Abdeljawad, ThabetAgarwal, Ravi P.Alzabut, JehadJarad, FahdOzbekler, AbdullahMathematics2024-07-052024-07-052018331029-242X10.1186/s13660-018-1731-x2-s2.0-85048879028https://doi.org/10.1186/s13660-018-1731-xhttps://hdl.handle.net/20.500.14411/2703Alzabut, Jehad/0000-0002-5262-1138; Abdeljawad, Thabet/0000-0002-8889-3768; Agarwal, Ravi P/0000-0003-0075-1704; Jarad, Fahd/0000-0002-3303-0623We state and prove new generalized Lyapunov-type and Hartman-type inequalities fora conformable boundary value problem of order alpha is an element of (1,2] with mixed non-linearities of the form ((T alpha X)-X-a)(t) + r(1)(t)vertical bar X(t)vertical bar(eta-1) X(t) + r(2)(t)vertical bar x(t)vertical bar(delta-1) X(t) = g(t), t is an element of (a, b), satisfying the Dirichlet boundary conditions x(a) = x(b) = 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) circle T-alpha(a), alpha is an element of (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.eninfo:eu-repo/semantics/openAccessLyapunov inequalityHartman inequalityConformable derivativeGreen's functionBoundary value problemMixed non-linearitiesArticleQ1WOS:00043601960000530137730