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On variational methods to non-instantaneous impulsive fractional differential equation
Accepted Manuscript On variational methods to non-instantaneous impulsive fractional differential equation Adnan Khaliq, Mujeeb ur Rehman
PII: DOI: Reference:
S0893-9659(18)30079-X https://doi.org/10.1016/j.aml.2018.03.014 AML 5462
To appear in:
Applied Mathematics Letters
Received date : 26 December 2017 Revised date : 15 March 2018 Accepted date : 15 March 2018 Please cite this article as: A. Khaliq, M.u. Rehman, On variational methods to non-instantaneous impulsive fractional differential equation, Appl. Math. Lett. (2018), https://doi.org/10.1016/j.aml.2018.03.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
On variational methods to non-instantaneous impulsive fractional differential equation Adnan Khaliq∗, Mujeeb ur Rehman School of Natural Sciences, National University of Sciences and Technology, Islamabad Pakistan
Abstract In this paper by using the variational methods for a class of impulsive differential equation of fractional order with non-instantaneous impulses, we setup sufficient conditions for the existence and uniqueness of weak solutions. The problem is reduced to an equivalent form such that the weak solutions of the problem are defined as the critical points of a functional. Main results of the present work are established by using Lax-Milgram Theorem. Keywords: Fractional differential equations, Variational methods, Lax-Milgram theorem, Non-instantaneous impulse.
1. Introduction Fractional calculus generalizes the integer order integration and differentiation concepts to an arbitrary (real or complex) order. Fractional calculus is the most well known and valuable branch of mathematics which gives a good framework for biological and physical phenomena, mathematical modeling of engineering etc. To get a couple of developments about the theory of fractional differential equations, one can allude to the monographs of Lakshmikanthem et al. [4], Kilbas et al. [2], Pudlubny [6], Tarasov [7] and the references in there. The impulsive differential equations have become very useful tools to portray the change of processes in which discontinuous jumps (impulsive conditions) abruptly appear. Such processes are normally seen in engineering, biology and physics. A maximum number of research papers are written on the existence theory of the solutions for the impulsive differential equations with instantaneous impulses [9, 16, 18]. But in many cases it has been noted that certain dynamics of evolution processes cannot delineate by the instantaneous impulses, for example: High or low level of glucose, Pharmacotherapy, this situation can be illustrated as an impulsive jump starting suddenly at any fixed point and remains continue on a finite interval of time. Such sort of systems are more useful to study the dynamics of evolution processes and are called non-instantaneous impulsive systems. Hernadez & O’Regan introduced non-instantaneous impulsive differential equations in [14]. Recently, many authors applied nonlinear analysis methods (e.g. fixed point theory etc), to obtain the results dealing with the impulsive differential equations with non-instantaneous impulses for the existence and multiplicity of their solutions [5, 8, 11, 12, 20]. However, the problems (e.g. discussing BVP) for which the equivalent integral equation is not easy to obtain, these ∗ Corresponding
author. Tel +92 322 6623874 Email addresses: [email protected] ( Adnan Khaliq), [email protected] ( Mujeeb ur Rehman)
Preprint submitted to Elsevier
March 15, 2018
popular techniques look not appropriate. Some times it becomes very difficult for the boundary value problem of the fractional order and especially for those containing both right and left fractional derivatives to construct a suitable space and equivalent integral equation. Contrary, for such problems where equivalent integral equation is not available there are other very useful approaches: Variational techniques and critical point theory. Jiao and Zhou [3] first time showed that these useful approaches are powerful tools while dealing with the such class of fractional boundary value problems for the existence and uniqueness of their solutions. Critical point theorems and the variational methods were also used to setup results for impulsive problems and their solutions. In this direction pioneering work was done by Nieto and O’Regan [1] by considering a class of impulsive problem of second order. In [10, 13, 17], authors setup results for the fractional impulsive differential equations with instantaneous impulses by applying variational methods and critical point theory. On the other hand, the existence and uniqueness of solutions to impulsive differential equations of fractional order containing non-instantaneous impulses by using critical point theory and variational methods is never been investigated so far to the best of our knowledge. So in this paper we solved this untouched problem. Recently L. Bai & J. J. Nieto [15] discussed 2nd order impulsive differential equation with non-instantaneous impulses. By using Lax-Milgram theorem, they guaranteed for the existence and uniqueness of the weak solutions of the problem. Getting motivation from work cited above, we consider following fractional impulsive boundary value problem with not instantaneous impulses: α c α t ∈ (si , ti+1 ], i = 0, 1, 2, · · · , n, t DT 0 Dt u(t) = fi (t), c α α−1 t ∈ (ti , si ], i = 1, 2, · · · , n, t DT 0 Dt u(t) = ci α−1 c α + − α−1 c α i = 1, 2, · · · , n, t DT 0 Dt u(si ) 0 Dt u(si ) =t DT t Dα−1 (c Dα u(0)) = c0 , u(0) = u(T ) = 0, T 0 t
(1.1)
here 0 = s0 < t1 < s1 < t2 < s2 < · · · < tn < sn < tn+1 = T , c0 Dαt and t DαT are left Caputo fractional derivative and right c α Riemann-Liouville fractional derivatives of the order 0 < α ≤ 1 respectively. Keeping the derivative t Dα−1 0 Dt u(t) T
constant on a finite interval (ti , si ], the impulses begin suddenly at the points ti . Here fi : (si , ti+1 ] → R are given α α−1 ± α ± functions, ci are given constants and t Dα−1 0 Dt u(s) . T (0 Dt u(si )) = lim s→si t DT Through out the paper, we assume following condition is satisfied. (M1 ). For each i = 0, 1, 2, · · · n, function fi ∈ L2 (si , ti+1) .
