Geodesic
Monotonicity in the theory of almost periodic solutions of nonlinear operator equations
Sbornik. Mathematics, Tome 19 (1973) no. 2, pp. 209-223 Cet article a éte moissonné depuis la source Math-Net.Ru

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In a Banach space with a strictly convex norm we consider a nonlinear equation $u'+A(t)u=0$ of general form. Suppose that a “monotonicity” condition is satisfied: for any two solutions $u_1(t)$ and $u_2(t)$ the function $g(t)=\|u_1(t)-u_2(t)\|$ is nonincreasing with respect to $t$; suppose $A(t)$ is almost periodic (in some sense) with respect to $t$. The basic theorem reads as follows: given strong (weak) continuity of the solutions with respect to the initial conditions and the coefficients, there exists at least one almost periodic solution if there exists a compact (weakly compact) solution on $t\geqslant0$. Bibliography: 26 titles. 
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V. V. Zhikov. Monotonicity in the theory of almost periodic solutions of nonlinear operator equations. Sbornik. Mathematics, Tome 19 (1973) no. 2, pp. 209-223. http://geodesic.mathdoc.fr/item/SM_1973_19_2_a4/

[1] B. M. Levitan, Pochti-periodicheskie funktsii, Gostekhizdat, Moskva, 1953

[2] V. V. Nemytskii, V. V. Stepanov, Kachestvennaya teoriya differentsialnykh uravnenii, Gostekhizdat, Moskva, 1949

[3] V. V. Zhikov, “Pochti-periodicheskie resheniya lineinykh i nelineinykh uravnenii v banakhovom prostranstve”, DAN SSSR, 195:2 (1970), 178–281

[4] S. L. Sobolev, “O pochti-periodichnosti reshenii volnovogo uravneniya”, DAN SSSR, 48:8 (1945), 570–574

[5] L. Amerio, G. Prouse, Almost-periodic functions and functional equations, van Norstand, New York, 1971 | MR | Zbl

[6] B. P. Demidovich, Lektsii po matematicheskoi teorii ustoichivosti, izd-vo «Nauka», Moskva, 1967 | MR

[7] V. M. Cheresiz, “O pochti-periodicheskikh resheniyakh nelineinykh sistem”, DAN SSSR, 165:2 (1965), 281–284 | Zbl

[8] V. A. Yakubovich, “Metod matrichnykh neravenstv v teorii ustoichivosti reguliruemykh sistem. Absolyutnaya ustoichivost vynuzhdennykh kolebanii”, Avtomatika i telemekhanika, XXV:7 (1964), 1017–1028

[9] M. A. Krasnoselskii, V. Sh. Burd, Yu. S. Kolesov, Nelineinye pochti-periodicheskie kolebaniya, izd-vo «Nauka», Moskva, 1970 | MR

[10] V. V. Zhikov, “Pochti-periodicheskie resheniya differentsialnykh uravnenii v banakhovom prostranstve”, Teoriya funktsii i ee prilozheniya, no. 4, Kharkov, 1967, 176–188 | Zbl

[11] V. V. Zhikov, “K probleme suschestvovaniya pochti-periodicheskikh reshenii differentsialnykh i operatornykh uravnenii”, Sb. nauchnykh trudov VVPI, 1969, no. 8, 94–188

[12] M. I. Kadets, “Ob integrirovanii pochti-periodicheskoi funktsii so znacheniyami v prostranstve Banakha”, Funkts. analiz, 3:3 (1969), 71–74 | Zbl

[13] V. V. Zhikov, “Nekotorye zamechaniya ob usloviyakh kompaktnosti v svyazi s rabotoi M. I. Kadetsa ob integrirovanii abstraktnykh pochti-periodicheskikh funktsii”, Funkts. analiz, 5:1 (1971), 30–36 | Zbl

[14] V. M. Millionschikov, “Rekurrentnye i pochti-periodicheskie traektorii neavtonomnykh sistem differentsialnykh uravnenii”, DAN SSSR, 161:1 (1965), 43–45

[15] V. M. Millionschikov, “K teorii lineinykh sistem obyknovennykh differentsialnykh uravnenii”, Matem. zametki, 4:4 (1968), 483–489

[16] M. I. Vishik, “O razreshimosti pervoi kraevoi zadachi dlya nelineinykh ellipticheskikh sistem differentsialnykh uravnenii”, DAN SSSR, 134:4 (1960), 749–752 | Zbl

[17] Yu. A. Dubinskii, “Kvazilineinye uravneniya lyubogo poryadka”, Uspekhi matem. nauk, XXIII:1 (139) (1968), 45–90 | MR

[18] R. N. Kachurovskii, “Nelineinye monotonnye operatory v banakhovom prostranstve”, Uspekhi matem. nauk, XXIII:2 (140) (1968), 121–168

[19] F. E. Browder, “Non-linear equations of evolution”, Ann. Math., 80:3 (1964), 485–523 | DOI | MR | Zbl

[20] L. V. Kantorovich, G. P. Akilov, Funktsionalnyi analiz v normirovannykh prostranstvakh, Fizmatgiz, Moskva, 1962

[21] S. Bochner, J. von Neuman, “On compact operational differential equations”, Ann. Math., 36 (1935), 155–291 | DOI | MR

[22] Z. Opial, “Sur les solutions presque periodiques des equations differentielles du premier et second ordre”, Ann. Polon. Math., 7:1 (1959), 51–71 | MR

[23] I. M. Gelfand, “Abstrakte Funktionen und lineare Operatoren”, Matem. sb., 4 (46) (1938), 235–286 | Zbl

[24] A. D. Wallace, “The structure of topological semigroups”, Bull. Amer. Math. Soc., 61:2 (1955), 95–112 | DOI | MR | Zbl

[25] P. Ellis, “Distal transformation groups”, Pacific J. Math., 8:3 (1958), 401–405 | MR | Zbl

[26] J. L. Lions, Equations differentielles-operationneles et problems aux limites, Springer-Verlag, 1961 | MR | Zbl