Resumen de On the Differential Inequality u + ku ≥ 0 and Applications to Eigenvalue Problems
Mohamed Jleli, Bessem Samet
In this work, we introduce and study the class of k-convex functions, that is, the class of functions u ∈ C2(I) satisfying the second-order differential inequality u(t) + ku (t) ≥ 0, t ∈ I, where I is an interval of R and k = 0 is a constant. Among many other results, a Fejér-type inequality for k-convex functions is established. Making use of the obtained inequality, a general Lyapunov-type inequality is obtained for the eigenvalue problem − u(t) + ku (t) = λw(t)u(t), a < t < b, where k > 0, λ > 0, and w ∈ C([a, b])is a positive function. Next, different boundary conditions are investigated. To the best of our knowledge, this work is the first one showing a connection between Fejér-type inequalities and Lyapunov-type inequalities.
