Monotone-Iterative Method for the Initial Value Problem with Initial Time Difference for Differential Equations with “ Maxima ”

and Applied Analysis 3 In the paper, we will study the differential equation with “maxima” 2.1 with an initial condition at two different initial points. For this purpose, we consider the differential equation 2.1 with the initial condition x t φ t − τ0 for t ∈ τ0 − r, τ0 , 2.3 where τ0 ≥ 0. Assume there exist a solution x t; t0, φ of the IVP 2.1 , 2.2 defined on t0 − r, t0 T , and there exists a solution y t; τ0, φ of the IVP 2.1 , 2.3 defined on τ0 − r, τ0 T , where T > 0 is a given constant. In the paper, we will compare the behavior of both solutions. Definition 2.1. The function α ∈ C t0−r, t0 T ,R ∪C1 t0, t0 T ,R is called a lower upper solution of the IVP 2.1 , 2.2 for t ≥ t0 − h if the following inequalities are satisfied: α′ t ≤ ≥ f ( t, α t , max s∈ t−r,t α s ) for t ∈ t0, t0 T , α t ≤ ≥ φ t − t0 for t ∈ t0 − r, t0 . 2.4 Note that the function α t is a lower upper solution of the IVP 2.1 , 2.3 for t ∈ τ0 − r, τ0 T if the point t0 in the inequalities 2.4 is replaced by τ0. 3. Comparison Results 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. Theorem 3.1. Let the following conditions be satisfied. 1 Let t0, τ0 ≥ 0 be fixed such that η τ0 − t0 > 0. 2 The function α ∈ C t0 − r, t0 T ,R ∪ C1 t0, t0 T ,R is a lower solution of the IVP 2.1 , 2.2 . 3 The function β ∈ C τ0 − r, τ0 T ,R ∪C1 τ0, τ0 T ,R is an upper solution of the IVP 2.1 , 2.3 . 4 The function f t, u, v : R × R × R → R is nondecreasing in its both first and third argument t for any u ∈ R, and there exist positive constants L1, L2 such that for t ∈ R and u1 ≥ u2 and v1 ≥ v2, the inequality f t, u1, v1 − f t, u2, v2 ≤ L1 u1 − u2 L2 v1 − v2 3.1 holds. 5 The function φ t ∈ C −r, 0 ,R . Then α t ≤ β t η for t ∈ t0, t0 T . 4 Abstract and Applied Analysis Proof. Choose a positive number M such that M > L1 L2 and a positive number . Define a functionw t β t η e for t ∈ t0, t0 T andw t β t η e0 for t ∈ t0 − r, t0 . Then for t ∈ t0, t0 T , the following inequalities


Introduction
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 1 .The object of regulation is a generator of constant current with parallel stimulation, and the quantity regulated is the voltage on the clamps of the generator feeding an electric circuit with different loads.An equation with maxima is used if the regulator is constructed such that the maximal deviation of the quantity is regulated on the segment t − r, t .The equation describing the work of the regulator has the form Tu t u t q max s∈ t−r,t u s f t , 1.1 where T and q are constant characterizing the object, u t is the voltage regulated, and f t is the perturbing effect.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., 2-14 .At the same time, the presence of maximum function in the equations causes impossibilities of direct application of almost all known results for delay functional differential equations see the monograph 15 .Also, in the most cases, differential equations with "maxima," including the some linear scalar equations, are not possible to be solved in an explicit form.That requires the application of approximate methods.In the recent years, several effective approximate methods, based on the upper and lower solutions of the given problem, was proved for various problems of differential equations in 16-21 .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 22-30 .Also some approximate methods for various types of differential equations with initial time difference are proved in 23, 26, 31-33 .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., 20, 34-42 .In our case, it causes troubles with obtaining the solutions, which are successive approximations.To avoid these difficulties, we consider linear problems with different initial functions at any step.It allows us to chose appropriate initial data.Additionally, in connection with the presence of the maximum function in the equation and computer application of the suggested algorithm, an appropriate computer program is realized and it is applied to some examples illustrating the advantages and the practical application of the considered algorithm.

Preliminary Notes and Definitions
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" 2.1 with an initial condition at two different initial points.For this purpose, we consider the differential equation 2.1 with the initial condition where τ 0 ≥ 0. Assume there exist a solution x t; t 0 , ϕ of the IVP 2.1 , 2.2 defined on t 0 − r, t 0 T , and there exists a solution y t; τ 0 , ϕ of the IVP 2.1 , 2.3 defined on τ 0 − r, τ 0 T , where T > 0 is a given constant.In the paper, we will compare the behavior of both solutions.
Definition 2.1.The function α ∈ C t 0 −r, t 0 T , R ∪C 1 t 0 , t 0 T , R is called a lower upper solution of the IVP 2.1 , 2.2 for t ≥ t 0 − h if the following inequalities are satisfied:

2.4
Note that the function α t is a lower upper solution of the IVP 2.1 , 2.3 for t ∈ τ 0 − r, τ 0 T if the point t 0 in the inequalities 2.4 is replaced by τ 0 .

