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A periodic boundary value problem for nonlinear singular differential systems with ‘maxima’
Boundary Value Problems volume 2015, Article number: 201 (2015)
Abstract
In this paper, by using the method of quasilinearization to discuss the periodic boundary value problem for a nonlinear singular differential system with ‘maxima’, we obtain monotone iterative sequences of approximate solutions which converge uniformly and quadratically to the solution of the nonlinear singular differential system with ‘maxima’.
1 Introduction
Convergence has an important role to play in the development of qualitative theory of various nonlinear system. The method of quasilinearization is a powerful technique for obtaining approximate solutions of nonlinear problems. A systematic development of the method to ordinary differential equations was provided by Bellman and Kalaba [1], Lakshmikantham and Vatsala [2]. Afterwards, some generalized results were achieved for various types of differential systems, referred to in the monographs [3, 4], for functional differential equations to [5, 6], for impulsive equations to [7, 8], for fractional differential equations to [9–11], and to the references cited therein.
Recently, singular differential systems introduced by Rosenbrock [12], are studied because they have many applications in practical fields, such as non-Newtonian fluid mechanics, optimal control problems, and electrical circuits and some population growth models. It is more complicated than the ordinary systems, and its qualitative analyses involve greater difficulty than those of the ordinary ones. We noticed that there were few applicable results of convergence to singular differential systems [13–16] or differential equations with ‘maxima’ [17, 18]. By using the method of quasilinearization, in [13], the authors investigated the uniform and quadratic convergence of the initial value problem for singular differential systems. Agarwal and Hristova [18] investigated the initial value problem for differential equations with ‘maxima’. Singular differential systems with ‘maxima’ have not been studied yet.
In this paper, we apply the method to the study of the convergence of periodic boundary value problem (PBVP) of singular differential systems with ‘maxima’. The arrangement of this paper is as follows. In Section 2, we prove some basic lemmas which are needed in succeeding sections. In Section 3, under suitable conditions, we prove quadratic convergence of monotone sequences to the solution of singular differential systems with ‘maxima’. In Section 4, under less restrictive assumptions, we prove quadratic convergence by using the method of generalized quasilinearization.
2 Some basic results
Consider the periodic boundary value problem for the nonlinear singular differential system with ‘maxima’ (PBVP)
where A is a singular \(n\times n\) matrix, \(x\in R^{n}\), \(f\in C([0,T]\times R^{n}\times R^{n},R^{n})\), h and T are positive constants.
Throughout this paper, the inequalities between vectors are understood component wise. Defining \(|x|^{2}=(|x_{1}|^{2},|x_{2}|^{2},\ldots,|x_{n}|^{2})^{T}\) for \(x\in R^{n}\), \(|A|=(|a_{ij}|)\) for \(A\in R^{n\times n}\), and
Let the functions \(\alpha_{0}, \beta_{0}\in C([-h,T],R^{n})\) be such that \(\alpha_{0}(t)\leq\beta_{0}(t)\), we introduce two sets:
Definition 2.1
Let \(\alpha_{0}, \beta_{0}\in C([-h,T],R^{n})\cup C^{1}([0,T],R^{n})\). The function \(\alpha_{0}\) is called a lower solution of PBVP (2.1), if the following inequalities are satisfied:
Analogously, the function \(\beta_{0}\) is called an upper solution of PBVP (2.1), if the inequalities hold in an opposite direction.
Now, we will prove the following results which are needed for our further investigations.
First, we consider the singular differential inequalities
where A, \(M(t)\) are \(n\times n\) matrices, A is singular and \(M(t)\) is continuous on \([0,T]\).
Lemma 2.1
Assume that the following conditions hold.
- (H2.1):
-
There exists a constant λ such that \(L(t)=[\lambda A+M(t)]^{-1}\) exists, \(\hat{A}=AL(t)\) is a constant matrix.
