Existence theorems for nonlinear differential equations having trichotomy in Banach spaces

Gomaa, Adel Mahmoud

Czechoslovak Mathematical Journal, Tome 67 (2017) no. 2, pp. 339-365

Abstract (VO)

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation $$ \dot {x}(t)=\mathcal {L}( t)x(t)+f(t,x(t)),\quad t\in \mathbb {R}\leqno {\rm (P)} $$ where $\{\mathcal {L}(t)\colon t\in \mathbb {R}\}$ is a family of linear operators from a Banach space $E$ into itself and $f\colon \mathbb {R}\times E\to E$. By $L(E)$ we denote the space of linear operators from $E$ into itself. Furthermore, for $a0$, we let $C([-d,0],E)$ be the Banach space of continuous functions from $[-d,0]$ into $E$ and $f^{d}\colon [a,b]\times C([-d,0],E)\rightarrow E$. Let $\widehat {\mathcal {L}}\colon [a,b]\to L(E)$ be a strongly measurable and Bochner integrable operator on $[a,b]$ and for $t\in [a,b]$ define $\tau _{t}x(s)=x(t+s)$ for each $s \in [-d,0]$. We prove that, under certain conditions, the differential equation with delay $$ \dot {x}(t)=\widehat {\mathcal {L}}(t)x(t)+f^{d}(t,\tau _{t}x)\quad \text {if }t\in [a,b],\leqno {\rm (Q)} $$ 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.

Détail

DOI : 10.21136/CMJ.2017.0592-15

Classification : 34D09, 35F31
Keywords: nonlinear differential equation; trichotomy; existence theorem

Bibliographie

[1] Banaś, J., Goebel, K.: Measure of Noncompactness in Banach Spaces. Lecture Notes in Pure Mathematics 60 Marcel Dekker, New York (1980). | MR | JFM

[2] Boudourides, M. A.: An existence theorem for ordinary differential equations in Banach spaces. Bull. Aust. Math. Soc. 22 (1980), 457-463. | DOI | MR | JFM

[3] Caraballo, T., Morillas, F., Valero, J.: On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete Contin. Dyn. Syst. 34 (2014), 51-77. | DOI | MR | JFM

[4] Cichoń, M.: On bounded weak solutions of a nonlinear differential equation in Banach space. Funct. Approximatio Comment. Math. 21 (1992), 27-35. | MR | JFM

[5] Cichoń, M.: A point of view on measures of noncompactness. Demonstr. Math. 26 (1993), 767-777. | MR | JFM

[6] Cichoń, M.: On measures of weak noncompactness. Publ. Math. 45 (1994), 93-102. | MR | JFM

[7] Cichoń, M.: Trichotomy and bounded solutions of nonlinear differential equations. Math. Bohem. 119 (1994), 275-284. | MR | JFM

[8] Cichoń, M.: Differential inclusions and abstract control problems. Bull. Aust. Math. Soc. 53 (1996), 109-122. | DOI | MR | JFM

[9] Cramer, E., Lakshmikantham, V., Mitchell, A. R.: On the existence of weak solutions of differential equations in nonreflexive Banach spaces. Nonlinear Anal., Theory, Methods Appl. 2 (1978), 169-177. | DOI | MR | JFM

[10] Dawidowski, M., Rzepecki, B.: On bounded solutions of nonlinear differential equations in Banach spaces. Demonstr. Math. 18 (1985), 91-102. | MR | JFM

[11] Elaydi, S., Hajek, O.: Exponential trichotomy of differential systems. J. Math. Anal. Appl. 129 (1988), 362-374. | DOI | MR | JFM

[12] Elaydi, S., Hájek, O.: Exponential dichotomy and trichotomy of nonlinear diffrerential equations. Differ. Integral Equ. 3 (1990), 1201-1224. | MR | JFM

[13] Gohberg, I. T., Goldenstein, L. S., Markus, A. S.: Investigation of some properties of bounded linear operators in connection with their $q$-norms. Uchen. Zap. Kishinevskogo Univ. 29 (1957), 29-36 Russian.

