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 anti-periodic 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 realization
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 [1-4], monograph ).
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 anti-periodic boundary condition. An improved algorithm of monotone-iterative techniques is suggested. The main advantage of this scheme is connected with the construction of the initial conditions.
2 Preliminary notes and definitions
Consider the following nonlinear differential equation with ‘maxima’:
with a boundary condition
and an initial condition
In this paper, we study boundary condition (2) in the case when the function is nondecreasing with respect to its second argument y. So, the anti-periodic boundary value problem is a partial case of boundary condition (2). Note that similar problems are investigated for ordinary differential equations , delay differential equations  and impulsive differential equations , and some approximate methods are suggested. The presence of the maximum of the unknown function requires additionally some new comparison results, existence results as well as a new algorithm for constructing successive approximations to the exact unknown solution.
In connection with the construction of successive approximations, we will introduce a couple of quasi-solutions of boundary value problem (1)-(3).
In the proof of our main results, we will use the following lemma.
Lemma 1 (Comparison result)
Let the following conditions be fulfilled:
Proof Assume the statement of Lemma 1 is not true. Consider the following two cases.
Inequality (10) contradicts (6).
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 .
Lemma 2 (Existence and uniqueness)
Let the following conditions be fulfilled:
Then the initial value problem for a linear differential equation with ‘maxima’
3 Monotone-iterative 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:
Proof We will give an algorithm for construction of successive approximations to the unknown exact solution of nonlinear boundary value problem (1)-(3).
From inequalities (17) and (18) it follows
Therefore, both sequences converge pointwisely and monotonically. Let and for . According to Dini’s theorem, both sequences converge uniformly and the functions , are continuous. Additionally, the claims (H1), (H2) prove .
For any , we introduce the notation . From condition (H1) it follows that for any the inequalities hold and thus, , , i.e., the sequence is monotone nondecreasing and bounded from above by for any . Therefore, there exists the limit .
Assume . According to the definition of the function , it follows that for the fixed number , we have . Then there exists a natural number N such that and . Therefore, . The obtained contradiction proves the assumption is not valid.
Therefore, the required equality (23) is fulfilled.
Let the function be Lipschitz. Then if (1) has a solution , it is unique (see ). In this case, and for ,
We will apply the given above algorithm for approximate solving of a nonlinear boundary value problem.
Consider the following nonlinear boundary value problem for a nonlinear differential equation with ‘maxima’:
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.
According to Lemma 2, initial value problems (28) and (29) have unique solutions and , respectively. Because of the presence of the maximum of the unknown function over a past time interval, there is no explicit formula for the exact solutions of (28) and (29). We use a computer program based on a modified numerical method to solve these problems (see ).
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 approximationsand,
From Table 1 and Figure 1, it is obvious that the sequence is increasing and the sequence is decreasing and both monotonically converge to the unique solution 0 of nonlinear boundary value problem (26), (27).
Figure 1. Graphic of the successive approximationsand,.
The authors declare that they have no competing interests.
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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