- (H2.2):
-
There exists a nonsingular matrix Q such that \(Q^{-1}\) exists and \(Q^{-1}, (LQ)\geq0\), satisfying
$${Q^{- 1}}\hat{A}Q = \left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} C & 0 \\ 0 & 0 \end{array}\displaystyle \right ), \qquad Q^{-1}\hat{M}Q = \left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} I_{1} - \lambda C & 0 \\ 0 & I_{2} \end{array}\displaystyle \right ), $$where C is a diagonal matrix with \(C^{-1}\geq0\), \((I_{1}-\lambda C)\leq0\), \(\hat{M}=M(t)L(t)=I-\lambda\hat{A}\).
- (H2.3):
-
The matrix \(D^{-1}=(I-e^{-\int_{0}^{T}C^{-1}(I_{1}-\lambda C)\, ds})^{-1}\) exists and is positive.
Then \(x(t)\leq0\) on \([0,T]\).
Proof
Using the transformation \(x(t)=L(t)y(t)\), we get from (2.3)
Letting \(y(t)=Qz(t)\) in (2.4) and multiplying (2.4) by \(Q^{-1}\geq0\), we have
which is equivalent to \(Cz'_{1}+[I_{1}-\lambda C]z_{1}\leq0\), and \(z_{2}\leq0\) on \(t\in[0,T]\). We see from \(Cz'_{1}+[I_{1}-\lambda C]z_{1}\leq0\) that
Then
Since \(D^{-1}\) is nonsingular and positive, we conclude that \(z_{1}(T)\leq0\). Thus, \(z_{1}(t)\leq0\), \(t\in[0,T]\). Due to the fact that \(x=LQz\) and \(LQ\geq0\), we have \(x(t)\leq0\) on \([0,T]\). The proof is complete. □
For the boundary value problem
We have the following well-known result.
Lemma 2.2
(See [14])
Assume that the condition (H2.1) of Lemma 2.1 holds, \(\operatorname{index}(A)=1\), and
- (H2.4):
-
the boundary condition satisfies the requirement that \(J=E-F\exp\{-\hat{A}^{D}\hat{M}T\}\) is invertible.
Then the unique solution of
is given by
where \(\xi_{1}=E(I-\hat{A}\hat{A}^{D})\hat{M}^{D}g(0)-F(I-\hat{A}\hat {A}^{D})\hat{M}^{D}g(T)\),
where \({\hat{A}}^{D}\), \({\hat{M}}^{D}\) mean the Drazin inverse of the matrices Â, M̂. Note that once we have \(y(t)\), we get \(x(t)=L(t)y(t)\), where \(x(t)\) is the solution of (2.5).
Consider the singular differential inequalities
where A, \(M(t)\) are \(n\times n\) matrices, A is singular and \(M(t)\), \(N(t)\) are continuous on \([0,T]\).
Lemma 2.3
Assume that the conditions (H2.1)-(H2.4) hold, and
- (H2.5):
-
there exists a matrix N such that \(N\leq N(t)\leq0\) on \([0,T]\), and the matrix \([I-M]^{-1}\) exists and is nonnegative, where
$$ M = \max_{s\in[0,T]} \biggl\{ - \bigl[\lambda A+M(s) \bigr]^{-1} \biggl[ \hat{A}\hat{A}^{D}\int _{0}^{T}G(t,\sigma)Nd\sigma + \bigl(I-\hat{A} { \hat{A}}^{D} \bigr){\hat{M}}^{D}N \biggr] \biggr\} . $$
Then \(x(t)\leq0\) on \([-h,T]\).
Proof
In view of the condition (H2.5), we can get from (2.7)
Lemma 2.1 shows that \(x(t)\leq y(t)\) on \([0,T]\), where \(y(t)\) is the solution of
Thus, for \(t\in[0,T]\), using the expression of \(x(t)\) in Lemma 2.2, we obtain
Hence, we have
using the condition (H2.5), we have \(\max_{s\in[0,T]}x(s)\leq0\). Then we obtain \(x(t)\leq0\) on \([0,T]\). Furthermore, we conclude that \(x(t)\leq0\) on \([-h,T]\). The proof is complete. □
Next, we will prove an existence result for the PBVP (2.1), which is vital for our main results.
