Abstract
General nonlocal boundary value problems are considered for systems of impulsive equations with finite and fixed points of impulses. Sufficient conditions are established for the solvability and unique solvability of these problems, among them effective spectral conditions.
MSC: 34B37.
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1 Statement of the problem and formulation of the results
In the present paper, we consider the system of nonlinear impulsive equations with a finite number of impulse points
with the general boundary value condition
where (we will assume and , if necessary), , is a natural number, f belongs to Carathéodory class , () are continuous operators, and is a continuous, vector functional, nonlinear, in general.
In the paper sufficient conditions (among them effective sufficient) are given for solvability and unique solvability of the general nonlinear impulsive boundary value problem (1.1), (1.2); (1.3). We established the Conti-Opial type theorems for the solvability and unique solvability of this problem. Analogous problems are investigated in [1]–[5] (see also the references therein) for the general nonlinear boundary value problems for ordinary differential and functional-differential systems, and in [6]–[10] (see also the references therein) for generalized ordinary differential systems.
Some results obtained in the paper are more general than known results even for the ordinary differential case.
Quite a number of issues of the theory of systems of differential equations with impulsive effect (both linear and nonlinear) have been studied sufficiently well (for a survey of the results on impulsive systems, see, e.g., [10]–[19] and references therein). But the above-mentioned works, as is well known, do not contain the results obtained in the present paper.
Throughout the paper the following notation and definitions will be used.
, ; () is a closed segment;
is the space of all real matrices with the norm
is the space of all real column n-vectors ; ;
if , then , detX and are, respectively, the matrix inverse to X, the determinant of X, and the spectral radius of X; is the identity matrix;
is the variation of the matrix function , i.e., the sum of variations of the latter’s components; , where , for ;
and are the left and the right limits of the matrix function at the point t (we will assume for and for , if necessary);
is the set of all matrix functions of bounded variation (i.e., such that );
, where , is the set of all continuous matrix functions ;
is the set of all matrix functions , having the one sided limits () and (), whose restrictions to an arbitrary closed interval from belong to ;
is the Banach space of all matrix functions with the norm ;
, where , is the set of all absolutely continuous matrix functions ;
is the set of all matrix functions , having the one sided limits () and (), whose restrictions to an arbitrary closed interval from belong to .
If and are normed spaces, then an operator (nonlinear, in general) is positive homogeneous if for every and .
The inequalities between the matrices are understood componentwise.
An operator is called nondecreasing if for every such that for the inequality holds.
A matrix function is said to be continuous, nondecreasing, integrable, etc., if each of its components is.
, where , is the set of all measurable and integrable matrix functions .
If and , then is the Carathéodory class, i.e., the set of all mappings such that:
-
(a)
the function is measurable for every ;
-
(b)
the function is continuous for almost all , and
is the set of all mappings such that the functions (; ) are measurable for every vector function with bounded variation.
By a solution of the impulsive system (1.1), (1.2) we understand a vector function , continuous from the left, satisfying both the system (1.1) a.e. on and the relation (1.2) for every .
Definition 1.1
Let be a linear continuous operator, and let be a positive homogeneous operator. We say that a pair , consisting of a matrix function and a finite sequence of continuous operators (), satisfy the Opial condition with respect to the pair if:
-
(a)
there exist a matrix function and constant matrices () such that
(1.4)
and
-
(b)
(1.6)
and the problem
has only the trivial solution for every matrix function and constant matrices () for which there exists a sequence () such that
and
Remark 1.1
Note that, due to the condition (1.5), the condition (1.6) holds if
Below, we will assume that and, in addition, can be arbitrary for and .
Theorem 1.1
Let the conditions
and
hold, whereandare, respectively, linear continuous and positive homogeneous continuous operators, the pairsatisfies the Opial condition with respect to the pair; is a function nondecreasing in the second variable, and () andare nondecreasing, respectively, functions and vector functions such that
Then the problem (1.1), (1.2); (1.3) is solvable.
Theorem 1.2
Let the conditions (1.12)-(1.14),
and
hold, where, (), (; ), andare, respectively, linear continuous and positive homogeneous continuous operators; is a function nondecreasing in the second variable, and () andare nondecreasing, respectively, functions and vector function such that the condition (1.15) holds. Let, moreover, the condition (1.6) hold and the problem (1.7), (1.8); (1.9) have only the trivial solution for every matrix functionand constant matrices () such that
and
Then the problem (1.1), (1.2); (1.3) is solvable.
Remark 1.2
Theorem 1.2 is interesting only in the case when , because the theorem immediately follows from Theorem 1.1 in the case when .
