Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 339-365, 2017


Existence theorems for nonlinear differential equations having trichotomy in Banach spaces

Adel Mahmoud Gomaa

Received October 31, 2015.  First published March 31, 2017.

Abstract:  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} \tag{P}where $\{\mathcal{L}(t) t\in\mathbb{R}\}$ is a family of linear operators from a Banach space $E$ into itself and $f \mathbb{R}\times E\to E$. 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 $f^d [a,b]\times C([-d,0],E)\rightarrow E$. Let $\widehat{\mathcal{L}} [a,b]\to L(E)$ be a strongly measurable and Bochner integrable operator on $[a,b]$ and for $t\in[a,b]$ define $\tau_tx(s)=x(t+s)$ 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_tx)\quad\text{if }t\in[a,b], \tag{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
Classification MSC:  35F31, 34D09
DOI:  10.21136/CMJ.2017.0592-15


References:
[1] J. Banaś, K. Goebel: Measure of Noncompactness in Banach Spaces. Lecture Notes in Pure Mathematics 60 Marcel Dekker, New York (1980). MR 0591679 | Zbl 0441.47056
[2] M. A. Boudourides: An existence theorem for ordinary differential equations in Banach spaces. Bull. Aust. Math. Soc. 22 (1980), 457-463. DOI 10.1017/S0004972700006766 | MR 0601651 | Zbl 0442.34057
[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. DOI 10.3934/dcds.2014.34.51 | MR 3072985 | Zbl 1323.34087
[4] M. Cichoń: On bounded weak solutions of a nonlinear differential equation in Banach space. Funct. Approximatio Comment. Math. 21 (1992), 27-35. MR 1296988 | Zbl 0777.34041
[5] M. Cichoń: A point of view on measures of noncompactness. Demonstr. Math. 26 (1993), 767-777. MR 1265840 | Zbl 0809.47049
[6] M. Cichoń: On measures of weak noncompactness. Publ. Math. 45 (1994), 93-102. MR 1291804 | Zbl 0829.47042
[7] M. Cichoń: Trichotomy and bounded solutions of nonlinear differential equations. Math. Bohem. 119 (1994), 275-284. MR 1305530 | Zbl 0819.34040
[8] M. Cichoń: Differential inclusions and abstract control problems. Bull. Aust. Math. Soc. 53 (1996), 109-122. DOI 10.1017/S0004972700016774 | MR 1371918 | Zbl 0849.34016
[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. DOI 10.1016/0362-546X(78)90063-9 | MR 0512280 | Zbl 0379.34041
[10] M. Dawidowski, B. Rzepecki: On bounded solutions of nonlinear differential equations in Banach spaces. Demonstr. Math. 18 (1985), 91-102. MR 0816022 | Zbl 0593.34062
[11] S. Elaydi, O. Hajek: Exponential trichotomy of differential systems. J. Math. Anal. Appl. 129 (1988), 362-374. DOI 10.1016/0022-247X(88)90255-7 | MR 0924294 | Zbl 0651.34052
[12] S. Elaydi, O. Hájek: Exponential dichotomy and trichotomy of nonlinear diffrerential equations. Differ. Integral Equ. 3 (1990), 1201-1224. MR 1073067 | Zbl 0722.34053
[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).
[14] A. M. Gomaa: Weak and strong solutions for differential equations in Banach spaces. Chaos Solitons Fractals 18 (2003), 687-692. DOI 10.1016/S0960-0779(02)00643-4 | MR 1984052 | Zbl 1058.34077
[15] A. M. Gomaa: Existence solutions for differential equations with delay in Banach spaces. Proc. Math. Phys. Soc. Egypt 84 (2006), 1-12. MR 2349563
[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. MR 2377955 | Zbl 1182.34080
[17] A. M. Gomaa: Existence and topological properties of solution sets for differential inclusions with delay. Commentat. Math. 48 (2008), 45-58. MR 2440748 | Zbl 1179.34072
[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. DOI 10.1142/S0219887810004336 | MR 2646768 | Zbl 1214.34068
[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. DOI 10.7146/math.scand.a-15242 | MR 3073456 | Zbl 1276.34063
[20] E. Hille, R. S. Phillips: Functional Analysis and Semigroups. Colloquium Publications 31, American Mathematical Society, Providence (1957). MR 0423094 | Zbl 0078.10004
[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. MR 1490000 | Zbl 0910.34027
[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. MR 1710255 | Zbl 0954.34050
[23] K. Kuratowski: Sur les espaces complets. Fundamenta 15 (1930), 301-309. (In French). JFM 56.1124.04
[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. DOI 10.1155/2013/409049 | MR 3111843 | Zbl 06281050
[25] M. Makowiak: On some bounded solutions to a nonlinear differential equation. Demonstr. Math. 30 (1997), 801-808. MR 1617273 | Zbl 0909.34049
[26] J. L. Massera, J. J. Schäffer: Linear Differential Equations and Function Spaces. Pure and Applied Mathematics 21, Academic Press, New York (1966). MR 0212324 | Zbl 0243.34107
[27] M. Megan, C. Stoica: On uniform exponential trichotomy of evolution operators in Banach spaces. Integral Equations Oper. Theory 60 (2008), 499-506. DOI 10.1007/s00020-008-1555-z | MR 2390441 | Zbl 1151.34051
[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), 387-403. DOI 10.1016/b978-0-12-434160-9.50028-x | MR 0502554 | Zbl 0452.34054
[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. MR 0147733 | Zbl 0173.35303
[30] N. S. Papageorgiou: Weak solutions of differential equations in Banach spaces. Bull. Aust. Math. Soc. 33 (1986), 407-418. DOI 10.1017/S0004972700003993 | MR 837487 | Zbl 0578.34039
[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. DOI 10.17114/j.aua.2014.39.28 | MR 3304423 | Zbl 06521157
[32] B. Przeradzki: The existence of bounded solutions for differential equations in Hilbert spaces. Ann. Pol. Math. 56 (1992), 103-121. MR 1159982 | Zbl 0805.47041
[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. MR 0211302 | Zbl 0165.49102
[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. DOI 10.1016/j.amc.2014.07.108 | MR 3260730 | Zbl 1335.39027
[35] B. Sasu, A. L. Sasu: Exponential trichotomy and $p$-admissibility for evolution families on the real line. Math. Z. 253 (2006), 515-536. DOI 10.1007/s00209-005-0920-8 | MR 2221084 | Zbl 1108.34047
[36] A. Szep: Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces. Stud. Sci. Math. Hung. 6 (1971), 197-203. MR 0330688 | Zbl 0238.34100
[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. MR 0718727 | Zbl 0532.34045

Affiliations:   Adel Mahmoud Gomaa, Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah, Universities Road, P.O. Box: 344 42353 Medina, Kingdom of Saudi Arabia, e-mail: mohameda59@yahoo.com

 
PDF available at: