Abstract
In this paper, we study the existence and multiplicity results of solutions for some class of fractional differential inclusions with boundary conditions. Some existence and multiplicity results of solutions are given by using the least action principle and minmax methods in nonsmooth critical point theory. Recent results in the literature are generalized and improved. Some examples are given in the paper to illustrate our main results.
Highlights
In this paper, we consider the fractional boundary value problem (BVP for short) for the following differential inclusion: ⎧ ⎨– d dt ( D–t β (u (t)) + t D–Tβ (t))) ∈∂ F (t, u(t)), a.e. t ∈ [, T],⎩u( ) = u(T) =
To the best of the authors’ knowledge, there are few results on the solutions to fractional BVP which were established by the nonsmooth critical point theory, since it is often very difficult to establish a suitable space and variational functional for fractional differential equations with boundary conditions
In Section, using variational methods, we prove two existence theorems for the solutions of problem ( . ) which generalize the results in [ ]
Summary
Where T > , D–t β and tD–Tβ are the left and right Riemann-Liouville fractional integrals of order ≤ β < respectively, F : [ , T] × RN → R satisfies the following assumptions: (A) F(t, x) is measurable in t for every x ∈ RN and locally Lipschitz in x for a.e. t ∈ [ , T], F(t, ) ∈ L ( , T) and there exist f , g ∈ L∞( , T; R+) and ν ∈ [ , ∞) such that ζ ∈ ∂F(t, x) ⇒ |ζ | ≤ f (t)|x|ν + g(t) for a.e. t ∈ [ , T] and all x ∈ RN . To the best of the authors’ knowledge, there are few results on the solutions to fractional BVP which were established by the nonsmooth critical point theory, since it is often very difficult to establish a suitable space and variational functional for fractional differential equations with boundary conditions.
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