#### Existence and multiplicity of positive solutions for a system of fractional boundary value problems

Boundary Value Problems
Existence and multiplicity of positive solutions for a system of fractional boundary value problems
Johnny Henderson 1
Rodica Luca 0
0 Department of Mathematics, Gh. Asachi Technical University , Iasi, 700506 , Romania
1 Department of Mathematics, Baylor University , Waco, TX 76798-7328 , USA
We study the existence and multiplicity of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations, subject to integral boundary conditions. The nonsingular and singular cases for the nonlinearities are investigated. MSC: 34A08; 45G15 We consider the system of nonlinear ordinary fractional differential equations
Riemann-Liouville fractional differential equation; integral boundary conditions; positive solutions
1 Introduction
Dα+u(t) + f (t, v(t)) = , t ∈ (, ), n – < α ≤ n,
β
D+v(t) + g(t, u(t)) = , t ∈ (, ), m – < β ≤ m,
with the integral boundary conditions
u() = u () = · · · = u(n–)() = ,
v() = v () = · · · = v(m–)() = ,
u() = u(s) dH(s),
v() = v(s) dK (s),
where n, m ∈ N, n, m ≥ , Dα+ and Dβ+ denote the Riemann-Liouville derivatives of orders
α and β, respectively, and the integrals from (BC) are Riemann-Stieltjes integrals.
Under sufficient conditions on functions f and g, which can be nonsingular or singular
in the points t = and/or t = , we study the existence and multiplicity of positive solutions
of problem (S)-(BC). We use the Guo-Krasnosel’skii fixed point theorem (see []) and some
theorems from the fixed point index theory (from [] and []). By a positive solution of
problem (S)-(BC) we mean a pair of functions (u, v) ∈ C([, ]) × C([, ]) satisfying (S)
and (BC) with u(t) ≥ , v(t) ≥ for all t ∈ [, ] and supt∈[,] u(t) > , supt∈[,] v(t) > .
The system (S) with α = n, β = m and the boundary conditions (BC) where H and K are
scale functions (that is, multi-point boundary conditions) has been investigated in [] (the
nonsingular case) and [] (the singular case). In [], the authors give sufficient conditions
for λ, μ, f , and g such that the system
Dα+u(t) + λf (t, u(t), v(t)) = , t ∈ (, ), n – < α ≤ n,
β
D+v(t) + μg(t, u(t), v(t)) = , t ∈ (, ), m – < β ≤ m,
with the boundary conditions (BC) with H and K scale functions, has positive solutions
(u(t) ≥ , v(t) ≥ for all t ∈ [, ], and (u, v) = (, )).
Fractional differential equations describe many phenomena in various fields of
engineering and scientific disciplines such as physics, biophysics, chemistry, biology, economics,
control theory, signal and image processing, aerodynamics, viscoelasticity,
electromagnetics, and so on (see [–]).
In Section , we present the necessary definitions and properties from the fractional
calculus theory and some auxiliary results dealing with a nonlocal boundary value problem
for fractional differential equations. In Section , we give some existence and multiplicity
results for positive solutions with respect to a cone for our problem (S)-(BC), where f and
g are nonsingular functions. The case when f and g are singular at t = and/or t = is
studied in Section . Finally, in Section , we present two examples which illustrate our
main results.
2 Preliminaries and auxiliary results
We present here the definitions, some lemmas from the theory of fractional calculus and
some auxiliary results that will be used to prove our main theorems.
provided the right-hand side is pointwise defined on (, ∞), where (α) is the Euler
gamma function defined by (α) = ∞ tα–e–t dt, α > .
The notation α stands for the largest integer not greater than α. We also denote the
Riemann-Liouville fractional derivative of f by Dα+f (t). If α = m ∈ N then Dm+f (t) = f (m)(t)
for t > , and if α = then D+f (t) = f (t) for t > .
Lemma . ([]) Let α > and n = α + for α ∈/ N and n = α for α ∈ N; that is, n is the
smallest integer greater than or equal to α. Then the solutions of the fractional differential
equation Dα+u(t) = , < t < , are
u(t) = ctα– + ctα– + · · · + cntα–n, < t < ,
where c, c, . . . , cn are arbitrary real constants.
