Quasilinearization for nonlinear boundary value problems for delay-type difference equations with maxima
© Hristova and Golev; licensee Springer. 2014
Received: 29 October 2013
Accepted: 27 January 2014
Published: 31 March 2014
The paper deals with an approximate method for solving a mixed boundary value problem for nonlinear difference equations containing a maximum of the unknown function over a past time interval. Every successive approximation to the unknown solution is the unique solution of an appropriately constructed initial value problem for a linear difference equation with maxima, and an algorithm for its explicit obtaining is suggested. Also, each approximation is a lower/upper solution of the given mixed problem. The rapid convergence of the successive approximations is proved. The suggested algorithm is realized as a computer program and it is applied to an example.
MSC: 39A23, 39A99, 65Q10.
In the last few decades great attention has been paid to automatic control systems and their applications to computational mathematics and modeling. Many problems in the control theory correspond to the maximal deviation of the regulated quantity. Discrete modeling of such kind of problems is done adequately by difference equations with maxima over a past time discrete interval. These difference equations are a part of the set of difference equations with delays. Meanwhile, the delays have recently been found crucial in many areas such as neuronal dynamics. For example, the effects of periodic subthreshold pacemaker activity and time-delayed coupling on stochastic resonance over scale-free neuronal networks are studied in ; the effects of spatiotemporal additive noise on the spatial dynamics of excitable neuronal media that is locally modeled by a two-dimensional map are considered in ; the discrete model of the movement of eukaryotic cells regulated by a process of phase separation of two competing enzymes on the cell membrane is studied in ; front propagation and synchronization transitions in dependence on the information transmission delay and coupling strength over scale-free neuronal networks with different average degrees and scaling exponents are investigated in .
The presence of the maximum function over a discrete past time interval in the discrete equation requires not only more complicated calculations but also a development of new methods for qualitative investigations of the behavior of their solutions as well as approximate methods for their solving. The character of the maximum function leads to a variety of different types of difference equations. The properties of solutions of some special types of difference equations with maxima are studied in [5–7]. In several papers various types of boundary value problems for difference equations have been studied and the monotone iterative method has been applied. For example, in [8, 9] first-order difference equations are studied; in  some criteria for existence and uniqueness results for nth order anti-periodic difference equations are developed; in  a generalized delay difference equation is studied by lower and upper solutions, but the problem consists only of a boundary condition, which does not get uniqueness of the solution; the global boundary value problem for difference equations without any kind of delay is well studied in ; in  a nonlinear boundary value problem for a delay difference equation with one delay is studied by the monotone iterative method; in  an approximate method with a rapid convergence is applied to an initial value problem for difference equations with maxima. Also, in , the monotone iterative technique is applied to a periodic boundary value problem for difference equations with maxima, but the successive approximation is solutions of periodic boundary value problems, which are practically difficult to be obtained. Approximate methods for various problems for differential equations with maxima are proved and applied in [16–18].
at each current time the value of unknown function is included in both parts of the equation, so the equation could not be solved recursively;
the delays are without any restrictions;
the boundary condition is set up in a very general way and it involves many other cases, which are studied in the literature (see, for example, the above-mentioned papers).
The main purpose of the paper is to establish comparison results which allow us to use upper and lower solutions in order to build two convergent monotonic sequences of functions with discrete domains. Each term of these sequences is a solution of an appropriately constructed initial value problem for a linear difference equation with maxima. An algorithm for solving these initial value problems is given. Also, each term of both sequences is a lower/upper solution of the given nonlinear boundary value problem. The limits of both sequences coincide to the solution of a given problem and the rapid convergence of both sequences is proved. Also, the algorithm is computerized and it is applied to a particular example to show the advantages of the suggested approximate method.
2 Statement of the problem
Let , ℤ be the set of all integers. For any : , we denote .
Let : , and : be fixed.
where , , , , : , , , , and .
Any solution of MBVP (1)-(3) is a finite sequence of real numbers, and we consider it as a real-valued function with a discrete domain.
The presence of delays generalizes the type of the considered difference equation since the function f could depend on different delays at any point k. In the case and , , the right-hand side of (1) is reduced to . For some other particular cases, see equation (39) of the paper.
The presence of both delays and the maximum function in the equation leads to a new statement of the problem, which involves both the boundary condition and the initial condition. Problem (1)-(3) covers many different problems for difference equations with delays and maxima such as the initial value problem, the periodic boundary value problem, the linear boundary value problem.
3 Preliminary notes, basic notations and definitions
Note and , where and , .
