The object of investigation of the paper is a special type of functional differential equations containing the maximum value of the unknown function over a past time interval. An improved algorithm of the monotone-iterative technique is suggested to nonlinear differential equations with “maxima.” The case when upper and lower solutions of the given problem are known at different initial time is studied. Additionally, all initial value problems for successive approximations have both initial time and initial functions different. It allows us to construct sequences of successive approximations as well as sequences of initial functions, which are convergent to the solution and to the initial function of the given initial value problem, respectively. The suggested algorithm is realized as a computer program, and it is applied to several examples, illustrating the advantages of the suggested scheme.

Equations with “maxima” find wide applications in the theory of automatic regulation. As a simple example of mathematical simulations by means of such equations, we shall consider the system of regulation of the voltage of a generator of constant current ([

Note that the above given model as well as all types of differential equations with “maxima” could be considered as delay functional differential equations. Differential equations with delay are studied by many authors (see, e.g., [

In the current paper, an approximate method for solving the initial value problem for nonlinear differential equations with “maxima” is considered. This method is based on the method of lower and upper solutions. Meanwhile, in studying the initial value problems the authors usually keep the initial time unchanged. But, in the real repeated experiments it is difficult to keep this time fixed because of all kinds of disturbed factors. It requires the changing of initial time to be taken into consideration. Note that several qualitative investigations of the solutions of ordinary differential equations with initial time difference are studied in [

In this paper, an improved algorithm of monotone-iterative techniques is suggested to nonlinear differential equations with “maxima.” The case when upper and lower solutions of the given problem are known at different initial time is studied. The behavior of the lower/upper solutions of the initial value problems with different initial times is studied. Also, some other improvements in the suggested algorithm are given. The main one is connected with the initial functions. In the known results for functional differential equations, the initial functions in the corresponding linear problems for successive approximations are the same (see, e.g., [

Consider the following initial value problem for the nonlinear differential equation with “maxima’’ (IVP):

In the paper, we will study the differential equation with “maxima” (

Assume there exist a solution

The function

Note that the function

Often in the real world applications, the lower and upper solutions of one and the same differential equation are obtained at different initial time intervals.

The following result is a comparison result for lower and upper solutions with initial conditions given on different initial time intervals.

Let the following conditions be satisfied.

Let

The function

The function

The function

The function

Then

Choose a positive number

Therefore, we obtain

Note that for

Assume the contrary, that is, there exists a point

Therefore,

Let the conditions of Theorem

The comparison results are true if the inequality

Let the following conditions be satisfied.

The conditions 2, 3, 5 of Theorem

Let

The function

Then

The proof of Theorem

In what follows, we shall need that the functions

Consider the sets:

In our further investigations, we will need the following comparison result on differential inequalities with “maxima.’’

Let the function

Then the inequality

In our further investigations, we will use the following result, which is a partial case of Theorem

Let the following conditions be fulfilled.

The functions

The functions

Then the initial value problem for the linear scalar equation

Denote

Let

Let the following conditions be fulfilled.

The points

The function

The function

The function

The function

Then there exist two sequences of functions

the functions

the functions

the sequence

the sequence

the inequalities

both sequences uniformly converge and

According to Theorem

Let

Therefore,

We consider the linear differential equation with “maxima’’

We will prove that

From the choice of

Now, let

We will prove that the function

From equality (

Next, according to Lemma

Let

Therefore,

We consider the linear differential equation with “maxima’’

There exists a unique solution

The function

Functions

Similarly, recursively, we can construct two sequences of functions

Now following exactly as for the case

Therefore, the sequence

Denote

Let

In this case, we could approximate again the solution of the given initial value problem, starting from lower and upper solutions given at two different initial points. Since the proofs are similar, we will set up only the results.

Let the following conditions be fulfilled.

The points

The conditions 2, 3, 4 of Theorem

The function

the functions

the functions

the sequence

the sequence

the inequalities

both sequences uniformly converge and

Since the set of differential equations, which could be solved in an explicit form, is very narrow, we will realize the above suggested algorithm numerically. At present, we cannot solve numerically differential equations with “maxima” by the existing ready-made systems, such as

Now we will illustrate the employment of the suggested above scheme to a particular nonlinear scalar differential equations with “maxima.’’

Initially we will consider the case of one and the same initial points of both initial value problems. By this way, we will emphasize our considerations to the advantages of the involved in the initial conditions constants.

Consider the following scalar nonlinear differential equation with “maxima’’:

In this case

The successive approximations

Now we will construct an increasing sequence of lower solutions and a decreasing sequence of upper solutions, which will be convergent to the zero solution.

The first lower approximation

The second lower approximation

The first upper approximation

The second upper approximation

It is obviouse that

Example

Now we will illustrate the suggested method in the case when lower and upper solutions are defined for different initial points and on different intervals.

Consider the following IVP for the scalar nonlinear differential equation with “maxima’’:

In this case,

Choose

Let

In this case,

Let

According to Theorem

The successive approximations

Note the above IVPs are linear, but we are not able to obtain their solutions in explicit form because of the presence of maximum function.

We will use the computer realization of the considered method, explained in Section

Values of the successive lower/upper approximations

1.50 | 1.55 | 1.60 | 1.65 | 1.70 | 1.75 | |
---|---|---|---|---|---|---|

0.2500000 | 0.2396285 | 0.2326388 | 0.2286608 | 0.2273177 | 0.2281468 | |

0.0625000 | 0.0795487 | 0.0954878 | 0.1138875 | 0.1307175 | 0.1464314 | |

0.0156250 | 0.0322634 | 0.0490684 | 0.0671041 | 0.0850914 | 0.1029515 | |

0.0039063 | 0.0211106 | 0.0383552 | 0.0558882 | 0.0734248 | 0.0909491 | |

0.0009766 | 0.0183982 | 0.0358232 | 0.0532996 | 0.0707523 | 0.0881732 | |

−0.0009766 | −0.0236676 | −0.0416446 | −0.0552493 | −0.0648509 | −0.0708277 | |

−0.0039063 | −0.0269991 | −0.0452915 | −0.0591247 | −0.0688717 | −0.0749110 | |

−0.0156250 | −0.0399773 | −0.0590208 | −0.0731539 | −0.0828101 | −0.0884511 | |

−0.0625000 | −0.0871172 | −0.1048556 | −0.1164338 | −0.1225992 | −0.1243204 | |

−0.2500000 | −0.2427252 | −0.2336358 | −0.2230384 | −0.2111877 | −0.1982954 |

(graphs of

This paper was partially supported by Fund “Scientific Research’’ NI11FMI004/30.05.2011, Plovdiv University and BG051PO001/3.3-05-001 Science and Business, financed by the Operative Program “Development of Human Resources,” European Social Fund.