A New Unification of Continuous, Discrete, and Impulsive Calculus through Stieltjes Derivatives

Stieltjes Bochner spaces and applications to the study of parabolic equations

“…In order to achieve this, and for convenience of the reader, we start by reviewing some concepts related to Bochner spaces [5][6][7][8]. Let us consider the measure space (R, M g , µ g ) induced by g [1] and V a real Banach space.…”

“…The study of this type of derivatives and its application to the field of ODEs appears in [1][2][3][4]. We use the notation established in previous works.…”

“…We further assume that k 1 > 0 is a positive constant, k 2 ≥ 0, u 0 ∈ L 2 (Ω) and f ∈ L 2 g ([0, T ], L 2 (Ω)), with ([0, T ], M g , µ g ) a suitable measure space associated to g [1]. It is important to mention that if u : A ⊂ R → H is g-continuous for every t 0 ∈ A in the sense of:…”

“…then f is constant in the same intervals as g [2, Proposition 3.2]. Moreover, continuity in the previous sense does not imply continuity in the classical sense, but if g is continuous at t 0 ∈ [0, T ], then so is f [1]. Taking into account that g es left-continuous, we observe that the spaces of bounded g-continuous functions BC g ([0, T ], L 2 (Ω)) and BC g ([0, T ), L 2 (Ω)) are basically the same since any function in BC g ([0, T ], L 2 (Ω)) must be continuous at T .…”

“…We will also prove new Lebesgue-differentiationtype results for the Stieltjes derivatives and some continuous injections. In Section 3 we will define the concept of solution of problem (1). In Section 4 we will prove an existence result for system (1) that generalizes some aspects of the classical theory of parabolic partial differential equations.…”