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

Abstract

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.

1. 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 form𝑇𝑢^′(𝑡)+𝑢(𝑡)+𝑞max_{[]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.

2. Preliminary Notes and Definitions

Consider the following initial value problem for the nonlinear differential equation with “maxima’’ (IVP): 𝑥^′=𝑓𝑡,𝑥(𝑡),max_{[]𝑠∈𝑡−𝑟,𝑡}𝑥(𝑠)for𝑡∈𝑡₀,𝑥(𝑡),∞=𝜑𝑡−𝑡₀for𝑡∈𝑡₀−𝑟,𝑡₀, (2.1)(2.2) where 𝑥∈ℝ, 𝑡₀≥0, 𝑓∶ℝ₊×ℝ×ℝ→ℝ, 𝜑(𝑡)∶[−𝑟,0]→ℝ, and 𝑟>0 is a fixed constant.

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 condition𝑥(𝑡)=𝜑𝑡−𝜏₀for𝑡∈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)≤(≥)𝑓𝑡,𝛼(𝑡),max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼(𝑠)for𝑡∈𝑡₀,𝑡₀,𝛼(𝑡)≤(≥)𝜑+𝑇𝑡−𝑡₀for𝑡∈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 𝑓𝑡,𝑢₁,𝑣₁−𝑓𝑡,𝑢₂,𝑣₂≤𝐿₁𝑢₁−𝑢₂+𝐿₂𝑣₁−𝑣₂ (3.1) holds.(5)The function 𝜑(𝑡)∈𝐶([−𝑟,0],ℝ).
Then 𝛼(𝑡)≤𝛽(𝑡+𝜂) for 𝑡∈[𝑡₀,𝑡₀+𝑇].

Proof. Choose a positive number 𝑀 such that 𝑀>𝐿₁+𝐿₂ and a positive number 𝜖. Define a function 𝑤(𝑡)=𝛽(𝑡+𝜂)+𝜖𝑒^𝑀𝑡 for 𝑡∈[𝑡₀,𝑡₀+𝑇] and 𝑤(𝑡)=𝛽(𝑡+𝜂)+𝜖𝑒^𝑀𝑡0for𝑡∈[𝑡₀−𝑟,𝑡₀). Then for 𝑡∈[𝑡₀,𝑡₀+𝑇], the following inequalities 𝑤(𝑡)>𝛽𝑡+𝜂,max_{[]𝑠∈𝑡−𝑟,𝑡}𝑤(𝑠)=max_{[]𝑠∈𝑡−𝑟,𝑡}𝛽𝑠+𝜂+𝜖𝑒^𝑀𝑠≥max_{𝑠∈[𝑡−𝑟,𝑡]}𝛽𝑠+𝜂=max_{𝑠∈[𝑡+𝜂−𝑟,𝑡+𝜂]}𝛽(𝑠),max_{[]𝑠∈𝑡−𝑟,𝑡}𝑤(𝑠)≤max_{[]𝑠∈𝑡−𝑟,𝑡}𝛽𝑠+𝜂+𝜖𝑒^𝑀𝑡=max_{𝑠∈[𝑡+𝜂−𝑟,𝑡+𝜂]}𝛽(𝑠)+𝜖𝑒^𝑀𝑡 (3.2) hold.
Therefore, we obtain 𝑤^′(𝑡)≥𝑓𝑡+𝜂,𝛽𝑡+𝜂,max_{𝑠∈[𝑡+𝜂−𝑟,𝑡+𝜂]}𝛽(𝑠)+𝜖𝑀𝑒^𝑀𝑡≥−𝐿₁𝑤(𝑡)−𝛽𝑡+𝜂−𝐿₂max_{[]𝑠∈𝑡−𝑟,𝑡}𝑤(𝑠)−max_{𝑠∈[𝑡+𝜂−𝑟,𝑡+𝜂]}𝛽(𝑠)+𝑓𝑡+𝜂,𝑤(𝑡),max_{[]𝑠∈𝑡−𝑟,𝑡}𝑤(𝑠)+𝜖𝑀𝑒^𝑀𝑡≥𝑓𝑡+𝜂,𝑤(𝑡),max_{[]𝑠∈𝑡−𝑟,𝑡}𝑤(𝑠)+𝜖𝑒^𝑀𝑡𝑀−𝐿₁−𝐿₂>𝑓𝑡+𝜂,𝑤(𝑡),max_{[]𝑠∈𝑡−𝑟,𝑡}𝑤(𝑠)≥𝑓𝑡,𝑤(𝑡),max_{[]𝑠∈𝑡−𝑟,𝑡}𝑤(𝑠). (3.3)
Note that for 𝑡∈[𝑡₀−𝑟,𝑡₀] we have 𝛼(𝑡)≤𝜑(𝑡−𝑡₀)=𝜑(𝑡+𝜂−𝜏₀)≤𝛽(𝑡+𝜂)<𝑤(𝑡). We will prove that 𝛼(𝑡)≤𝑤(𝑡)for𝑡∈𝑡₀−𝑟,𝑡₀+𝑇. (3.4)
Assume the contrary, that is, there exists a point 𝑡₁>𝑡₀ such that 𝛼(𝑡)<𝑤(𝑡) for 𝑡∈[𝑡₀−𝑟,𝑡₁), 𝛼(𝑡₁)=𝑤(𝑡₁) and 𝛼(𝑡)≥𝑤(𝑡) for 𝑡∈(𝑡₁,𝑡₂), where 𝑡₂ is sufficiently close to 𝑡₁. Therefore 𝛼^′(𝑡₁)≥𝑤^′(𝑡₁). Then 𝑤^′(𝑡₁)≤𝛼^′(𝑡₁)≤𝑓(𝑡₁,𝛼(𝑡₁),max_{𝑠∈[𝑡1−𝑟,𝑡1]}𝛼(𝑠))≤𝑓(𝑡₁,𝑤(𝑡₁),max_{𝑠∈[𝑡1−𝑟,𝑡1]}𝑤(𝑠))<𝑤^′(𝑡₁). The obtained contradiction proves the claim.
Therefore, 𝛼(𝑡)≤𝑤(𝑡)<𝛽(𝑡+𝜂) for 𝑡∈[𝑡₀−𝑟,𝑡₀+𝑇].

