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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 851691, 25 pages
Study of Solutions to Some Functional Differential Equations with Piecewise Constant Arguments
1Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain
2Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Received 12 October 2011; Revised 6 January 2012; Accepted 9 January 2012
Academic Editor: Josef Diblík
Copyright © 2012 Juan J. Nieto and Rosana Rodríguez-López. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We provide optimal conditions for the existence and uniqueness of solutions to a nonlocal boundary value problem for a class of linear homogeneous second-order functional differential equations with piecewise constant arguments. The nonlocal boundary conditions include terms of the state function and the derivative of the state function. A similar nonhomogeneous problem is also discussed.
In the study of second-order functional differential equations, there is a wide range of works dealing with periodic boundary value problems and piecewise continuous functional dependence. We mention, for instance, , where an equation independent of the first derivative is analyzed, and many other works as [2–14], where existence and stability results are provided.
Most of works on this field deal with nonconstructive existence results. However, in [7, 11], explicit solutions are found for second-order functional differential equations with piecewise constant arguments, through the calculus of the Green’s function. Other works in relation with first and higher-order differential equations with delay are [15–18].
In , a class of linear second-order differential equations with piecewise constant arguments is considered under the nonlocal conditions , , where , that is, an initial position is assumed and the boundary conditions are independent of the derivative of the function.
In this paper, we study the following nonlocal boundary value problem for homogeneous linear second-order functional differential equations: for , and , where the functional dependence is given by the greatest integer part , and the nonlocal boundary conditions involve both the state function and its derivative, which is the main difference from the study in . A discussion for nonhomogeneous equations is also included. We consider the existence and uniqueness of solution to this problem, providing optimal conditions and calculating the exact expression of solutions.
To better illustrate the significant differences between the results included in this paper and those in , we remark that  is devoted to the study of the same class of linear second-order differential equations with piecewise constant arguments but considering a nonlocal boundary value problem where the value of the unknown function is fixed at the initial instant (i.e., an initial condition is imposed) and the boundary condition also involves the value of the sought solution at the right endpoint of the interval and an intermediate point . None of the conditions imposed in  involve the value of the rate of change of the solution at any points. Thus, the nonlocal conditions in  can be reduced, by using the expression of the solution, to a boundary value problem affecting only the state of the solution, being independent of the rate of change of the state of the system. On the other hand, the nonlocal problem considered in this paper does not fix a certain initial position and, moreover, the boundary condition introduces the dependence, not only on the state, but also on the variation of the state of the system. Indeed, the value of the unknown function and its derivative is determined by the value of the corresponding magnitudes at an intermediate point of the interval of interest. Moreover, since the results in these two papers are also extensible to the special case where the intermediate point is identical to the initial instant (), we compare the consequences of both works to explain better the implications of the study of each problem. If we consider , the results in  are applicable to obtain the solution to a class of linear second-order differential equations with piecewise constant arguments subject to the boundary conditions , and, therefore, we solve a problem with fixed values of the function at the end-points of the interval. However, again for , the results presented in this paper allow to characterize the existence of solution (and provide its explicit expression) imposing boundary conditions of the type , , which include, as a particular case, periodic boundary conditions (on and ).
The main results are stated after a few preliminary results are recalled, and, finally, examples are included to show the applicability of these results.
Consider the equation where and . The following concepts and results come from .
For the constants , we define as Consider also given by In the definitions of functions and , we denote, for ,
It is easy to check  that .
Theorem 2.2 (see [7, Theorem 2.1]). The initial value problem for , has the solution where ,
To simplify calculus, for , one denotes
3. Main Results
First, we consider the nonlocal boundary value problem for , and .
If , the condition is reduced to a periodic boundary condition of Dirichlet type. On the other hand, if , we obtain , and, moreover, if , we deduce the periodicity of the derivative of the function.
Theorem 3.1. If and , then problem (3.1) is solvable for each in the image of the mapping
Hence, there exists a unique solution to (3.1) for every if and only if the matrix
is nonsingular. In this case, the solution is given by
where is the inverse image of by .
If the matrix (3.3) is singular, then there exists an infinite number of solutions to problem (3.1) for every in the image of , taking as initial position and initial slope in (3.4) any preimage of by , and there exist no solutions for the remaining pairs .
Proof. We consider the initial value problem
for , whose solution is, by Theorem 2.2 [7, Theorem 2.1],
where . We analyze under which conditions the boundary conditions are fulfilled, taking into account that
First, we consider the case , and . To obtain the solution to (3.1), we calculate and from the expressions of (3.6) and (3.7), in order to satisfy and . Hence the boundary conditions are written as that is, The properties of functions and produce that the boundary conditions, in the cases where , or (or both), can be derived from the expression obtained in the case studied , and .
Therefore, this system has a solution only for in the image of the mapping Therefore, the solution is unique if the matrix (3.3) is nonsingular, in which case, the initial condition and initial slope are given as On the other hand, if the matrix (3.3) is singular, for in the image of , the infinitely many solutions are calculated from (3.6) taking as initial conditions any preimage of by , which proves the result.
Remark 3.2. The existence and uniqueness condition (3.3) is reduced, if , to the nonsingularity of the matrices and On the other hand, if , it is just reduced to the nonsingularity of the matrix .