Theorem 1.3
Let the conditions (1.14),
and
hold, where, , and () are constant matrices, andare, respectively, linear continuous and positive homogeneous continuous operators; is a vector function nondecreasing in the second variable, and () andare nondecreasing vector functions such that
Let, moreover, the conditions
and
hold and the system of impulsive inequalities
have only the trivial solution under the condition (1.9). Then the problem (1.1), (1.2); (1.3) is solvable.
Corollary 1.1
Let the conditions
and
hold, where, () are constant matrices, is a linear continuous operator, is a function nondecreasing in the second variable, and () andare nondecreasing functions such that
Let, moreover,
and the impulsive system
have only the trivial solution under the condition
Then the problem (1.1), (1.2); (1.3) is solvable.
For every matrix function and a sequence of constant matrices () we introduce the operators
Corollary 1.2
Let the conditions (1.26)-(1.30) hold, where
, are constant matrices, , is a function nondecreasing in the second variable, and () andare nondecreasing functions. Let, moreover, there exist natural numbers k and m such that the matrix
is nonsingular and
where the operators () are defined by (1.33), and
Then the problem (1.1), (1.2); (1.3) is solvable.
Corollary 1.3
Let the conditions (1.26)-(1.30) hold, where
, () are constant matrices, and (), is a function nondecreasing in the second variable, and () andare nondecreasing functions. Let, moreover, the constant matrices () be pairwise permutable, and let the matrix function P satisfy the Lappo-Danilevskicondition, i.e.
and
Then the condition
guarantees the solvability of the problem (1.1), (1.2); (1.3).
Corollary 1.4
Let the conditions (1.26)-(1.30) and (1.35) hold, where, () are constant matrices, and (), is a function nondecreasing in the second variable, and () andare nondecreasing functions. Let, moreover, there exist natural numbers k and m such that the matrix
is nonsingular and the inequality (1.34) holds, where
Then the problem (1.1), (1.2); (1.3) is solvable.
Corollary 1.5
Let the conditions (1.26)-(1.30) and (1.35) hold, where, () are constant matrices, and (), is a function nondecreasing in the second variable, and () andare nondecreasing functions. Let, moreover,
hold and
where
Then the problem (1.1), (1.2); (1.3) is solvable.
Theorem 1.4
Let the conditions (1.22), (1.23),
and
hold, where, , and () are constant matrices, andare, respectively, linear continuous and positive homogeneous continuous operators. Let, moreover, the system of impulsive inequalities (1.24), (1.25) have only the trivial solution under the condition (1.9). Then the problem (1.1), (1.2); (1.3) is uniquely solvable.
2 Auxiliary propositions
Lemma 2.1
Let () be such that
and
where, as, andis a nondecreasing vector function. Then
The proof of Lemma 2.1 is given in [9].
Lemma 2.2
(Lemma on a priori estimates)
Let the subsetsand, and a positive homogeneous continuous operatorbe such that:
-
(a)
there exist a matrix function and a constant matrix such that
for every, and
-
(b)
the condition (1.6) holds and the system (1.7), (1.8) has only the trivial solution under the condition
(2.1)
for every matrix functionand constant matricessuch that;
-
(c)
if (), (), and are such that
and
thenand. Then there exists a positive numbersuch that
Proof
Let us assume that the statement of the lemma is not true. Then for every natural k there exist a matrix function , a constant matrix and a vector function such that
Let
and
Let, moreover,
Then
and
On the other hand, by the estimate (a) we have
Therefore, by the Arzelá-Ascoli lemma we can assume without loss of generality that the sequence () converges uniformly on , and the sequence () converges for every .
Let
It is evident that the matrix function B is absolutely continuous. Therefore,
where . From this and (2.6), by the condition (c) we have and .
According to (2.3) we can assume that the sequence () converges. It is evident that the function is a solution of the system
for every natural k. Using now Theorem 1.2 of the paper [2], from the conditions (a), (2.5), and (2.6) it follows that
where x is a solution of the system (1.7), (1.8) under the condition
and
Take into account (2.4) and (2.7), we conclude . So that x is a solution of the problem (1.7), (1.8); (2.1). Consequently, by the condition (b) we have . But this contradicts the condition (2.3). The lemma is proved. □
3 Proof of the main results
Proof of Theorem 1.1
Let , and be, respectively, the sets of all matrix functions and constant matrix-vectors , satisfying the condition (1.6), such that the conditions (1.10) and (1.11) hold for some sequence (). By virtue of Definition 1.1 the conditions (a), (b), (c) of Lemma 2.2 are fulfilled for the sets and .