Corollary 3.2. Let the conditions of Theorem 3.1 be fulfilled. Then 𝛼(𝑡−𝜂)≤𝛽(𝑡) for 𝑡∈[𝜏₀−𝑟,𝜏₀+𝑇].

The comparison results are true if the inequality 𝜏₀<𝑡₀ holds.

Theorem 3.3 (comparison result). Let the following conditions be satisfied.(1)The conditions 2, 3, 5 of Theorem 3.1 are satisfied.(2)Let 𝑡₀,𝜏₀≥0 be fixed such that 𝜂=𝜏₀−𝑡₀<0.(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 𝐿₁,𝐿₂ such that for 𝑡∈ℝ₊ and 𝑢₁≥𝑢₂ and 𝑣₁≥𝑣₂ the inequality𝑓𝑡,𝑢₁,𝑣₁−𝑓𝑡,𝑢₂,𝑣₂≤𝐿₁𝑢₁−𝑢₂+𝐿₂𝑣₁−𝑣₂ (3.5) holds.
Then 𝛼(𝑡)≤𝛽(𝑡+𝜂) for 𝑡≥𝑡₀.

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∈𝐶([𝑡₀−𝑟,𝑇+𝑡₀],ℝ) are such that 𝛼(𝑡)≤𝛽(𝑡+𝜂).

Consider the sets: 𝑆𝛼,𝛽,𝑡₀,𝜂=𝑢∈𝐶𝑡₀−𝑟,𝑇+𝑡₀,ℝ∶𝛼(𝑡)≤𝑢(𝑡)≤𝛽𝑡+𝜂for𝑡∈𝑡₀−𝑟,𝑇+𝑡₀,𝑆𝛼,𝛽,𝜏₀,𝜂=𝑢∈𝐶𝜏₀−𝑟,𝑇+𝜏₀,ℝ∶𝛼𝑡−𝜂≤𝑢(𝑡)≤𝛽(𝑡)for𝑡∈𝜏₀−𝑟,𝑇+𝜏₀,Ω𝛼,𝛽,𝑡₀=𝑡,𝑥,𝑦∈𝑡₀,𝑇+𝑡₀×ℝ²∶𝛼(𝑡)≤𝑥≤𝛽(𝑡),max_{𝑠∈[𝑡−𝑟,𝑡]}𝛼(𝑠)≤𝑦≤max_{𝑠∈[𝑡−𝑟,𝑡]}𝛽(𝑠). (3.6)

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

Lemma 3.4 (see [43, Lemma  2.1]). Let the function 𝑚∈𝐶([𝑡₀−𝑟,𝑇+𝑡₀],ℝ)∪𝐶¹([𝑡₀,𝑇+𝑡₀],ℝ) satisfies the inequalities 𝑚^′≤−𝐿₁𝑚(𝑡)−𝐿₂min_{[]𝑠∈𝑡−𝑟,𝑡}𝑚(𝑠)for𝑡∈𝑡₀,𝑇+𝑡₀,𝑚(𝑡)≤0for𝑡∈𝑡₀−𝑟,𝑡₀, (3.7) where the positive constants 𝐿₁,𝐿₂ are such that (𝐿₁+𝐿₂)𝑇≤1.
Then the inequality 𝑚(𝑡)≤0 holds on [𝑡₀−𝑟,𝑇+𝑡₀].

