An Application of the Fixed Point Theory to the Study of Monotonic Solutions for Systems of Differential Equations
Abstract
:1. Introduction
2. Preliminaries
- (1)
- , for all ;
- (2)
- , for all ;
- (3)
- , for all .
- (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.
- (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.
- (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.
- (i)
- is a complete normed linear space, where , ;
- (ii)
- is a complete metric space, where , .
3. The Existence and Uniqueness of the Monotonic Solutions
- (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 .
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Branga, A.N.; Olaru, I.M. An Application of the Fixed Point Theory to the Study of Monotonic Solutions for Systems of Differential Equations. Mathematics 2020, 8, 1183. https://doi.org/10.3390/math8071183
Branga AN, Olaru IM. An Application of the Fixed Point Theory to the Study of Monotonic Solutions for Systems of Differential Equations. Mathematics. 2020; 8(7):1183. https://doi.org/10.3390/math8071183
Chicago/Turabian StyleBranga, Adrian Nicolae, and Ion Marian Olaru. 2020. "An Application of the Fixed Point Theory to the Study of Monotonic Solutions for Systems of Differential Equations" Mathematics 8, no. 7: 1183. https://doi.org/10.3390/math8071183