[14] Gomaa, A. M.: Weak and strong solutions for differential equations in Banach spaces. Chaos Solitons Fractals 18 (2003), 687-692. | DOI | MR | JFM

[15] Gomaa, A. M.: Existence solutions for differential equations with delay in Banach spaces. Proc. Math. Phys. Soc. Egypt 84 (2006), 1-12. | MR

[16] Gomaa, A. M.: 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. | MR | JFM

[17] Gomaa, A. M.: Existence and topological properties of solution sets for differential inclusions with delay. Commentat. Math. 48 (2008), 45-58. | MR | JFM

[18] Gomaa, A. M.: 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. | DOI | MR | JFM

[19] Gomaa, A. M.: On bounded weak and strong solutions of non linear differential equations with and without delay in Banach spaces. Math. Scand. 112 (2013), 225-239. | DOI | MR | JFM

[20] Hille, E., Phillips, R. S.: Functional Analysis and Semigroups. Colloquium Publications 31, American Mathematical Society, Providence (1957). | MR | JFM

[21] Ibrahim, A.-G., Gomaa, A. M.: Strong and weak solutions for differential inclusions with moving constraints in Banach spaces. PU.M.A., Pure Math. Appl. 8 (1997), 53-65. | MR | JFM

[22] Krzyśka, S., Kubiaczyk, I.: On bounded pseudo and weak solutions of a nonlinear differential equation in Banach spaces. Demonstr. Math. 32 (1999), 323-330. | MR | JFM

[23] Kuratowski, K.: Sur les espaces complets. Fundamenta 15 (1930), 301-309 French \99999JFM99999 56.1124.04. | MR

[24] Lupa, N., Megan, M.: Generalized exponential trichotomies for abstract evolution operators on the real line. J. Funct. Spaces Appl. 2013 (2013), Article ID 409049, 8 pages. | DOI | MR | JFM

[25] Makowiak, M.: On some bounded solutions to a nonlinear differential equation. Demonstr. Math. 30 (1997), 801-808. | DOI | MR | JFM

[26] Massera, J. L., Schäffer, J. J.: Linear Differential Equations and Function Spaces. Pure and Applied Mathematics 21, Academic Press, New York (1966). | MR | JFM

[27] Megan, M., Stoica, C.: On uniform exponential trichotomy of evolution operators in Banach spaces. Integral Equations Oper. Theory 60 (2008), 499-506. | DOI | MR | JFM

[28] Mitchell, A. R., Smith, C.: 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), 387-403. | DOI | MR | JFM

[29] Olech, O.: 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. | MR | JFM

[30] Papageorgiou, N. S.: Weak solutions of differential equations in Banach spaces. Bull. Aust. Math. Soc. 33 (1986), 407-418. | DOI | MR | JFM

[31] Popa, I.-L., Megan, M., Ceauşu, T.: On $h$-trichotomy of linear discrete-time systems in Banach spaces. Acta Univ. Apulensis, Math. Inform. 39 (2014), 329-339. | DOI | MR | JFM

[32] Przeradzki, B.: The existence of bounded solutions for differential equations in Hilbert spaces. Ann. Pol. Math. 56 (1992), 103-121. | DOI | MR | JFM

[33] Sadovski\uı, B. N.: On a fixed-point principle. Funct. Anal. Appl. 1 (1967), 151-153 translation from Funkts. Anal. Prilozh. 1 1967 74-76. | MR | JFM

[34] Sasu, A. L., Sasu, B.: A Zabczyk type method for the study of the exponential trichotomy of discrete dynamical systems. Appl. Math. Comput. 245 (2014), 447-461. | DOI | MR | JFM

[35] Sasu, B., Sasu, A. L.: Exponential trichotomy and $p$-admissibility for evolution families on the real line. Math. Z. 253 (2006), 515-536. | DOI | MR | JFM

[36] Szep, A.: Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces. Stud. Sci. Math. Hung. 6 (1971), 197-203. | MR | JFM

[37] Szufla, S.: On the existence of solutions of differential equations in Banach spaces. Bull. Acad. Pol. Sci., Sér. Sci. Math. 30 (1982), 507-515. | MR | JFM