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.(1)The functions 𝑄∈𝐶([𝑡₀,𝑡₀+𝑇],ℝ),𝑞∈𝐶([𝑡₀,𝑡₀+𝑇],ℝ).(2)The functions 𝜑∈𝐶([−𝑟,0],ℝ).
Then the initial value problem for the linear scalar equation 𝑢^′=𝐿₁𝑢(𝑡)+𝐿₂max_{[]𝑠∈𝑡−𝑟,𝑡}𝑢(𝑠)+𝑄(𝑡)for𝑡∈𝑡₀,𝑇+𝑡₀,𝑢(𝑡)=𝜑𝑡−𝑡₀for𝑡∈𝑡₀−𝑟,𝑡₀ (3.8) has a unique solution 𝑢(𝑡−𝑡₀) on the interval [𝑡₀−𝑟,𝑇+𝑡₀].

4. Main Results

Denote 𝐽₁=[𝑡₀−𝑟,𝑇+𝑡₀] and 𝐽₂=[𝜏₀−𝑟,𝑇+𝜏₀], where 𝑡₀,𝜏₀≥0 are fixed numbers.

Case 1. Let 𝑡₀≤𝜏₀.
Theorem 4.1. Let the following conditions be fulfilled.(1)The points 𝑡₀,𝜏₀≥0 are such that 𝜂=𝜏₀−𝑡₀≥0.(2)The function 𝛼₀(𝑡)∈𝐶([𝑡₀−𝑟,𝑇+𝑡₀],ℝ)∪𝐶¹([𝑡₀,𝑇+𝑡₀],ℝ) is a lower solution of the IVP (2.1), (2.2) in 𝐽₁.(3)The function 𝛽₀(𝑡)∈𝐶([𝜏₀−𝑟,𝑇+𝜏₀],ℝ)∪𝐶¹([𝜏₀,𝑇+𝜏₀],ℝ) is an upper solution of the IVP (2.1), (2.3) in 𝐽₂.(4)The function 𝜑∈𝐶([−𝑟,0],ℝ).(5)The function 𝑓∈𝐶(Ω(𝛼₀,𝛽₀,𝑡₀),ℝ) is nondecreasing in its first argument and satisfies the one side Lipschitz condition 𝑓𝑡,𝑢₁,𝑣₁−𝑓𝑡,𝑢₂,𝑣₂≥𝐿₁𝑢₂−𝑢₁+𝐿₂𝑣₂−𝑣₁, (4.1) for 𝑢₁≤𝑢₂ and 𝑣₁≤𝑣₂, where 𝐿₁,𝐿₂>0 are such that𝐿₁+𝐿₂𝑇≤1. (4.2)
Then there exist two sequences of functions {𝛼_𝑛(𝑡)}^∞₀ and {𝛽_𝑛(𝑡)}^∞₀ such that(a)the functions 𝛼_𝑛(𝑡)=1,2,…  are lower solutions of the IVP (2.1), (2.2) on [𝑡₀−𝑟,𝑇+𝑡₀];(b)the functions 𝛽_𝑛(𝑡)=1,2,…  are upper solutions of the IVP (2.1), (2.3) on [𝜏₀−𝑟,𝑇+𝜏₀]; (c)the sequence {𝛼_𝑛(𝑡)}^∞₀ is increasing;(d)the sequence {𝛽_𝑛(𝑡)}^∞₀ is decreasing;(e)the inequalities 𝛼₀(𝑡)≤⋯≤𝛼_𝑛(𝑡)≤𝛽_𝑛𝑡+𝜂≤⋯≤𝛽₀𝑡+𝜂,𝑡∈𝑡₀−𝑟,𝑇+𝑡₀,𝛼₀𝑡−𝜂≤⋯≤𝛼_𝑛𝑡−𝜂≤𝛽_𝑛(𝑡)≤⋯≤𝛽₀(𝑡),𝑡∈𝜏₀−𝑟,𝑇+𝜏₀ (4.3) hold;(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(𝛼₀,𝛽₀,𝑡₀,𝜂), and 𝑦(𝑡)=lim_𝑛→∞𝛽_𝑛(𝑡) is a solution of the IVP (2.1), (2.3) in 𝑆(𝛼₀,𝛽₀,𝜏₀,𝜂).Proof. According to Theorem 3.1, the inequality 𝛼₀(𝑡)≤𝛽₀(𝑡+𝜂) holds on [𝑡₀−𝑟,𝑇+𝑡₀].
Let 𝐿₀=min_{𝑠∈[𝑡0−𝑟,𝑡0]}(𝜑(𝑠−𝑡₀)−𝛼₀(𝑠))≥0. Choose a number 𝑘₀∈[0,1) such that 𝑘₀≤𝐿₀. (4.4)
Therefore, 𝜑(𝑡−𝑡₀)−𝛼₀(𝑡)≥𝐿₀>𝑘₀𝐿₀ or 𝛼₀(𝑡)≤𝜑(𝑡−𝑡₀)−𝑘₀𝐿₀ on [𝑡₀−𝑟,𝑡₀].
We consider the linear differential equation with “maxima’’ 𝑥^′(𝑡)=𝑓𝑡,𝛼₀(𝑡),max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼₀(𝑠)−𝐿₁𝑥(𝑡)−𝛼₀(𝑡)−𝐿₂max_{[]𝑠∈𝑡−𝑟,𝑡}𝑥(𝑠)−max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼₀(𝑠)for𝑡∈𝑡₀,𝑇+𝑡₀, (4.5) with the initial condition 𝑥(𝑡)=𝜑𝑡−𝑡₀−𝑘₀𝐿₀for𝑡∈𝑡₀−𝑟,𝑡₀. (4.6) According to Lemma 3.5, the linear initial value problem (4.5), (4.6) has a unique solution 𝛼₁(𝑡) in 𝐽₁.