Comparison Results
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.Theorem 3.1.Let the following conditions be satisfied.
4 The function f t, u, v : R × R × R → R is nondecreasing in its both first and third argument t for any u ∈ R, and there exist positive constants L 1 , L 2 such that for t ∈ R and

3.4
Assume the contrary, that is, there exists a point t 1 > t 0 such that α t < w t for The comparison results are true if the inequality τ 0 < t 0 holds.

Theorem 3.3 comparison result .
Let the following conditions be satisfied.
3 The function f t, u, v : R × R × R → R is nonincreasing in its first argument and is nondecreasing in its third argument for any u ∈ R, and there exist positive constants L 1 , L 2 such that for t ∈ R and u 1 ≥ u 2 and v 1 ≥ v 2 the inequality The proof of Theorem 3.3 is similar to the proof of Theorem 3.1 and we omit it.
In what follows, we shall need that the functions α, β Consider the sets:

3.6
In our further investigations, we will need the following comparison result on differential inequalities with "maxima." where the positive constants Then the inequality m t ≤ 0 holds on t 0 − r, T t 0 .
In our further investigations, we will use the following result, which is a partial case of Theorem 3.1 44 .Lemma 3.5 existence and uniqueness result .Let the following conditions be fulfilled.

Then the initial value problem for the linear scalar equation
has a unique solution u t − t 0 on the interval t 0 − r, T t 0 .
Theorem 4.1.Let the following conditions be fulfilled.
5 The function f ∈ C Ω α 0 , β 0 , t 0 , R is nondecreasing in its first argument and satisfies the one side Lipschitz condition Then there exist two sequences of functions {α n t } ∞ 0 and {β n t } ∞ 0 such that a the functions α n t f both sequences uniformly converge and x t lim n → ∞ α n t is a solution of the IVP 2.1 , 2.2 in S α 0 , β 0 , t 0 , η , and y t lim n → ∞ β n t is a solution of the IVP 2.1 , 2.3 in S α 0 , β 0 , τ 0 , η .
Let L 0 min s∈ t 0 −r,t 0 ϕ s − t 0 − α 0 s ≥ 0. Choose a number k 0 ∈ 0, 1 such that We consider the linear differential equation with "maxima" with the initial condition According to Lemma 3.5, the linear initial value problem 4.5 , 4.6 has a unique solution α 1 t in J 1 .
We will prove that α 0 t ≤ α 1 t on J 1 .
From the choice of L 0 , k 0 and equality 4.6 , it follows that Now, let t ∈ t 0 , T t 0 .Consider the function u t α 0 t − α 1 t defined on J 1 .It is clear that u t ≤ 0 on t 0 − r, t 0 .From the definition of the function α 0 t and 4.5 , we have Now from inequality 4.7 and we obtain According to Lemma 3.4, the inequality u t ≤ 0 holds on J 1 , that is, α 0 t ≤ α 1 t .We will prove that the function α 1 t is a lower solution of the IVP 2.1 , 2.2 on J 1 .From equality 4.6 , it follows the validity of the inequality α 1 t ≤ ϕ t−t 0 on t 0 −r, t 0 .Now, let t ∈ t 0 , T t 0 .Then, since α 0 t ≤ α 1 t , according to the one side Lipschitz condition 2 of Theorem 4.1, we have where Q t f t, α 0 t , max s∈ t−r,t α 0 s .Next, according to Lemma 3.4 for the function u t α 1 t − β 0 t − η , the inequality α 1 t ≤ β 0 t η holds on J 1 , that is, the inclusion α 1 ∈ S α 0 , β 0 , t 0 , η is valid.

4.13
There exists a unique solution β 1 t of the IVP 4.12 , 4.13 , which is defined on J 2 .The function β 1 t is an upper solution of the IVP 2.1 , 2.3 on J 2 , and the inclusion β 1 ∈ S α 0 , β 0 , τ 0 , η is valid.The proofs are similar to the ones about the function α 1 .We omit the proofs.
Functions α 1 t , β 1 t are a lower and an upper solution of the IVP 2.1 , 2.2 and 2.1 , 2.3 correspondingly.According to Lemma 3.4 the inequality α 1 t ≤ β 1 t η on J 1 holds.
Similarly, recursively, we can construct two sequences of functions {α n t } ∞ 0 and {β n t } ∞ 0 .In fact, if the functions α n t and β n t are known, and L n min s∈ −r,0 ϕ s − α n s , C n min s∈ −r,0 β n s − ϕ s , and the numbers k n , p n ∈ 0, 1 are such that Now following exactly as for the case n 0, it can be proved that the function α n 1 t is a lower solution of the IVP 2.1 , 2.2 on J 1 , the function β n 1 t is an upper solution of the IVP 2.1 , 2.3 on J 2 , the inclusions α n 1 ∈ S α n , β n , t 0 , η , β n 1 ∈ S α n , β n , τ 0 , η are valid, and the inequalities 4.3 hold.
Therefore, the sequence {α n t } ∞ 0 is uniformly convergent on J 1 and From the uniform convergence and the definition of the functions α n t and β n t , it follows that the following inequalities hold 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.Theorem 4.2.Let the following conditions be fulfilled.