Lemma . ([, ]) Let α > , n be the smallest integer greater than or equal to α (n – <
α ≤ n) and y ∈ L(, ). The solutions of the fractional equation Dα+u(t) + y(t) = , < t < ,
are
where c, c, . . . , cn are arbitrary real constants.
We consider now the fractional differential equation
< t < , n – < α ≤ n,
with the integral boundary conditions
u() = u () = · · · = u(n–)() = ,
u() =
Lemma . If H : [, ] → R is a function of bounded variation, = – sα– dH(s) =
and y ∈ C([, ]), then the solution of problem ()-() is given by
Proof By Lemma ., the solutions of equation () are
≤ t ≤ .
So, we obtain
c =
Therefore, we get the expression () for the solution of problem ()-().
value problem ()-() is given by
G(t, s) = g(t, s) +
(t, s) ∈ [, ] × [, ],
g(t, s) =
≤ s ≤ t ≤ ,
≤ t ≤ s ≤ .
Proof By Lemma . and relation (), we conclude
u(t) =
G(t, s)y(s) ds,
where g and G are given in () and (), respectively. Hence u(t) = G(t, s)y(s) ds for all
t ∈ [, ].
for all s ∈ [, ],
Proof By using the assumptions of this lemma, we have G(t, s) ≥ for all (t, s) ∈ [, ] ×
[, ], and so u(t) ≥ for all t ∈ [, ].
Lemma . Assume that H : [, ] → R is a nondecreasing function and
Green’s function G of the problem ()-() satisfies the inequalities:
(a) G(t, s) ≤ J(s), ∀(t, s) ∈ [, ] × [, ], where
s ∈ [, ].
(b) For every c ∈ (, /), we have
G(t, s) ≥ γJ(s) ≥ γG t , s ,
∀t , s ∈ [, ].
> . Then the
G(t, s) ≥ cα–g θ(s), s +
g(τ , s) dH(τ ) = γJ(s) ≥ γG t , s .
Therefore, we obtain the inequalities (b) of this lemma.
Lemma . Assume that H : [, ] → R is a nondecreasing function and > , c ∈
(, /), and y ∈ C([, ]), y(t) ≥ for all t ∈ [, ]. Then the solution u(t), t ∈ [, ] of
problem ()-() satisfies the inequality mint∈[c,–c] u(t) ≥ γ maxt ∈[,] u(t ).
u(t) =
G t , s y(s) ds = γu t .
Then we deduce the conclusion of this lemma.
< t < , m – < β ≤ m,
with the integral boundary conditions
v() = v () = · · · = v(m–)() = ,
v() =
3 The nonsingular case
We consider the Banach space X = C([, ]) with supremum norm
cone P ⊂ X by P = {u ∈ X, u(t) ≥ , ∀t ∈ [, ]}.
We also define the operators A : P → X by
· and define the
(Au)(t) =
G(t, s)f s,
G(s, τ )g τ , u(τ ) dτ ds, t ∈ [, ],
and B : P → X, C : P → X by
(Bu)(t) =
G(t, s)u(s) ds,
(Cu)(t) =
G(t, s)u(s) ds, t ∈ [, ].
Under the assumptions (H) and (H), using also Lemma ., it is easy to see that A,
B, and C are completely continuous from P to P. Thus the existence and multiplicity of
positive solutions of the system (S)-(BC) are equivalent to the existence and multiplicity
of fixed points of the operator A.
(i) f ∞i = lim inf inf f (t, u)
u→∞ t∈[c,–c] up
∈ (, ∞];
(ii) g∞i = lim inf inf g(t, u)
u→∞ t∈[c,–c] u/p = ∞.
(i) fs = lim sup sup f (t, u) ∈ [, ∞);
u→+ t∈[,] uq
(ii) gs = lim sup sup g(t, u)
u→+ t∈[,] u/q = ,
Proof Because the proof of the theorem is similar to that of Theorem . from [], we will
sketch some parts of it. From assumption (i) of (H), we deduce that there exist C, C >
such that
f (t, u) ≥ Cup – C,
∀(t, u) ∈ [c, – c] × [, ∞).
(Au)(t) ≥ C
where C = C c–c J(s) ds.
∀t ∈ [, ], ()
For c given in (H), we define the cone P = {u ∈ P, inft∈[c,–c] u(t) ≥ γ u }, where γ =
min{γ, γ}. From our assumptions and Lemma ., for any y ∈ P, we can easily show that
u = By ∈ P and v = Cy ∈ P, that is, B(P) ⊂ P and C(P) ⊂ P.