For any function , , , we denote by the first derivative of with respect to its j th argument, and by the second derivative of with respect to its j th and k th arguments.
Using the above notations, the right-hand sides of equation (1) and boundary condition (3) could be written in a simpler way: and .
We will use the norm .
We will use lower/upper solutions of MBVP (1)-(3) which are defined in a well-known way.
Definition 1 
4 Linear delay difference inequalities with maxima
where K is a nonnegative constant.
The inequalities (4) are the base of the main proof, and we will obtain a solution of (4) in an explicit form.
The delays , are such that , .
- 2.The functions , , and(5)
The function satisfies inequalities (4).
The function is nondecreasing and , , for , .
By mathematical induction from (8), condition 2 and the definition of , we prove the claim. □
In the case when all delays for , condition 2 of Lemma 1 could be changed by a simpler one.
The delays for all and .
- 2.The functions , , and(9)
The function satisfies inequalities (4).
Corollary 1 Let conditions 1 and 2 of Lemma 1/Lemma 2 be fulfilled, and let the function satisfy inequalities (4) for .
Then for all .
5 Linear delay difference equations with maxima
where , for , and : for .
We will consider two cases with respect to the type of delays .
Case 1: Let the inequality hold for all , . Additionally, we assume that inequality (9) is satisfied for all .
Assume that the values of the unknown solution are obtained for all , where . Now let .
Case 2: Let there exist at least one and such that . Additionally, we assume that inequality (5) is satisfied for all .
Assume that the values of the unknown solution are obtained for all , where . Now let .
Case 2.2: Let inequality (17) be not satisfied for all .
Therefore, there exists such that .
6 Method of quasilinearization
We will apply the method of quasilinearization to obtain the approximate solution of MBVP (1)-(3).
The delays for all and .
- 2.The function and for any and , the equality holds, where the functions , are twice continuously differentiable with respect to any component of V and the following inequalities are valid for , :(20)(21)(22)
The function is nondecreasing with respect to all its arguments.
The functions , is a lower solution, is an upper solution of MBVP (1)-(3), and for .
() are lower solutions of MBVP (1)-(3).
() are upper solutions of MBVP (1)-(3).
Both sequences are convergent on and their limits and are solutions of MBVP (1)-(3) in . In the case of uniqueness, both limits coincide with this solution.
- (e)The convergence is semi-quadratic, i.e., there exist , , such that
Proof We will give an algorithm for construction of successive approximations to the exact unknown solution of MBVP (1)-(3).
Assume that the functions , , are constructed so that the following conditions are satisfied:
(H1) and for ;
(H2) for ;
(H3) for ;
(H4) functions , are lower and upper solutions of MBVP (1)-(3), respectively.
From inequalities (20) and , , , it follows that . From inequalities (20), (22) we get that the functions , , satisfy (5) for . IVPs (23), (24) and (25), (26) have unique solutions and , respectively, which are defined on the interval .
From condition (H3), for , the constants follow.
Define a function by .
According to Corollary 1, from inequalities (27)-(29) it follows that for , i.e., for .
Similarly, we prove for , i.e., condition (H1) is satisfied for .
Now, we will prove that the function is a lower solution of MBVP (1)-(3).
Let . From (24) we get .
Let . From the proved above and condition 3 of Theorem 1, the inequality holds.
Similarly, we prove that is an upper solution of MBVP (1)-(3).
Now, we will prove (H2) for . Define a function by the equality .
Let . Then .
Also, from condition 3 of Theorem 1 and (H2) for , we get .
According to Corollary 1, the inequality holds for , i.e., for .
From claim (c) of Theorem 1 it follows that . Taking limits as in IVPs (23), (24) and (25), (26), we obtain that V and W are solutions of MBVP (1)-(3) in .
Now, we will study the power of convergence of the sequences of functions and .
Let and .
where is defined by (6) for , i.e., the sequence semi-quadratically converges to the exact solution of (1)-(3).
Similarly, we prove the semi-quadratic convergence of . □
Then all the claims of Theorem 1 are true.
Now we will give an example of a generalized difference equation to illustrate the advantage of both the introducing delay functions in the equation and the suggested above scheme for approximate obtaining of a solution.
MBVP (39) is of type (1)-(3), where , , , , , , , , .
The functions and , are lower and upper solutions of (39). Inequalities (37), (38) and condition 3 of Theorem 1 are satisfied. According to Theorem 2, MBVP (39) has a solution, and we will obtain it as a limit of two sequences of successive approximations.
Values of successive lower approximations and upper approximations ,
Research was partially supported by Fund Scientific Research MU13FMI002, Plovdiv University.
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