Remark 1.1. It should be noted that for α = 1, one has t DαT u(t) = −u′ (t) and c0 Dαt u(t) = u′ (t), and our problem (1.1) reduces to [15]. So our problem (1.1) gives generalization of [15].
We organized rest of the article as follows. According to the requirements of paper, some basic definitions and preliminary results are given in Section 2. After establishing variational structure of BVP (1.1), we prove our main results in Section 3. In the end an example is given to highlight the effectiveness of our results.
2
2. Preliminaries From the fractional calculus, some of the definitions and basic results are given in this section which will be utilized as a part of the evidence later in this paper. Definition 2.1. ([2, 6]) The right Riemann-Liouville fractional integral for a function u defined on [a, b] of order α > 0 is denoted by t D−α b u(t) and is defined by −α t Db u(t)
1 Γ(α)
=
Z
b t
(v − t)α−1 u(v)dv,
t ∈ [a, b],
the right hand side is required to be pointwise defined on [a, b]. Definition 2.2. ([2, 6]) The left Riemann-Liouville fractional integral for a function u defined on [a, b] of order α > 0 is denoted by a D−α t u(t) and is defined as −α a Dt u(t)
1 = Γ(α)
Z
a
t
(t − v)α−1 u(v)dv,
t ∈ [a, b],
here Γ is the gamma function and the right hand side is required to be pointwise defined on [a, b]. Definition 2.3. ([2, 6]) The left and right Riemann-Liouville fractional derivatives for a function u defined on [a, b] of order α > 0 are denoted by a Dαt u(t) and t Dαb u(t), respectively and are defined by α a Dt u(t)
and α t Db u(t)
dn −(n−α) u(t) aD dtn t Z dn t 1 n−α−1 = (t − v) u(v)dv , Γ(n − α) dtn a =
dn −(n−α) u(t) tD dtn b n Z t 1 n d (−1) n (v − t)n−α−1 u(v)dv , = Γ(n − α) dt a
= (−1)n
respectively where t ∈ [a, b], n − 1 < α ≤ n and n ∈ N.
Definition 2.4. ([2, 6]) Let u ∈ AC m ([a, b], R) and m − 1 < α ≤ m, then the right and left Caputo fractional derivatives of order α are denoted as ct Dαb u(t) and ca Dαt u(t) respectively, are given by c α t Db u(t)
and c α a Dt u(t)
respectively where t ∈ [a, b].
dm = (−1)mt Db−(m−α) m u(t) dt Z t (−1)m (v − t)m−α−1 u(m) (v)dv, = Γ(m − α) a dm u(t) dtm Z t 1 (t − v)m−α−1 u(m) (v)dv, = Γ(m − α) a = a Dt−(m−α)
3
Lemma 2.5. ([2, 6]) (a). Let m − 1 < α ≤ m and u, v ∈ L2 (a, b) then Z b Z b 1−α u(t) t D1−α D u(t) v(t)dt = a t b v(t) dt. a
(2.1)
a
(b). Let m − 1 < α ≤ m, v ∈ AC([a, b], RN ), v′ ∈ L2 ([a, b], RN ), ca Dαt u(t) ∈ L2 ([a, b], RN ) and t DαT (c0 Dαt u(t)) ∈ AC([a, b], RN ), then Z
a
b
(ca Dαt u(t))(ca Dαt v(t))dt = =
Z
b
a
Z
(ca Dαt u(t))(ca Dα−1 v′ (t))dt t
b
′ α−1 c α t DT 0 Dt u(t) v (t)dt
a
c α t=b = t Dα−1 T (0 Dt u(t))v(t)|t=a −
=
α−1 c α t=b t DT (0 Dt u(t))v(t)|t=a
+
Z
(2.2)
b
a
Z
b
a
d (t Dα−1 (c Dα u(t)))v(t)dt dt T 0 t α−1 c α t DT (0 Dt u(t))v(t)dt.
Fractional Derivative Space A fractional derivative space is required in order to use the critical point theory for the existence of solutions to the problem (1.1). We define fractional derivative space E0α by the closure of C0∞ ([0, T ], RN ) with respect to norm ||u||α = (
Z
T
0
(|0 Dαt u(t)|2 + |u(t)|2 )dt)1/2 ,
(2.3)
where 0 Dαt denotes the left Riehmann-Liouville derivative of fractional order. Clearly E0α is space of the functions u for which u ∈ L2 [0, T ], u(0) = u(T ) = 0 and α order fractional derivative 0 Dαt u(t) ∈ L2 [0, T ]. This space is similar to
the fractional derivative space introduced in [3]. First give us a chance to review that
||u||L p = (
||u||∞ = max |u(t)|, t∈[0,T ]
Z
T
0
p
1 p
|u(s)| ds) ,
for any fixed t ∈ [0, T ] , T > 0 and 1 ≤ p < ∞. Lemma 2.6. ([3]) For all u ∈ E0α and 1/2 < α ≤ 1, we have ||u||L2 ≤
Tα ||0 Dαt u(t)||L2 , Γ(α + 1)
(2.4)
1
Remark 2.7. If we define ||u||α as
T α− 2 ||u||∞ ≤ ||0 Dαt u(t)||L2 . √ Γ(α) 2α − 1 ||u||α = (
Z
0
T
|0 Dαt u(t)|2 dt)
1 2
,
(2.5)
(2.6)
then in the view of (2.4) and (2.5), we can conclude that ||u||α defined in (2.3) and (2.6) are equivalent to the norms.
Therefore E0α can be considered with the norm defined in (2.6) in the remaining sequel.
