# Existence theorems for nonlinear differential equations having trichotomy in banach spaces

Article

- First Online:

- Received:

- 22 Downloads

## Abstract

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation
where {ℒ(
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.

$$\dot x\left( t \right) = \mathcal{L}\left( t \right)x\left( t \right) + f\left( {t,x\left( t \right)} \right),t \in \mathbb{R}$$

(P)

*t*):*t*∈ R} is a family of linear operators from a Banach space*E*into itself and*f*: R ×*E*→*E*. 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*f*^{d}: [*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*τ*_{t}*x*(*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)

### Keywords

nonlinear differential equation trichotomy existence theorem### MSC 2010

35F31 34D09## Preview

Unable to display preview. Download preview PDF.

### References

- [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]
*M. A. Boudourides*: An existence theorem for ordinary differential equations in Banach spaces. Bull. Aust. Math. Soc.*22*(1980), 457–463.MathSciNetCrossRefMATHGoogle Scholar - [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]
*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]
*M. Cichoń*: A point of view on measures of noncompactness. Demonstr. Math.*26*(1993), 767–777.MathSciNetMATHGoogle Scholar - [6]
*M. Cichoń*: On measures of weak noncompactness. Publ. Math.*45*(1994), 93–102.MathSciNetMATHGoogle Scholar - [7]
*M. Cichoń*: Trichotomy and bounded solutions of nonlinear differential equations. Math. Bohem.*119*(1994), 275–284.MathSciNetMATHGoogle Scholar - [8]
*M. Cichoń*: Differential inclusions and abstract control problems. Bull. Aust. Math. Soc.*53*(1996), 109–122.MathSciNetCrossRefMATHGoogle Scholar - [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]
*M. Dawidowski, B. Rzepecki*: On bounded solutions of nonlinear differential equations in Banach spaces. Demonstr. Math.*18*(1985), 91–102.MathSciNetMATHGoogle Scholar - [11]
*S. Elaydi, O. Hajek*: Exponential trichotomy of differential systems. J. Math. Anal. Appl.*129*(1988), 362–374.MathSciNetCrossRefMATHGoogle Scholar - [12]
*S. Elaydi, O. Hájek*: Exponential dichotomy and trichotomy of nonlinear diffrerential equations. Differ. Integral Equ.*3*(1990), 1201–1224.MATHGoogle Scholar - [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]
*A. M. Gomaa*: Weak and strong solutions for differential equations in Banach spaces. Chaos Solitons Fractals*18*(2003), 687–692.MathSciNetCrossRefMATHGoogle Scholar - [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]
*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]
*A. M. Gomaa*: Existence and topological properties of solution sets for differential inclusions with delay. Commentat. Math.*48*(2008), 45–58.MathSciNetMATHGoogle Scholar - [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]
*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]
*E. Hille, R. S. Phillips*: Functional Analysis and Semigroups. Colloquium Publications 31, American Mathematical Society, Providence, 1957.Google Scholar - [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]
*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]
*K. Kuratowski*: Sur les espaces complets. Fundamenta*15*(1930), 301–309. (In French.)MATHGoogle Scholar - [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]
*M. Makowiak*: On some bounded solutions to a nonlinear differential equation. Demonstr. Math.*30*(1997), 801–808.MathSciNetMATHGoogle Scholar - [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]
*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]
*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]
*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]
*N. S. Papageorgiou*: Weak solutions of differential equations in Banach spaces. Bull. Aust. Math. Soc.*33*(1986), 407–418.MathSciNetCrossRefMATHGoogle Scholar - [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]
*B. Przeradzki*: The existence of bounded solutions for differential equations in Hilbert spaces. Ann. Pol. Math.*56*(1992), 103–121.MathSciNetMATHGoogle Scholar - [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]
*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]
*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]
*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]
*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