Abstract
In this paper, we investigate the impulsive fractional q-difference equation with antiperiodic conditions. The existence and uniqueness results of solutions are established via the theorem of nonlinear alternative of Leray-Schauder type and the Banach contraction mapping principle. Two examples are given to illustrate our results.
Highlights
We investigate the impulsive fractional q-difference equation with antiperiodic conditions
We are concerned with the existence and uniqueness of solutions for the following impulsive fractional q-difference equation with antiperiodic boundary conditions cDαq u (t) = f (t, u (t), Tu (t), Su (t)), t ∈ J = J \ {t1, t2, . . . , tm}, Δu|t=tk = Ik (u (t−k )), ΔDqut=tk = Ik∗ (u (t−k )), (1)
Zhang and Wang [24] have applied cone contraction fixed point theorem to establish the existence of solutions to nonlinear fractional differential equation with impulses and antiperiodic boundary conditions cDαu (t) = f (t, u (t)), 1 < α ≤ 2, t ∈ J \ {t1, t2, . . . , tm}, J = [0, T], Δu|t=tk = Ik (u), Δut=tk = Ik∗ (u), k = 1, 2, . . . , p, u (0) = −u (T), u (0) = −u (T), (3)
Summary
We are concerned with the existence and uniqueness of solutions for the following impulsive fractional q-difference equation with antiperiodic boundary conditions cDαq u (t) = f (t, u (t) , Tu (t) , Su (t)) , t ∈ J = J \ {t1, t2, . Zhang and Wang [24] have applied cone contraction fixed point theorem to establish the existence of solutions to nonlinear fractional differential equation with impulses and antiperiodic boundary conditions cDαu (t) = f (t, u (t)) ,. Ahmad et al [28] studied existence of solutions for the following antiperiodic boundary value problem (BVP for short) of impulsive fractional q-difference equation. In this paper we are concerned with the existence and uniqueness of solutions for impulsive fractional q-difference equation antiperiodic BVP. Some ideas of this paper are from [29, 30]
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