http://dx.doi.org/10.7494/OpMath.2014.34.4.691
This paper is dedicated with great esteem and admiration to Professor Leon Mikołajczyk.
REMARKS FOR ONE-DIMENSIONAL
FRACTIONAL EQUATIONS
Massimiliano Ferrara and Giovanni Molica Bisci
Communicated by Marek Galewski
Abstract.In this paper we study a class of one-dimensional Dirichlet boundary value prob-lems involving the Caputo fractional derivatives. The existence of infinitely many solutions for this equations is obtained by exploiting a recent abstract result. Concrete examples of applications are presented.
Keywords: fractional differential equations, Caputo fractional derivatives, variational methods.
Mathematics Subject Classification:34A08, 26A33, 35A15.
1. INTRODUCTION
The aim of this short note is to study nonlinear fractional boundary value problems whose general form is given by
(
∆F,αu(t) +f(t, u(t)) = 0 a.e.t∈[0, T],
u(0) =u(T) = 0,
where
∆F,αu(t) := d
dt
0Dαt−1(c0Dtαu(t))−tDTα−1(ctDαTu(t))
,
α ∈ (1/2,1], 0Dtα−1 and tDTα−1 are the left and right Riemann-Liouville fractional integrals of order 1−α respectively, c
0Dαt and ctDTα are the left and right Caputo fractional derivatives of orderαrespectively, andf : [0, T]×R→Ris a continuous
function.
c
Following [12], denote by C∞
0 ([0, T],R)the set of all functions g∈C∞([0, T],R)
with g(0) =g(T) = 0. The fractional derivative Hilbert spaceEα
0 is defined by the
closure ofC∞
0 ([0, T],R)with respect to the norm
kuk:=
T Z
0
|c0Dtαu(t)|2dt+ T Z
0
|u(t)|2dt
!1/2
for everyu∈Eα
0.
For basic facts and usual notation on the variational setting adopted here we refer the reader to [12, 18]. Let us denote
κ:= T
α−1 2
Γ(α)√2α−1,
C(T, α) :=
T /4
Z
0
t2−2αdt+ 3ZT /4
T /4
t1−α−t−T41−α
2 dt
+
T Z
3T /4
t1−α−t−T41−α−t−34T1−α
2 dt,
and
B0:= lim sup
ξ→0+
3ZT /4
T /4
F(t, ξ)dt
ξ2 ,
whereF is the potential off defined by
F(t, ξ) :=
ξ Z
0
f(t, x)dx, (t, ξ)∈[0, T]×R.
With the above notation, in [2, Theorem 3.2], exploiting a quoted critical point the-orem established by Ricceri in [17], the following result has been shown.
Theorem 1.1. Let f : [0, T]×R→Rbe a continuous function such that
(f1) F(t, ξ)≥0for every (t, ξ)∈([0,T4]∪[34T, T])×R.
Assume that there exist two real sequences {cn} and {dn} in [0,+∞), with
limn→∞dn= 0, satisfying the conditions:
(h3) for somen0∈None has cn<
T|cos(πα)|Γ(2−α) 4κpC(T, α) dn
(h4) A0:= lim
n→∞ϕ(cn, dn, α, T)<
B0
16κ2C(T, α), where
ϕ(cn, dn, α, T) := RT
0 max|ξ|≤dnF(t, ξ)dt−
R3T /4
T /4 F(t, cn)dt T2|cos(πα)|2Γ2(2−α)d2
n−16κ2c2nC(T, α)
.
Then, for each
λ∈
16C(T, α)
T2Γ2(2−α)|cos(πα)|B0,
1
κ2T2Γ2(2−α)|cos(πα)|A 0
,
problem (
∆F,αu(t) +λf(t, u(t)) = 0 a.e.t∈[0, T],
u(0) =u(T) = 0,
admits a sequence of non-zero solutions which strongly converges to zero inEα
0.
The aim of this paper is to prove the following remarkable consequence of Theo-rem 1.1.
