Positive solutions to a system of semipositone fractional boundary value problems
Rodica Luca
0
Alexandru Tudorache
1
0
Department of Mathematics, Gh. Asachi Technical University
,
Iasi, 700506
,
Romania
1
Faculty of Computer Engineering and Automatic Control, Gh. Asachi Technical University
,
Iasi, 700050
,
Romania
We study the existence of positive solutions for a system of nonlinear RiemannLiouville fractional differential equations with signchanging nonlinearities, subject to integral boundary conditions. MSC: 34A08; 45G15

with the integral boundary conditions
D+u(t) + f (t, u(t), v(t)) = , t (, ), n < n,
D+v(t) + g(t, u(t), v(t)) = , t (, ), m < m,
u() = u () = = u(n)() = ,
v() = v () = = v(m)() = ,
where n, m N, n, m , D+, and D+ denote the RiemannLiouville derivatives of orders
and , respectively, the integrals from (BC) are RiemannStieltjes integrals and f , g are
signchanging continuous functions (that is, we have a socalled system of semipositone
boundary value problems). These boundary conditions include multipoint and integral
boundary conditions and the sum of these in a single framework.
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 []). Integral boundary conditions arise in thermal conduction
problems, semiconductor problems, and hydrodynamic problems.
By using a nonlinear alternative of LeraySchauder type, we present intervals for
parameters and such that the above problem (S)(BC) has at least one positive
solution. 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
u(t) > , v(t) > for all t (, ). In the case when f and g are nonnegative, the above
problem (S)(BC) has been investigated in [] by using the GuoKrasnoselskii fixed point
theorem. The system (S) with = = , and with f (t, u, v) and g(t, u, v) replaced by f (t, v)
and g(t, u), respectively, with the boundary conditions (BC), was studied in []. In [], the
authors obtained the existence and multiplicity of positive solutions (u(t) , v(t) for
all t [, ], supt[,] u(t) > , supt[,] v(t) > ) by applying some theorems from the fixed
point index theory. We would also like to mention the paper [], where the authors
investigated the existence and multiplicity of positive solutions of the semipositone system (S)
with = and the boundary conditions u(i)() = v(i)() = , i = , . . . , n , u() = av( ),
v() = bv(), , (, ), and < ab < .
The paper is organized as follows. Section contains some preliminaries and lemmas.
The main theorem is presented in Section and, finally, in Section , two examples are
given to support the new result.
2 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 theorem.
Definition . The (leftsided) fractional integral of order > of a function f : (, )
R is given by
provided the righthand side is pointwise defined on (, ), where () is the Euler
gamma function defined by () = tet dt, > .
Definition . The RiemannLiouville fractional derivative of order for a function
f : (, ) R is given by
The notation stands for the largest integer not greater than . We also denote the
RiemannLiouville fractional derivative of f by D+f (t). If = m N then Dm+f (t) = f (m)(t)
for t > , and if = then D+f (t) = f (t) for t > .
Lemma . ([]) Let > and n = + for / N and n = for N; that is, n is the
smallest integer greater than or equal to . Then the solutions of the fractional differential
equation D+u(t) = , < t < , are
u(t) = ct + ct + + cntn, < t < ,
where c, c, . . . , cn are arbitrary real constants.
Lemma . ([, ]) Let > , n be the smallest integer greater than or equal to (n <
n) and y L(, ). The solutions of the fractional equation D+u(t) + y(t) = , < t < ,
u(t) =
where c, c, . . . , cn are arbitrary real constants.
We consider now the fractional differential equation
with the integral boundary conditions
u() = u () = = u(n)() = ,
where n N, n , and H : [, ] R is a function of the bounded variation.
By using Lemma ., after some computations, we obtain the following lemma.