Lemma 2.8. ([3]) The fractional derivative space E0α for 0 < α ≤ 1 is a reflexive as well as separable Banach space. 4
Remark 2.9. Obviously, with respect to the norm ||u||α, E0α for 0 < α ≤ 1 is a reflexive and separable Hilbert space. Theorem 2.10. ([19])(Lax-Milgram theorem) Let b : K × K → R be a bounded bilinear form where K be a real
Hilbert space. If b is coercive, then there exists a unique v ∈ K such that
for every t ∈ K and for any g ∈ K ′ (the dual of K).
b(v, t) = hg, ti,
In addition, if b is symmetric, then the functional ψ : K → R defined by 1 ψ(t) = b(t, t) − hg, ti, 2 has its minimum value at v. 3. Variational structure and main result This section is devoted to the formation of variational structure to the problem. we have following lemma for the equivalent form of problem (1.1). Lemma 3.1. For u ∈ E0α , the following form is an equivalent form of the problem (1.1): Z T n n Z ti+1 X X fi (t)(v(t) − v(ti+1 ))dt + (c0 Dαt u(t))(c0 Dαt v(t))dt = [ci−1 − ci ]v(ti ), 0
Proof. Let u ∈
i=0
E0α ,
then for each v ∈
si
C0∞ [0, T ]
∀v ∈ C0∞ [0, T ].
i=1
Using the method of variational approach to impulsive differential
equations with non-instantaneous [15], to instantaneous impulsive fractional differential equations [10], (2.1) and
(2.2), we have, Z Z T Z t ′ (c0 Dαt u(t))(c0 Dα−1 v (t))dt = (c0 Dαt u(t))(c0 Dαt v(t))dt = t
0
0
0
=
Z
t1
0
+
Z
T
+
i=1
T
sn
′ α−1 c α t DT (0 Dt u(t))v (t)dt
n Z X
ti
α−1 c α ′ t DT (0 Dt u(t))v (t)dt si
′ α−1 c α t DT (0 Dt u(t))v (t)dt
+
n−1 Z X i=1
ti +1 si
′ α−1 c α t DT (0 Dt u(t))v (t)dt
′ α−1 c α t DT (0 Dt u(t))v (t)dt
Z
t1
d α−1 c α t DT (0 Dt u(t)) v(t)dt 0 dt n n Z X X c α − α−1 c α + + [t Dα−1 ( D u(s ))v(s ) − D ( D u(t ))v(t )] − i t T i i i T 0 t 0 t − c α = t Dα−1 T (0 Dt u(t1 ))v(t1 ) −
si
d α−1 c α t DT (0 Dt u(t)) v(t)dt dt i=1 i=1 ti n−1 Z ti+1 n−1 X X d α−1 c α + c α − α−1 c α ))v(s )] − + ( D u(s ( D u(t ))v(t ) − D [t Dα−1 i t DT (0 Dt u(t)) v(t)dt i+1 t i T T 0 t 0 t i+1 dt i=1 si i=1 Z T d + c α α−1 c α − t Dα−1 t DT (0 Dt u(t)) v(t)dt T (0 Dt u(sn ))v(sn ) − sn dt Z T n X + α c α + α−1 c α c α = [t Dα−1 t DT (0 Dt u(t))v(t)dt + T (0 Dt u(ti )) − t DT (0 Dt u(ti ))]v(ti ) 0
+
n X i=1
i=1
+ α−1 c α c α + [t Dα−1 T (0 Dt u(si )) − t DT (0 Dt u(si ))]v(si ).
5
Using the given conditions of problem (1.1), we get, Z
0
T
α c α t DT (0 Dt u(t))v(t)dt
=
Z
t 0
(c0 Dαt u(t))(c0 Dαt v(t))dt
Also from problem (1.1), we have, Z T n Z X α c α t DT (0 Dt u(t))v(t)dt = 0
i=0
= =
n Z X
si ti+1
i=0 si n Z ti+1 X si
i=0
=
ti+1
n Z X i=0
ti+1
si
i=1
[ci−1 − ci ]v(ti ) +
α c α t DT (0 Dt u(t))v(t)dt
fi (t)v(t)dt + fi (t)v(t)dt +
n Z X
+
n Z X i=1
si
i=1 ti n Z si X i=1
ti
− −
si
ti
n−1 Z X i=0
ti+1 si
fi+1 (t)v(ti+1 )dt.
i=0
ti+1 si
(3.1)
α c α t DT (0 Dt u(t))v(t)dt
d α−1 c α t DT (0 Dt u(t)) v(t)dt dt d ci v(t)dt dt
fi (t)v(t)dt.
Using (3.1), (3.2) and v(tn+1 ) = v(T ) = 0, we have Z T n Z X c α c α (0 Dt u(t))(0 Dt v(t))dt = 0
−
n X
(3.2)
fi (t)(v(t) − v(ti+1 ))dt +
n X i=1
[ci−1 − ci ]v(ti ).
(3.3)
The proof is completed. Now we can give the concept of the weak solution for (1.1) after considering the equivalent form (3.3) of the problem. Definition 3.2. Let u ∈ E0α , then the function u is called weak solution of (1.1) if (3.3) is satisfied for each v ∈ C0∞ . We define a functional ψ : E0α → R as ψ(u) =
1 2
Z
0
T
|c0 Dαt u(t)|2 dt −
n Z X i=0
ti+1 si
n
X fi (t) u(t) − u(ti+1 ) dt − [ci−1 − ci ]u(ti ).
(3.4)
i=1
Then in the view of (M1 ), it is clear that ψ ∈ C 1 (E0α , R) and ′
hψ (u), vi =
Z
0
T
(c0 Dαt u(t))(c0 Dαt v(t))dt
−
n Z X i=0
ti+1 si
fi (t)(v(t) − v(ti+1 ))dt −
n X i=1
[ci−1 − ci ]v(ti ).