4.22
Then there exist two sequences of functions {α n t } ∞ 0 and {β n t } ∞ 0 such that a the functions α n t f both sequences uniformly converge and x t lim n → ∞ α n t is a solution of the IVP 2.1 , 2.2 in S α 0 , β 0 , t 0 , η and y t lim n → ∞ β n t is a solution of the IVP 2.1 , 2.3 in S α 0 , β 0 , τ 0 , η .

Computer Realization
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 Mathematica, Mathlab, and so forth, because of the maximum of the unknown function over a past time interval.It requires proposed and computer realized new algorithms for solving such kind of equations.In 43 , an algorithm for solving a class of differential equations with "maxima," based on the trapezoid's method, is presented.The main problem in the suggested algorithm is obtaining the maximum of the unknown function over a past time interval.It is based on the idea that the local maximum depends on values that are initially inserted and later they are removed within the range.A special structure is applied to help quicker obtaining the local maximum 43 .The suggested scheme could be based on any numerical method for solving differential equations.In this paper, we will use Euler method in the application of the algorithm 43 for solving differential equations with "maxima."We combine the algorithm of Euler method and obtaining of maximum of the function, and we get the following modification: where the values max y n,k max s∈ t k −r,t k y n s and max y n 1,k max s∈ t k −r,t k y n 1 s are obtained by the algorithm suggested in 43 , and the initial values of y n 1 t 0 are found by initial conditions 4.16 and 4.18 .

Applications
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.Example 6.1.Consider the following scalar nonlinear differential equation with "maxima": with the initial condition where x ∈ R. It is easy to check that the initial value problem 6.1 , 6.2 has a zero solution.

6.3
That is, L 1 0.64, L 2 2, and the inequality L 1 L 2 T < 1 holds.The successive approximations α n 1 t and β n 1 t will be the unique solutions of the linear problems x s for t ∈ 0, 0.

6.13
That is, L 1 3.2, L 2 0.5, and the inequality L 1 L 2 T < 1 holds.According to Theorem 4.1, there exist two convergent monotone sequences of functions {α n t } ∞ 0 and {β n t } ∞ 0 such that 4.3 holds and their limits are solutions of the IVP 6.9 and 6.10 .where α 0 t −1/4, β 0 t 1/4, L n C n 0.25, k n 0.25 n , p n 0.25 n , and n 1, 2, 3, . . . .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 4, to solve the initial value problems 6.14 and 6.15 for n 1, 2, 3, 4, 5.In this case at any step, the function y n t k in 5.1 is replaced by the functions α n t k or β n t k , respectively, the function y n 1 t k is the next approximation α n 1 t k or β n 1 t k , respectively, and the function f is the right part of 6.14 or 6.15 , respectively, and h 0.00001.Some of the numerical values of the successive approximations {α n t } ∞ 0 and {β n t } ∞ 0 , which are lower/upper solutions of the given problem, are shown in the Table 1.Also, the obtained successive approximations are graped on the Figure 1.Both, Table 1 and Figure 1, illustrate us the monotonicity of the sequences {α n t } ∞ 0 and {β n t } ∞ 0 and the validity of the inequalities
4.14then the function α n 1 t is the unique solution of the initial value problem for the linear differential equation with "maxima" n C n for t ∈ τ 0 − r, τ 0 .4.18 1, 2, . . .are lower solutions of the IVP 2.1 , 2.2 on t 0 − r, T t 0 ; b the functions β n t 1, 2, . . .are upper solutions of the IVP 2.1 , 2.3 on τ 0 − r, T τ 0 ; min s∈ −0.1,0 −α n s ≥ 0, C n min s∈ −0.1,0 β n s ≥ 0, and the numbers k n , p n ∈ 0, 1 are chosen such that k n ≤ L n , p n ≤ C n , n 1, 2, 3, ....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 α 1 t is a solution of the IVP Choose k 1 19/33.Then the IVP 6.6 has an exact solution α 2 t −57/725.It is clear thatα 0 t ≤ α 1 t ≤ α 2 t < 0 x t .The first upper approximation β 1 t is a solution of the IVP illustrates that the presence of the constants p n and k n allow us easily to construct each successive approximation in an explicit form.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.
Figure 1: graphs of α k t and β k t .85,k 1, 2, 3, 4 .The successive approximations α n 1 t will be the unique solutions of the linear problemsx t 1 α n t − 0.5α n t − 2 − 3.2 x − α n t − 0.5 max n s − 2 − 3.2x − 0.5 max s∈ t−0.1,t x s for t ∈ 1.5, 1.75 , x t −k n L n for t ∈ 1.4, 1.5 , 6.14and the successive approximations β n 1 t will be the unique solutions of the linear problems

Table 1 :
Values of the successive lower/upper approximations α k t and β k t .85 .