Lemma . ([]) If H : [, ] R is a function of bounded variation, =
s dH(s) = and z C([, ]), then the solution of problem ()() is u(t) = G(t, s)
z(s) ds, where
G(t, s) = g(t, s) +
(t, s) [, ] [, ],
g(t, s) =
Lemma . The function g given by () has the properties
(a) g : [, ] [, ] R+ is a continuous function, g(t, s) for all
(t, s) [, ] [, ] and g(t, s) > for all (t, s) (, ) (, ).
(b) g(t, s) h(s) for all (t, s) [, ] [, ], where h(s) = s(s)
() .
(c) g(t, s) k(t)h(s) for all (t, s) [, ] [, ], where
k(t) = min
Proof The first part (a) is evident. For the second part (b), see [].
For part (c), for s t, we obtain
g(t, s) =
g(t, s) =
t s( s) t
() t( s) s ( )
Lemma . ([]) If H : [, ] R is a nondecreasing function and > , then the Greens
function G of problem ()() given by () is continuous on [, ] [, ] and satisfies
G(t, s) for all (t, s) [, ] [, ], G(t, s) > for all (t, s) (, ) (, ). Moreover,
if z C([, ]) satisfies z(t) for all t [, ], then the unique solution u of problem
()() satisfies u(t) for all t [, ].
dH( ) = + H() H() .
Proof (a) We have
G(t, s) = g(t, s) +
h(s) +
(b) For the second inequality, we obtain
G(t, s) k(t)h(s) +
Lemma . Assume that H : [, ] R is a nondecreasing function, > and z
C([, ]), z(t) for all t [, ]. Then the solution u(t), t [, ] of problem ()() satisfies
the inequality u(t) (t) maxt [,] u(t ) for all t [, ].
Proof For t [, ], we obtain
u(t) =
G(t, s)z(s) ds
J(s)z(s) ds
t, t [, ].
We can also formulate similar results as Lemmas .. above for the fractional
differential equation
with the integral boundary conditions
v() = v () = = v(m)() = ,
where m N, m , K : [, ] R is a nondecreasing function and z C([, ]). We
denote by , g, G, h, k, , J, and the corresponding constants and functions for
problem ()() defined in a similar manner as , g, G, h, k, , J, and , respectively.
In the proof of our main result we shall use the following nonlinear alternative of
LeraySchauder type (see []).
3 Main result
D+x(t) + (f (t, [x(t) q(t)], [y(t) q(t)]) + p(t)) = ,
D+y(t) + (g(t, [x(t) q(t)], [y(t) q(t)]) + p(t)) = ,
with the integral boundary conditions
z(t) =
x() = x () = = x(n)() = ,
y() = y () = = y(m)() = ,
and (q, q) with q(t) = G(t, s)p(s) ds, q(t) = G(t, s)p(s) ds is the solution of the
system of fractional differential equations
with the integral boundary conditions
q() = q() = = q(n)() = ,
q() = q() = = q(m)() = ,
By (H), we have q(t) > , q(t) > for all t (, ).
We shall prove that there exists a solution (x, y) for the boundary value problem ()()
with x(t) q(t) and y(t) q(t) for all t [, ]. In this case, the functions u(t) = x(t) q(t)
and v(t) = y(t) q(t), t [, ], represent a nonnegative solution, positive on (, ) of the
boundary value problem (S)(BC). Indeed, by ()() and ()(), we have
t (, ),
D+v(t) = D+y(t) D+q(t) = g t, x(t) q(t) , y(t) q(t)
t (, ),
u() = x() q() = ,
v() = y() q() = ,
u(n)() = x(n)() q(n)() = ,
v(m)() = y(m)() q(m)() = ,
u() = x() q() =
x(s) dH(s)
q(s) dH(s) =
v() = y() q() =
y(s) dK (s)
q(s) dK (s) =
Therefore, in what follows, we shall investigate the boundary value problem ()().
