In this paper we consider the fractional Hamiltonian system given by(0.1)                       −tDα T(0Dα t u(t)) = ∇F(t,u(t)), a.e t ∈ [0,T]                                 u(0) = u(T) = 0.where α ∈ (1/2,1), t ∈ [0,T], u ∈Rn, F : [0,T]×Rn →R is a given function and ∇F(t,u) is the gradient of F at u. The novelty of this paper is that, using a modified version of mountain pass theorem for functional bounded from below we prove the existence of at least three solutions for (0.2). 

In this paper we consider the fractional Hamiltonian system given by(0.1)                       −tDα T(0Dα t u(t)) = ∇F(t,u(t)), a.e t ∈ [0,T]                                 u(0) = u(T) = 0.where α ∈ (1/2,1), t ∈ [0,T], u ∈Rn, F : [0,T]×Rn →R is a given function and ∇F(t,u) is the gradient of F at u. The novelty of this paper is that, using a modified version of mountain pass theorem for functional bounded from below we prove the existence of at least three solutions for (0.2). 

Digital literacy represents a key challenge in closing the gaps in access to and use of technologies among Amazonian Indigenous, Andean, and Afro-Peruvian populations in Peru. This article analyzes the evolution of access to and use of digital technologies within these groups using data from the National Household Survey (ENAHO) between 2019 and 2024. Significant progress is evident, such as the increase in internet use and the expansion of mobile telephony; however, challenges to digital inclusion persist, linked to geographic, socioeconomic, cultural, and linguistic factors. The main initiatives implemented by the Peruvian State to promote more equitable access to information and public services are also reviewed. Finally, the findings reveal that populations whose mother tongue is Amazonian exhibit the lowest levels of digital connectivity, with internet access 35–40% lower than And...