Abstract
Existence and controllability results for nonlinear Hilfer fractional differential equations are studied. Sufficient conditions for existence and approximate controllability for Sobolev-type impulsive fractional differential equations are established, where the time fractional derivative is the Hilfer derivative. An example for Sobolev-type Hilfer fractional delay partial differential equation with impulsive condition is considered.
Highlights
1 Introduction Nonlinear fractional differential equations can be observed in many areas such as population dynamics, heat conduction in materials with memory, seepage flow in porous media, autonomous mobile robots, fluid dynamics, traffic models, electro magnetic, aeronautics, economics, and so on [1,2,3,4,5,6,7,8,9,10]
Controllability results for linear and nonlinear integer order differential systems were studied by several authors
4 Existence and approximate controllability First, we study existence and uniqueness for Sobolev-type neutral Hilfer fractional differential equation with impulsive condition in the following form:
Summary
Nonlinear fractional differential equations can be observed in many areas such as population dynamics, heat conduction in materials with memory, seepage flow in porous media, autonomous mobile robots, fluid dynamics, traffic models, electro magnetic, aeronautics, economics, and so on [1,2,3,4,5,6,7,8,9,10]. State x(·) takes values in the Banach space E, and h : J × J → R is a continuous function, x|t=tk = Ik(x(tk–)), where x(tk+) and x(tk–) represent the right and left limits of x(t) at t = tk, respectively, and the nonlinear operators f : J × E × E → E, g : J × E → E are given. We will establish a set of sufficient conditions for controllability of impulsive delay Hilfer fractional differential equation in the following form: Bu(t) tk , f System (3.4) is controllable on J. where the delay γi(t) : J → J, i = 1, 2, 3, are continuous functions, the state x(·) takes values in the Banach space E, the symbols A and Z are linear operators on E. (2) If 0 < β < γ ≤ 1, D((AZ–1)γ ) → D((AZ–1)β ) and the embedding is compact whenever the resolvent operator of (AZ–1) is compact
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