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On the Unique Solvability of a Nonlinear Nonlocal Boundary-Value Problem for Systems of Second-Order Functional Differential Equations

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We establish some optimal, in a certain sense, general conditions sufficient for the unique solvability of the boundary-value problem for a system of nonlinear second-order functional differential equations. The considered class of equations covers, in particular, equations of the neutral type. Specific example is presented to illustrate the general theory.

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References

  1. N. Azbelev, V. Maksimov, and L. Rakhmatullina, Introduction to the Theory of Linear Functional Differential Equations, GA: World Federation Publ. Com., Atlanta (1995).

  2. N. Dilna, “Unique solvability of second order functional differential equations with nonlocal boundary conditions,” Electron. J. Qualit. Theory Different. Equat., 14, 1–13 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  3. N. Dilnaya and A. Ronto, “Multistage iterations and solvability of linear Cauchy problems,” Miskolc Math. Notes, 4, No. 2, 89–102 (2003).

    MathSciNet  MATH  Google Scholar 

  4. N. Z. Dil’naya and A. N. Ronto, “New solvability conditions for the Cauchy problem for systems of linear functional differential equations,” Ukr. Math. J., 56, No. 7, 867–884 (2004).

    MATH  Google Scholar 

  5. N. Dilna and A. Ronto, “Unique solvability of a nonlinear nonlocal boundary-value problem for systems of nonlinear functional differential equations,” Math. Slovaca, 60, No. 3, 327–338 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  6. N. Dilna, “On the solvability of the Cauchy problem for linear integral differential equations,” Miskolc Math. Notes, 5, No. 2, 161–171 (2004).

    MathSciNet  MATH  Google Scholar 

  7. A. Domoshnitsky, R. Hakl, and B. Půža, “On the dimension of the solution set to the homogeneous linear functional differential equation of the first order,” Czechoslovak Math. J., 62(137), 1033–1053 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Hakl, A. Lomtatidze, and J. Šremr, “Some boundary value problems for first order scalar functional differential equations,” in: Folia Fac. Sci. Natur. Univ. Masar. Brun. Mathematica, Masaryk Univ., Brno (2002).

  9. U. van der Heiden, A. Longtin, M. C. Mackey, J. G. Milton, and R. Scholl, “Oscillatory modes in a nonlinear second order differential equation with delays,” J. Dynam. Different. Equat., 2, No. 4, 423–449 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  10. I. Y. Karaca, “On positive solutions for the second-order boundary value problems for functional differential equations,” Appl. Math. Comput., 219, No. 10, 5433–5439 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  11. M. A. Krasnoselskii, Positive Solutions of Operator Equations, Wolters-Noordhoff Sci. Publ., Groningen (1964).

    Google Scholar 

  12. M. A. Krasnosel’skii, E. A. Lifshits, Yu. V. Pokornyi, and Ya. V. Stetsenko, “Positive-invertible linear operators and solvability of nonlinear equations,” Dokl. Akad. Nauk Tadzhik. SSR, 17, No. 1, 12–14 (1974).

    Google Scholar 

  13. M. A. Krasnoselskii and P. P. Zabreiko, Geometrical methods of nonlinear analysis, New York: Springer, Berlin (1984).

  14. A. Lomtatidze and H. Štěpánková, “On sign constant and monotone solutions of second order linear functional differential equations,” Mem. Different. Equat. Math. Phys., 35, 65–90 (2005).

    MathSciNet  MATH  Google Scholar 

  15. R. Ma, “Positive solutions for boundary value problems of functional differential equations,” Appl. Math. Comput., 193, No. 1, 66–72 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Mukhigulashvili and J. Šremr, “On a two-point boundary value problem for the second order linear functional differential equations with monotone operators,” Funct. Different. Equat., 13, No. 3–4, 519–537 (2006).

    MathSciNet  Google Scholar 

  17. Z. A. Petrova, “Oscillations of some equations and inequalities of second order and applications in mechanics,” in: M. D. Todorov (editor), Appl. Math. Eng. Econ., Proc. 33rd Summer School. (2007), pp. 224–234.

  18. I. Rachůnková, S. Staněk, E.Weinmüller, and M. Zenz, Limit Properties of Solutions of Singular Second-Order Differential Equations, Boundary Value Problems (2009), Art. No. 905769.

  19. A. N. Ronto, “Some exact conditions for the solvability of an initial value problem for systems of linear functional-differential equations,” Nonlin. Oscillat., 7, No. 4, 521–537 (2004).

    Article  MathSciNet  Google Scholar 

  20. A. Ronto, V. Pylypenko, and N. Dilna, “On the unique solvability of nonlocal boundary-value problems for linear functional differential equations,” Math. Modelling Anal., 13, No. 2, 241–250 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Ronto and N. Dilna, “Unique solvability conditions of the initial value problem for linear differential equations with argument deviations,” Nonlin. Oscillat., 9, No. 4, 535–547 (2006).

    Article  MATH  Google Scholar 

  22. S. Staněk, “Boundary-value problems for systems of second-order functional differential equations,” Electron. J. Qualit. Theory Different. Equat., 28, 1–14 (2000).

    MATH  Google Scholar 

  23. J. Šremr, “On the Cauchy-type problem for systems of functional differential equations,” Nonlin. Anal., 67, No. 12, 3240–3260 (2007).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova Str., 49, 814 73, Bratislava, Slovakia

    N. Dilna

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  1. N. Dilna
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Correspondence to N. Dilna.

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Published in Neliniini Kolyvannya, Vol. 19, No. 2, pp. 203–216, April–June, 2016.

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Dilna, N. On the Unique Solvability of a Nonlinear Nonlocal Boundary-Value Problem for Systems of Second-Order Functional Differential Equations. J Math Sci 223, 257–272 (2017). https://doi.org/10.1007/s10958-017-3352-1

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  • DOI: https://doi.org/10.1007/s10958-017-3352-1

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