Delay differential equations: a partially ordered sets approach in vectorial metric spaces
© Bachar and Khamsi; licensee Springer. 2014
Received: 13 July 2014
Accepted: 8 September 2014
Published: 19 September 2014
In this paper we develop a fixed point theorem in the partially ordered vector metric space by using vectorial norm. Then we use it to prove the existence of periodic solutions to nonlinear delay differential equations.
MSC:06F30, 46B20, 47E10, 34K13, 34K05.
where , where is the space of a continuous function from into . Moreover, on compact intervals, depends continuously on φ and f. The existence, uniqueness and continuous dependency results for delay differential equations motivate the definition of the state of the system to the pair , which completely describes the past history of the solution at time .
This type of problems was already investigated for delay or ordinary differential equations in real-valued spaces by Drici et al. and also Nieto and Rodríguez-López [2–4]; see also [5–10]. The papers cited therein provide additional reading on this topic. Recently, many authors such as Dhage  started to investigate the Krasnosel’skii theorem in partially ordered spaces with applications to nonlinear fractional differential equations in real-valued spaces.
The aim of this paper is to extend the fixed point results of contraction mappings in vectorial partially ordered sets by using vectorial norms introduced by Agarwal [12, 13] which will allow us to investigate the existence of periodic solutions of vectorial delay differential equations. For more on fixed point theory, the reader may consult the books [14, 15]. To be able to prove such results, we will need some monotonicity result of the flow. The main difficulty encountered comes from the fact that we are not working on a classical normed vector space. In fact, consider the example of the evolution of a burning zone. The velocity of its evolution is not the same if the burning area is narrow or if it is wide. In this case we cannot study the trajectory of one point without taking into account the others. Therefore we need to consider mappings defined on the subsets of which have nice properties with respect to a metric distance other than the regular norm.
for any and .
for any such that .
2 Fixed point theorems in
The first version of the classical Banach contraction principle in partially ordered metric spaces was given by Ran and Reurings . Their result was used in applications to linear and nonlinear matrix equations.
Theorem 2.1 
for any . If there exists with or , then for any , the sequence of iterates is convergent and its limit is independent of x.
Nieto and Rodríguez-López  extended the above theorem to obtain the following.
Theorem 2.2 
f is continuous and there exists with or ;
is such that for any convergent nondecreasing sequence , we have for any ;
is such that for any convergent nonincreasing sequence , we have for any .
Then f has a fixed point. Moreover, if every pair of elements of X has an upper or a lower bound, then f has a unique fixed point and for any , the orbit converges to the fixed point of f.
for any .
if and only if ;
for any ;
for any , where the order in is the usual pointwise order inherited from ℝ.
The above fixed point theorems may be stated in terms of vector-valued distances. To the best of our knowledge, this is the first attempt to do such generalization.
for any . Assume that there exists with or . Then T has a unique fixed point , and for any , the sequence of iterates converges to h.
The uniqueness of the fixed point follows from the above conclusion. □
3 Application to delay differential equations
The aim of this section is to give an existence result of periodic solutions of the delay differential equations (1.1)-(1.3). The following lemma will be needed.
where is defined by (3.5). □
Before stating the main result of this section, we will need the following definition.
An upper solution for (1.1) satisfies the reversed inequalities.
The following theorem gives a sufficient condition for the existence of periodic solutions of a delay differential equation assuming the existence of a lower or an upper solution.
Theorem 3.1 Consider problem (1.1)-(1.2) with f continuous and satisfying (1.4). Then the existence of a lower solution or an upper solution for (1.1)-(1.2) provides the existence of a unique solution of (1.1)-(1.2).
Otherwise, if is an upper solution, we will get for any . Theorem 2.3 will enable us to prove the existence and uniqueness of the periodic solution. □
Then our vectorial condition (1.4) is also satisfied for the degenerated delay differential equation. Moreover, if we choose a constant vector v such that , then can be considered as a lower solution of the delay differential equation (3.13). This will guarantee the existence of periodic solutions.
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this Research group No. (RG-1435-079).
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