Czechoslovak Mathematical Journal

, Volume 67, Issue 2, pp 339–365

Existence theorems for nonlinear differential equations having trichotomy in banach spaces

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Abstract

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation
$$\dot x\left( t \right) = \mathcal{L}\left( t \right)x\left( t \right) + f\left( {t,x\left( t \right)} \right),t \in \mathbb{R}$$
(P)
where {ℒ(t): t ∈ R} is a family of linear operators from a Banach space E into itself and f: R × EE. By L(E) we denote the space of linear operators from E into itself. Furthermore, for a < b and d > 0, we let C([−d, 0],E) be the Banach space of continuous functions from [−d, 0] into E and fd: [a, b] × C([−d, 0],E) → E. Let \(\hat {\mathcal{L}}:[a,b] \to L(E)\) be a strongly measurable and Bochner integrable operator on [a, b] and for t ∈ [a, b] define τtx(s) = x(t + s) for each s ∈ [−d, 0]. We prove that, under certain conditions, the differential equation with delay
$$\dot x\left( t \right) = \hat {\mathcal{L}}\left( t \right)x\left( t \right) + {f^d}\left( {t,{\tau _t}x} \right),ift \in \left[ {a,b} \right],$$
(Q)
has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too.

Keywords

nonlinear differential equation trichotomy existence theorem 

MSC 2010

35F31 34D09 

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References

  1. [1]
    J. Banaś, K. Goebel: Measure of Noncompactness in Banach Spaces, Lecture Notes in Pure Mathematics 60. Marcel Dekker, New York, 1980.MATHGoogle Scholar
  2. [2]
    M. A. Boudourides: An existence theorem for ordinary differential equations in Banach spaces. Bull. Aust. Math. Soc. 22 (1980), 457–463.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    T. Caraballo, F. Morillas, J. Valero: On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete Contin. Dyn. Syst. 34 (2014), 51–77.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    M. Cichoń: On bounded weak solutions of a nonlinear differential equation in Banach space. Funct. Approximatio Comment. Math. 21 (1992), 27–35.MathSciNetMATHGoogle Scholar
  5. [5]
    M. Cichoń: A point of view on measures of noncompactness. Demonstr. Math. 26 (1993), 767–777.MathSciNetMATHGoogle Scholar
  6. [6]
    M. Cichoń: On measures of weak noncompactness. Publ. Math. 45 (1994), 93–102.MathSciNetMATHGoogle Scholar
  7. [7]
    M. Cichoń: Trichotomy and bounded solutions of nonlinear differential equations. Math. Bohem. 119 (1994), 275–284.MathSciNetMATHGoogle Scholar
  8. [8]
    M. Cichoń: Differential inclusions and abstract control problems. Bull. Aust. Math. Soc. 53 (1996), 109–122.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    E. Cramer, V. Lakshmikantham, A. R. Mitchell: On the existence of weak solutions of differential equations in nonreflexive Banach spaces. Nonlinear Anal., Theory, Methods Appl. 2 (1978), 169–177.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    M. Dawidowski, B. Rzepecki: On bounded solutions of nonlinear differential equations in Banach spaces. Demonstr. Math. 18 (1985), 91–102.MathSciNetMATHGoogle Scholar
  11. [11]
    S. Elaydi, O. Hajek: Exponential trichotomy of differential systems. J. Math. Anal. Appl. 129 (1988), 362–374.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    S. Elaydi, O. Hájek: Exponential dichotomy and trichotomy of nonlinear diffrerential equations. Differ. Integral Equ. 3 (1990), 1201–1224.MATHGoogle Scholar
  13. [13]
    I. T. Gohberg, L. S. Goldenstein, A. S. Markus: Investigation of some properties of bounded linear operators in connection with their q-norms. Uchen. Zap. Kishinevskogo Univ. 29 (1957), 29–36. (In Russian.)Google Scholar
  14. [14]
    A. M. Gomaa: Weak and strong solutions for differential equations in Banach spaces. Chaos Solitons Fractals 18 (2003), 687–692.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    A. M. Gomaa: Existence solutions for differential equations with delay in Banach spaces. Proc. Math. Phys. Soc. Egypt 84 (2006), 1–12.MathSciNetGoogle Scholar
  16. [16]
    A. M. Gomaa: On theorems for weak solutions of nonlinear differential equations with and without delay in Banach spaces. Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 47 (2007), 179–191.MathSciNetMATHGoogle Scholar