We now consider the function u(t) = G(t, s) ds = (By)(t) ≥ , t ∈ [, ], with y(t) =
for all t ∈ [, ]. We define the set
u(t) = (Au)(t) + λu(t) ≥ (Au)(t)
–c –c
where C = C + CCmmγγp > .
Hence, inft∈[c,–c] u(t) ≥ inft∈[c,–c] u(t) – C, and so
u(t) ≤ C,
∀u ∈ M.
Now from relations () and (), one obtains u ≤ (inft∈[c,–c] u(t))/γ ≤ C/γ , for all
u ∈ M, that is, M is a bounded subset of X.
u ≤ γ t∈[icn,f–c] u(t),
∀u ∈ M.
g(t, u) p ≥ εpu – C,
∀(t, u) ∈ [c, – c] × [, ∞),
i(A, BL ∩ P, P) = .
f (t, u) ≤ Muq,
∀(t, u) ∈ [, ] × [, ];
where ε = min{/M, (/(MMMq))/q} > , M = J(s) ds > , M = J(s) ds > .
Hence, for any u ∈ Bδ ∩ P and t ∈ [, ], we obtain
Besides, there exists a sufficiently large L > such that
J(s) u(s) /q ds ≤ εM u /q ≤ .
(Au)(t) ≤ M
J(s) ds = MεqMMq u ≤ u .
This implies that Au ≤ u / for all u ∈ ∂Bδ ∩ P. From [], we conclude that the fixed
point index of the operator A over Bδ ∩ P with respect to P is
Combining () and (), we obtain
Using similar arguments as those used in the proofs of Theorem . and Theorem .
in [], we also obtain the following results for our problem (S)-(BC).
(ii) g∞s = lim sup sup g(t, u)
u→∞ t∈[,] u/r = .
(ii) gi = lim inf inf
u→+ t∈[c,–c]
∈ (, ∞];
= ∞,
∀t ∈ [, ],
where m = max{K, K}, K = maxs∈[,] J(s), K = maxs∈[,] J(s), and J, J are
defined in Section , then the problem (S)-(BC) has at least two positive solutions
(u(t), v(t)), (u(t), v(t)), t ∈ [, ].
4 The singular case
f (t, x) ≤ p(t)q(x),
g(t, x) ≤ p(t)q(x),
∀t ∈ (, ), x ∈ R+.
(Au)(t) =
G(t, s)f s,
(Au)(t) =
G(t, s)f s,
g(t, s) +
Proof We denote by α = J(s)p(s) ds and β = J(s)p(s) ds. Using (H), we deduce that
< α < ∞ and < β < ∞. By Lemma . and the corresponding lemma for G, we see that
A maps P into P.
We shall prove that A maps bounded sets into relatively compact sets. Suppose D ⊂ P
is an arbitrary bounded set. Then there exists M > such that u ≤ M for all u ∈ D.
By using (H) and Lemma ., we obtain Au ≤ αM for all u ∈ D, where M =
supx∈[,βM] q(x), and M = supx∈[,M] q(x). In what follows, we shall prove that A(D) is
equicontinuous. By using Lemma ., we have
∀t ∈ [, ].
(Au) (t) =
So, for any t ∈ (, ), we deduce
(Au) (t) ≤
h(t) =
≤ M
g(τ , s) dH(τ ) p(s) ds, t ∈ (, ).
h(t) dt =
h(t) dt +
( – s)α–p(s) + H() – H()
(Au)(t) – (Au)(t) =
(Au) (t) dt ≤ M
Therefore A is a compact operator. Besides, we can easily show that A is continuous on P.
Hence A : P → P is completely continuous.
(i) q∞ = lim sup q(x)
s x→∞ xα ∈ [, ∞);
(ii) qs∞ = lixm→s∞up qxα(x) = .
