This manuscript is based on the standard fractional calculus iteration procedure on conformable derivatives. We introduce new fractional integration and differentiation operators. We define spaces and present some theorems related to these operators.

Generalization of fractional differential operators was subjected to an intense debate in the last few years in order to contribute to a deep understanding of the behavior of complex systems with memory effect. In this article, a Caputo-type modification of Hadamard fractional derivatives is introduced. The properties of the modified derivatives are studied.

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This paper is devoted to the study of Caputo modification of the Hadamard fractional derivatives. From here and after, by Caputo-Hadamard derivative, we refer to this modified fractional derivative (Jarad et al. in Adv. Differ. Equ. 2012:142, 2012. We present the generalization of the fundamental theorem of fractional calculus (FTFC) in the Caputo-Hadamard setting. Also, several new related results are presented.

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We prove a generalized Lyapunov-type inequality for a conformable boundary value problem (BVP) of order α ∈ (1, 2]. Indeed, it is shown that if the boundary valuehas a nontrivial solution, where r is a real-valued continuous function on [c, d]Moreover, a Lyapunov type inequality of the formis obtained for a sequential conformable BVP. Some examples are given and an application to conformable Sturm-Liouville eigenvalue problem is analyzed. MSC: 34A08; 26D15Keywords: Lyapunov inequality; conformable derivative; Green's function; boundary value problem; Sturm-Liouville eigenvalue problem BackgroundIn [], it was proved that if the boundary value problem (BVP)has a nontrivial solution, where r is a real-valued continuous function, thenUndoubtedly, the Lyapunov inequality () proves cooperative and supportive in differential equations. Indeed, it is frequently used to dominate certain quantities for the sake of

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