  17. [17]
    A. M. Gomaa: Existence and topological properties of solution sets for differential inclusions with delay. Commentat. Math. 48 (2008), 45–58.MathSciNetMATHGoogle Scholar
  18. [18]
    A. M. Gomaa: On bounded weak and pseudo-solutions of nonlinear differential equations having trichotomy with and without delay in Banach spaces. Int. J. Geom. Mathods Mod. Phys. 7 (2010), 357–366.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    A. M. Gomaa: On bounded weak and strong solutions of non linear differential equations with and without delay in Banach spaces. Math. Scand. 112 (2013), 225–239.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    E. Hille, R. S. Phillips: Functional Analysis and Semigroups. Colloquium Publications 31, American Mathematical Society, Providence, 1957.Google Scholar
  21. [21]
    A.-G. Ibrahim, A. M. Gomaa: Strong and weak solutions for differential inclusions with moving constraints in Banach spaces. PU. M. A., Pure Math. Appl. 8 (1997), 53–65.MathSciNetMATHGoogle Scholar
  22. [22]
    S. Krzyśka, I. Kubiaczyk: On bounded pseudo and weak solutions of a nonlinear differential equation in Banach spaces. Demonstr. Math. 32 (1999), 323–330.MathSciNetMATHGoogle Scholar
  23. [23]
    K. Kuratowski: Sur les espaces complets. Fundamenta 15 (1930), 301–309. (In French.)MATHGoogle Scholar
  24. [24]
    N. Lupa, M. Megan: Generalized exponential trichotomies for abstract evolution operators on the real line. J. Funct. Spaces Appl. 2013 (2013), Article ID 409049, 8 pages.Google Scholar
  25. [25]
    M. Makowiak: On some bounded solutions to a nonlinear differential equation. Demonstr. Math. 30 (1997), 801–808.MathSciNetMATHGoogle Scholar
  26. [26]
    J. L. Massera, J. J. Schäffer: Linear Differential Equations and Function Spaces. Pure and Applied Mathematics 21, Academic Press, New York, 1966.Google Scholar
  27. [27]
    M. Megan, C. Stoica: On uniform exponential trichotomy of evolution operators in Banach spaces. Integral Equations Oper. Theory 60 (2008), 499–506.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    A. R. Mitchell, C. Smith: An existence theorem for weak solutions of differential equations in Banach spaces. Nonlinear Equations in Abstract Spaces. Proc. Int. Symp., Arlington, 1977, Academic Press, New York, 1978, pp. 387–403.Google Scholar
  29. [29]
    O. Olech: On the existence and uniqueness of solutions of an ordinary differential equation in the case of Banach space. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 8 (1960), 667–673.MathSciNetMATHGoogle Scholar
  30. [30]
    N. S. Papageorgiou: Weak solutions of differential equations in Banach spaces. Bull. Aust. Math. Soc. 33 (1986), 407–418.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    I.-L. Popa, M. Megan, T. Ceauşu: On h-trichotomy of linear discrete-time systems in Banach spaces. Acta Univ. Apulensis, Math. Inform. 39 (2014), 329–339.MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    B. Przeradzki: The existence of bounded solutions for differential equations in Hilbert spaces. Ann. Pol. Math. 56 (1992), 103–121.MathSciNetMATHGoogle Scholar
  33. [33]
    B. N. Sadovskiĭ: On a fixed-point principle. Funct. Anal. Appl. 1 (1967), 151–153; translation from Funkts. Anal. Prilozh. 1 (1967), 74–76.MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    A. L. Sasu, B. Sasu: A Zabczyk type method for the study of the exponential trichotomy of discrete dynamical systems. Appl. Math. Comput. 245 (2014), 447–461.MathSciNetMATHGoogle Scholar
  35. [35]
    B. Sasu, A. L. Sasu: Exponential trichotomy and p-admissibility for evolution families on the real line. Math. Z. 253 (2006), 515–536.MathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    A. Szep: Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces. Stud. Sci. Math. Hung. 6 (1971), 197–203.MathSciNetMATHGoogle Scholar
  37. [37]
    S. Szufla: On the existence of solutions of differential equations in Banach spaces. Bull. Acad. Pol. Sci., Sér. Sci. Math. 30 (1982), 507–515.MATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceTaibah University, Al-MadinahMedinaKingdom of Saudi Arabia

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