(i) fi = lixm→in+f t∈[icn,f–c] f (xtβ,x) ∈ (, ∞];
(ii) gi = lixm→in+f t∈[icn,f–c] g(xtβ,x) = ∞,
Proof Because the proof of this theorem is similar to that of Theorem in [], we will
sketch some parts of it. For c given in (H), we consider the cone P = {u ∈ X, u(t) ≥
, ∀t ∈ [, ], mint∈[c,–c] u(t) ≥ γ u }, where γ = min{γ, γ}. Under assumptions
(H)(H), we obtain A(P) ⊂ P. By (H), we deduce that there exist C, C, C > and ε ∈
(, (α Cαβα )–/α ) such that
∀x ∈ [, ∞).
(Au)(t) ≤
G(t, s)p(s)q
≤ C
G(t, s)p(s)
≤ C
≤ C
J(s)p(s) ds
J(s)p(s)
J(s)p(s) ds
≤ Cα εα αβα u αα + Cα αβα Cα + αC,
∀t ∈ [, ].
Au ≤ u ,
∀u ∈ ∂BR ∩ P.
∀(t, x) ∈ [c, – c] × [, x],
J(τ )p(τ )q u(τ ) dτ ≤ x.
(Au)(t) ≥ C
≥ C
By (), (), and the Guo-Krasnosel’skii fixed point theorem, we deduce that A has at
least one fixed point u ∈ (BR \ Bε ) ∩ P. Then our problem (S)-(BC) has at least one
positive solution (u, v) ∈ P × P where v(t) = G(t, s)g(s, u(s)) ds. The proof of
Theorem . is completed.
Using similar arguments as those used in the proof of Theorem in [] (see also [] for
a particular case of the problem studied in []), we also obtain the following result for our
problem (S)-(BC).
(i) qs = lim sup q(x)
x→+ xr ∈ [, ∞);
(ii) g∞i = lim inf inf g(t, x)
x→∞ t∈[c,–c] xl
= ∞,
and K (t) = t for all t ∈ [, ]. Then
sv(s) ds.
We consider the system of fractional differential equations
u(s) dH(s) = u( ) + u( ) and
v(s) dK (s) =
with the boundary conditions
D/+u(t) + f (t, v(t)) = ,
D/+v(t) + g(t, u(t)) = ,
u() = u () = , u() = u( ) + u( ),
v() = , v() = sv(s) ds.
Then we obtain = – sα– dH(s) = – ( )/ – ( )/ = (–√) ≈ . > ,
– sβ– dK (s) = – s/ ds = = . > . We also deduce
=
θ(s) = –s+s and θ(s) = s for all s ∈ [, ].
For the functions J and J, we obtain
J(s) = g θ(s), s +
= g
⎪⎪⎧ √π { (s–(–s+s)s/)/ + (–√) [(( – s)/ – ( – s)/)
= ⎨⎪⎪⎪⎪⎪ √π+{((s–√(–s+s()s/)–/ s+)/(––√() –[(s)–/s))]}/,
⎪⎪⎪⎪⎪⎪⎪⎩ √π+[((s–√(–s+s)s(/)–/ s+)/(+–√((–)√(––)s)s/)]/,)]},
≤ s < ,
J(s) = g θ(s), s +
= g(s, s) +
Example We consider the functions
f (t, u) = a uα + uβ ,
g(t, u) = b uγ + uδ , t ∈ [, ], u ≥ ,
f ∞i = ul→im∞ a(uαu+/uβ ) = ∞,
fi = ul→im+ a(uαu+ uβ ) = ∞,
g∞i = ul→im∞ b(uγu+ uδ ) = ∞,
gi = ul→im+ b(uγu+ uδ ) = ∞.
Example We consider the functions
We have < p(s) ds < ∞, < p(s) ds < ∞.
In (H), for r < a, r < b and rr ≥ , we obtain
lixm→su+p qx(rx) = ,
lixm→su+p qx(rx) = .
lixm→i∞nf t∈[icn,f–c] f (xtl,x) = ∞,
lixm→i∞nf t∈[icn,f–c] g(xtl,x) = ∞.
q(x) = xa,
q(x) = xb.
For example, if a = /, b = , r = , r = /, l = , l = /, the above conditions are
satisfied. Then, by Theorem ., we deduce that the problem (S)-(BC) has at least one
positive solution.
Acknowledgements
The work of R Luca was supported by the CNCS grant PN-II-ID-PCE-2011-3-0557, Romania.
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Cite this article as: Henderson and Luca: Existence and multiplicity of positive solutions for a system of fractional boundary value problems . Boundary Value Problems 2014 , 2014 : 60