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 theory1 Introduction
The theory of impulsive differential equations has been emerging as an important area of investigation in recent years [13]. There is a vast literature on the existence of solutions by using topological methods, including fixed point theorems, LeraySchauder degree theory, and fixed point index theory [415]. But it is quite difficult to apply the variational approach 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 [1620]. Motivated by [17], 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 secondorder nonlinear impulsive differential equation:
where , , , and () are any real numbers, is a real function defined on , where R denotes the set of all real numbers, and is continuous on , left continuous at , i.e.
for any (), and the right limit at exists, i.e.
(denoted by ) exists for any (). denotes the jump of at , i.e.
where and represent the right and left limits of at , respectively. Similarly,
where and represent the right and left limits of at , respectively. Let = { is a real function on J such that is continuous at , left continuous at , and exists, } and = { is continuous at and , exist, }. A function is called a solution of BVP (1) if satisfies (1).
Let us list some conditions.
(H_{1}) There exist , and such that
(H_{2}) There exist and such that
Lemma 1is a solution of BVP (1) if and only ifis a solution of the integral equation
where
and
Proof For , we have the formula (see [21], Lemma 1(b))
So, if is a solution of BVP (1), then, by (1) and (6), we have
It is clear, by (5), that
so
Substituting (9) into (7), we get
By virtue of (5), we see that (in fact, ) and
Substituting (11) into (10), we get
so is a solution of the integral equation (2).
Conversely, suppose that is a solution of (2), i.e.
By (4), it is clear that is continuous on , so differentiation of (12) gives
Differentiating again, we get
From (13) we see that and () exist and
It follows from (4), (5), (12), (14), and (15) that and satisfies (1). □
Lemma 2Let condition (H_{1}) be satisfied. Ifis a solution of the integral equation (2), then.
Proof It is clear, for function defined by (5),
By (4), (5), (16), and condition (H_{1}), we have
so,
where
It is clear that satisfies the Caratheodory condition, i.e. is measurable with respect to t on J for every and is continuous with respect to v on R for almost (in fact, is discontinuous only at ()), so (17) implies [22,23] that the operator g defined by
is bounded and continuous from into , where ().
Let be a solution of the integral equation (2). Then by the Hölder inequality,
which implies by virtue of the uniform continuity of on that . □
2 Variational approach
Theorem 1If conditions (H_{1}) and (H_{2}) are satisfied, then BVP (1) has at least one solution.
Proof By Lemma 1 and Lemma 2, we need only to show that the integral equation (2) has a solution . The integral equation (2) can be written in the form
where G is the linear integral operator defined by
and the nonlinear operator g is defined by (18), which is bounded and continuous from into (). It is well known that is a positivedefinite kernel with eigenvalues () and, by the continuity of , we have
so [22,23] the linear operator G defined by (20) is completely continuous from into and also from into , and , where (the positive squareroot operator of G) is completely continuous from into and denotes the adjoint operator of H, which is completely continuous from into . We now show that (19) has a solution is equivalent to the equation
has a solution . In fact, if is a solution of (19), i.e., then , so, and u is a solution of (22). Conversely, if is a solution of (22), then , so, and v is a solution of (19). Consequently, we need only to show that (22) has a solution . It is well known [22,23] that the functional Φ defined by
is a functional on and its Fréchet derivative is
Hence we need only to show that there exists a such that (θ denotes the zero element of ), i.e.u is a critical point of functional Φ.
By (4), (5), (16), and condition (H_{1}), we have
and
So, (25), (26), and condition (H_{2}) imply
It is well known [24],
where G is defined by (20) and is regarded as a positivedefinite operator from into , and denotes the largest eigenvalue of G. It follows from (23), (27), and (28) that
which implies by virtue of (see condition (H_{2})) that
It is well known [22,23] that the ball is weakly closed and weakly compact and the functional is weakly lower semicontinuous, so, there exists such that
It follows from (31) and (32) that
Hence and the theorem is proved. □
Example 1 Consider the BVP
where , , , and () are any real numbers.
Conclusion BVP (33) has at least one solution .
Proof Evidently, (33) is a BVP of the form (1) with
It is clear that . By (34), we have
Moreover, it is well known that
So, (35) and (36) imply that
and consequently, condition (H_{1}) is satisfied for , and . On the other hand, choose such that
By (35), we have
It follows from (38) and (39) that
Since, by virtue of (37),
we see that (40) implies that condition (H_{2}) is satisfied for and . Hence, our conclusion follows from Theorem 1. □
By using the Mountain Pass Lemma and the Minimax Principle established by Ambrosetti and Rabinowitz [25,26], we have obtained in [23] the existence of a nontrivial solution and the existence of infinitely many nontrivial solutions for a class of nonlinear integral equations. Since (2) is a special case of such nonlinear integral equations, we get the following result for (2).
Lemma 3 (Special case of Theorem 1 and Theorem 2 in [23])
Suppose the following.
(c) asuniformly forandasuniformly for.
Then the integral equation (2) has at least one nontrivial solution in. If, in addition,
Then the integral equation (2) has infinite many nontrivial solutions in.
Let us list more conditions for the function .
(H_{3}) There exist and such that
(H_{4}) as uniformly for , and as uniformly for .
Theorem 2Suppose that conditions (H_{1}), (H_{3}), and (H_{4}) are satisfied. Then BVP (1) has at least one solution. If, in addition, condition (H_{5}) is satisfied, then BVP (1) has infinitely many solutions ().
Proof In the proof of Lemma 2, we see that condition (H_{1}) implies condition (a) of Lemma 3 (see (17)). On the other hand, it is clear that conditions (H_{3}), (H_{4}), (H_{5}) are the same as conditions (b), (c), (d) in Lemma 3, respectively. Hence the conclusion of Theorem 2 follows from Lemma 3, Lemma 2, and Lemma 1. □
Example 2 Consider the BVP
Conclusion BVP (41) has infinite many solutions ().
Proof Obviously, (41) is a BVP of form (1). In this situation, , , , , , , and
It is clear that is continuous on , left continuous at , and the right limit exists. By (42), we have
so, condition (H_{1}) is satisfied for , and . By (5), we have
and, consequently, (H_{3}) is satisfied for and any . On the other hand, from (44) we see that conditions (H_{4}) and (H_{5}) are all satisfied. Hence, our conclusion follows from Theorem 2. □
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).
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