Hence the critical points of ψ are the weak solutions of (1.1). Theorem 3.3. For each fi ∈ L2 (si , ti+1 ), non-instantaneous fractional impulsive problem (1.1) has a weak solution u ∈ E0α which is unique and also the functional ψ(u) has minimum value in E0α . Proof. If we define the operators b : E0α × E0α → R and l(v) : E0α → R as Z T (c0 Dαt u(t))(c0 Dαt v(t))dt, b(u, v) = 0
and
l(v) =
n Z X i=0
ti+1 si
fi (t)(v(t) − v(ti+1 ))dt + 6
n X i=1
[ci−1 − ci ]v(ti ),
(3.5)
(3.6)
then we note that finding u ∈ E0α of the following problem is equivalent to finding the weak solutions of (1.1), ∀v ∈ E0α .
b(u, v) = l(v),
(3.7)
From (3.5), it is obvious that b is bilinear, coercive, symmetric and bounded. Now for l(v), using (2.5) and Holder ¨ inequality, we have |l(v)| ≤
n X
i=0 n X
≤2 ≤
|| fi ||L2
i=0
Z
ti+1 si
|v(t) − v(ti+1 )|2 dt 1
|| fi ||L2 (ti+1 − si ) 2 ||v||∞ +
n X i=0
12
+
n X i=0
|ci−1 − ci ||v(ti )|
|ci−1 − ci | ||v||∞
X X 1 T α− 2 2 || fi ||L2 (ti+1 − si ) 2 + |ci−1 − ci | ||v||α , √ Γ(α) 2α − 1 i=0 i=0 1
n
n
which shows that l(v) is bounded. Also from (3.6), it is clear that l is linear. Thus by Lax-Milgram theorem 2.10 non-instantaneous impulsive fractional boundary value problem (1.1) has a weak solution u ∈ E0α which is unique and
minimizes the functional ψ. The proof is completed.
Example 3.4. Consider the following boundary value problem,
where 0 = s0 <
1 7
= t1 <
9 7
3 3 4 c 4 t ∈ (si , ti+1 ], i = 0, 1, 2, t D2 0 Dt u(t) = fi (t), − 41 c 34 t ∈ (t1 , s1 ], t D2 0 Dt u(t) = 0.2, 1 3 1 3 − − − + 4 c 4 4 c 4 t D2 0 Dt u(s1 ) =t D2 0 Dt u(s1 ) 3 1 t D− 4 (c D 4 u(0)) = 0.1, u(0) = u(2) = 0, 2 0 t
= s1 < 2 = t2 and fi (t) =
(3.8)
√ ti cos it. Since fi (t) ∈ L2 (si , ti+1 ) for each i = 0, 1, therefore it
follows from Theorem 3.3 that the above non-instantaneous impulsive problem of fractional order has a unique weak solution. References [1] J. J. Nieto, D. O’Regan, Variational approach to impulsive differential equations, Nonlinear Anal., Real World Appl., 2(2009), 680-690. [2] A. A. Kilbas, M. H. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science, Amesterdam, (2006). [3] F. Jiao, Y. Zhou, Existence results for fractional boundary value problem via critical point theory, Internet. J. Bifur. Chous Appl. Sci. Engrg., 22(2012), 17 pages. [4] V. Lashmikantham, S. Leela, J. V. Devi, Theory of fractional dynomic systems, Cambridge Scientificc Publishers, Cambridge, (2009).
7
[5] J. R. Wang, X. Li, Periodic BVP for integer/fractional order nonlinear differential equations with non instantaneous impulses, J. Appl. Math. Comput., 46, 1-2, (2014), 321-334. [6] I. Pudlubny, Fractional differential equations, Math. Sci. Eng., Academic Press, New York, (1999). [7] V. E. Tarasov, Fractional dynamics: application of frcational calculus to dynamics of particals, fields and media, Springer, Beijing, (2011). [8] A. Sood, S. K. Srivastava, On Stability of differential system with noninstantaneous impulses, MAth. Probl. Eng., 2015, Article ID 691687. [9] J. R. Wang, M. Feckan, A general class of impulsive evolution equations, Topological Math. Nonlinear Anal., 46(2015), 915-934. [10] G. Bonanno, R. Rodriguez-Lopez, S. Tersian, Existence of solutions to boundary value problem for impulsive fractional differential equations, Fract. Calc. Appl. Anal., 17(4), (2014), 1016-1038. [11] M. Pierri, D. O’Regan, V. Rolnik, Existence of solutions for semi-linear abstract differential equaitions with not instantaneous impulses, Appl. MAth. Comput., 219, (2013), 6743-6749. [12] R. Agarwal, D. O’Regan, S. Hristova, Stability by Lyapunov functions of nonlinear differential equations with non-instantaneous impulses, J. Appl. Math. Comput., (2015), 9-22, DOI 10.1007/s12190-015-0961-z. [13] P. Li, H. Wang, Z. Li, Solutions for impulsive fractional differential equations via variational methods, J. Funct. Spaces, (2016), Article ID 2941368. [14] E. Hernandez, D. O’Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141(5)(2013), 1641-1649. [15] L. Bai, J. J. Nieto, Variational approach to differential equations with not instantaneous impulses, Appl. Math. Lett., 73(2017), 44-48. [16] J. R. Wang, M. Feckan, Y. Zhou, Ulams type stability of impulsive ordinary differential equations, J. Math. Anal. Appl., 395(2012), 258-264. [17] N. Nyamoradi, R. Rodriguez-Lopez, On boundary value problems for impulsive fractional differential equations, J. Appl. Math. Comput., 271(2015), 874-892. [18] J. R. Wang, A. G. Ibrahim, M. Feckan, Nonlocal impulsive fractional differrential inclusions withe fractional sectorial operators on Banach spaces, Applied Math. Comput., 257(2015), 103-108. [19] M. Chipot, Elements of Nonlinear Analysis, Birkhauser Verlag, Basel, (2000). [20] M. Feckan, J. R. Wang, Y. Zhou, Periodic solutions for nonlinear evolution equations with non-instantaneous impulses, Nonauton. Dyn. Syst., 1(2014), 93-101.
