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Citation - WoS: 12
Citation - Scopus: 13
On Defining the Distributions Δ<sup>r</Sup> and (δ′)<sup>r</Sup> by Conformable Derivatives
(Springeropen, 2018) Abdeljawad, Thabet; Jarad, Fahd; Adjabi, Yassine; Baleanu, Dumitru
In this paper, starting from a fixed delta-sequence, we use the generalized Taylor's formula based on conformable derivatives and the neutrix limit to find the powers of the Dirac delta function delta(r) and (delta')(r) for any r is an element of R.
Citation - WoS: 71
Citation - Scopus: 94
A Generalized Lyapunov-Type Inequality in the Frame of Conformable Derivatives
(Springeropen, 2017) Abdeljawad, Thabet; Alzabut, Jehad; Jarad, Fahd
We prove a generalized Lyapunov-type inequality for a conformable boundary value problem (BVP) of order alpha is an element of (1, 2]. Indeed, it is shown that if the boundary value problem (T(alpha)(c)x)(t) + r(t) x(t) = 0, t is an element of (c, d), x(c) = x(d) = 0 has a nontrivial solution, where r is a real-valued continuous function on [c, d], then integral(d)(c) vertical bar r(t)vertical bar dt > alpha(alpha)/(alpha - 1)(alpha-1) (d - c)(a-1). (1) Moreover, a Lyapunov type inequality of the form integral(d)(c)vertical bar r(t)vertical bar dt > 3 alpha - 1/(d - c)(2 alpha-1) (3 alpha - 1/2 alpha - 1)(2 alpha-1/a), 1/2 < alpha <= 1, (2) is obtained for a sequential conformable BVP. Some examples are given and an application to conformable Sturm-Liouville eigenvalue problem is analyzed.

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