Let be the positive number appearing in the conclusion of Lemma 2.2. According to the condition (1.15) there exists a positive number such that
Assume
and consider the auxiliary boundary value problem
for every .
According to the Opial condition the problem
has only the trivial solution for every .
Therefore, in view of Theorem 3.1 from [10] the problem (3.5), (3.6); (3.7) has a unique solution . In addition, by (3.5), (3.6), and (3.7), it follows from Lemma 2.2 that
From this, due to (1.12), (1.13), and (3.2)-(3.4) we have
On the other hand, taking into account the inequalities (1.14) and (3.1), the condition (3.8) implies
and
Thus . Further, due to Theorem 1 from [13] we conclude that the operator is continuous.
By (1.4), (1.5), (1.12), (1.13), and (3.2) we have
if , where and . So that, using the Arzelá-Ascoli lemma on the every closed interval () we conclude that the set is precompact.
According to the Schauder principle there exists such that
From this, by virtue of (1.14) and (3.2)-(3.4), it follows that the function x is a solution of the system (1.1), (1.2) satisfying the conditions
and
Due to Lemma 2.2 and inequalities (1.12), (1.13), (3.1), and (3.10) we have
In fact, the first estimate immediately follows from Lemma 2.2 with regard to the conditions (1.12), (1.13), and (3.10). Now, if we assume that then by (3.1), for , it will be
The obtained inequality contradicts the first estimate of (3.11).
In view of the estimate (3.11) from (3.3) and (3.4) we have . Consequently, by (3.9) we conclude that the vector function x satisfies the condition (1.3). The theorem is proved. □
Proof of Theorem 1.2
Let be the set of all matrix functions satisfying the inequalities (1.18), and let be the set all constant matrices satisfying the condition (1.6) and the inequalities (1.19). It is evident that the conditions of Lemma 2.2 hold for these sets and the operator .
Let be the number such that the conclusion of Lemma 2.2 is true. In view of (1.15) there exists a positive number such that the estimate (3.1) holds. Consider the impulsive system
where χ is the function defined by (3.3). According to Theorem 1.1 the problem (3.12), (3.13); (1.3) is solvable since the pair satisfies the Opial condition with respect to the pair . Let x be an arbitrary solution of this problem. Then
where
and
On the other hand, by (1.16), (1.17), and (3.3) the matrix function A and constant matrices () satisfy, respectively, the inequalities (1.18) and (1.19). Therefore we have and . Therefore, due to Lemma 2.2 and the inequalities (1.12)-(1.14) and (3.1), the estimate (3.11) is valid. But by (3.3) every solution of the system (3.12), (3.13) satisfying such an estimate is a solution of the system (1.1), (1.2), too. The theorem is proved. □
Proof of Theorem 1.3
Let
Assuming
and
in view of (1.20) and (1.21), respectively, we find
and
where
and
In addition, . On the other hand, the problem (1.7), (1.8); (1.9) has only the trivial solution for every matrix function and constant matrices (), satisfying, respectively, the inequalities (1.18) and (1.19), since the problem (1.24), (1.25); (1.9) has only the trivial solution. Therefore, the theorem follows from Theorem 1.2. □
Corollary 1.1 immediately follows from Theorem 1.3 if we assume therein and ().
To prove Corollaries 1.2-1.5 it is sufficient to show that the problem (1.31), (1.32) has only the trivial solution under the condition . But this fact is valid, respectively, due to Theorem 3.2, Theorem 3.4, Theorem 3.5, and Corollary 3.2 from [10].
Proof of Theorem 1.4
The solvability of the problem (1.1), (1.2); (1.3) follows from Theorem 1.3, because its conditions are fulfilled for
Let now x and y be two solutions of the problem (1.1), (1.2); (1.3). Then by (1.36)-(1.38) the vector function will be a solution of the problem (1.24), (1.25); (1.9). But this problem has only the trivial solution. Therefore, . The theorem is proved. □
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Acknowledgements
The present paper was supported by the Shota Rustaveli National Science Foundation (Grant # FR/182/5-101/11).
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The main results have been obtained by MA, and the corollaries have been obtained by GE and NK.
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Ashordia, M., Ekhvaia, G. & Kekelia, N. On the solvability of general boundary value problems for systems of nonlinear impulsive equations with finite and fixed points of impulse actions. Bound Value Probl 2014, 157 (2014). https://doi.org/10.1186/s13661-014-0157-8
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DOI: https://doi.org/10.1186/s13661-014-0157-8