Skip to main content
SpringerLink
Log in

System of Urysohn Integral Equations: Existence of a Constant-Sign Solution

  • Chapter
  • First Online:
Constant-Sign Solutions of Systems of Integral Equations

  • 708 Accesses

Abstract

In this chapter we shall consider the following system of Urysohn integral equations

$$\displaystyle{ u_{i}(t) =\int _{ 0}^{1}g_{ i}(t,s,u_{1}(s),u_{2}(s),\cdots \,,u_{n}(s))ds,\ \ t \in [0,1],\ 1 \leq i \leq n. }$$

A solution \(u = (u_{1},u_{2},\cdots \,,u_{n})\) of (18.1.1) will be sought in \({(C[0,1])}^{n} = C[0,1] \times \cdots \times C[0,1]\) (n times). We are particularly interested in achieving a constant-sign solution u, i.e., for each 1 ≤ i ≤ n, we have \(\theta _{i}u_{i}(t) \geq 0\) for t ∈ [0,1], where \(\theta _{i} \in \{ 1,-1\}\) is fixed.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. R.P. Agarwal, J. Henderson, P.J.Y. Wong, On superlinear and sublinear (n,p) boundary value problems for higher order difference equations. Nonlinear World 4, 101–115 (1997)

    MathSciNet  MATH  Google Scholar 

  2. R.P. Agarwal, D. O’Regan, P.J.Y. Wong, Positive Solutions of Differential, Difference and Integral Equations (Kluwer, Dordrecht, 1999)

    Book  MATH  Google Scholar 

  3. R.P. Agarwal, D. O’Regan, P.J.Y. Wong, Constant-sign solutions of a system of Fredholm integral equations. Acta Appl. Math. 80, 57–94 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. R.P. Agarwal, D. O’Regan, P.J.Y. Wong, Constant-sign periodic and almost periodic solutions for a system of integral equations. Acta Appl. Math. 89, 177–216 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. R.P. Agarwal, D. O’Regan, P.J.Y. Wong, Constant-sign solutions of a system of Urysohn integral equations. Numer. Funct. Anal. Optim. 29, 1205–1239 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. R.P. Agarwal, P.J.Y. Wong, Advanced Topics in Difference Equations (Kluwer, Dordrecht, 1997)

    Book  MATH  Google Scholar 

  7. J. Appell, E. De Pascale, J.V. Lysenko, P.P. Zabrejko, New results on Newton–Kantorovich approximations with applications to nonlinear integral equations. Numer. Funct. Anal. Optim. 18, 1–17 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Appell, E. De Pascale, P.P. Zabrejko, On the application of the Newton–Kantorovich method to nonlinear integral equations of Uryson type. Numer. Funct. Anal. Optim. 12, 271–283 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. M.A. Darwish, On integral equations of Urysohn–Volterra type. Appl. Math. Comput. 136, 93–98 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. W.G. El-Sayed, A.A. El-Bary, M.A. Darwish, Solvability of Urysohn integral equation. Appl. Math. Comput. 145, 487–493 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. L.H. Erbe, S. Hu, H. Wang, Multiple positive solutions of some boundary value problems. J. Math. Anal. Appl. 184, 640–648 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. L.H. Erbe, H. Wang, On the existence of positive solutions of ordinary differential equations. Proc. Am. Math. Soc. 120, 743–748 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. D. Franco, G. Infante, D. O’Regan, Positive and nontrivial solutions for the Urysohn integral equation. Acta Math. Sin. (Engl. Ser.) 22, 1745–1750 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. D. Franco, G. Infante, D. O’Regan, Nontrivial solutions in abstract cones for Hammerstein integral systems. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 14, 837–850 (2007)

    MathSciNet  MATH  Google Scholar 

  15. C. Georgescu, Existence results for Urysohn integral equations (preprint)

    Google Scholar 

  16. G. Infante, J.R.L. Webb, Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations. Proc. Edinb. Math. Soc. (2) 49, 637–656 (2006)

    Google Scholar 

  17. M. Joshi, Existence theorems for Urysohn’s integral equation. Proc. Am. Math. Soc. 49, 387–392 (1975)

    MATH  Google Scholar 

  18. J. Kolomý, The solvability of nonlinear integral equations. Comment. Math. Univ. Carolinae 8, 273–289 (1967)

    MathSciNet  MATH  Google Scholar 

  19. M.A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations (Pergamon Press, Oxford, 1964)

    Google Scholar 

  20. K.Q. Lan, Multiple positive solutions of semilinear differential equations with singularities. J. Lond. Math. Soc. 63, 690–704 (2001)

    Article  MATH  Google Scholar 

  21. K.Q. Lan, Positive solutions of semi-positone Hammerstein integral equations and applications. Commun. Pure Appl. Anal. 6, 441–451 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. F. Li, Y. Li, Z. Liang, Existence of solutions to nonlinear Hammerstein integral equations and applications. J. Math. Anal. Appl. 323, 209–227 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. W. Lian, F. Wong, C. Yeh, On the existence of positive solutions of nonlinear second order differential equations. Proc. Am. Math. Soc. 124, 1117–1126 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. H. Liang, H. Pang, W. Ge, Triple positive solutions for boundary value problems on infinite intervals. Nonlinear Anal. 67, 2199–2207 (2007)

    Article  MathSciNet  Google Scholar 

  25. D. O’Regan, Volterra and Urysohn integral equations in Banach spaces. J. Appl. Math. Stoch. Anal. 11, 449–464 (1998)

    Article  MATH  Google Scholar 

  26. D. O’Regan, M. Meehan, Existence Theory for Nonlinear Integral and Integrodifferential Equations (Kluwer, Dordrecht, 1998)

    Book  MATH  Google Scholar 

  27. B.G. Pachpatte, Method of upper and lower solutions for nonlinear integral equations of Urysohn type. J. Integr. Equat. 6, 119–125 (1984)

    MATH  Google Scholar 

  28. P.J.Y. Wong, Solutions of constant signs of a system of Sturm–Liouville boundary value problems. Math. Comput. Model. 29, 27–38 (1999)

    Article  MATH  Google Scholar 

  29. P.J.Y. Wong, A system of (n i ,p i ) boundary value problems with positive/nonpositive nonlinearities. J. Math. Anal. Appl. 243, 293–312 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Texas A&M University – Kingsville, Kingsville, TX, USA

    Ravi P. Agarwal

  2. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Galway, Ireland

    Donal O’Regan

  3. School of Electrical & Electronic Engineering, Nanyang Technological University, Singapore, Singapore

    Patricia J. Y. Wong

Authors
  1. Ravi P. Agarwal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Donal O’Regan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Patricia J. Y. Wong
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Agarwal, R.P., O’Regan, D., Wong, P.J.Y. (2013). System of Urysohn Integral Equations: Existence of a Constant-Sign Solution. In: Constant-Sign Solutions of Systems of Integral Equations. Springer, Cham. https://doi.org/10.1007/978-3-319-01255-1_18

Download citation

Publish with us

Policies and ethics

Search

Navigation