8
PII: DOI: Reference:
S0893-9659(18)30079-X https://doi.org/10.1016/j.aml.2018.03.014 AML 5462
To appear in:
Applied Mathematics Letters
Received date : 26 December 2017 Revised date : 15 March 2018 Accepted date : 15 March 2018 Please cite this article as: A. Khaliq, M.u. Rehman, On variational methods to non-instantaneous impulsive fractional differential equation, Appl. Math. Lett. (2018), https://doi.org/10.1016/j.aml.2018.03.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
On variational methods to non-instantaneous impulsive fractional differential equation Adnan Khaliq∗, Mujeeb ur Rehman School of Natural Sciences, National University of Sciences and Technology, Islamabad Pakistan
Abstract In this paper by using the variational methods for a class of impulsive differential equation of fractional order with non-instantaneous impulses, we setup sufficient conditions for the existence and uniqueness of weak solutions. The problem is reduced to an equivalent form such that the weak solutions of the problem are defined as the critical points of a functional. Main results of the present work are established by using Lax-Milgram Theorem. Keywords: Fractional differential equations, Variational methods, Lax-Milgram theorem, Non-instantaneous impulse.
1. Introduction Fractional calculus generalizes the integer order integration and differentiation concepts to an arbitrary (real or complex) order. Fractional calculus is the most well known and valuable branch of mathematics which gives a good framework for biological and physical phenomena, mathematical modeling of engineering etc. To get a couple of developments about the theory of fractional differential equations, one can allude to the monographs of Lakshmikanthem et al. [4], Kilbas et al. [2], Pudlubny [6], Tarasov [7] and the references in there. The impulsive differential equations have become very useful tools to portray the change of processes in which discontinuous jumps (impulsive conditions) abruptly appear. Such processes are normally seen in engineering, biology and physics. A maximum number of research papers are written on the existence theory of the solutions for the impulsive differential equations with instantaneous impulses [9, 16, 18]. But in many cases it has been noted that certain dynamics of evolution processes cannot delineate by the instantaneous impulses, for example: High or low level of glucose, Pharmacotherapy, this situation can be illustrated as an impulsive jump starting suddenly at any fixed point and remains continue on a finite interval of time. Such sort of systems are more useful to study the dynamics of evolution processes and are called non-instantaneous impulsive systems. Hernadez & O’Regan introduced non-instantaneous impulsive differential equations in [14]. Recently, many authors applied nonlinear analysis methods (e.g. fixed point theory etc), to obtain the results dealing with the impulsive differential equations with non-instantaneous impulses for the existence and multiplicity of their solutions [5, 8, 11, 12, 20]. However, the problems (e.g. discussing BVP) for which the equivalent integral equation is not easy to obtain, these ∗ Corresponding
author. Tel +92 322 6623874 Email addresses: [email protected] ( Adnan Khaliq), [email protected] ( Mujeeb ur Rehman)
Preprint submitted to Elsevier
March 15, 2018
popular techniques look not appropriate. Some times it becomes very difficult for the boundary value problem of the fractional order and especially for those containing both right and left fractional derivatives to construct a suitable space and equivalent integral equation. Contrary, for such problems where equivalent integral equation is not available there are other very useful approaches: Variational techniques and critical point theory. Jiao and Zhou [3] first time showed that these useful approaches are powerful tools while dealing with the such class of fractional boundary value problems for the existence and uniqueness of their solutions. Critical point theorems and the variational methods were also used to setup results for impulsive problems and their solutions. In this direction pioneering work was done by Nieto and O’Regan [1] by considering a class of impulsive problem of second order. In [10, 13, 17], authors setup results for the fractional impulsive differential equations with instantaneous impulses by applying variational methods and critical point theory. On the other hand, the existence and uniqueness of solutions to impulsive differential equations of fractional order containing non-instantaneous impulses by using critical point theory and variational methods is never been investigated so far to the best of our knowledge. So in this paper we solved this untouched problem. Recently L. Bai & J. J. Nieto [15] discussed 2nd order impulsive differential equation with non-instantaneous impulses. By using Lax-Milgram theorem, they guaranteed for the existence and uniqueness of the weak solutions of the problem. Getting motivation from work cited above, we consider following fractional impulsive boundary value problem with not instantaneous impulses: α c α t ∈ (si , ti+1 ], i = 0, 1, 2, · · · , n, t DT 0 Dt u(t) = fi (t), c α α−1 t ∈ (ti , si ], i = 1, 2, · · · , n, t DT 0 Dt u(t) = ci α−1 c α + − α−1 c α i = 1, 2, · · · , n, t DT 0 Dt u(si ) 0 Dt u(si ) =t DT t Dα−1 (c Dα u(0)) = c0 , u(0) = u(T ) = 0, T 0 t
(1.1)
here 0 = s0 < t1 < s1 < t2 < s2 < · · · < tn < sn < tn+1 = T , c0 Dαt and t DαT are left Caputo fractional derivative and right c α Riemann-Liouville fractional derivatives of the order 0 < α ≤ 1 respectively. Keeping the derivative t Dα−1 0 Dt u(t) T
constant on a finite interval (ti , si ], the impulses begin suddenly at the points ti . Here fi : (si , ti+1 ] → R are given α α−1 ± α ± functions, ci are given constants and t Dα−1 0 Dt u(s) . T (0 Dt u(si )) = lim s→si t DT Through out the paper, we assume following condition is satisfied. (M1 ). For each i = 0, 1, 2, · · · n, function fi ∈ L2 (si , ti+1) .
Remark 1.1. It should be noted that for α = 1, one has t DαT u(t) = −u′ (t) and c0 Dαt u(t) = u′ (t), and our problem (1.1) reduces to [15]. So our problem (1.1) gives generalization of [15].
