R E S E A R C H
Open Access
Variational approach to a class of impulsive
differential equations
Dajun Guo
**Correspondence:
[email protected] Department of Mathematics, Shandong University, Jinan, Shandong 250100, People’s Republic of China
Abstract
In this article, the author discusses the existence of solutions for a class of impulsive differential equations by means of a variational approach different from earlier approaches.
MSC: 34B37; 45G10; 47H30; 47J30
Keywords: impulsive differential equation; integral equation; variational method; critical point theory
1 Introduction
The theory of impulsive differential equations has been emerging as an important area of investigation in recent years [–]. There is a vast literature on the existence of solutions by using topological methods, including fixed point theorems, Leray-Schauder degree the-ory, and fixed point index theory [–]. But it is quite difficult to apply the variational ap-proach to an impulsive differential equation; therefore, there was no result in this area for a long time. Only in the recent five years, there appeared a few articles which dealt with some impulsive differential equations by using variational methods [–]. Motivated by [], in this article we shall use a different variational approach to discuss the existence of solutions for a class of impulsive differential equations and we only deal with classical solutions.
Consider the boundary value problem (BVP) for the second-order nonlinear impulsive differential equation:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
–u(t) =f(t,u(t)), ∀t∈J,
u|t=tk=ck (k= , , , . . . ,m),
u|t=tk=dk (k= , , , . . . ,m), u() =u() = ,
()
where J= [, ], <t<· · ·<tk <· · ·<tm < , J=J\{t, . . . ,tk, . . . ,tm}, ck and dk (k=
, , . . . ,m) are any real numbers,f(t,u) is a real function defined onJ×R, whereR de-notes the set of all real numbers, and f(t,u) is continuous onJ×R, left continuous at t=tk,i.e.
lim
t→tk–,w→u
f(t,w) =f(tk,u)
for anyu∈R(k= , , . . . ,m), and the right limit att=tkexists,i.e.
lim
t→tk+,w→u f(t,w)
(denoted byf(t+
k,u)) exists for anyu∈R(k= , , . . . ,m).u|t=tkdenotes the jump ofu(t) att=tk,i.e.
u|t=tk=u
tk+–ut–k,
whereu(t+k) andu(t–
k) represent the right and left limits ofu(t) att=tk, respectively.
Sim-ilarly,
u|t=tk=u
t+
k
–utk–,
whereu(tk+) andu(tk–) represent the right and left limits ofu(t) att=tk, respectively. Let
PC[J,R] ={u:uis a real function onJsuch thatu(t) is continuous att=tk, left continuous
att=tk, andu(tk+) exists,k= , , . . . ,m}andPC[J,R] ={u∈PC[J,R] :u(t) is continuous
att=tkandu(tk+),u(t–k) exist,k= , , . . . ,m}. A functionu∈PC[J,R]∩C[J,R] is called
a solution of BVP () ifu(t) satisfies (). Let us list some conditions.
(H) There existp> ,a> andb> such that
f(t,u)≤a+b|u|p–, ∀t∈J,u∈R.
(H) There exist <c<π
andd> such that
u
f(t,v)dv≤cu+d, ∀t∈J,u∈R.
