On boundary value problems for systems of nonlinear generalized ordinary differential equations

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Abstract

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx = dA(t) · f(t, x), h(x) = 0 is established, where f: [a, b]×Rn → Rn is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A: [a, b] → Rn×n with bounded total variation components, and h: BVs([a, b],Rn) → Rn is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x(t1(x)) = B(x) · x(t2(x))+c0, where ti: BVs([a, b],Rn) → [a, b] (i = 1, 2) and B: BVs([a, b], Rn) → Rn are continuous operators, and c0 ∈ Rn.

Keywords

system of nonlinear generalized ordinary differential equations Kurzweil-Stieltjes integral general boundary value problem solvability principle of a priori bound-edness 

MSC 2010

34K10 

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References

  1. [1]
    M. T. Ashordiya: On solvability of quasilinear boundary value problems for systems of generalized ordinary differential equations. Soobshch. Akad. Nauk Gruz. SSR 133 (1989), 261–264. (In Russian. English summary.)MathSciNetMATHGoogle Scholar
  2. [2]
    M. Ashordia: On the correctness of linear boundary value problems for systems of generalized ordinary differential equations. Georgian Math. J. 1 (1994), 343–351.CrossRefMATHGoogle Scholar
  3. [3]
    M. Ashordia: On the stability of solutions of a multipoint boundary value problem for a system of generalized ordinary differential equations. Mem. Differ. Equ. Math. Phys. 6 (1995), 1–57.MathSciNetMATHGoogle Scholar
  4. [4]
    M. T. Ashordiya: Criteria for the existence and uniqueness of solutions to nonlinear boundary value problems for systems of generalized ordinary differential equations. Differ. Equations 32 (1996), 442–450 (In English. Russian original.); translation from Differ. Uravn. 32 (1996), 441–449.MathSciNetMATHGoogle Scholar
  5. [5]
    M. Ashordia: Criteria of correctness of linear boundary value problems for systems of generalized ordinary differential equations. Czech. Math. J. 46 (1996), 385–404.MathSciNetMATHGoogle Scholar
  6. [6]
    M. T. Ashordiya: A solvability criterion for a many-point boundary value problem for systems of generalized ordinary differential equations. Differ. Equations 32 (1996), 1300–1308 (In English. Russian original.); translation from Differ. Uravn. 32 (1996), 1303–1311.MathSciNetMATHGoogle Scholar
  7. [7]
    M. Ashordia: On the correctness of nonlinear boundary value problems for systems of generalized ordinary differential equations. Georgian Math. J. 3 (1996), 501–524.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    M. Ashordia: Conditions for existence and uniqueness of solutions to multipoint boundary value problems for systems of generalized ordinary differential equations. Georgian Math. J. 5 (1998), 1–24.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    M. Ashordia: On the solvability of linear boundary value problems for systems of generalized ordinary differential equations. Funct. Differ. Equ. 7 (2000), 39–64.MathSciNetMATHGoogle Scholar
  10. [10]
    M. Ashordia: On the general and multipoint boundary value problems for linear systems of generalized ordinary differential equations, linear impulse and linear difference systems. Mem. Differ. Equ. Math. Phys. 36 (2005), 1–80.MathSciNetMATHGoogle Scholar
  11. [11]
    R. Conti: Problèmes linéaires pour les équations différentielles ordinaires. Math. Nachr. 23 (1961), 161–178. (In French.)MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    J. Groh: A nonlinear Volterra-Stieltjes integral equation and a Gronwall inequality in one dimension. Ill. J. Math. 24 (1980), 244–263.MathSciNetMATHGoogle Scholar
  13. [13]
    T. H. Hildebrandt: On systems of linear differentio-Stieltjes-integral equations. Ill. J. Math. 3 (1959), 352–373.MathSciNetMATHGoogle Scholar
  14. [14]
    I. T. Kiguradze: Boundary-value problems for systems of ordinary differential equations. J. Sov. Math. 43 (1988), 2259–2339 (In English. Russian original.); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Novejshie Dostizh. 30 (1987), 3–103.CrossRefMATHGoogle Scholar
  15. [15]
    I. T. Kiguradze, B. Půža: On boundary value problems for functional-differential equations. Mem. Differ. Equ. Math. Phys. 12 (1997), 106–113.MathSciNetMATHGoogle Scholar
  16. [16]
    I. T. Kiguradze, B. Půža: Theorems of Conti-Opial type for nonlinear functionaldifferential equations. Differ. Equations 33 (1997), 184–193 (In English. Russian original.); translation from Differ. Uravn. 33 (1997), 185–194.MathSciNetMATHGoogle Scholar
  17. [17]
    I. T. Kiguradze, B. Půža: On the solvability of nonlinear boundary value problems for functional-differential equations. Georgian Math. J. 5 (1998), 251–262.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    I. T. Kiguradze, B. Půža: Conti-Opial type existence and uniqueness theorems for nonlinear singular boundary value problems. Funct. Differ. Equ. 9 (2002), 405–422.MathSciNetMATHGoogle Scholar
  19. [19]
    I. T. Kiguradze, B. Půža: Boundary Value Problems for Systems of Linear Functional Differential Equations. Folia Facultatis Scientiarum Naturalium Universitatis Masarykianae Brunensis. Mathematica 12. Brno: Masaryk University, 2003.Google Scholar
  20. [20]
    J. Kurzweil: Generalized ordinary differential equations and continuous dependence on a parameter. Czech. Math. J. 7 (1957), 418–449.MathSciNetMATHGoogle Scholar
  21. [21]
    Z. Opial: Linear problems for systems of nonlinear differential equations. J. Differ. Equations 3 (1967), 580–594.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    Š. Schwabik: Generalized Ordinary Differential Equations. Series in Real Analysis 5, World Scientific, Singapore, 1992.CrossRefMATHGoogle Scholar
  23. [23]
    Š. Schwabik, M. Tvrdý: Boundary value problems for generalized linear differential equations. Czech. Math. J. 29 (1979), 451–477.MathSciNetMATHGoogle Scholar
  24. [24]
    Š. Schwabik, M. Tvrdý, O. Vejvoda: Differential and Integral Equations. Boundary Value Problems and Adjoints. Reidel, Dordrecht, in co-ed. with Academia, Publishing House of the Czechoslovak Academy of Sciences, Praha, 1979.Google Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State UniversityTbilisiGeorgia
  2. 2.Sukhumi State UniversityTbilisiGeorgia

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