Remark 3.3. In Theorem 3.1, if we consider , then the boundary conditions are , and the condition of existence and uniqueness of solution is reduced to the nonsingularity of which coincides with condition (15) in [7, Theorem 2.2]. Moreover, if , the above-mentioned nonsingularity condition provides that the unique solution to the homogeneous equation subject to periodic boundary value conditions is the trivial solution (see [7, Theorem 2.2]).
Remark 3.5. Summarizing, the study of the solvability of (3.1) is reduced to the discussion of system (3.9). In the case of existence of an infinite number of solutions, we must analyze the rank of the matrix in (3.3) to determine whether any pair is admissible as initial position and slope or the space of initial conditions is one-dimensional. If the rank of (3.3) is zero, problem (3.1) is solvable only for , that is, problem for , and , has an infinite number of solutions, given by (3.4), for any initial point .
On the other hand, and denoting by the matrix in (3.3), if and depends linearly on each column of , then we have a one-dimensional space of solutions, whose starting conditions are determined from the row of the system corresponding to a nonzero minor of .
Next, with the purpose of extending Theorem 3.1 to the nonhomogeneous case, we consider the following nonlocal boundary value problem for a nonhomogeneous equation: for , , , and . Using the expression of the solution for the corresponding initial value problem provided by [19, Theorem 5.2], we prove the following existence (and uniqueness) result.
Theorem 3.6. Consider , , , and . Then problem (3.19) has a unique solution if and only if the matrix (3.3) is nonsingular. Under this assumption, the unique solution to (3.19) is given by with defined as and taking, as the initial condition , where On the other hand, if the matrix (3.3) is singular, the number of solutions to (3.19) is determined by the discussion of the linear system and, for each solution to this system (in case it exists), the expression (3.20) provides a solution to (3.19).
Proof. The result follows from the expression of the solution (3.20) for the corresponding initial value problem, whose derivative is given by and the fact that the restrictions represented by the boundary conditions produce, respectively,
We present some examples where different situations are analyzed in order to decide if the existence and uniqueness condition (nonsingularity of (3.3)) holds and, in case of nonuniqueness, the dimension of the space of solutions. The exact expression of the solution (or solutions) is given explicitly.
We recall that, for , is given, depending on the values of the coefficients as follows.(i)If , thus Note that the determinant of the matrix is equal to (ii)If , , thus The determinant of the matrix is equal to (iii)If , thus The determinant of the matrix is equal to (iv)If , and denoting thus In this case, the determinant of the matrix is equal to (v)If , and denoting ,thusThe determinant of is given by
Example 4.1. Consider the problem
where . In this case, , , and . Therefore, we get so that matrix is reduced towhose determinant is calculated as
Hence, is nonsingular if and only if and . In this case, problem (4.16) has a unique solution for every , given by (see (3.4))
For the particular case where , , and , then
and hence the unique solution to
is given by
See Figure 1.
If , then and the matrix is reduced to , which has rank one. Since is singular, there exists an infinite number of solutions to problem (4.16) for every in the image of the mapping given by the matrix , that is, for every , where . On each of these cases, the solutions are given by (3.4) taking as initial position and initial slope any preimage of , that is, the initial slope can be taken as and the initial position can be chosen as any real number .
Hence, the solutions to the problem where , are given by and the same expression where and . Hence, for each fixed, all the vertical shifts of the function are solutions to (4.27). In Figure 2, we show the graph of a solution which generates, by vertical shifting, all the solutions to problem (4.27) for .
If , there is no solution for problem (4.16) with .
Finally, if , then has rank one. Hence problem (4.16), for , has an infinite number of solutions for in the image of the mapping given by the matrix , that is, for the values of , and satisfying the system for some values of . For the existence of these values of , it is necessary and sufficient that the rank of the matrix of the system (equal to one) coincides with the rank of the matrix that is, for instance, equivalent to Under this assumption, there exists an infinite number of solutions to problem which are given by where and satisfy (one of) the equations in (4.31).
On the other hand, if (4.34) fails, there is no solution to (4.35).
Example 4.2. Next, consider the problem
where . In this case, , , and . Hence, the matrix
is reduced to
whose determinant is . If and , then is invertible and problem (4.37) has a unique solution for every . Since
the unique solution to (4.37), for fixed, is given by
For instance, for and , then and , which is invertible with inverse . Then the unique solution to (4.37), for and and fixed, is given by See Figure 3 for the solution to (4.37) with , , , and :
On the other hand, if , then is singular with rank 1, since if and only if and if and only if or .
Hence, there exists an infinite number of solutions to problem (4.37) with , for every in the image of the mapping given by the matrix , that is, for every such that . Under this assumption, the solutions to (4.37) with are given by where the initial position can be chosen as any real number and the initial slope must satisfy
For instance, if and , then , so that . Therefore, there exists an infinite number of solutions to problem (4.37) with and , for every such that . In this case, the solutions to (4.37) with and are given by where is any real number. Note that . All the solutions are obtained as vertical shifts of the function
See Figure 4 for the graph of function for , which generates by vertical shifting the one-dimensional space of solutions to problem (4.37) with , , , and .
Finally, if , then which has rank 1. Note that for and ; for and ; for ; and for and .
Therefore, problem (4.37) for has solution (an infinite number of solutions) if and only if