Theorem 1.2. Let f :R→Rbe a continuous function such that f|(−∞,0] ≡0 and infξ≥0F(ξ) = 0. Further, leth∈C0([0, T]) with
(a0) min
t∈[0,T]h(t)>0.
Suppose that there exist two sequences {cn} and {dn} in (0,+∞), withcn < dn for
every n≥ν, and lim
n→∞dn = 0, such that:
(a1) lim
n→∞ dn
cn
= +∞, (a2) max
x∈[cn,dn]
f(x)≤0 for everyn≥ν, (a3) 16C(T, α)
T2Γ2(2−α)|cos(πα)|
3T
4
Z
T
4
h(t)dt
<lim sup
ξ→0+
F(ξ)
ξ2 <+∞.
Then, the following problem
(
∆F,αu(t) +h(t)f(u(t)) = 0 a.e.t∈[0, T],
u(0) =u(T) = 0
admits a sequence of non-zero solutions which strongly converges to zero inEα
0.
For several results on fractional differential equations, one can see, for example, the monographs of Miller and Ross [15], Samkoet al.[18], Podlubny [16], Hilfer [11], Kilbaset al.[13] and the papers [1, 3–7]
2. PROOF OF THE MAIN RESULT
Our aim is to apply Theorem 1.1. First of all observe that, by (a0), condition (f1)
holds. Further, if {cn} and{dn} are two real sequences satisfying our assumptions, we have that there existsn0≥ν such that
c2
n
d2
n
< T|cos(πα)|Γ(2−α) 4κpC(T, α) ,
for everyn≥n0. Hence the hypothesis(h3)in Theorem 1.1 is verified. We will prove
that
A0:= lim
n→∞
khkL1([0,T]) max
|ξ|≤dn F(ξ)−
3T 4 Z T 4
h(t)dt
F(cn)dt
T2|cos(πα)|2Γ2(2−α)d2n−16κ2c2nC(T, α)
= 0.
Set
hn:=khkL1([0,T])
max |ξ|≤dn
F(ξ)
c2 n − 3T 4 Z T 4
h(t)dt
F(cc2n)
n
for everyn≥n0 and observe that hypothesis(a2)yields
max |ξ|≤dn
F(ξ) = max |ξ|≤cn
F(ξ). (2.1)
Thus, since 3T 4 Z T 4
h(t)dt
khkL1([0,T]) ≤
1 and F(cn)≥0,
by (2.1), we can write
max |ξ|≤dn
F(ξ)
c2
n
= max |ξ|≤cn
F(ξ)
c2
n ≥
F(cn)
c2 n ≥ 3T 4 Z T 4
h(t)dt
khkL1([0,T])
F(cn)
c2
n
.
for everyn≥n0.
Since hn≥0 for everyn≥n0, one easily gets
0≤lim sup
n→∞ hn.
Further, by(a3)we have
0<lim sup
ξ→0+
F(ξ)
and consequently (note thatcn ց0+ as n→ ∞) we obtain
0≤lim sup
n→∞ F(cn)
c2
n
<+∞. (2.3)
Now, letξn ∈(0, cn]be a sequence such thatF(ξn) := max
|ξ|≤cn
F(ξ)for everyn≥n0.
Thus
lim sup
n→∞ max |ξ|≤dn
F(ξ)
c2
n
= lim sup
n→∞ max |ξ|≤cn
F(ξ)
c2
n
= lim sup
n→∞ F(ξn)
c2
n
≤lim sup
n→∞ F(ξn)
ξ2
n
.
The above inequalities and (2.2) yield
0≤lim sup
n→∞ max |ξ|≤dn
F(ξ)
c2
n
≤lim sup
n→∞ F(ξn)
ξ2
n
<+∞.
Hence, there exists a constantβ such that
0≤lim sup
n→∞ hn=β. (2.4)
Then, by(a1)and (2.4), one has
A0= lim sup
n→∞
hn
T2|cos(πα)|2Γ2(2−α)d 2
n
c2
n −
16κ2C(T, α)
= 0.
