Abstract
In this paper, we establish some conditions for the existence and uniqueness of the monotonic solutions for nonhomogeneous systems of first-order linear differential equations, by using a result of the fixed points theory for sequentially complete gauge spaces.
1. Introduction
Demidovich [] proved an important result regarding the boundedness property of the monotonic solutions for homogeneous systems of linear differential equations of first order. Iseki [] extended this result for nonhomogeneous systems and showed that under certain conditions any monotonic solution is bounded and its limit exists to . The weak point of the papers of Demidovich and Iseki consists of the fact that they demonstrate the boundedness of monotonous solutions, without specifying whether such solutions exist. Regarding the existence of monotonic solutions of differential equations, important results were obtained by Rovderová [], Tóthová and Palumbíny [], Rovder [], Li and Fan [], Yin [], Ertem and Zafer [], Aslanov [], Chu [], and Sanhan et al. [].
The purpose of this article is to study the existence and uniqueness of the monotonic solutions for nonhomogeneous systems of first-order linear differential equations with variable coefficients. The novelty and originality of our article consists of us proving the existence and uniqueness of the monotonic solution, and finding the conditions under which these properties take place. To prove the theorem of existence and uniqueness of a monotonic solution we rely on the theory of gauge spaces. Dugundji [] showed that any family of pseudometrics (gauge structure) on a nonempty set induces a uniform structure on that set, and conversely, any uniform structure on a nonempty set is generated by a family of pseudometrics. Moreover, the uniform structure is separating (Hausdorff) if and only if the gauge structure is separating. In this way, the gauge spaces (separating gauge structures) can be identified with Hausdorff uniform spaces. Colojoara [] and Gheorghiu [] extended the Banach contraction principle to the gauge spaces. Similar fixed point results were obtained by Knill [] and Tarafdar [] in the case of Hausdorff uniform spaces.
2. Preliminaries
Throughout this paper we follow the standard terminology and notation for systems of ordinary differential equations.
Further, we denote by the real interval .
Let us consider a nonhomogeneous system of first-order linear differential equations with variable coefficients:
where , and .
Definition 1.
([]) Let X be a nonempty set. A map is called a pseudometric (gauge) on X if the following conditions are satisfied:
- (1)
- , for all ;
- (2)
- , for all ;
- (3)
- , for all .
Definition 2.
([]) Let X be a nonempty set. We say that:
- (i)
- A family of pseudometrics on X is named a gauge structure on X;
- (ii)
- A gauge structure on X is called separating if for each pair of points , with , there is such that ;
- (iii)
- A pair of a nonempty set X and a separating gauge structure on X is named a gauge space.
Definition 3.
([]) Let be a gauge space, where . We say that:
- (i)
- A sequence is called convergent in X if: there exists a point with the property that for every and there is a number such that for all we have ;
- (ii)
- A sequence is named a Cauchy sequence if: for every and there is a number such that for all and we have ;
- (iii)
- The gauge space is called sequentially complete if: any Cauchy sequence of points in X is convergent in X.
Theorem 1.
([,]) Let be a sequentially complete gauge space, where , and is an operator. We suppose that: for every there exists such that
Then, T has a unique fixed point on X.
Theorem 2.
([]) Let X be a nonempty set, a metric space, and , , a sequence of functions. Then, the following statements are true:
- (i)
- If , is uniformly convergent on X to a function , then , , is a uniformly Cauchy sequence;
- (ii)
- If , , is a uniformly Cauchy sequence and is complete, then there is a function such that , , is uniformly convergent on X to f.
Theorem 3.
([]) Let be a topological space, a complete metric space, and , , a sequence of functions. If , , is uniformly convergent on X to a function , and every function , , is continuous on X, then f is continuous on X.
Theorem 4.
([]) Let us consider . The following properties are valid:
- (i)
- is a complete normed linear space, where , ;
- (ii)
- is a complete metric space, where , .
3. The Existence and Uniqueness of the Monotonic Solutions
Theorem 5.
Let us consider . Then the following statements are valid:
- (i)
- The maps , , , are pseudometrics on , where ;
- (ii)
- is a sequentially complete gauge space, where ;
- (iii)
- If the functions , are continuous on and there is a number such that , , (A is bounded on ), then the system of first-order linear differential equations ((Equation 1)), with initial condition , has a unique solution for .
Proof.
(i) Let be an arbitrary number.
We choose arbitrary elements. We deduce that is a continuous function on . On the other hand, the norm is a continuous map on . Consequently, the function , is continuous on . Additionally, the function , is continuous on . Therefore, the function , is continuous on . It follows that is a continuous function on . Applying the Weierstrass extreme value theorem we find that is bounded on and there exists such that ( attains its supremum in ). Therefore, the map is well-defined.
