On the Differential Inequality u + ku ≥ 0 and Applications to Eigenvalue Problems

• Mohamed Jleli ^[1] ; Bessem Samet ^[1]
  1. [1] King Saud University

     King Saud University

     Arabia Saudí

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 3, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • 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.

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