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

Abstract

We state and prove new generalized Lyapunov-type and Hartman-type inequalities for a conformable boundary value problem of order $\alpha \in (1,2]$ with mixed non-linearities of the form $\left(T_{\alpha }^{a}x\right)\left(t\right)+r_1\left(t\right)\left|x\right(t\left){|}^{\eta -1}x\right(t)+r_2(t\left)\right|x\left(t\right){|}^{\delta -1}x\left(t\right)=g\left(t\right),t\in (a,b),$ 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 $T_{\alpha }^a\circ T_{\alpha }^a$ , $\alpha \in (1/2,1]$ . 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.