Lemma 2.4
Assume that the conditions (H2.1)-(H2.5) hold, and
- (H2.6):
-
the functions \(\alpha_{0}, \beta_{0}\in C([-h,T],R^{n})\cup C^{1}([0,T],R^{n})\) are lower and upper solutions of PBVP (2.1), and \(\alpha_{0}(t)\leq\beta_{0}(t)\) on \([-h,T]\);
- (H2.7):
-
there exists a function \(f\in C(\Omega(\alpha_{0},\beta_{0}),R^{n})\) that satisfies the inequality
$$f(t,y,v)-f(t,x,u)\leq-M_{0}(y-x)-N_{0}(v-u), $$where \(y\leq x\), \(v\leq u\), \(M(t_{0})=M_{0}\), \(N(t_{0})=N_{0}\), \(t_{0}\in[0,T]\).
Then there exists a solution x of PBVP (2.1) such that \(\alpha_{0}(t)\leq x(t)\leq\beta_{0}(t)\) on \([-h,T]\).
Proof
Consider the iterative scheme
where \(Q_{n+1}(t)=f(t,U_{n}(t), \max_{s\in[t-h,t]}U_{n}(s))+M_{0}U_{n}(t)+N_{0} \max_{s\in[t-h,t]}U_{n}(s)\). According to the iterative scheme, the sequences \(\{\alpha_{n}(t)\}\) and \(\{\beta_{n}(t)\}\) were generated by \(\alpha_{0}(t)\) and \(\beta_{0}(t)\), respectively.
Now, we will prove that
For this purpose, setting \(p_{1}(t)=\alpha_{0}(t)-\alpha_{1}(t)\), using the condition (H2.6), we have
According to Lemma 2.3, we have \(p_{1}(t)\leq0\) on \([-h,T]\). Similarly, setting \(p_{2}(t)=\beta_{1}(t)-\beta_{0}(t)\), we can obtain \(p_{2}(t)\leq0\) on \([-h,T]\).
Letting \(p_{3}(t)=\alpha_{1}(t)-\beta_{1}(t)\), from the condition (H2.7), we have
By Lemma 2.3, we have \(p_{3}(t)\leq0\) on \([-h,T]\).
The process can be continued to obtain
It is easy to see that the sequence \(\{\alpha_{n}(t)\}\) is uniformly bounded and equicontinuous, employing the Ascoli-Arzela theorem, the nondecreasing sequence \(\{\alpha_{n}(t)\}\) converges pointwise to a function \(x(t)\) that satisfies \(\alpha_{0}(t)\leq x(t)\leq\beta_{0}(t)\). In view of PBVP (2.8) and the Dominated Convergence Theorem, we see that \(x(t)\) is a solution of
that is, \(x(t)\) is a solution of PBVP (2.1). Therefore, we conclude that there exists a solution \(x(t)\) of PBVP (2.1) that satisfies \(\alpha_{0}(t)\leq x(t)\leq\beta_{0}(t)\) on \([-h,T]\). The proof is complete. □
3 Quasilinearization technique
We apply the method of quasilinearization to nonlinear singular differential system with ‘maxima’. By assuming suitable conditions on the function f, we see that there exist two monotone sequences which converge quadratically to the solution of PBVP (2.1).
Theorem 3.1
Assume that the conditions (H2.1)-(H2.7) hold, and
- (A3.1):
-
there exists a function \(f\in C^{0,2,2}(\Omega(\alpha_{0},\beta_{0}),R^{n})\) such that \(f_{y}\geq0\) and \(H(f)\geq0\), where
$$\begin{aligned} H(f) =&\int_{0}^{1} \biggl[ \bigl(x(t)-y(t) \bigr) \frac{\partial}{\partial x}+ \Bigl( \max_{s\in[t-h,t]}x(s)- \max _{s\in[t-h,t]}y(s) \Bigr) \frac{\partial}{\partial y} \biggr]^{2} \\ &{}\times f \Bigl(t,\sigma x(t)+(1-\sigma)y(t),\sigma \max_{s\in[t-h,t]}x(s) +(1-\sigma) \max_{s\in[t-h,t]}y(s) \Bigr)\,d\sigma \end{aligned}$$for \(\alpha_{0}(t)\leq x(t)\), \(y(t)\leq\beta_{0}(t)\) on \([-h,T]\).