Lemma u∈PC[J,R]∩C[J,R]is a solution of BVP()if and only if v∈C[J,R]is a solution of the integral equation
v(t) =
G(t,s)gs,v(s)ds, ∀t∈J, ()
where
G(t,s) =
s( –t), ∀≤s≤t≤;
t( –s), ∀≤t<s≤, () g(t,v) =ft,v+a(t) –a()t, ∀t∈J,v∈R ()
and
v(t) =u(t) –a(t) +a()t, a(t) = <tk<t
ck+ (t–tk)dk
Proof Foru∈PC[J,R]∩C[J,R], we have the formula (see [], Lemma (b))
u(t) =u() +tu() +
t
(t–s)u(s)ds
+
<tk<t
utk+–u(tk)
+ (t–tk)
utk+–utk–, ∀t∈J. ()
So, ifu∈PC[J,R]∩C[J,R] is a solution of BVP (), then, by () and (), we have
u(t) =tu() –
t
(t–s)fs,u(s)ds+ <tk<t
ck+ (t–tk)dk
=tu() –
t
(t–s)fs,u(s)ds+a(t), ∀t∈J. ()
It is clear, by (), that
a(t) = , ∀≤t≤t; a() =
m
k=
ck+ ( –tk)dk
, ()
so
v() =u() +a(). ()
Substituting () into (), we get
v(t) =tv() –
t
(t–s)fs,u(s)ds
=tv() –
t
(t–s)fs,v(s) +a(s) –a()sds
=tv() –
t
(t–s)gs,v(s)ds, ∀t∈J. ()
By virtue of (), we see thatv∈C[J,R] (in fact,v∈C[J,R]) and v() =u() –a() +a() =u() = ,
so, lettingt= in (), we find
v() =
( –s)gs,v(s)ds. ()
Substituting () into (), we get
v(t) =
t
t( –s)gs,v(s)ds+
t
s( –t)gs,v(s)ds
=
G(t,s)gs,v(s)ds, ∀t∈J,
Conversely, suppose thatv∈C[J,R] is a solution of (),i.e.
v(t) = ( –t)
t
sgs,v(s)ds+t
t
( –s)gs,v(s)ds, ∀t∈J. ()
By (), it is clear thatg(t,v(t)) is continuous onJ, so differentiation of () gives
v(t) = –
t
sgs,v(s)ds+ ( –t)tgt,v(t)
+
t
( –s)gs,v(s)ds–t( –t)gt,v(t)
= –
t
sgs,v(s)ds+
t
( –s)gs,v(s)ds, ∀t∈J. ()
Differentiating again, we get
v(t) = –tgt,v(t)– ( –t)gt,v(t)= –gt,v(t), ∀t∈J. ()
From () we see thatv(t+
k) andv(t–k) (k= , , . . . ,m) exist and
vtk+=vt–k= –
tk
sgs,v(s)ds+
tk
( –s)gs,v(s)ds. ()
It follows from (), (), (), (), and () that u∈PC[J,R]∩C[J,R] andu(t)
satis-fies ().
Lemma Let condition(H)be satisfied.If v∈Lp[J,R]is a solution of the integral equation (),then v∈C[J,R].
Proof It is clear, for functiona(t) defined by (),
a(t)≤a, ∀t∈J; a=
m
k=
|ck|+ ( –tk)|dk|
. ()
By (), (), (), and condition (H), we have
g(t,v)≤a+bv+a(t) –a()tp–≤a+b|v|+ a p–
≤a+bmax|v|, a p–≤
a+bp–|v|p–+ (a)p–
, ∀t∈J,v∈R,
so,
g(t,v)≤a+b|v|p–, ∀t∈J,v∈R, ()
where
It is clear thatg(t,v) satisfies the Caratheodory condition,i.e. g(t,v) is measurable with respect totonJfor everyv∈Rand is continuous with respect tovonRfor almostt∈J (in fact,g(t,v) is discontinuous only att=tk(k= , , . . . ,m)), so () implies [, ] that
the operatorgdefined by
(gv)(t) =gt,v(t), ∀t∈J ()
is bounded and continuous fromLp[J,R] intoLq[J,R], where
p+
q= (q> ).
Letv∈Lp[J,R] be a solution of the integral equation (). Then by the Hölder inequality, v(t) –v(t)≤
G(t,s) –G(t,s)
p
ds
p
gs,v(s)qds
q
, ∀t,t∈J,
which implies by virtue of the uniform continuity ofG(t,s) onJ×Jthatv∈C[J,R]. 2 Variational approach
Theorem If conditions(H)and(H)are satisfied,then BVP()has at least one solution u∈PC[J,R]∩C[J,R].