We will prove that 𝛼₀(𝑡)≤𝛼₁(𝑡) on 𝐽₁.
From the choice of 𝐿₀,𝑘₀ and equality (4.6), it follows that 𝛼₀(𝑡)≤𝜑(𝑡−𝑡₀)−𝑘₀𝐿₀=𝛼₁(𝑡) for 𝑡∈[𝑡₀−𝑟,𝑡₀].
Now, let 𝑡∈[𝑡₀,𝑇+𝑡₀]. Consider the function 𝑢(𝑡)=𝛼₀(𝑡)−𝛼₁(𝑡) defined on 𝐽₁. It is clear that 𝑢(𝑡)≤0 on [𝑡₀−𝑟,𝑡₀]. From the definition of the function 𝛼₀(𝑡) and (4.5), we have 𝑢^′(𝑡)≤−𝐿₁𝑢(𝑡)−𝐿₂max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼₀(𝑠)−max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼₁(𝑠)for𝑡∈𝑡₀,𝑇+𝑡₀. (4.7) Now from inequality (4.7) and max_{[]𝑠∈𝑡−ℎ,𝑡}𝛼₀(𝑠)−max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼₁(𝑠)=max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼₀(𝑠)−𝛼₁(𝜉)≥𝛼₀(𝜉)−𝛼₁(𝜉)≥min_{[]𝑠∈𝑡−𝑟,𝑡}𝛼₀(𝑠)−𝛼₁(𝑠)=min_{[]𝑠∈𝑡−𝑟,𝑡}𝑢(𝑠), (4.8) we obtain 𝑢^′(𝑡)≤−𝐿₁𝑢(𝑡)−𝐿₂min_{[]𝑠∈𝑡−𝑟,𝑡}𝑢(𝑠),𝑡∈𝑡₀,𝑇+𝑡₀. (4.9) According to Lemma 3.4, the inequality 𝑢(𝑡)≤0 holds on 𝐽₁, that is, 𝛼₀(𝑡)≤𝛼₁(𝑡).
We will prove that the function 𝛼₁(𝑡) is a lower solution of the IVP (2.1), (2.2) on 𝐽₁.
From equality (4.6), it follows the validity of the inequality 𝛼₁(𝑡)≤𝜑(𝑡−𝑡₀) on [𝑡₀−𝑟,𝑡₀]. Now, let 𝑡∈[𝑡₀,𝑇+𝑡₀]. Then, since 𝛼₀(𝑡)≤𝛼₁(𝑡), according to the one side Lipschitz condition 2 of Theorem 4.1, we have 𝛼^′₁(𝑡)=𝑄(𝑡)−𝐿₁𝛼₁(𝑡)−𝛼₀(𝑡)−𝐿₂max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼₁(𝑠)−max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼₀(𝑠)=𝑓𝑡,𝛼₁(𝑡),max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼₁(𝑠)+𝑓𝑡,𝛼₀(𝑡),max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼₀(𝑠)−𝑓𝑡,𝛼₁(𝑡),max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼₁(𝑠)−𝐿₁𝛼₁(𝑡)−𝛼₀(𝑡)−𝐿₂max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼₁(𝑠)−max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼₀(𝑠)≤𝑓𝑡,𝛼₁(𝑡),max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼₁(𝑠)for𝑡∈𝑡₀,𝑇+𝑡₀, (4.10) where 𝑄(𝑡)=𝑓(𝑡,𝛼₀(𝑡),max_{𝑠∈[𝑡−𝑟,𝑡]}𝛼₀(𝑠)).
