Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions
Bai et al. Boundary Value Problems
Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions
Zhanbing Bai 0 1 2 3 4 6
Xiaoyu Dong 0 1 2 3 4 6
Chun Yin 0 1 2 4 5
0 Technology , Qianwangang Road
1 University of Science
2 System Science , Shandong
3 College of Mathematics
4 Qingdao , 266590 , P.R. China
5 School of Automation Engineering, University of Electronic Science and Technology of China , Chengdu, 611731 , P.R. China
6 College of Mathematics and System Science, Shandong University of Science and Technology , Qianwangang Road, Qingdao, 266590 , P.R. China
In this paper, the existence and uniqueness of solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order are obtained. Our results are based on some fixed point theorems. Some examples are also presented to illustrate the main results.
fractional differential equations; impulse; mixed boundary value

problem; fixed point theorem
1 Introduction
Recently, boundary value problems of nonlinear fractional differential equations have
been addressed by several researchers. Fractional differential equations arise in many
engineering and scientific disciplines as the mathematical modeling of systems and processes
in the fields of physics, chemistry, control theory, biology, economics, blood flow
phenomena, signal and image processing, biophysics, aerodynamics, fitting of experimental data,
etc. For details, see [–] and the references therein.
Impulsive differential equations, which provide a natural description of observed
evolution processes, are regarded as important mathematical tools for a better understanding
of several real world problems in the applied sciences. Recently, the boundary value
problems of impulsive differential equations of integer order have been studied extensively in
the literature (see [, –, –]). In [, ], Wang et al. gave a new concept of some
impulsive differential equations with fractional derivative, which is a correction of that of
piecewise continuous solutions used in [, , –].
This paper is strongly motivated by the above research papers. We investigate the
existence and uniqueness of solutions for a mixed boundary value problem of nonlinear
impulsive differential equations of fractional order given by
u(tk) = Ik(u(tk)),
⎧ CDq+ u(t) = f (t, u(t)), t ∈ J ,
⎨⎪
⎪⎩ u() + u () = ,
u() + u () = ,
u (tk) = Jk(u(tk)), k = , , . . . , p,
(.)
where CDq+ is the Caputo fractional derivative of order q ∈ (, ), f ∈ C(J × R, R). Ik, Jk ∈
C(R, R), J = [, ], J = J \ {t, t, . . . , tp}, the {tk} satisfy = t < t < t < · · · < tp < tp+ = ,
p ∈ N , u(tk) = u(tk+) – u(tk–), u (tk) = u (tk+) – u (tk–), where u(tk+) and u(tk–) represent the
right and left limits of u(t) at t = tk .
A function u ∈ PC(J, R) is said to be a solution of problem (.) if u(t) = uk(t) for
t ∈ (tk, tk+) and uk ∈ C([, tk+], R) satisfies CDq+ u(t) = f (t, u(t)) a.e. on (, tk+) with the
restriction that uk(t) on [, tk) is just uk–(t) and the conditions u(tk) = Ik(u(tk)), u (tk) =
Jk(u(tk)), k = , , . . . , p with u() + u () = , u() + u () = .
The rest of this paper is organized as follows. In Section , we give some notations, recall
some concepts and preparation results. In Section , we give the main results, the first
result based on Banach contraction principle, the second result based on Krasnoselskii’s
fixed point theorem. Two examples are given in Section to demonstrate the application
of our main results.
2 Preliminaries
In this section, we introduce preliminary facts which are used throughout this paper.
Let J = [, t], J = (t, t], . . . , Jp– = (tp–, tp], Jp = (tp, ]. We have
PC(J) = u : [, ] → R  u ∈ C J , and u tk+ , u tk– exist, and
u tk– = u(tk), ≤ k ≤ p .
Obviously, PC(J) is a Banach space with the norm
u PC = sup u(t) .
≤t≤
Iq+f (t) =
t f (s)
(q) (t – s)–q ds, t > , q > ,
(.)
Definition . The fractional integral of order q of a function f : [, ∞) → R is defined as
provided the right side is pointwise defined on (, ∞), where (·) is the gamma function.
