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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

In this paper, we state and prove a new discrete q-fractional version of the Gronwall inequality. Based on this result, a particular version expressed by means of the q-Mittag-Leffler function is provided. To apply the proposed results, we prove the uniqueness and obtain an estimate for the solutions of nonlinear delay Caputo q-fractional difference system. We examine our results by providing a numerical example. MSC: 26A33; 39A11Keywords: generalized q-fractional Gronwall inequality; delay q-fractional difference system; uniqueness of solution; estimate for the solution BackgroundThe study of q-difference equations has gained intensive interest in the last years. It has been shown that these types of equations have numerous applications in diverse fields and thus have evolved into multidisciplinary subjects [-]. For more details on q-calculus, we refer the reader to the remarkable monograph []. On the other hand, the fractional differential equations have recently received considerable attention in the last two decades. Indeed, many researchers have investigated these types of equations due to their significant applications in various fields of science and engineering; see for instance the monographs [-] and the references therein.The corresponding theory of fractional difference equations is considered to be at its first stages of progress; we suggest [-] whose authors have taken the lead to promote and develop this theory. The q-fractional calculus and differential equations have been recently studied in many papers; we recommend the monograph [] and the papers cited therein. For the q-fractional difference equations which serve as a bridge between fractional difference equations and q-difference equations there have appeared some papers which study the qualitative properties of solutions [, -]. However, less attention has been paid to these types of equations in the literature.The differential and integral inequalities, which are considered as an effective tools for studying solutions properties, have also been under consideration. Due to its benefit in the determination of uniqueness, boundedness and stability of solutions, in particular, the

A newly proposed p-Laplacian nonperiodic boundary value problem is studied in this research paper in the form of generalized Caputo fractional derivatives. The existence and uniqueness of solutions are fully investigated for this problem using some fixed point theorems such as Banach and Schauder. This work is supported with an example to apply all obtained new results and validate their applicability.

By employing the contraction mapping principle and applying Gronwall-Bellman's inequality, sufficient conditions are established to prove the existence and exponential stability of positive almost periodic solution for nonlinear impulsive delay model of hematopoiesis.