We organized rest of the article as follows. According to the requirements of paper, some basic definitions and preliminary results are given in Section 2. After establishing variational structure of BVP (1.1), we prove our main results in Section 3. In the end an example is given to highlight the effectiveness of our results.
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2. Preliminaries From the fractional calculus, some of the definitions and basic results are given in this section which will be utilized as a part of the evidence later in this paper. Definition 2.1. ([2, 6]) The right Riemann-Liouville fractional integral for a function u defined on [a, b] of order α > 0 is denoted by t D−α b u(t) and is defined by −α t Db u(t)
1 Γ(α)
=
Z
b t
(v − t)α−1 u(v)dv,
t ∈ [a, b],
the right hand side is required to be pointwise defined on [a, b]. Definition 2.2. ([2, 6]) The left Riemann-Liouville fractional integral for a function u defined on [a, b] of order α > 0 is denoted by a D−α t u(t) and is defined as −α a Dt u(t)
1 = Γ(α)
Z
a
t
(t − v)α−1 u(v)dv,
t ∈ [a, b],
here Γ is the gamma function and the right hand side is required to be pointwise defined on [a, b]. Definition 2.3. ([2, 6]) The left and right Riemann-Liouville fractional derivatives for a function u defined on [a, b] of order α > 0 are denoted by a Dαt u(t) and t Dαb u(t), respectively and are defined by α a Dt u(t)
and α t Db u(t)
dn −(n−α) u(t) aD dtn t Z dn t 1 n−α−1 = (t − v) u(v)dv , Γ(n − α) dtn a =
dn −(n−α) u(t) tD dtn b n Z t 1 n d (−1) n (v − t)n−α−1 u(v)dv , = Γ(n − α) dt a
= (−1)n
respectively where t ∈ [a, b], n − 1 < α ≤ n and n ∈ N.
Definition 2.4. ([2, 6]) Let u ∈ AC m ([a, b], R) and m − 1 < α ≤ m, then the right and left Caputo fractional derivatives of order α are denoted as ct Dαb u(t) and ca Dαt u(t) respectively, are given by c α t Db u(t)
and c α a Dt u(t)
respectively where t ∈ [a, b].
dm = (−1)mt Db−(m−α) m u(t) dt Z t (−1)m (v − t)m−α−1 u(m) (v)dv, = Γ(m − α) a dm u(t) dtm Z t 1 (t − v)m−α−1 u(m) (v)dv, = Γ(m − α) a = a Dt−(m−α)
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Lemma 2.5. ([2, 6]) (a). Let m − 1 < α ≤ m and u, v ∈ L2 (a, b) then Z b Z b 1−α u(t) t D1−α D u(t) v(t)dt = a t b v(t) dt. a
(2.1)
a
(b). Let m − 1 < α ≤ m, v ∈ AC([a, b], RN ), v′ ∈ L2 ([a, b], RN ), ca Dαt u(t) ∈ L2 ([a, b], RN ) and t DαT (c0 Dαt u(t)) ∈ AC([a, b], RN ), then Z
a
b
(ca Dαt u(t))(ca Dαt v(t))dt = =
Z
b
a
Z
(ca Dαt u(t))(ca Dα−1 v′ (t))dt t
b
′ α−1 c α t DT 0 Dt u(t) v (t)dt
a
c α t=b = t Dα−1 T (0 Dt u(t))v(t)|t=a −
=
α−1 c α t=b t DT (0 Dt u(t))v(t)|t=a
+
Z
(2.2)
b
a
Z
b
a
d (t Dα−1 (c Dα u(t)))v(t)dt dt T 0 t α−1 c α t DT (0 Dt u(t))v(t)dt.
Fractional Derivative Space A fractional derivative space is required in order to use the critical point theory for the existence of solutions to the problem (1.1). We define fractional derivative space E0α by the closure of C0∞ ([0, T ], RN ) with respect to norm ||u||α = (
Z
T
0
(|0 Dαt u(t)|2 + |u(t)|2 )dt)1/2 ,
(2.3)
where 0 Dαt denotes the left Riehmann-Liouville derivative of fractional order. Clearly E0α is space of the functions u for which u ∈ L2 [0, T ], u(0) = u(T ) = 0 and α order fractional derivative 0 Dαt u(t) ∈ L2 [0, T ]. This space is similar to
the fractional derivative space introduced in [3]. First give us a chance to review that
||u||L p = (
||u||∞ = max |u(t)|, t∈[0,T ]
Z
T
0
p
1 p
|u(s)| ds) ,
for any fixed t ∈ [0, T ] , T > 0 and 1 ≤ p < ∞. Lemma 2.6. ([3]) For all u ∈ E0α and 1/2 < α ≤ 1, we have ||u||L2 ≤
Tα ||0 Dαt u(t)||L2 , Γ(α + 1)
(2.4)
1
Remark 2.7. If we define ||u||α as
T α− 2 ||u||∞ ≤ ||0 Dαt u(t)||L2 . √ Γ(α) 2α − 1 ||u||α = (
Z
0
T
|0 Dαt u(t)|2 dt)
1 2
,
(2.5)
(2.6)
then in the view of (2.4) and (2.5), we can conclude that ||u||α defined in (2.3) and (2.6) are equivalent to the norms.
Therefore E0α can be considered with the norm defined in (2.6) in the remaining sequel.
