A note on the qualitative behavior of some nonlinear local improper conformable differential equations

Abstract

In this paper, we present two qualitative results concerning the solutions of nonlinear generalized differential equations, with a local derivative defined by the authors in previous works. The first result covers the boundedness of solutions while the second one discusses when all the solutions are in L$^{2}$.

On the generalized laplace transform and applications

“…This generalized differential operator contains many of the known local operators (for example, the conformable derivative of 1 and the non-conformable of 2 )) and has shown its usefulness in various applications, as it can be consulted, for example, in. 3,4,8,10,11,[32][33][34] One of the most required properties of a derivative operator is the Chain Rule, 9 to calculate the derivative of compound functions, which does not exist in the case of classical fractional derivatives…”

Analytical Solutions of Fuzzy Linear Differential Equations in the Conformable Setting

“…We have infinite choices of derivative orders to consider by considering fractional derivatives, and with this, we can determine what the fractional differential equation is that we can use to model our phenomenon. For more research works and details on relevant concept, see for example ( [19], [20], [21], [22], [23]).…”

Note on the conformable boundary value problems: Sturm’s theorems and Green’s function

“…The difference between conformable fractional derivative and non-conformable fractional derivative is that the tangent line angle is conserved in the conformable one, while it is not conserved in the sense of nonconformable one [25] (see also [26,34,36,37] for more new related results about this newly proposed definition of non-conformable fractional derivative). The definition of non-conformable fractional derivative has been investigated and applied in various research studies and applications of physics and natural sciences such as the stability analysis, oscillatory character, and boundedness of fractional Liénard-type systems [27,28,33], analysis of the local fractional Drude model [29], Hermite-Hadamard inequalities [30], fractional Laplace transform [31], fractional logistic growth models [32], oscillatory character of fractional Emden-Fowler equation [35], asymptotic behavior of fractional nonlinear equations [38], and qualitative behavior of nonlinear differential equations [39]. This paper is organized as follows: In Sec.…”