Definition . The Caputo derivative of fractional order q for a function f : [, ∞) → R
is defined as
CDq+ f (t) =
dn
(n – q) dtn
t f (s) – kn=– skk! f (k)()
(t – s)q–n+
where [q] denotes the integer part of the real number q.
ds, t > , n = –[–q],
(.)
Remark . In the case f (t) ∈ Cn[, +∞), there is CDq+ f (t) = In+–qf (n)(t). That is to say that
Definition . is just the usual Caputo’s fractional derivative. In this paper, we consider an
impulsive problem, so Definition . is appropriate.
Lemma . ([]) Let M be a closed, convex, and nonempty subset of a Banach space X,
and A, B the operators such that
() Ax + By ∈ M whenever x, y ∈ M;
() A is compact and continuous;
() B is a contraction mapping.
Then there exists z ∈ M such that z = Az + Bz.
Lemma . ([]) The set F ⊂ PC([, T ], Rn) is relatively compact if and only if:
(i) F is bounded, that is, x ≤ C for each x ∈ F and some C > ;
(ii) F is quasiequicontinuous in [, T ]. That is to say that for any > there exists δ >
such that if x ∈ F ; k ∈ N ; τ, τ ∈ (tk–, tk ], and τ – τ < δ, we have x(τ) – x(τ) < .
Lemma . ([]) For q > , the general solution of the fractional differential equation
C Dq+ u(t) = is given by
(.)
(.)
(.)
u(t) = c + ct + ct + · · · + cn–tn–,
where ci ∈ R, i = , , , . . . , n – , n = –[–q].
In view of Lemma ., it follows that
q C Dq+ u (t) = u(t) + c + ct + ct + · · · + cn–tn–,
I+
where ci ∈ R, i = , , , . . . , n – , n = –[–q].
Lemma . Let q ∈ (, ) and h : J → R be continuous. A function u given by
t
(q) (t – s)q–h(s) ds + –(qt) ( – s)q–h(s) ds
+ (q––t) ( – s)q–h(s) ds, t ∈ [, t];
t
(q) (t – s)q–h(s) ds + –(qt) ( – s)q–h(s) ds
–t
+ (q–) ( – s)q–h(s) ds + ( – t)
+ ( – t) jp= Ij(u(tj)) – (t – tj) jp=k+ Jj(u(tj)) –
t ∈ (tk , tk+], k = , , . . . , p – ;
t
(q) (t – s)q–h(s) ds + –(qt) ( – s)q–h(s) ds
–t
+ (q–) ( – s)q–h(s) ds + ( – t)
is a unique solution of the following impulsive problem:
⎪⎪⎨⎧ C Dq+ u(t) = h(t),
u(tk ) = Ik (u(tk )),
⎪⎪⎩ u() + u () = ,
t ∈ J ,
u (tk ) = Jk (u(tk )),
u() + u () = .
k = , , . . . , p,
Proof With Lemma ., a general solution u of the equation C Dq+ u(t) = h(t) on each
interval (tk , tk+] (k = , , , . . . , p) is given by
u(t) =
(q)
t
(t – s)q–h(s) ds + ak + bk t,
for t ∈ (tk , tk+],
where t = and tp+ = . Then we have
u (t) =
So applying the boundary conditions (.), we have
(t – s)q–h(s) ds + bk, for t ∈ (tk, tk+].
(.)
a + b = ,
Combining (.), (.), (.) with (.) yields
(q – )
Hence for k = , , , . . . , p – , (.) and (.) imply
ak + bk t =
( – s)q–h(s) ds +
j=
p
j=k+
Ij u(tj)
(.)
(.)
(.)
(.)
(.)
– t
(q)
For k = p, (.) and (.) imply
ak + bk t =
( – s)q–h(s) ds +
3 Main results
This section deals with the existence and uniqueness of solutions to problem (.).
