Recent Fixed Point Techniques in Fractional Set-Valued Dynamical Systems
Dynamical Systems - Analytical and Computational Techniques
In this chapter, we present a recollection of fixed point theorems and their applications in fractional set-valued dynamical systems. In particular, the fractional systems are used in describing many natural phenomena and also vastly used in engineering. We consider mainly two conditions in approaching the problem. The first condition is about the cyclicity of the involved operator and this one takes place in ordinary metric spaces. In the latter case, we develop a new fundamental theorem in
... ental theorem in modular metric spaces and apply to show solvability of fractional set-valued dynamical systems. & In this case, we assume that f : ½0, T · R ! R is continuous and u ∈ C 1 ð½0, TÞ. From simple calculus, we may see that this system is equivalent to the following integral equation: This is where Banach got the idea to solve the problem. He proposed his famous fixed point theorem known today as the contraction principle in 1922 , mainly to solve this Cauchy problem effectively. Recall that the contraction principle states that if X is a complete metric space and T : X ! X is Lipschitz continuous with constant 0 < L < 1, then T has a unique fixed point. Let us consider a map Λ : C 1 ð½0, TÞ ! C 1 ð½0, TÞ given by ½0, t f ðs, uðsÞÞds, ∀u ∈ C 1 ð½0, TÞ, ∀t ∈ ½0, T One can notice that u ∈ C 1 ð½0, TÞ solves Eq. (1) if and only if it is a fixed point of Λ. With this approach, by considering C 1 ð½0, TÞ with the supremum norm ∥ Á ∥ ∞ , we end up with the local solvability of the Cauchy problem. To obtain the global solution, we have to apply some techniques to extend the boundary of the local solution. It is not very obvious that renorming by the L-weighted norm ∥f ∥ L :¼ sup t ∈ ½0, T e −Lt f ðtÞ, with L > 0, will resolve such difficulty. We shall give the short solvability result of the Cauchy problem with the contraction principle here, to illustrate the concept of how we apply fixed point theorem to continuous dynamical systems. Under the assumption that f must be Lipschitz in the second variable with constant L > 0, we have for any x, y ∈ C 1 ð½0, TÞ the following: ½0, t f ðs, xðsÞÞ−f ðs, yðsÞÞdsj ≤ e −Lt ð ½0, t jf ðs, xðsÞÞ−f ðs, yðsÞÞjds ≤ e −Lt ð ½0, t Le Ls e −Ls jxðsÞ−yðsÞjds ≤ e −Lt ∥x−y∥ L ð ½0, t Le Ls e −Ls ds ≤ e −Lt ðe Lt −1Þ∥x−y∥ L ≤ ð1−e −LT Þ∥x−y∥ L : Taking supremum over t ∈ ½0, T yields the result and the solvability thus follows.