Next, according to Lemma 3.4 for the function 𝑢(𝑡)=𝛼₁(𝑡)−𝛽₀(𝑡−𝜂), the inequality 𝛼₁(𝑡)≤𝛽₀(𝑡+𝜂) holds on 𝐽₁, that is, the inclusion 𝛼₁∈𝑆(𝛼₀,𝛽₀,𝑡₀,𝜂) is valid.
Let 𝐶₀=min_{𝑠∈[𝜏0−𝑟,𝜏0]}(𝛽₀(𝑠)−𝜑(𝑠−𝜏₀))≥0. Choose a number 𝑝₀∈[0,1) such that 𝑝₀≤𝐶₀. (4.11)
Therefore, 𝛽₀(𝑡)−𝜑(𝑡−𝜏₀)≥𝐶₀>𝑝₀𝐶₀ or 𝛽₀(𝑡)≥𝜑(𝑡−𝜏₀)+𝑝₀𝐶₀ on [𝜏₀−𝑟,𝜏₀].
We consider the linear differential equation with “maxima’’ 𝑥^′(𝑡)=𝑓𝑡,𝛽₀(𝑡),max_{[]𝑠∈𝑡−𝑟,𝑡}𝛽₀(𝑠)−𝐿₁𝑥(𝑡)−𝛽₀(𝑡)−𝐿₂max_{[]𝑠∈𝑡−𝑟,𝑡}𝑥(𝑠)−max_{[]𝑠∈𝑡−𝑟,𝑡}𝛽₀(𝑠)for𝑡∈𝜏₀,𝑇+𝜏₀ (4.12) with the initial condition 𝑥(𝑡)=𝜑𝑡+𝜏₀+𝑝₀𝐶₀for𝑡∈𝜏₀−𝑟,𝜏₀. (4.13)
There exists a unique solution 𝛽₁(𝑡) of the IVP (4.12), (4.13), which is defined on 𝐽₂.
The function 𝛽₁(𝑡) is an upper solution of the IVP (2.1), (2.3) on 𝐽₂, and the inclusion 𝛽₁∈𝑆(𝛼₀,𝛽₀,𝜏₀,𝜂) is valid. The proofs are similar to the ones about the function 𝛼₁. We omit the proofs.
Functions 𝛼₁(𝑡),𝛽₁(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 𝛼₁(𝑡)≤𝛽₁(𝑡+𝜂) on 𝐽₁ 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 𝑘_𝑛≤𝐿_𝑛,𝑝_𝑛≤𝐶_𝑛, (4.14) then the function 𝛼_𝑛+1(𝑡) is the unique solution of the initial value problem for the linear differential equation with “maxima’’ 𝑥′=𝑓𝑡,𝛼_𝑛(𝑡),max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼_𝑛(𝑠)−𝐿₁𝑥−𝛼_𝑛(𝑡)−𝐿₂max_{[]𝑠∈𝑡−𝑟,𝑡}𝑥(𝑠)−max_{[]𝑠∈𝑡−𝑟,𝑡}𝛼_𝑛(𝑠)for𝑡∈𝑡₀,𝑇+𝑡₀,𝑥(𝑡)=𝜑(𝑡)−𝑘_𝑛𝐿_𝑛for𝑡∈𝑡₀−𝑟,𝑡₀, (4.15)(4.16) and the function 𝛽_𝑛+1(𝑡) is the unique solution of the initial value problem 𝑥^′(𝑡)=𝑓𝑡,𝛽_𝑛(𝑡),max_{[]𝑠∈𝑡−𝑟,𝑡}𝛽_𝑛(𝑠)−𝐿₁𝑥−𝛽_𝑛(𝑡)−𝐿₂max_{[]𝑠∈𝑡−ℎ,𝑡}𝑥(𝑠)−max_{[]𝑠∈𝑡−𝑟,𝑡}𝛽_𝑛(𝑠)for𝑡∈𝜏₀,𝑇+𝜏₀,𝑥(𝑡)=𝜑(𝑡)+𝑝_𝑛𝐶_𝑛for𝑡∈𝜏₀−𝑟,𝜏₀. (4.17)(4.18)
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 𝐽₁, the function 𝛽_𝑛+1(𝑡) is an upper solution of the IVP (2.1), (2.3) on 𝐽₂, the inclusions 𝛼_𝑛+1∈𝑆(𝛼_𝑛,𝛽_𝑛,𝑡₀,𝜂), 𝛽_𝑛+1∈𝑆(𝛼_𝑛,𝛽_𝑛,𝜏₀,𝜂) are valid, and the inequalities (4.3) hold.
Therefore, the sequence {𝛼_𝑛(𝑡)}^∞₀ is uniformly convergent on 𝐽₁ and {𝛽_𝑛(𝑡)}^∞₀ is uniformly convergent on 𝐽₂.
Denote lim_𝑛→∞𝛼_𝑛(𝑡)=𝑢(𝑡),𝑡∈𝑡₀−𝑟,𝑇+𝑡₀,lim_𝑛→∞𝛽_𝑛(𝑡)=𝑣(𝑡),𝑡∈𝜏₀−𝑟,𝑇+𝜏₀. (4.19) From the uniform convergence and the definition of the functions 𝛼_𝑛(𝑡) and 𝛽_𝑛(𝑡), it follows that the following inequalities hold 𝛼₀(𝑡)≤𝑢(𝑡)≤𝑣𝑡+𝜂≤𝛽₀𝑡+𝜂,𝑡∈𝑡₀−𝑟,𝑇+𝑡₀. (4.20)

