Abstract
In this paper, a fixed point theorem of nonexpansive mapping is established to study the existence and sufficient conditions for the controllability of nonlinear fractional control systems in reflexive Banach spaces. The result so obtained have been modified and developed in arbitrary space having Opial’s condition by using fixed point theorem deals with nonexpansive mapping defined on a set has normal structure. An application is provided to show the effectiveness of the obtained result.
Highlights
Many systems in physics, chemistry, biology, stochastic, and control theory are represented by fractional control systems (FCS)
Since every (HS) are reflexive Banach space (RBS) and the contraction mapping is nonexpansive mapping, but the converse in general is not true (7), the purpose of this paper is to study the controllability of (FCS) in arbitrary (RBS) by using fixed point theorem that deals with nonexpansive mapping
Theorem 2 (8, 11): Let Y be nonexpansive from Winto W, where W is a nonempty weakly compact convex subset having a normal structure in a Banach space X, Y has fixed point in W
Summary
Chemistry, biology, stochastic, and control theory are represented by fractional control systems (FCS). Theorem 2 (8, 11): Let Y be nonexpansive from Winto W, where W is a nonempty weakly compact convex subset having a normal structure in a Banach space X, Y has fixed point in W. Since w(r) is closed, convex, bounded subset of (RBS) Qn and w(r) is weakly compact having normal structure (see example of Definition 3 and example 1), by theorem 2, there exists a fixed point z ∈ w(r) such that (Yz)(t) = z(t) and this fixed point is a solution of Eq.[4] on I, which satisfied z(h) = z1, the nonlinear control system Eq.[4] is controllable on I. Remark 3 : Let Qn be only a Banach space with 0 ≤ λ < 1 in condition [a3], by using the same manner of theorem 4, the operator Y being a contraction mapping.
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