Theorem . Let f : J × R → R be a continuous function. Suppose there exist positive
constants L, L, L, M, M such that
(A) f (t, x) – f (t, y) ≤ Lx – y, for all t ∈ J , x, y ∈ R;
(A) Ik (x) – Ik (y) ≤ Lx – y, Jk (x) – Jk (y) ≤ Lx – y, Ik (x) ≤ M, Jk (x) ≤ M,
x, y ∈ R, k = , , . . . , p,
with
( – s)q– f s, u(s) – f (s, ) ds
( – s)q– f (s, ) ds +
(qM+ ) + L(qr) + M(q)
( – s)q– f s, x(s) – f s, y(s) ds
+
+
p
j=
p
j=k+
+
+
j=
p
j=k+
p
j=
p
j=k+
Jj x(tj) – Jj y(tj) +
Ij x(tj) – Ij y(tj)
Jj x(tj) – Jj y(tj) +
Ij x(tj) – Ij y(tj)
≤ (qL+ ) x – y PC +
p
(qL+ ) x – y PC +
p
L x – y PC +
L x – y PC
L(q) x – y PC
j=
p
k=j+
L x – y PC +
L x – y PC
≤ (qL+ ) x – y PC +
L(q) x – y PC + pL x – y PC
+ pL x – y PC + pL x – y PC + pL x – y PC
T is a contraction mapping. Thus, the conclusion follows by the contraction mapping
principle.
Define the operators P and Q on Br as
r ≥ μ L σ (J)
(q)( q––σσ )–σ +
(Qu)(t) = ( – t)
Jk u(tk) ( – tk) + ( – t)
Ik u(tk)
– (t – tk)
Jk u(tk) –
Ik u(tk) .
p
k=
p
k=j+
p
k=j+
p
k=
For any u, v ∈ Br and t ∈ J , using the condition that f (t, u) ≤ μ(t) and the Hölder
inequality,
t
≤
t
≤
t
≤
(t – s)q–f s, u(s) ds
Therefore,
t
t
t
μ(s) σ ds
μ(s) σ ds
μ(s) σ ds
σ
σ
σ
μ
≤ ( q––σσL)σ–(σJ) ,
μ
≤ ( q––σσL)σ–(σJ) ,
μ
L σ (J) .
≤ ( q––σσ– )–σ
Pu + Qv PC
μ L σ (J)
≤ (q)( q––σσ )–σ +
μ L σ (J)
(q – )( q––σσ– )–σ
+ pM + pM + pM + pM
= μ L σ (J)
(q)( q––σσ )–σ +
fmax
(q)
( – s)q–f s, u(s) ds –
( – s)q–f s, u(s) ds –
(τ – s)q–f s, u(s) ds
which tends to zero as τ → τ. This shows that P is quasiequicontinuous on the interval
(tk, tk+]. It is obvious that P is compact by Lemma ., so P is relatively compact on Br.
Thus all the assumptions of Lemma . are satisfied and problem (.) has at least one
solution on J .
4 Example
Example . Consider the following impulsive fractional boundary value problem:
⎪⎪⎩⎪ u() + u () = ,
⎪⎪⎧⎪⎨ CDu(+u)(=t)=+u((ut+())) ,si+nuu((tt)) , u (t ∈) [=,+]u,(ut()= ) ,
u() + u () = .
Obviously, L = /, L = /, L = /, M = /, M = /, p = ,
((q++q)) = √π ,
L (q+ ) + (q)
L < ((q++q)) ,
+ p(L + L) = √ π + < .
Thus, all the assumptions in Theorem . are satisfied. Hence, the impulsive fractional
boundary value problem (.) has a unique solution on [, ].
Example . Consider the following impulsive fractional boundary value problem:
Set
Obviously,
f (t, u) ≤ (t +et) .
⎪⎧⎪⎪⎨ CDu(+u)(=t)=++(utu(+e(t))) ,+uu(t()t) , ut(∈ )[=, +],uut((=)) ,
⎪⎪⎪⎩ u() + u () = , u() + u () = .
f (t, u) = (t +et) +uu , (t, u) ∈ [, ] × [, ∞).
(.)
(.)
Set
L = L = ,
and
et
μ(t) = (t + ) ∈ L [, ], R .
Thus, all the assumptions in Theorem . are satisfied. Hence, the impulsive fractional
boundary value problem (.) has at least one solution on [, ].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors express their sincere thanks to the anonymous reviews for their valuable suggestions and corrections for
improving the quality of the paper. This work is supported by NSFC (11571207, 61503064), the Taishan Scholar project.
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