The analytical solution of Van der Pol and Lienard differential equations within conformable fractional operator by retarded integral inequalities

Abstract: In this study we introduced and tested retarded conformable fractional integral inequalities utilizing non-integer order derivatives and integrals. In line with this purpose, we used the Katugampola type conformable fractional calculus which has several practical properties. These inequalities generalize some famous integral inequalities which provide explicit bounds on unknown functions. The results provided here had been implemented to the global existence of solutions to the conformable fractional differential equations with time delay.


Introduction
Being important tools in the analysis of differential equations, integral equations and integro-differential equations, a number of generalizations of Gronwall inequality and their utilizations have greatly attracted the interests of several mathematicians. In 1995, Pachpatte [1] provided a generalization of an interesting integral inequality thanks to Ou-Iang [2]. Later on Lipovan [3] proposed a retarded type of Pachpatte and Ou-Iang integral inequalities. Then Sun [4] made a generalization for results given by Lipovan. A number of new definitions have been proposed in academia to provide a better method for fractional calculus such as Riemann-Liouville, Caputo, Hadamard, Erdelyi-Kober, Grunwald-Letnikov, Marchaud and Riesz among others. [5,6]. Now, all these definitions satisfy the property that the fractional derivative is linear. This is the only property inherited from the first derivative by all the definitions. However, all definitions do not provide some properties such as Product Rule (Leibniz Rule), Quotient Rule, Chain Rule, Rolls Theorem and Mean Value Theorem. In addition most of the fractional derivatives (except Caputo-type derivatives), do not satisfy D α (f) (1) = if α is not a natural number. Recently a new local, limit-based definition of a conformable derivative has been introduced in [7]. Among the others, we refer the readers to [8]- [12] and references therein. This new idea was quickly generalized by Katugampola [13], whose definition forms the basis for this work and is referred to here as the Katugampola derivative (D α will henceforth be referred to the Katugampola derivative). This definition has several practical properties which are summarized below In this study, we presented a retarded conformable fractional integral inequalities using the Katugampola conformable fractional calculus. The remainder of this study is organized as follows: In Section 2, the related definitions and theorems are reviewed. In Section 3, the general versions of retarded integral inequalities utilizing conformable fractional calculus are obtained while some conclusions and remarks are discussed in Section 4.
We will also take advantage of the following significant consequences, which can be derived from the results above.
Lemma 1. Let the conformable differential operator D α be given as in (2.1), where α ∈ (0, 1] and η ≥ 0, and assume the maps f and g are α-differentiable as needed. Then In this manuscript, by using the Katugampola type conformable fractional calculus, we introduced retarded conformable fractional integrals inequalities. The results provided here can be implemented to the global existence of solutions to the conformable fractional differential equations with time delay.

Main findings and cumulative results
where m is a non-negative constant, then Proof. Let us first assume that m > 0. Define the non-decreasing positive function z(η) by the right-hand side of (3.1). Then ν(η) ≤ z(η) and z(0) = m, and Then from the definition of G we have Then by taking the α−th order of conformable derivative of G(z(η)), we have Then by taking integration from 0 to φ(η), we get Then we obtain Since ν(η) ≤ z(η) and G −1 is increasing on Dom(G −1 ), we get the desired inequality, that is
where m is a non-negative constant. Then with G as in Theorem 2.
If we define an arbitrary number T such that 0 ≤ T ≤ η, then we have Now an application of Theorem 2 given above, we get By using the fact that ν(η) ≤ √︀ z(η) and taking η = T in the above inequality, we obtain Thus we get the desired result.

Remark 3. If we take φ(η) = η, then the inequality given by Theorem 3 reduces to Pachpatte's generalization of Ou-Iang type inequality for conformable integrals.
where m is a non-negative constant. Then ∫︁ 0 g(s)dαs.

Remark 5. Corollary 3 is called a retarded version of a conformable fractional integral inequality whose classical version given in
Assume that φ and k are non-decreasing with φ(η) ≤ η for η ≥ 0 and k(ν) > 0. If ν ∈ C (︀ R + , R + )︀ satisfies where m is a non-negative constant. Then Proof. By following the similar steps of the proof of Theorem 3, we obtain By setting φ(η) = s and integrating from 0 to φ(η) with respect to s, we get [m + f(s)k( √︀ z(s)) + g(s)k( √︀ z(s))]dαs.
By using the fact that ν(η) ≤ √︀ z(η) in the above inequality, we obtain Thus we get the desired result.

Applications
In this part, we give some applications of our results to obtain the solution of several specific non-linear conformable fractional differential equations.

Concluding remark
In this paper, retarded conformable fractional integral inequalities is proposed and tested with the help of Katugampola type conformable fractional calculus. To verify the results given here, we applied them to the global existence of solutions to conformable fractional differential equations with time delay.