Then there exist two sequences \(\{\alpha_{n}(t)\}\) and \(\{\beta_{n}(t)\}\) converging uniformly to the solution \(x(t)\) of PBVP (2.1) and the convergence is quadratic, that is, there exist positive matrices \(K_{1}\), \(K_{2}\) such that the solution \(x(t)\) of PBVP (2.1) in \(S(\alpha_{0},\beta_{0})\), the inequalities
hold, where
Proof
From the condition (A3.1), we obtain
and
where
with \(\alpha_{0}(t)\leq x(t)\), \(y(t)\leq\beta_{0}(t)\), \(\sum_{j=1}^{n}|f_{x_{j}x}|\leq M_{11}\), \(\sum_{j=1}^{n}|f_{x_{j}y}|\leq M_{12}\), \(\sum_{j=1}^{n}|f_{y_{j}x}|\leq M_{13}\), \(\sum_{j=1}^{n}|f_{y_{j}y}|\leq M_{14}\) for \(t\in[0,T]\), \(M_{11}\), \(M_{12}\), \(M_{13}\), and \(M_{14}\) are positive matrices, and \(M_{1}=M_{11}+M_{12}+M_{13}+M_{14}\).
Now, we consider the singular differential system with ‘maxima’
We shall now show that \(\alpha_{0}(t)\) and \(\beta_{0}(t)\) are lower and upper solutions of PBVP (3.3), respectively. In fact, the condition (H2.6) and inequality (3.1) imply
and
Hence, by Lemma 2.4, there exists a solution \(\alpha_{1}(t)\) of PBVP (3.3) such that \(\alpha_{0}(t)\leq\alpha_{1}(t)\leq\beta_{0}(t)\) on \([-h,T]\).
Similarly, consider the singular differential system with ‘maxima’
We show that \(\alpha_{1}(t)\) and \(\beta_{0}(t)\) are lower and upper solutions of PBVP (3.4), respectively. Using the conclusion that \(\alpha_{1}(t)\) is a solution of PBVP (3.3), the condition (H2.6), and inequalities (3.1), (3.2), we obtain
and
These imply that \(\alpha_{1}(t)\) and \(\beta_{0}(t)\) are lower and upper solutions of PBVP (3.4), respectively. Thus, as before, there exists a solution \(\beta_{1}(t)\) of PBVP (3.4) such that \(\alpha_{1}(t)\leq\beta_{1}(t)\leq\beta_{0}(t)\) on \([-h,T]\).
Next, we will show that \(\alpha_{1}(t)\) and \(\beta_{1}(t)\) are lower and upper solutions of PBVP (2.1), respectively. Since \(\alpha_{1}(t)\) is a solution of PBVP (3.3), using the inequality (3.1), we get
This shows that \(\alpha_{1}(t)\) is a lower solution of PBVP (2.1) on \([-h,T]\). Similarly, since \(\beta_{1}(t)\) is a solution of PBVP (3.4), it follows from the inequality (3.2) that
This proves that \(\beta_{1}(t)\) is an upper solution of PBVP (2.1) on \([-h,T]\). As a result, we get
Continuing this process by induction, we obtain
where \(\alpha_{n+1}(t)\) and \(\beta_{n+1}(t)\) are solutions of the singular differential systems with ‘maxima’
and
By induction we have, for all n,
Clearly, the sequences \(\{\alpha_{n}(t)\}\), \(\{\beta_{n}(t)\}\) are equicontinuous and uniformly bounded. Thus, employing the Ascoli-Arzela theorem, we see that the monotone sequences \(\{\alpha_{n}(t)\}\) and \(\{\beta_{n}(t)\}\) converge uniformly and monotonically on \([-h,T]\) with
We shall prove that the convergence of \(\{\alpha_{n}(t)\}\) and \(\{\beta_{n}(t)\}\) is quadratic. To do this, let \(x(t)\) be a solution of PBVP (2.1) in \(S(\alpha_{0},\beta_{0})\). Define
Case 1. \(t\in[-h,0]\), \(a_{n+1}(t)=a_{n+1}(0)=a_{n+1}(T)\).