We now prove that the function verifies the properties of a pseudometric. Let be arbitrary functions. By using the properties of the norm we get
hence
(ii) The family of pseudometrics defines on a gauge structure. We remark that this gauge structure is separating because for each pair of elements , with , and every , we have . Consequently, is a gauge space.
We choose and arbitrary elements.
We now show that the gauge space is sequentially complete. Let us consider an arbitrary Cauchy sequence. It follows that for and there is a number such that for all and we have , i.e., . As for all , we get for all ; hence, . Consequently, for all and we have , i.e., , which implies that for all . As the number was chosen arbitrarily, it follows that for every there is a number such that for all and we have for all . Therefore, , , is a uniformly Cauchy sequence. According to Theorem 4 (ii), is a complete metric space and using Theorem 2 (ii) we find that there is a function such that , , is uniformly convergent on to x. Since every function , , is continuous on and is a complete metric space (Theorem 4 (ii)), by applying Theorem 3 we deduce that x is continuous on . Consequently, we proved that the sequence , is uniformly convergent on to a continuous function . As the number was chosen arbitrarily, it follows that the sequence of continuous functions , , is uniformly convergent on to a continuous function .
Since the sequence , , is uniformly convergent on to a function , it follows that for there is a number such that for all we have for all , i.e., for all . Therefore, for all we get for all , which implies that ; i.e., . As the elements and were arbitrarily selected, we deduce that for every and there is a number such that for all we have . Consequently, we proved that there exists a function with the property that for every and there is a number such that for all we have . Therefore, the sequence of functions is convergent in to a function . Since the Cauchy sequence of functions was chosen arbitrarily, we find that any Cauchy sequence of functions in is convergent in . Consequently, the gauge space is sequentially complete.
(iii) As the functions , , are continuous for , we deduce that the system of first-order linear differential Equations (1), with initial condition , is equivalent to the system of integral equations
Using relation (2), we can define an operator ,
For every , , , , we have, successively:
Hence, for every , , , we get
i.e.,
which is equivalent to
Therefore, for every , , we obtain
i.e.,
Consequently, for and denoting , we have
Thus, is a sequentially complete gauge space, where , and an operator with the property that: for every there exists such that
Applying Theorem 1 it follows that T has a unique fixed point in ; i.e., there exists an unique element such that . Therefore, the system of integral Equation (2) has a unique solution for . Consequently, the system of first-order linear differential Equations (1), with initial condition , has a unique solution for . □
Theorem 6.
If the functions , are continuous on and there is a number such that , , , , , , then the system of first-order linear differential Equations (1), with initial condition , has a unique solution for and this solution is monotonic for .
Proof.
Similarly to the proof of Theorem 5, the system of first-order linear differential Equations (1), with initial condition , has a unique solution for . Let us denote by this solution. Therefore,
It follows that is a monotonically increasing function on . Consequently, each function , , is monotonic on ; i.e., is monotonic for . □
Example 1.
Let us consider the matrix function ,
and the vector function ,
We remark that the functions A, b are continuous on and there is a number such that , , , , , ; therefore, the conditions of Theorem 6 are fulfilled. Considering the vector and applying Theorem 6 it follows that the system of first-order linear differential equations
with initial condition , has a unique solution on and this solution is monotonic for .
4. Conclusions
In this article we studied the existence and uniqueness of the monotonic solutions for nonhomogeneous systems of first-order linear differential equations with variable coefficients. The novelty and originality of our article consists of us proving the existence and uniqueness of the monotonic solution, and finding the conditions under which these properties take place. An example was presented at the end of the paper which reinforces that our theory is correct. Additionally, the paper established conditions for the existence and uniqueness of the solution of the systems of first-order linear differential equations, with initial condition, defined over an unbounded interval (the positive real axis).
Author Contributions
Conceptualization, A.N.B. and I.M.O.; formal analysis, I.M.O.; funding acquisition, A.N.B.; methodology, A.N.B. and I.M.O.; supervision, A.N.B.; validation, A.N.B. and I.M.O.; visualization, I.M.O.; writing—original draft, A.N.B. and I.M.O.; writing—review and editing, A.N.B. All authors contributed equally and significantly to the creation of this article. All authors have read and agreed to the published version of the manuscript.
Funding
This research was funded by Lucian Blaga University of Sibiu grant number LBUS-IRG-2018-04.
Acknowledgments
The authors thank the anonymous editors and reviewers for their valuable comments and suggestions which helped us to improve the content of this paper.
Conflicts of Interest
The authors declare no conflict of interest.
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