Case 2. Let 𝑡₀>𝜏₀.
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.(1)The points 𝑡₀,𝜏₀≥0 are such that 𝜂=𝑡₀−𝜏₀≥0.(2)The conditions 2, 3, 4 of Theorem 4.1 are satisfied.(3)The function 𝑓∈𝐶(Ω(𝛼₀,𝛽₀,𝜏₀),ℝ) is nonincreasing in its first argument and satisfies the one side Lipschitz condition 𝑓𝑡,𝑢₁,𝑣₁−𝑓𝑡,𝑢₂,𝑣₂≥𝐿₁𝑢₂−𝑢₁+𝐿₂𝑣₂−𝑣₁ (4.21) for 𝑢₁≤𝑢₂ and 𝑣₁≤𝑣₂, where 𝐿₁,𝐿₂>0 are such that 𝐿₁+𝐿₂𝑇≤1. (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)=1,2,…  are lower solutions of the IVP (2.1), (2.2) on [𝑡₀−𝑟,𝑇+𝑡₀];(b)the functions 𝛽_𝑛(𝑡)=1,2,…  are upper solutions of the IVP (2.1), (2.3) on [𝜏₀−𝑟,𝑇+𝜏₀]; (c)the sequence {𝛼_𝑛(𝑡)}^∞₀ is increasing;(d)the sequence {𝛽_𝑛(𝑡)}^∞₀ is decreasing;(e)the inequalities 𝛼₀(𝑡)≤⋯≤𝛼_𝑛(𝑡)≤𝛽_𝑛𝑡+𝜂≤⋯≤𝛽₀𝑡+𝜂,𝑡∈𝑡₀−𝑟,𝑇+𝑡₀,𝛼₀𝑡−𝜂≤⋯≤𝛼_𝑛𝑡−𝜂≤𝛽_𝑛(𝑡)≤⋯≤𝛽₀(𝑡),𝑡∈𝜏₀−𝑟,𝑇+𝜏₀ (4.23) hold;(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(𝛼₀,𝛽₀,𝑡₀,𝜂) and 𝑦(𝑡)=lim_𝑛→∞𝛽_𝑛(𝑡) is a solution of the IVP (2.1), (2.3) in 𝑆(𝛼₀,𝛽₀,𝜏₀,𝜂).

