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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

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

### Abstract

We state and prove new generalized Lyapunov-type and Hartman-type inequalities for a conformable boundary value problem of order $\alpha \in \left(1,2\right]$ 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 $x\left(a\right)=x\left(b\right)=0$ , where ${r}_{1}$ , ${r}_{2}$ , and g are real-valued integrable functions, and the non-linearities satisfy the conditions $0<\eta <1<\delta <2$ . Moreover, Lyapunov-type and Hartman-type inequalities are obtained when the conformable derivative ${T}_{\alpha }^{a}$ is replaced by a sequential conformable derivative ${T}_{\alpha }^{a}\circ {T}_{\alpha }^{a}$ , $\alpha \in \left(1/2,1\right]$ . The potential functions ${r}_{1}$ , ${r}_{2}$ as well as the forcing term g require no sign restrictions. The obtained inequalities generalize some existing results in the literature.

#### KEYWORDS:

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.