Proof By Lemma and Lemma , we need only to show that the integral equation () has a solutionv∈Lp[J,R]. The integral equation () can be written in the form
v=Ggv, ()
whereGis the linear integral operator defined by
(Gv)(t) =
G(t,s)v(s)ds, ∀t∈J, ()
and the nonlinear operatorgis defined by (), which is bounded and continuous from Lp[J,R] intoLq[J,R] (
p+
q= ). It is well known thatG(t,s) is aL
positive-definite kernel
with eigenvalues{nπ}(n= , , , . . .) and, by the continuity ofG(t,s), we have
G(t,s)pds dt<∞, ()
so [, ] the linear operatorGdefined by () is completely continuous fromL[J,R] intoL[J,R] and also fromLq[J,R] intoLp[J,R], andG=HH∗, whereH=G (the positive square-root operator ofG) is completely continuous fromL[J,R] intoLp[J,R] andH∗
de-notes the adjoint operator ofH, which is completely continuous fromLq[J,R] intoL[J,R]. We now show that () has a solutionv∈Lp[J,R] is equivalent to the equation
u=H∗gHu ()
has a solutionu∈L[J,R]. In fact, ifv∈Lp[J,R] is a solution of (),i.e. v=HH∗gv, then
solution of (). Consequently, we need only to show that () has a solutionu∈L[J,R]. It is well known [, ] that the functionaldefined by
(u) = (u,u) –
dt
(Hu)(t)
g(t,v)dv, ∀u∈L[J,R] ()
is aCfunctional onL[J,R] and its Fréchet derivative is
(u) =u–H∗gHu, ∀u∈L[J,R]. ()
Hence we need only to show that there exists au∈L[J,R] such that(u) =θ(θdenotes the zero element ofL[J,R]),i.e. uis a critical point of functional.
By (), (), (), and condition (H), we have
u
g(t,v)dv=
u+a(t)–a()t
f(t,w)dw–
a(t)–a()t
f(t,w)dw, ∀t∈J,u∈R ()
and
a(t)–a()tf(t,w)dw≤a(t) –a()ta+ba(t) –a()tp–
≤a
a+bp–ap–=a, ∀t∈J. () So, (), (), and condition (H) imply
(Hu)(t)
g(t,v)dv≤
(Hu)(t)+a(t)–a()t
f(t,w)dw+a
≤c(Hu)(t) +a(t) –a()t+d+a
≤c(Hu)(t)+a(t) –a()t+d+a
≤c(Hu)(t)+ ca+d+a, ∀u∈L[J,R],t∈J. ()
It is well known [],
G=λ=
π, ()
whereGis defined by () and is regarded as a positive-definite operator fromL[J,R] into L[J,R], andλ
denotes the largest eigenvalue ofG. It follows from (), (), and () that
(u)≥
(u,u) – c(Hu,Hu) – ca
–d–a
=
(u,u) – c(Gu,u) – ca
–d–a≥ (u,u) –
c
π(u,u) – ca
–d–a
=
–
c
π
u– ca–d–a, ∀u∈L[J,R], ()
which implies by virtue of <c<π (see condition (H)) that
lim
So, there exists ar> such that
(u) >(θ) = , ∀u∈L[J,R],u>r. () It is well known [, ] that the ballT(θ,r) ={u∈L[J,R] :u ≤r}is weakly closed and weakly compact and the functional(u) is weakly lower semicontinuous, so, there exists u∗∈T(θ,r) such that
u∗= inf
u∈T(θ,r)(u)≤(θ). ()
It follows from () and () that
u∗= inf
u∈L[J,R](u).
Hence(u∗) =θand the theorem is proved.
Example Consider the BVP
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
–u(t) =
u(t)sin(t–u(t)) –t
, ∀t∈J, u|t=tk=ck (k= , , . . . ,m),
u|t=tk=dk (k= , , . . . ,m), u() =u() = ,
()
where J = [, ], <t<· · ·<tk <· · ·<tm < , J =J\{t, . . . ,tk, . . . ,tm}, ck anddk (k=
, , . . . ,m) are any real numbers.
Conclusion BVP () has at least one solutionu∈PC[J,R]∩C[J,R]. Proof Evidently, () is a BVP of the form () with
f(t,u) =
usin(t–u) –t
. ()
It is clear thatf ∈C[J×R,R]. By (), we have f(t,u)≤
|u|+ , ∀t∈J,u∈R. ()
Moreover, it is well known that
|u| ≤
+u, ∀u∈R. ()
So, () and () imply that
f(t,u)≤ u
+
, ∀t∈J,u∈R,
and consequently, condition (H) is satisfied forp= ,a= andb=. On the other hand, choosesuch that
<<
For|u| ≥
, we have|u| ≤u , so,
|u| ≤u+
, ∀u∈R. ()
By (), we have
u
f(t,v)dv≤ u
+|u|, ∀t∈J,u∈R. ()
It follows from () and () that
u
f(t,v)dv≤
+
u+
, ∀t∈J,u∈R. ()
Since, by virtue of (),
< +<
π
,
we see that () implies that condition (H) is satisfied forc=+andd=. Hence,
our conclusion follows from Theorem .
