Abstract
In this paper, we apply critical point theory and variational methods to study the multiple solutions of boundary value problems for an impulsive fractional differential equation with p-Laplacian. Some new criteria guaranteeing the existence of multiple solutions are established for the considered problem.
Highlights
Definitions and c aLet C ∞([ , T], R) be the set of all functions x ∈ C∞([ , T], R) with x( ) = x(T) = and the norm x ∞ = max[ ,T] |x(t)|
Introduction and main resultsConsidering the following impulsive fractional differential equations:⎧ ⎪⎪⎨tDαT p(c Dαt u(t)) + |u(t)|p– u(t) = f (t, u(t)), < t < T, t = tj,⎪⎪⎩u(( tD) =αT–u (Tp)(c =D αt,u))(tj) = Ij(u(tj)), j =, . . . , m, ( . )where < α ≤, p(s) = |s|p– s (s = ), p( ) =, p > and tDαT denotes the right Riemann-Liouville fractional derivative of order α; = t < t < · · · < tm+ = T and t DαT– p cDαt u = tDαT– tj+ – tDαT–tj, where t DαT– p c Dαt u tj+ =
With the impulsive effects and p-Laplacian operator taken into consideration, the corresponding variational functional φ will be more complicated
Summary
Let C ∞([ , T], R) be the set of all functions x ∈ C∞([ , T], R) with x( ) = x(T) = and the norm x ∞ = max[ ,T] |x(t)|. Denote the norm of the space Lp([ , T], R) for ≤ p < ∞. The fractional derivative space E α,p( , T) (denoted by Eα,p for short) is defined by the closure of C ∞([ , T], R), with respect to the following norm:. It is well known that the space Eα,p is a reflexive and separable. Q = p(p – )– > are two positive constants. For any u ∈ Eα,p, the imbedding of Eα,p in C([ , T], R) is compact
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