\(\sigma \)-Increasing Positive Solutions for Systems of Linear Functional Differential Inequalities of Non-Metzler Type

Abstract

Consider the system of functional differential inequalities:

$$\begin{aligned} \mathcal {D}\big (\sigma \big )\big [u'(t)-\ell (u)(t)\big ]\ge 0\qquad \text{ for } \text{ a. } \text{ e. } \,t\in [a,b],\quad \varphi (u)\ge 0, \end{aligned}$$

where \(\ell :C\big ([a,b];\mathbb {R}^n\big )\rightarrow L\big ([a,b];\mathbb {R}^n\big )\) is a linear bounded operator, \(\varphi :C\big ([a,b];\mathbb {R}^n\big )\rightarrow \mathbb {R}^n\) is a linear bounded functional, \(\sigma =(\sigma _i)_{i=1}^n\), where \(\sigma _i\in \{-1,1\}\), and \(\mathcal {D}\big (\sigma \big )={\text {diag}}(\sigma _1,\dots ,\sigma _n)\). In the present paper, we establish conditions guaranteeing that every absolutely continuous vector-valued function u satisfying the above-mentioned inequalities admits also the inequalities \(u(t)\ge 0\) for \(t\in [a,b]\) and \(\mathcal {D}\big (\sigma \big )u'(t)\ge 0\) for a. e. \(t\in [a,b]\).

This is a preview of subscription content, log in to check access.

Notes

  1. 1.

    By a solution to (3.2), we understand a function \(u\in AC\big ([a,b];\mathbb {R}^n\big )\) that satisfies the differential equality in (3.2) almost everywhere in [ab] and it satisfies the boundary condition \(\varphi (u)=c\).

References

  1. 1.

    Afonso, S.M., Rontó, A.: Measure functional differential equations in the space of functions of bounded variation. Abstr. Appl. Anal. (2013). https://doi.org/10.1155/2013/582161. (Art. ID 582161)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Dilna, N., Rontó, A.: General conditions guaranteeing the solvability of the Cauchy problem for functional differential equations. Math. Bohem. 133(4), 435–445 (2008)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Dilna, N., Ronto, A.: Unique solvability of a non-linear non-local boundary-value problem for systems of non-linear functional differential equations. Math. Slovaca 60(3), 327–338 (2010). https://doi.org/10.2478/s12175-010-0015-9

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Domoshnitsky, A., Hakl, R., Šremr, J.: Component-wise positivity of solutions to periodic boundary problem for linear functional differential system. J. Inequal. Appl. 112, 23 (2012). https://doi.org/10.1186/1029-242X-2012-112

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Hakl, R., Kiguradze, I., Půža, B.: Upper and lower solutions of boundary value problems for functional differential equations and theorems on functional differential inequalities. Georgian Math. J. 7(3), 489–512 (2000)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Hakl, R., Vacková, J.: Bounded solutions to systems of nonlinear functional differential equations. Funct. Differ. Equ. 25(1–2), 65–89 (2018)

    MathSciNet  Google Scholar 

  7. 7.

    Kiguradze, I., Šremr, J.: Solvability conditions for non-local boundary value problems for two-dimensional half-linear differential systems. Nonlinear Anal. 74(17), 6537–6552 (2011). https://doi.org/10.1016/j.na.2011.06.038

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Lomtatidze, A., Opluštil, Z., Šremr, J.: On a nonlocal boundary value problem for first order linear functional differential equations. Mem. Differ. Equ. Math. Phys. 41, 69–85 (2007)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Lomtatidze, A., Opluštil, Z., Šremr, J.: Nonpositive solutions to a certain functional differential inequality. Nelīnīĭnī Koliv 12(4), 461–494 (2009). https://doi.org/10.1007/s11072-010-0090-4

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Lomtatidze, A., Opluštil, Z., Šremr, J.: Solvability conditions for a nonlocal boundary value problem for linear functional differential equations. Fasc. Math. 41, 81–96 (2009)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Opluštil, Z., Šremr, J.: On a non-local boundary value problem for linear functional differential equations. Electron. J. Qual. Theory Differ. Equ. (2009). https://doi.org/10.14232/ejqtde.2009.1.36

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Rontó, A., Šremr, J.: Abstract differential inequalities and the Cauchy problem for infinite-dimensional linear functional differential equations. J. Inequal. Appl. 3, 235–250 (2005). https://doi.org/10.1155/JIA.2005.235

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Rontó, A., Šremr, J.: Equivalent solutions of nonlinear equations in a topological vector space with a wedge. J. Inequal. Appl. (2007). https://doi.org/10.1155/2007/46041. (Art. ID 46041)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Šremr, J.: A note on two-dimensional systems of linear differential inequalities with argument deviations. Miskolc Math. Notes 7(2), 171–187 (2006)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Šremr, J.: On systems of linear functional differential inequalities. Georgian Math. J. 13(3), 539–572 (2006)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Šremr, J.: On the Cauchy type problem for systems of functional differential equations. Nonlinear Anal. 67(12), 3240–3260 (2007). https://doi.org/10.1016/j.na.2006.10.008

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Robert Hakl.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Robert Hakl was supported by RVO: 67985840.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Aguerrea, M., Hakl, R. \(\sigma \)-Increasing Positive Solutions for Systems of Linear Functional Differential Inequalities of Non-Metzler Type. Mediterr. J. Math. 17, 181 (2020). https://doi.org/10.1007/s00009-020-01639-8

Download citation

Keywords

  • Functional differential inequality
  • Boundary value problem

Mathematics Subject Classification

  • 34K10