A fixed point theorem for L-contractions

Our goal is to introduce a new fixed point theorem for operators acting on the space 𝐶([0, 𝑇];𝑋). This result can be considered as a generalization of the celebrated Banach Contraction Principle. Let be a Banach space, > 0 and consider the space 𝐶([0, 𝑇];𝑋) of continuous 𝑋-valued functions from the segment = [0, 𝑇] to equipped with the uniform norm: ‖𝑢‖ = max 𝑡∈[0,𝑇 ]‖𝑢(𝑡)‖𝑋. Let be a closed subset of 𝐶([0, 𝑇];𝑋). Consider a continuous non-linear operator : → that maps to itself. We say that the operator is 𝐿-contraction on if for any 𝑢, ∈ it satisfies the so called 𝐿-condition: ‖𝑁(𝑢)(𝑡) - 𝑁(𝑣)(𝑡)‖𝑋 ≤ 𝐿(‖𝑢(𝑡) - 𝑣(𝑡)‖𝑋), where 𝐿: 𝐶[0, 𝑇] → 𝐶[0, 𝑇] is a linear positive monotone operator acting on the space 𝐶([0, 𝑇];R) of the real-valued continuous functions and having the spectral radius (𝐿)