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J Inequal Appl. 2018;2018(1):143. doi: 10.1186/s13660-018-1731-x. Epub 2018 Jun 20.

Lyapunov-type inequalities for mixed non-linear forced differential equations within conformable derivatives.

Author information

1Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia.
2Department of Mathematics, Texas A&M University-Kingsville, Kingsville, USA.
3Department of Mathematics, Çankaya University, Ankara, Turkey.
4Department of Mathematics, Atilim University, Ankara, Turkey.


We state and prove new generalized Lyapunov-type and Hartman-type inequalities for a conformable boundary value problem of order α(1,2] with mixed non-linearities of the form (Tαax)(t)+r1(t)|x(t)|η-1x(t)+r2(t)|x(t)|δ-1x(t)=g(t),t(a,b), satisfying the Dirichlet boundary conditions x(a)=x(b)=0 , where r1 , r2 , and g are real-valued integrable functions, and the non-linearities satisfy the conditions 0<η<1<δ<2 . Moreover, Lyapunov-type and Hartman-type inequalities are obtained when the conformable derivative Tαa is replaced by a sequential conformable derivative TαaTαa , α(1/2,1] . The potential functions r1 , r2 as well as the forcing term g require no sign restrictions. The obtained inequalities generalize some existing results in the literature.


Boundary value problem; Conformable derivative; Green’s function; Hartman inequality; Lyapunov inequality; Mixed non-linearities

Conflict of interest statement

The authors declare that they have no competing interests.

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