By using Lemma ., the system ()() is equivalent to the system
x(t) = G(t, s)(f (s, [x(s) q(s)], [y(s) q(s)]) + p(s)) ds,
y(t) = G(t, s)(g(s, [x(s) q(s)], [y(s) q(s)]) + p(s)) ds,
We consider the Banach space X = C([, ]) with supremum norm and the Banach
space Y = X X with the norm (x , y) Y = x + y . We also define the cones
P = x X, x(t) (t)x , t [, ] X,
P = y X, y(t) (t)y , t [, ] X,
and P = P P Y .
t ,
t .
Lemma . If (H) and (H) hold, then the operator Q : P P is a completely continuous
operator.
(t)J(s) g s, x(s) q(s) , y(s) q(s) + p(s) ds
(t)J(s) f s, x(s) q(s) , y(s) q(s) + p(s) ds
for all t [, ], and
for all t, t [, ]. Therefore, we obtain
t [, ],
It is clear that (x, y) P is a solution of problem ()() if and only if (x, y) is a fixed point
of Q.
Theorem . Assume that (H)(H) hold. Then there exist constants > and >
such that for any (, ] and (, ], the boundary value problem (S)(BC) has at
least one positive solution.
t [, ], u, v [, R].
f(R) =
g(R) =
c =
f (t, u, v) + p(t) max f (t, , ) + p(t) > ,
t
g(t, u, v) + p(t) max g(t, , ) + p(t) > ,
t
J(s) ds > ,
c =
J(s) ds > ,
then, for all t [, ], we obtain
Hence x R/ and y R /. Then R = (x , y) = x
is a contradiction.
+ y
R + R = R , which
G(t, s)p(s) ds = q(t),
t (, ),
G(t, s)p(s) ds = q(t),
t (, ).
Therefore, x(t) > q(t) > and y(t) > q(t) > for all t (, ).
Let u(t) = x(t) q(t) and v(t) = y(t) q(t) for all t [, ], with u(t) > , v(t) >
on (, ). Then (u, v) is a positive solution of the boundary value problem (S)(BC).
4 Examples
H(t) = ,, tt [[/,,//),),
/, t [/, ],
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) =
D/+u(t) + f (t, u(t), v(t)) = ,
D+/v(t) + g(t, u(t), v(t)) = ,
with the boundary conditions
u() = u () = , u() = u( ) + u( ),
v() = v () = v () = , v() = sv(s) ds.
Then we obtain = s/ dH(s) = ( )/ ( )/ = . > ,
s/ dK (s) = s/ ds = . > .
We also deduce
=
g(t, s) =
Example We consider the functions
t [, ], u, v ,
where b > a > , d > c > , , > .
There exists M > such that f (t, u, v) + M , g(t, u, v) + M (p(t) = p(t) = M,
t [, ]) for all t [, ], u, v . Indeed, M = max{ (ba) + , (dc) + } satisfies the
above inequalities.
Let = min{ a((abb+a)) , c(dcdc) } < and R = min{ a , c , , }. Then
for all t [, ], u, v [, R]. Besides,
f(R) =
g(R) =
f (t, u, v) + p(t) = ab + + M,
Example We consider the functions
t [, ], u, v ,
where a, b, , > .
There exists M > (M = ) such that f (t, u, v) + M , g(t, u, v) + M (p(t) =
p(t) = M, t [, ]) for all t [, ], u, v .
Let = < and R = min{ , }. Then
f (t, u, v) f (t, , ) = ,
g(t, u, v) g(t, , ) = ,
t [, ], u, v [, R].
f(R) =
g(R) =
f (t, u, v) + p(t) = Ra + ,
g(t, u, v) + p(t) = Rb + .
+ , = (+)c ., and = (+)c
has a positive solution (u, v), with (u , v)
Competing interests
The authors declare that they have no competing interests.
Authors contributions
Both authors contributed equally to this paper. Both authors read and approved the final manuscript.
Acknowledgements
This work was supported by the CNCS grant PNIIIDPCE201130557, Romania.