5. 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: 𝑡_𝑘+1=𝑡_𝑘𝑦+ℎ,_𝑛+1𝑡_𝑘+1=𝑦_𝑛+1𝑡_𝑘+ℎ𝑓𝑡_𝑘,𝑦_𝑛+1𝑡_𝑘,𝑦_𝑛𝑡_𝑘,max𝑦_𝑛,𝑘,max𝑦_𝑛+1,𝑘, (5.1) where the values max𝑦_𝑛,𝑘=max_{𝑠∈[𝑡𝑘−𝑟,𝑡𝑘]}𝑦_𝑛(𝑠) and max𝑦_𝑛+1,𝑘=max_{𝑠∈[𝑡𝑘−𝑟,𝑡𝑘]}𝑦_𝑛+1(𝑠) are obtained by the algorithm suggested in [43], and the initial values of 𝑦_𝑛+1(𝑡₀) are found by initial conditions (4.16) and (4.18).

6. 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’’: 𝑥^′=11−𝑥(𝑡)−2max_{[]𝑠∈𝑡−0.1,𝑡}𝑥(𝑠)−1,for𝑡∈[0,0.35], (6.1) with the initial condition 𝑥(𝑡)=0,𝑡∈[−0.1,0], (6.2) where 𝑥∈ℝ. It is easy to check that the initial value problem (6.1), (6.2) has a zero solution.
In this case 𝑓(𝑡,𝑥,𝑦)≡1/(1−𝑥)−2𝑦−1, 𝑡₀=𝜏₀=0, 𝑇=0.35, 𝑟=0.1, 𝐽₁=𝐽₂=[−0.1,0.35]. Choose 𝛼₀(𝑡)=−1/4 and 𝛽₀(𝑡)=1/4. Then 𝛼₀(𝑡) is a lower solution and 𝛽₀(𝑡) is an upper solution of the IVP (6.1), (6.2). In this case, Ω(𝛼₀,𝛽₀,𝑡₀)={(𝑡,𝑢,𝑣)∈[0,0.35]×ℝ²∶−1/4≤𝑢,𝑣≤1/4}. Let (𝑡,𝑢₁,𝑣₁),(𝑡,𝑢₂,𝑣₂)∈Ω(𝛼₀,𝛽₀,𝑡₀) and 𝑢₁≤𝑢₂ and 𝑣₁≤𝑣₂. Then 𝑓𝑡,𝑢₁,𝑣₁−𝑓𝑡,𝑢₂,𝑣₂=11−𝑢₁−11−𝑢₂+2𝑣₂−𝑣₁=11−𝑢₁1−𝑢₂𝑢₂−𝑢₁+2𝑣₂−𝑣₁≥16𝑢25²−𝑢₁+2𝑣₂−𝑣₁. (6.3) That is, 𝐿₁=0.64,𝐿₂=2, and the inequality (𝐿₁+𝐿₂)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^′=11−𝛼_𝑛(𝑡)+0.64𝛼_𝑛(𝑡)−1−0.64𝑥−2max_{[]𝑠∈𝑡−0.1,𝑡}𝑥(𝑠)[],𝑥(𝑡)for𝑡∈0,0.35=−𝑘_𝑛𝐿_𝑛[],𝑥for𝑡∈−0.1,0_′(𝑡)=11−𝛽_𝑛(𝑡)−1+0.64𝛽_𝑛(𝑡)−0.64𝑥−2max_{[]𝑠∈𝑡−0.1,𝑡}𝑥(𝑠)[],𝑥(𝑡)for𝑡∈0,0.35=𝑝_𝑛𝐶_𝑛for𝑡∈[−0.1,0], (6.4) where 𝐿_𝑛=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 𝛼₁(𝑡) is a solution of the IVP 𝑥^′9=−25−0.64𝑥−2max_{𝑠∈[𝑡−0.1,𝑡]}𝑥(𝑠)[],𝑥(𝑡)𝑘for𝑡∈0,0.35=−⁰4,𝑡∈[−0.1,0]. (6.5) Choose 𝑘₀=6/11. Then the IVP (6.5) has an exact solution 𝛼₁(𝑡)=−3/22.
The second lower approximation 𝛼₂(𝑡) is a solution of the IVP 𝑥^′=−57275−0.64𝑥−2max_{𝑠∈[𝑡−0.1,𝑡]}𝑥(𝑠)[],𝑥(𝑡)for𝑡∈0,0.35=−𝑘₁3[].22,𝑡∈−0.1,0 (6.6) Choose 𝑘₁=19/33. Then the IVP (6.6) has an exact solution 𝛼₂(𝑡)=−57/725. It is clear that 𝛼₀(𝑡)≤𝛼₁(𝑡)≤𝛼₂(𝑡)<0=𝑥(𝑡).
The first upper approximation 𝛽₁(𝑡) is a solution of the IVP 𝑥^′(𝑡)=3775−0.64𝑥−2max_{[]𝑠∈𝑡−0.1,𝑡}𝑥(𝑠)[],𝑥(𝑡)=𝑝for𝑡∈0,0.35₀4,𝑡∈[−0.1,0]. (6.7) Choose 𝑝₀=74/99. Then the IVP (6.7) has an exact solution 𝛽₁(𝑡)=37/198≈0.1869.
The second upper approximation 𝛽₂(𝑡) is a solution of the IVP 𝑥^′=−114481398475−0.64𝑥−2max_{𝑠∈[𝑡−0.1,𝑡]}𝑥(𝑠)[],𝑥(𝑡)for𝑡∈0,0.35=𝑝₁37[].198,𝑡∈−0.1,0 (6.8) Choose 𝑝₁=114481/196581. Then the IVP (6.8) has an exact solution 𝛽₂(𝑡)=114481/1051974≈0.108825.
It is obviouse that 𝛽₀(𝑡)>𝛽₁(𝑡)>𝛽₂(𝑡)>0=𝑥(𝑡)>𝛼₂(𝑡)>𝛼₁(𝑡)>𝛼₀(𝑡) on the interval [−0.1,0.35].
Example 6.1 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.

