Abstract
An algorithm for constructing two sequences of successive approximations of a solution of the nonlinear boundary value problem for a nonlinear differential equation with ‘maxima’ is given. The case of a boundary condition of antiperiodic type is investigated. This algorithm is based on the monotone iterative technique. Two sequences of successive approximations are constructed. It is proved both sequences are monotonically convergent. Each term of the constructed sequences is a solution of an initial value problem for a linear differential equation with ‘maxima’ and it is a lower/upper solution of the given problem. A computer realization of the algorithm is suggested and it is illustrated on a particular example.
MSC: 34K10, 34K25, 34B15.
Keywords:
differential equations with ‘maxima’; nonlinear boundary value problem; approximate solution; computer realization1 Introduction
Differential equations with ‘maxima’ are adequate models of real world problems, in which the present state depends significantly on its maximum value on a past time interval (see [14], monograph [5]).
Note that usually differential equations with ‘maxima’ are not possible to be solved in an explicit form and that requires the application of approximate methods. In the current paper, the monotone iterative technique [6,7], based on the method of lower and upper solutions, is theoretically proved to a boundary value problem for a nonlinear differential equation with ‘maxima’. The case when the nonlinear boundary function is a nondecreasing one with respect to its second argument is studied. This type of the boundary function covers the case of an antiperiodic boundary condition. An improved algorithm of monotoneiterative techniques is suggested. The main advantage of this scheme is connected with the construction of the initial conditions.
2 Preliminary notes and definitions
Let
Consider the following nonlinear differential equation with ‘maxima’:
with a boundary condition
and an initial condition
where
In this paper, we study boundary condition (2) in the case when the function
Let
Definition 1 The function
Definition 2 The function
In connection with the construction of successive approximations, we will introduce a couple of quasisolutions of boundary value problem (1)(3).
Definition 3 We will say that the functions
Definition 4 We will say that the functions
and
In the proof of our main results, we will use the following lemma.
Lemma 1 (Comparison result)
Let the following conditions be fulfilled:
1. The functions
2. The function
Then
Proof Assume the statement of Lemma 1 is not true. Consider the following two cases.
Case 1: Let
Denote
According to the mean value theorem, it follows that there exists
From inequalities
Inequality (10) contradicts (6).
Case 2: Let
Therefore,
In our further investigations, we will use the following result for differential equations with ‘maxima’ which is a partial case of Theorem 3.1.1 [5].
Lemma 2 (Existence and uniqueness)
Let the following conditions be fulfilled:
1. The function
2. The functions
Then the initial value problem for a linear differential equation with ‘maxima’
has a unique solution
3 Monotoneiterative method
We will give an algorithm for obtaining an approximate solution of the boundary value problem for a nonlinear differential equation with ‘maxima’ (1)(3).
Theorem 1Let the following conditions be fulfilled:
1. The functions
2. The function
3. The function
holds, where the functions
Then there exist two sequences
(a) The functions
(b) The sequence
(c) The sequence
(d) For
hold.
(e) Both sequences are uniformly convergent on
(f) If additionally the function
Proof We will give an algorithm for construction of successive approximations to the unknown exact solution of nonlinear boundary value problem (1)(3).
Assume the functions
and
where
and
According to Lemma 2, initial value problems (12), (13) and (14), (15) have unique
solutions
So, step by step we can construct two sequences of functions
Now, we will prove by induction that for
(H1)
(H2)
(H3)
Assume the claims (H1)(H3) are satisfied for
We will prove (H1) for
Define the function
Let
Let
Note that for any
From inequalities (17) and (18) it follows
According to Lemma 1, we get
Define the function
From equation (14), the inductive assumption, the definition of the functions
According to Lemma 1, we get
Define the function
Let
Let
According to Lemma 1, it follows
Now, we will prove the claim (H3) for
Let
From (H1) for
Let
Similarly, we prove the function
For any fixed
Therefore, both sequences converge pointwisely and monotonically. Let
Now, we will prove that for any
For any
From the monotonicity of the sequence of the quasilower solutions
Assume
Assume
Therefore, the required equality (23) is fulfilled.
In a similar way, we can prove that for any
holds.
Take a limit as
From (25) for
Taking a limit in the integral equation equivalent to (12), we obtain the function
In a similar way, we can prove that
Let the function
□
4 Applications
We will apply the given above algorithm for approximate solving of a nonlinear boundary value problem.
Example
Consider the following nonlinear boundary value problem for a nonlinear differential equation with ‘maxima’:
Boundary value problem (26), (27) is of type (1)(3), where
Let
Let
where
The function
The above given problem has a zero solution. We will apply the procedure given in Theorem 1 to obtain two sequences, which are monotonically convergent to 0.
The function
The function
According to Lemma 2, initial value problems (28) and (29) have unique solutions
Also, by a computer realization of the scheme given in Theorem 1 and applied to problems (28) and (29), we obtain the values in Table 1.
Table 1. Values of the successive approximations
From Table 1 and Figure 1, it is obvious that the sequence
Figure 1. Graphic of the successive approximations
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Each of the authors SH, AG and KS contributed to each part of the work equally and read and proved the final version of the manuscript.
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