Lemma 2.8. ([3]) The fractional derivative space E0α for 0 < α ≤ 1 is a reflexive as well as separable Banach space. 4
Remark 2.9. Obviously, with respect to the norm ||u||α, E0α for 0 < α ≤ 1 is a reflexive and separable Hilbert space. Theorem 2.10. ([19])(Lax-Milgram theorem) Let b : K × K → R be a bounded bilinear form where K be a real
Hilbert space. If b is coercive, then there exists a unique v ∈ K such that
for every t ∈ K and for any g ∈ K ′ (the dual of K).
b(v, t) = hg, ti,
In addition, if b is symmetric, then the functional ψ : K → R defined by 1 ψ(t) = b(t, t) − hg, ti, 2 has its minimum value at v. 3. Variational structure and main result This section is devoted to the formation of variational structure to the problem. we have following lemma for the equivalent form of problem (1.1). Lemma 3.1. For u ∈ E0α , the following form is an equivalent form of the problem (1.1): Z T n n Z ti+1 X X fi (t)(v(t) − v(ti+1 ))dt + (c0 Dαt u(t))(c0 Dαt v(t))dt = [ci−1 − ci ]v(ti ), 0
Proof. Let u ∈
i=0
E0α ,
then for each v ∈
si
C0∞ [0, T ]
∀v ∈ C0∞ [0, T ].
i=1
Using the method of variational approach to impulsive differential
equations with non-instantaneous [15], to instantaneous impulsive fractional differential equations [10], (2.1) and
(2.2), we have, Z Z T Z t ′ (c0 Dαt u(t))(c0 Dα−1 v (t))dt = (c0 Dαt u(t))(c0 Dαt v(t))dt = t
0
0
0
=
Z
t1
0
+
Z
T
+
i=1
T
sn
′ α−1 c α t DT (0 Dt u(t))v (t)dt
n Z X
ti
α−1 c α ′ t DT (0 Dt u(t))v (t)dt si
′ α−1 c α t DT (0 Dt u(t))v (t)dt
+
n−1 Z X i=1
ti +1 si
′ α−1 c α t DT (0 Dt u(t))v (t)dt
′ α−1 c α t DT (0 Dt u(t))v (t)dt
Z
t1
d α−1 c α t DT (0 Dt u(t)) v(t)dt 0 dt n n Z X X c α − α−1 c α + + [t Dα−1 ( D u(s ))v(s ) − D ( D u(t ))v(t )] − i t T i i i T 0 t 0 t − c α = t Dα−1 T (0 Dt u(t1 ))v(t1 ) −
si
d α−1 c α t DT (0 Dt u(t)) v(t)dt dt i=1 i=1 ti n−1 Z ti+1 n−1 X X d α−1 c α + c α − α−1 c α ))v(s )] − + ( D u(s ( D u(t ))v(t ) − D [t Dα−1 i t DT (0 Dt u(t)) v(t)dt i+1 t i T T 0 t 0 t i+1 dt i=1 si i=1 Z T d + c α α−1 c α − t Dα−1 t DT (0 Dt u(t)) v(t)dt T (0 Dt u(sn ))v(sn ) − sn dt Z T n X + α c α + α−1 c α c α = [t Dα−1 t DT (0 Dt u(t))v(t)dt + T (0 Dt u(ti )) − t DT (0 Dt u(ti ))]v(ti ) 0
+
n X i=1
i=1
+ α−1 c α c α + [t Dα−1 T (0 Dt u(si )) − t DT (0 Dt u(si ))]v(si ).
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Using the given conditions of problem (1.1), we get, Z
0
T
α c α t DT (0 Dt u(t))v(t)dt
=
Z
t 0
(c0 Dαt u(t))(c0 Dαt v(t))dt
Also from problem (1.1), we have, Z T n Z X α c α t DT (0 Dt u(t))v(t)dt = 0
i=0
= =
n Z X
si ti+1
i=0 si n Z ti+1 X si
i=0
=
ti+1
n Z X i=0
ti+1
si
i=1
[ci−1 − ci ]v(ti ) +
α c α t DT (0 Dt u(t))v(t)dt
fi (t)v(t)dt + fi (t)v(t)dt +
n Z X
+
n Z X i=1
si
i=1 ti n Z si X i=1
ti
− −
si
ti
n−1 Z X i=0
ti+1 si
fi+1 (t)v(ti+1 )dt.
i=0
ti+1 si
(3.1)
α c α t DT (0 Dt u(t))v(t)dt
d α−1 c α t DT (0 Dt u(t)) v(t)dt dt d ci v(t)dt dt
fi (t)v(t)dt.
Using (3.1), (3.2) and v(tn+1 ) = v(T ) = 0, we have Z T n Z X c α c α (0 Dt u(t))(0 Dt v(t))dt = 0
−
n X
(3.2)
fi (t)(v(t) − v(ti+1 ))dt +
n X i=1
[ci−1 − ci ]v(ti ).
(3.3)
The proof is completed. Now we can give the concept of the weak solution for (1.1) after considering the equivalent form (3.3) of the problem. Definition 3.2. Let u ∈ E0α , then the function u is called weak solution of (1.1) if (3.3) is satisfied for each v ∈ C0∞ . We define a functional ψ : E0α → R as ψ(u) =
1 2
Z
0
T
|c0 Dαt u(t)|2 dt −
n Z X i=0
ti+1 si
n
X fi (t) u(t) − u(ti+1 ) dt − [ci−1 − ci ]u(ti ).
(3.4)
i=1
Then in the view of (M1 ), it is clear that ψ ∈ C 1 (E0α , R) and ′
hψ (u), vi =
Z
0
T
(c0 Dαt u(t))(c0 Dαt v(t))dt
−
n Z X i=0
ti+1 si
fi (t)(v(t) − v(ti+1 ))dt −
n X i=1
[ci−1 − ci ]v(ti ).
