsive ordinary or fractional differential equations [25-36]. To the best of our knowledge, there is no literature on noninstantaneous impulsive inequalities. The main goal of the paper is to establish some new noninstantaneous impulsive inequalities using the conformable fractional calculus. The main results are presented in Sect. 2. In Sect. 3, the maximum principle and boundedness of solutions for noninstantaneous impulse problems are illustrated.

Main results

Assume that the independent variable t is the time defined on the half-line . Let and be two increasing sequences such that for and . In addition, we define subsets of by , and . Note that . Set  = { is continuous on , and exists for },  = {; is continuous for , and exists for },  = { is continuous everywhere for , and exists for },  = { is continuous everywhere for all , and exists for }, and .

Let the maximums of impulsive points less than or equal to t be defined by In addition, we define Note that and where are positive integers.

Throughout this paper, we assume that the unknown function is left-continuous at and (). Now, we are in the position to establish noninstantaneous impulsive differential inequalities.

Theorem 2.1

Let , , be given constants such that and ,  . Suppose that and Then and

Proof

For , the conformable fractional differential inequality can be written as By taking the conformable fractional integral of order α from to , we obtain which implies that (2.5) holds for .

For , we define the function Note that and Then, taking the derivative with respect to t, we have Multiplying this inequality by the integrating factor , we get which implies that By (2.7) with we have This shows that the bound in (2.6) is true for .

Now, we assume that inequality (2.5) holds for , . By mathematical induction we will show that (2.6) is true for . Let Then and . Using the above method, we have which leads to Substituting the bound of and inequality (2.5) with , it follows that by using formula (2.2). Therefore (2.6) is satisfied for .

Finally, we suppose that estimate (2.6) is fulfilled for , where . Next, we will prove that inequality (2.5) holds for . By using the above method, we get the inequality Using (2.6) with and applying (2.3), we obtain Therefore inequality (2.5) is valid on . This completes the proof. □

The following corollary can be obtained by replacing the given functions and by constants M and N, respectively.

Corollary 2.1

Let and ,  , be constants. If , , and then and where , , and .

Let be the Heaviside function. We define two functions and Next, we establish some new noninstantaneous impulsive integral inequalities.

Theorem 2.2

Let , constants , ,  , and . If where and are defined by (2.1), then we have for ,  , and for ,  .

To prove inequalities (2.13) and (2.14), for , we define the function which yields for all and . For any ,  , we get Also, taking the conformable fractional derivative of order α, we have For ,  , we obtain An application of Theorem 2.1 to (2.15) and (2.16) yields for ,  , and for ,  . From , , we get the desired results in (2.13) and (2.14). The proof is completed. □

Theorem 2.3

Let , let h be a positive fractional integrable function of order α, and let and ,  , be constants. If where and are defined by (2.1), then we have and where the constants ,  , are defined by .

For , setting we have for ,  , and for ,  . Since , , this reduces to and Now Theorem 2.1, together with the inequality , yields estimates (2.18) and (2.19), completing the proof. □

Next, we obtain the following corollary by putting constant values and .

Corollary 2.2

Let constants and ,  . If where and are defined by (2.1), then we have and where and are defined as in Corollary 2.1, and .

Applications

In this section, we establish two applications of noninstantaneous impulsive differential and integral inequalities. Let with and with for some . The first purpose is accomplished by considering two problems that have the end points at and , respectively. Now, we consider and where , , , and . Let us state the following conditions:

,

,

,

, where is defined by

Corollary 3.1

Let u and v be unknown functions satisfying (3.1) and (3.2), respectively. If (H1)–(H2) hold, then for . If (H3)–(H4) hold, then for .

Applying Theorem 2.1 to the first two inequalities in problem (3.1), we have Since for all and all constants are positive, it is sufficient to show that . At the end point , we obtain By conditions (H1)–(H2) we have which yields . Therefore for .

Next, we will show that for . The application of Theorem 2.1 for the first two inequalities in problem (3.2) leads to Substituting the end point at , we have which implies by conditions (H3)–(H4). This means that . In the same way, we can conclude that for . The proof is completed. □

Finally, we apply the noninstantaneous impulsive inequality to the initial value problem of the form where , , , , and the given function satisfies

, , for all .

Corollary 3.2

If (H5) holds, then the solution of problem (3.3) is estimated as and

Taking the conformable fractional integral of order α to the first equation of problem (3.3), we obtain From condition (H5) it follows that Since , by Theorem 2.2 inequalities (3.4)–(3.5) hold, and the proof is completed. □