By using the Mountain Pass Lemma and the Minimax Principle established by Am-brosetti and Rabinowitz [, ], we have obtained in [] the existence of a nontrivial solution and the existence of infinitely many nontrivial solutions for a class of nonlinear integral equations. Since () is a special case of such nonlinear integral equations, we get the following result for ().
Lemma (Special case of Theorem and Theorem in []) Suppose the following.
(a) There existp> anda> ,b> such that g(t,v)≤a+b|v|p–, ∀t∈J,v∈R.
(b) There exist≤τ< andM> such that
v
g(t,w)dw≤τvg(t,v), ∀t∈J,|v| ≥M.
(c) g(tv,v)→asv→uniformly fort∈Jandg(tv,v)→ ∞as|v| → ∞uniformly fort∈J. Then the integral equation()has at least one nontrivial solution in Lp[J,R].If,in
addi-tion,
(d) g(t, –v) = –g(t,v),∀t∈J,v∈R.
Then the integral equation()has infinite many nontrivial solutions in Lp[J,R].
Let us list more conditions for the functionf(t,u).
(H) There exist≤τ<andM> such that
u
(H) f(t,u+a(ut)–a()t) →asu→uniformly fort∈J, andf(t,u+a(ut)–a()t)→ ∞as|u| → ∞ uniformly fort∈J.
(H) f(t, –u+a(t) –a()t) = –f(t,u+a(t) –a()t),∀t∈J,u∈R.
Theorem Suppose that conditions(H), (H),and(H)are satisfied.Then BVP()has at least one solution u∈PC[J,R]∩C[J,R].If,in addition,condition(H
)is satisfied,then BVP()has infinitely many solutions un∈PC[J,R]∩C[J,R] (n= , , , . . .).
Proof In the proof of Lemma , we see that condition (H) implies condition (a) of Lem-ma (see ()). On the other hand, it is clear that conditions (H), (H), (H) are the same as conditions (b), (c), (d) in Lemma , respectively. Hence the conclusion of Theorem
follows from Lemma , Lemma , and Lemma .
Example Consider the BVP
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
–u(t) =
[u(t) –t], ∀≤t< ; [u(t) + t– ], ∀
<t≤,
u|t= = ,
u|t= = –, u() =u() = .
()
Conclusion BVP () has infinite many solutionsun∈PC[J,R]∩C[J,R] (n= , , , . . .).
Proof Obviously, () is a BVP of form (). In this situation,J= [, ],m= ,t=,J= [, ]\{
},c= ,d= –, and f(t,u) =
(u–t), ∀≤t≤ ; (u+ t– ), ∀
<t≤.
()
It is clear thatf(t,u) is continuous onJ×R, left continuous att=t, and the right limit f(t+,u) exists. By (), we have
f(t,u)≤
|u|+
≤
max
|u|,
≤
|u|+
= |u|+ , ∀t∈J,u∈R,
so, condition (H) is satisfied forp= ,a= andb= . By (), we have
a(t) =
, ∀≤t≤;
– t, ∀<t≤, ()
so,a() = – and () and () imply
ft,u+a(t) –a()t=u, ∀t∈J,u∈R, () and, consequently, (H) is satisfied forτ=and anyM> . On the other hand, from () we see that conditions (H) and (H) are all satisfied. Hence, our conclusion follows from
Competing interests
The author declares that they have no competing interests.
Acknowledgements
Research was supported by the National Nature Science Foundation of China (No. 10671167).
Received: 16 December 2013 Accepted: 14 January 2014 Published:07 Feb 2014
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Cite this article as:Guo:Variational approach to a class of impulsive differential equations.Boundary Value Problems