Hence the critical points of ψ are the weak solutions of (1.1). Theorem 3.3. For each fi ∈ L2 (si , ti+1 ), non-instantaneous fractional impulsive problem (1.1) has a weak solution u ∈ E0α which is unique and also the functional ψ(u) has minimum value in E0α . Proof. If we define the operators b : E0α × E0α → R and l(v) : E0α → R as Z T (c0 Dαt u(t))(c0 Dαt v(t))dt, b(u, v) = 0
and
l(v) =
n Z X i=0
ti+1 si
fi (t)(v(t) − v(ti+1 ))dt + 6
n X i=1
[ci−1 − ci ]v(ti ),
(3.5)
(3.6)
then we note that finding u ∈ E0α of the following problem is equivalent to finding the weak solutions of (1.1), ∀v ∈ E0α .
b(u, v) = l(v),
(3.7)
From (3.5), it is obvious that b is bilinear, coercive, symmetric and bounded. Now for l(v), using (2.5) and Holder ¨ inequality, we have |l(v)| ≤
n X
i=0 n X
≤2 ≤
|| fi ||L2
i=0
Z
ti+1 si
|v(t) − v(ti+1 )|2 dt 1
|| fi ||L2 (ti+1 − si ) 2 ||v||∞ +
n X i=0
12
+
n X i=0
|ci−1 − ci ||v(ti )|
|ci−1 − ci | ||v||∞
X X 1 T α− 2 2 || fi ||L2 (ti+1 − si ) 2 + |ci−1 − ci | ||v||α , √ Γ(α) 2α − 1 i=0 i=0 1
n
n
which shows that l(v) is bounded. Also from (3.6), it is clear that l is linear. Thus by Lax-Milgram theorem 2.10 non-instantaneous impulsive fractional boundary value problem (1.1) has a weak solution u ∈ E0α which is unique and
minimizes the functional ψ. The proof is completed.
Example 3.4. Consider the following boundary value problem,
where 0 = s0 <
1 7
= t1 <
9 7
3 3 4 c 4 t ∈ (si , ti+1 ], i = 0, 1, 2, t D2 0 Dt u(t) = fi (t), − 41 c 34 t ∈ (t1 , s1 ], t D2 0 Dt u(t) = 0.2, 1 3 1 3 − − − + 4 c 4 4 c 4 t D2 0 Dt u(s1 ) =t D2 0 Dt u(s1 ) 3 1 t D− 4 (c D 4 u(0)) = 0.1, u(0) = u(2) = 0, 2 0 t
= s1 < 2 = t2 and fi (t) =
(3.8)
√ ti cos it. Since fi (t) ∈ L2 (si , ti+1 ) for each i = 0, 1, therefore it
follows from Theorem 3.3 that the above non-instantaneous impulsive problem of fractional order has a unique weak solution. References [1] J. J. Nieto, D. O’Regan, Variational approach to impulsive differential equations, Nonlinear Anal., Real World Appl., 2(2009), 680-690. [2] A. A. Kilbas, M. H. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science, Amesterdam, (2006). [3] F. Jiao, Y. Zhou, Existence results for fractional boundary value problem via critical point theory, Internet. J. Bifur. Chous Appl. Sci. Engrg., 22(2012), 17 pages. [4] V. Lashmikantham, S. Leela, J. V. Devi, Theory of fractional dynomic systems, Cambridge Scientificc Publishers, Cambridge, (2009).
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[5] J. R. Wang, X. Li, Periodic BVP for integer/fractional order nonlinear differential equations with non instantaneous impulses, J. Appl. Math. Comput., 46, 1-2, (2014), 321-334. [6] I. Pudlubny, Fractional differential equations, Math. Sci. Eng., Academic Press, New York, (1999). [7] V. E. Tarasov, Fractional dynamics: application of frcational calculus to dynamics of particals, fields and media, Springer, Beijing, (2011). [8] A. Sood, S. K. Srivastava, On Stability of differential system with noninstantaneous impulses, MAth. Probl. Eng., 2015, Article ID 691687. [9] J. R. Wang, M. Feckan, A general class of impulsive evolution equations, Topological Math. Nonlinear Anal., 46(2015), 915-934. [10] G. Bonanno, R. Rodriguez-Lopez, S. Tersian, Existence of solutions to boundary value problem for impulsive fractional differential equations, Fract. Calc. Appl. Anal., 17(4), (2014), 1016-1038. [11] M. Pierri, D. O’Regan, V. Rolnik, Existence of solutions for semi-linear abstract differential equaitions with not instantaneous impulses, Appl. MAth. Comput., 219, (2013), 6743-6749. [12] R. Agarwal, D. O’Regan, S. Hristova, Stability by Lyapunov functions of nonlinear differential equations with non-instantaneous impulses, J. Appl. Math. Comput., (2015), 9-22, DOI 10.1007/s12190-015-0961-z. [13] P. Li, H. Wang, Z. Li, Solutions for impulsive fractional differential equations via variational methods, J. Funct. Spaces, (2016), Article ID 2941368. [14] E. Hernandez, D. O’Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141(5)(2013), 1641-1649. [15] L. Bai, J. J. Nieto, Variational approach to differential equations with not instantaneous impulses, Appl. Math. Lett., 73(2017), 44-48. [16] J. R. Wang, M. Feckan, Y. Zhou, Ulams type stability of impulsive ordinary differential equations, J. Math. Anal. Appl., 395(2012), 258-264. [17] N. Nyamoradi, R. Rodriguez-Lopez, On boundary value problems for impulsive fractional differential equations, J. Appl. Math. Comput., 271(2015), 874-892. [18] J. R. Wang, A. G. Ibrahim, M. Feckan, Nonlocal impulsive fractional differrential inclusions withe fractional sectorial operators on Banach spaces, Applied Math. Comput., 257(2015), 103-108. [19] M. Chipot, Elements of Nonlinear Analysis, Birkhauser Verlag, Basel, (2000). [20] M. Feckan, J. R. Wang, Y. Zhou, Periodic solutions for nonlinear evolution equations with non-instantaneous impulses, Nonauton. Dyn. Syst., 1(2014), 93-101.
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