Case 2. \(t\in[0,T]\). In view of the assumption \(f_{y}\geq0\) and the inequality (3.2), we have
where \(f_{x}(t,x,y)\leq-M(\bar{t})\), \(f_{y}(t,x,y)\leq-N(\bar{t})\) for \((t,x,y)\in\Omega(\alpha_{0},\beta_{0})\), \(\bar{t}\in[0,T]\). Then we obtain
According to Lemma 2.1, we see that \(a_{n+1}(t)\leq u(t)\) on \([0,T]\), where \(u(t)\) is the solution of
By using the expression of \(x(t)\) in Lemma 2.2 and taking suitable estimates, we conclude that
where \(K_{1}\) is a positive matrix. This shows that the convergence of \(\{\alpha_{n}(t)\}\) is quadratic.
Similarly, consider
Case 1. If \(t\in[-h,0]\), \(b_{n+1}(t)=b_{n+1}(0)=b_{n+1}(T)\).
Case 2. If \(t\in[0,T]\). The assumption \(f_{y}\geq0\) and the inequality (3.1) yield
Then we have
By Lemma 2.1, we see that \(b_{n+1}(t)\leq u(t)\) on \([0,T]\), where \(u(t)\) is the solution of
Hence, applying the expression of \(x(t)\) in Lemma 2.2 and taking suitable estimates, we obtain
where \(K_{2}\) is a positive matrix. Thus, the convergence of \(\{\beta_{n}(t)\}\) is quadratic. The proof is complete. □
4 Generalized quasilinearization technique
We apply the method of generalized quasilinearization to nonlinear singular differential system with ‘maxima’. By assuming less restrictive conditions on the function f, we see that there exist two monotone sequences which converge quadratically to the solution of PBVP (2.1).
Theorem 4.1
Assume that the conditions (H2.1)-(H2.7) hold, and
- (A4.1):
-
there exist functions \(F, \phi\in C^{0,2,2}(\Omega(\alpha_{0},\beta_{0}),R^{n})\) such that \(f_{y}\geq0\) and \(H(f+\phi)\geq0\) and \(H(\phi)\geq0\).
Then there exist two sequences \(\{\alpha_{n}(t)\}\) and \(\{\beta_{n}(t)\}\) converging uniformly to the solution \(x(t)\) of PBVP (2.1) and the convergence is quadratic, that is, there exist positive matrices \(K_{3}\), \(K_{4}\) such that for the solution \(x(t)\) of PBVP (2.1) in \(S(\alpha_{0},\beta_{0})\), the inequalities
hold, where
Proof
In view of (A4.1), we see that
and
where
with \(\alpha_{0}(t)\leq x(t)\), \(y(t)\leq\beta_{0}(t)\), \(\sum_{j=1}^{n}|\phi_{x_{j}x}|\leq N_{1}\), \(\sum_{j=1}^{n}|\phi_{x_{j}y}|\leq N_{2}\), \(\sum_{j=1}^{n}|\phi_{y_{j}x}|\leq N_{3}\), \(\sum_{j=1}^{n}|\phi_{y_{j}y}|\leq N_{4}\) for \(t\in[0,T]\), \(N_{1}\), \(N_{2}\), \(N_{3}\), and \(N_{4}\) are positive matrices, \(\bar{N}=N_{1}+N_{2}+N_{3}+N_{4}\). We have
with \(\alpha_{0}(t)\leq x(t)\), \(y(t)\leq\beta_{0}(t)\), \(\sum_{j=1}^{n}|F_{x_{j}x}|\leq M_{1}\), \(\sum_{j=1}^{n}|F_{x_{j}y}|\leq M_{2}\), \(\sum_{j=1}^{n}|F_{y_{j}x}|\leq M_{3}\), \(\sum_{j=1}^{n}|F_{y_{j}y}|\leq M_{4}\) for \(t\in[0,T]\), \(M_{1}\), \(M_{2}\), \(M_{3}\), and \(M_{4}\) are positive matrices, and \(\bar{M}=M_{1}+M_{2}+M_{3}+M_{4}\).