Concluding, hypothesis(h4)holds. Finally, bearing in mind condition(a3), one has
1∈
16C(T, α)
T2Γ2(2−α)|cos(πα)|B0,+∞
.
Thanks to Theorem 1.1, the thesis is achieved. The next result is a direct consequence of Theorem 1.2.
Proposition 2.1. Let h ∈ C0([0, T]) satisfying condition (a0). Also let {c
n} and
{dn} be two sequences in (0,+∞) such that dn+1 < cn < dn for every n ≥ ν,
limn→∞dn= 0, andlimn→∞cdnn = +∞. Moreover, letϕ∈C
1([0,1])be a nonnegative function such thatϕ(0) =ϕ(1) =ϕ′(0) =ϕ′(1) = 0 and
max
s∈[0,1]ϕ(s)>
16C(T, α)
T2Γ2(2−α)|cos(πα)|
3T
4
Z
T
4
h(t)dt .
Further, let g:R→Rbe the function defined by
g(t) :=
ϕ t−dn+1 cn−dn+1
ift∈Sn≥ν[dn+1, cn],
Then, problem
(
∆F,αu(t) +h(t)y(u(t)) = 0 a.e.t∈[0, T],
u(0) =u(T) = 0,
where
y(u(t)) :=|u(t)|(2g(u(t)) +ug′(u(t))),
admits a sequence of non-zero solutions which strongly converge to zero inEα
0. Proof. Let{cn} and {dn} be two positive sequences satisfying our assumptions. We claim that all the hypotheses of Theorem 1.2 are verified. Indeed, one has
F(ξ) :=
ξ Z
0
y(t)dt=ξ2g(ξ) for all ξ∈R+.
Moreover, direct computations ensure that
max
x∈[cn+1,dn+1]
y(x) = 0
for everyn≥ν, and
lim sup
ξ→0+
F(ξ)
ξ2 = maxs∈[0,1]ϕ(s)>
16C(T, α)
T2Γ2(2−α)|cos(πα)|
3T
4
Z
T
4
h(t)dt .
The assertion follows by Theorem 1.2.
In conclusion we present a concrete example of the application of Proposition 2.1.
Example 2.2. Let h ∈ C0([0, T]) satisfying condition (a0). Take the positive real
sequences
an:= 1
n!n and bn:= 1 n!
for everyn≥2. Now, defineϕ∈C1([0,1])as follows
ϕ(s) :=ζe 1
s(s−1)+ 4 (for alls∈[0,1]),
and set
b
g(t) :=
ϕ t−1/(n+ 1)! 1/(n!n)−1/(n+ 1)!
if t∈A,
0 otherwise,
where
A:= [
n≥2
h 1
(n+ 1)!, 1 (n!n)
i
If
ζ > 16C(T, α)
T2Γ2(2−α)|cos(πα)|
3T
4
Z
T
4
h(t)dt ,
then problem
∆F,αu(t) +h(t)y(u(t)) = 0 a.e.t∈[0, T],
u(0) =u(T) = 0,
where
y(u(t)) :=|u(t)|(2bg(u(t)) +ubg′(u(t))),
admits a sequence of non-zero solutions which strongly converges to zero inEα
0.
We just mention, for completeness, that related variational arguments have been used recently in [8] proving the existence of at least one non-zero solution for one dimensional fractional equations. See also [9, 10] for related topics.
Acknowledgments
The paper is realized with the auspices of the INdAM – GNAMPA Project 2014 titled: Proprietà geometriche ed analitiche per problemi non-locali.
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Massimiliano Ferrara
University of Reggio Calabria
and CRIOS University Bocconi of Milan Via dei Bianchi presso Palazzo Zani 89127 Reggio Calabria, Italy
Giovanni Molica Bisci [email protected]
Dipartimento P.A.U.
Università degli Studi Mediterranea di Reggio Calabria Salita Melissari - Feo di Vito