Example 6.2. Consider the following IVP for the scalar nonlinear differential equation with “maxima’’: 𝑥^′=𝑡1+𝑥(𝑡)−0.5max_{[]𝑠∈𝑡−0.1,𝑡}𝑥(𝑠)[],𝑥(𝑡)[],𝑥−2for𝑡∈1.5,1.75=0,𝑡∈1.4,1.5_′=𝑡1+𝑥(𝑡)−0.5max_{[]𝑠∈𝑡−0.1,𝑡}𝑥(𝑠)[],𝑥(𝑡)[],−2for𝑡∈2.35,2.6=0,𝑡∈2.25,2.35 (6.9)(6.10) where 𝑥∈ℝ.
In this case, 𝑓(𝑡,𝑥,𝑦)≡𝑡/(1+𝑥)−0.5𝑦−2, 𝑡₀=1.5, 𝜏₀=2.35, 𝜂=𝜏₀−𝑡₀=0.85, 𝑇=0.25, 𝑟=0.1, 𝐽₁=[1.4,1.75], 𝐽₂=[2.25,2.6].
Choose 𝛼₀(𝑡)=−1/4. Then for 𝑡∈[1.5,1.75] the inequality 𝑓𝑡,𝛼₀(𝑡),max_{[]𝑠∈𝑡−0.1,𝑡}𝛼₀(𝑠)=𝑡+11−1/48−2=4𝑡3−1584≥1.53−158=0.125>0 (6.11) holds. Therefore, 𝛼₀(𝑡) is a lower solution of the IVP (6.9) in [1.4,1.75].
Let 𝛽₀(𝑡)=1/4. Then for 𝑡∈[2.35,2.6] the inequality 𝑓𝑡,𝛽₀(𝑡),max_{[]𝑠∈𝑡−0.1,𝑡}𝛽₀(𝑠)=𝑡−11+1/48−2=4𝑡5−1784≤2.65−178=−0.045<0 (6.12) holds. Therefore, 𝛽₀(𝑡) is an upper solution of the IVP (6.10) on [2.25,2.6].
In this case, 𝜏₀>𝑡₀ and Ω(𝛼₀,𝛽₀,𝜏₀)={(𝑡,𝑢,𝑣)∈[1.5,1.75]×ℝ²∶−1/4≤𝑢,𝑣≤1/4}.
Let 𝑡∈[1.5,1.76], −1/4≤𝑢₁≤𝑢₂≤1/4 and −1/4≤𝑣₁≤𝑣₂≤1/4. Then from the inequality 𝑡/((1+𝑢₁)(1+𝑢₂))≤(16/9)𝑡≤3.1111<3.2, we obtain 𝑓𝑡,𝑢₁,𝑣₁−𝑓𝑡,𝑢₂,𝑣₂=𝑡1+𝑢₁−𝑡1+𝑢₂+0.5𝑣₂−𝑣₁=𝑡1+𝑢₁1+𝑢₂𝑢₂−𝑢₁+0.5𝑣₂−𝑣₁≤3.2𝑢₂−𝑢₁+𝑣₂−𝑣₁. (6.13) That is, 𝐿₁=3.2,𝐿₂=0.5, and the inequality (𝐿₁+𝐿₂)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).
The successive approximations 𝛼_𝑛+1(𝑡) will be the unique solutions of the linear problems 𝑥^′=𝑡1+𝛼_𝑛(𝑡)−0.5𝛼_𝑛(𝑡)−2−3.2𝑥−𝛼_𝑛(𝑡)−0.5max_{[]𝑠∈𝑡−0.1,𝑡}𝑥(𝑠)−max_{[]𝑠∈𝑡−0.1,𝑡}𝛼_𝑛(𝑠)=𝑡1+𝛼_𝑛(𝑡)+2.7𝛼_𝑛(𝑡)+0.5max_{[]𝑠∈𝑡−0.1,𝑡}𝛼_𝑛(𝑠)−2−3.2𝑥−0.5max_{[]𝑠∈𝑡−0.1,𝑡}𝑥(𝑠)[],𝑥(𝑡)for𝑡∈1.5,1.75=−𝑘_𝑛𝐿_𝑛for𝑡∈[1.4,1.5], (6.14) and the successive approximations 𝛽_𝑛+1(𝑡) will be the unique solutions of the linear problems (𝑡)=𝑡𝑥′1+𝛽_𝑛(𝑡)+2.7𝛽_𝑛(𝑡)+0.5max_{[]𝑠∈𝑡−0.1,𝑡}𝛽_𝑛(𝑠)−2−3.2𝑥−0.5max_{[]𝑠∈𝑡−0.1,𝑡}𝑥(𝑠)[],𝑥(𝑡)for𝑡∈2.35,2.6=𝑝_𝑛𝐶_𝑛for𝑡∈[2.25,2.35], (6.15) where 𝛼₀(𝑡)=−1/4, 𝛽₀(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 inequalities𝛼₀(𝑡)≤𝛼₁(𝑡)≤⋯≤𝛼_𝑛(𝑡)≤𝛽_𝑛(𝑡+.85)≤⋯≤𝛽₀(𝑡+.85),𝑡∈[1.5,1.75]. (6.16)

Table 1 

Values of the successive lower/upper approximations 𝛼_𝑘(𝑡) and 𝛽_𝑘(𝑡+.85).

Figure 1 

(graphs of 𝛼𝑘(𝑡) and 𝛽𝑘(𝑡+.85), 𝑘=1,2,3,4).

Acknowledgments

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.