We consider the singular differential system with ‘maxima’
Similar to the proof of Theorem 3.1, we can show that \(\alpha_{0}(t)\) and \(\beta_{0}(t)\) are lower and upper solutions of PBVP (4.3), respectively. Consequently, by Lemma 2.4, there exists a solution \(\alpha_{1}(t)\) of PBVP (4.3) such that \(\alpha_{0}(t)\leq\alpha_{1}(t)\leq\beta_{0}(t)\) on \([-h,T]\).
Next, we consider the singular differential system with ‘maxima’
Similar to the proof of PBVP (3.4), we see that there exists a solution \(\beta_{1}(t)\) of (4.4) such that \(\alpha_{1}(t)\leq\beta_{1}(t)\leq\beta_{0}(t)\) on \([-h,T]\).
Now, we shall show that \(\alpha_{1}(t)\) and \(\beta_{1}(t)\) are lower and upper solutions of PBVP (2.1), respectively. In fact, utilizing the conclusion that \(\alpha_{1}(t)\) is a solution of (4.3) and the inequality (4.1), we get
It proves that \(\alpha_{1}(t)\) is a lower solution of PBVP (2.1) on \([-h,T]\). Analogously, we see that \(\beta_{1}(t)\) is an upper solution of PBVP (2.1). Consequently, these results yield
Proceeding as before, we can get
where \(\alpha_{n+1}(t)\) and \(\beta_{n+1}(t)\) are solutions of the singular differential systems with ‘maxima’,
and
Using mathematical induction, we can show that, for all n,
It is clear that both sequences \(\{\alpha_{n}(t)\}\) and \(\{\beta_{n}(t)\}\) are uniformly bounded and equicontinuous on \([-h,T]\). In view of the Ascoli-Arzela theorem, we have
It remains to show the quadratic convergence. For this purpose, set
Case 1. \(t\in[-h,0]\), \(a_{n+1}(t)=a_{n+1}(0)=a_{n+1}(T)\).
Case 2. \(t\in[0,T]\). Using the assumption \(f_{y}\geq0\) and the inequality (4.2), we obtain
Then we obtain
According to Lemma 2.1, we obtain \(a_{n+1}(t)\leq u(t)\) on \([0,T]\), where \(u(t)\) is the solution of
Thus, using the expression of \(x(t)\) in Lemma 2.2 and taking suitable estimates, we arrive at
where \(K_{3}\) is a positive matrix.
Similarly, consider
Case 1. \(t\in[-h,0]\), \(b_{n+1}(t)=b_{n+1}(0)=b_{n+1}(T)\).
Case 2. \(t\in[0,T]\). It follows from the assumption \(f_{y}\geq0\) and the inequality (4.1) that
Then we obtain
According to Lemma 2.1, we obtain \(b_{n+1}(t)\leq u(t)\) on \([0,T]\), where \(u(t)\) is the solution of
Thus, utilizing the expression of \(x(t)\) in Lemma 2.2, we obtain after taking suitable estimates
Hence, the convergence is quadratic. The proof of the theorem is complete. □
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Acknowledgements
This paper is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).
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Wang, P., Liu, X. A periodic boundary value problem for nonlinear singular differential systems with ‘maxima’. Bound Value Probl 2015, 201 (2015). https://doi.org/10.1186/s13661-015-0463-9
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DOI: https://doi.org/10.1186/s13661-015-0463-9