Boundary value problems on infinite intervals

Andres, Jan; Gabor, Grzegorz; Górniewicz, Lech

doi:10.1090/S0002-9947-99-02297-7

Boundary value problems on infinite intervals
HTML articles powered by AMS MathViewer

by Jan Andres, Grzegorz Gabor and Lech Górniewicz PDF

Trans. Amer. Math. Soc. 351 (1999), 4861-4903 Request permission

Abstract:

We present two methods, both based on topological ideas, to the solvability of boundary value problems for differential equations and inclusions on infinite intervals. In the first one, related to the rich family of asymptotic problems, we generalize and extend some statements due to the Florence group of mathematicians Anichini, Cecchi, Conti, Furi, Marini, Pera, and Zecca. Thus, their conclusions for differential systems are as well true for inclusions; all under weaker assumptions (for example, the convexity restrictions in the Schauder linearization device can be avoided). In the second, dealing with the existence of bounded solutions on the positive ray, we follow and develop the ideas of Andres, Górniewicz, and Lewicka, who considered periodic problems. A special case of these results was previously announced by Andres. Besides that, the structure of solution sets is investigated. The case of l.s.c. right hand sides of differential inclusions and the implicit differential equations are also considered. The large list of references also includes some where different techniques (like the Conley index approach) have been applied for the same goal, allowing us to envision the full range of recent attacks on the problem stated in the title.

References

Kh. Abduvaitov, Some sufficient conditions for the existence of periodic and of bounded solutions of second-order nonlinear differential equations, Differentsial′nye Uravneniya 21 (1985), no. 12, 2027–2036, 2202 (Russian). MR 821843
Ravi P. Agarwal, Boundary value problems for higher order differential equations, World Scientific Publishing Co., Inc., Teaneck, NJ, 1986. MR 1021979, DOI 10.1142/0266
Shair Ahmad, A nonstandard resonance problem for ordinary differential equations, Trans. Amer. Math. Soc. 323 (1991), no. 2, 857–875. MR 1010407, DOI 10.1090/S0002-9947-1991-1010407-9
Ján Andres, Boundedness results of solutions to the equation $x''’+ax''+g(x)x’+h(x)=p(t)$ without the hypothesis $h(x)\,\textrm {sgn}\,x\geq 0$ for $|x|>R$, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 80 (1986), no. 7-12, 533–539 (1987) (English, with Italian summary). MR 976947
J. Andres, Four-point and asymptotic boundary value problems via a possible modification of Poincaré’s mapping, Math. Nachr. 149 (1990), 155–162. MR 1124801, DOI 10.1002/mana.19901490112
J. Andres, Note to the asymptotic behaviour of solutions of damped pendulum equations under forcing, Nonlinear Anal. 18 (1992), no. 8, 705–712. MR 1160114, DOI 10.1016/0362-546X(92)90166-C
Ján Andres, Asymptotic properties of solutions to quasi-linear differential systems, J. Comput. Appl. Math. 41 (1992), no. 1-2, 57–64. Asymptotic methods in analysis and combinatorics. MR 1181709, DOI 10.1016/0377-0427(92)90238-S
Jan Andres, Existence of periodic and bounded solutions of the generalized Liénard equation under forcing, Rep. Math. Phys. 39 (1997), no. 1, 91–98. MR 1475452, DOI 10.1016/S0034-4877(97)81474-7
Jan Andres, Large-period forced oscillations to higher-order pendulum-type equations, Differential Equations Dynam. Systems 3 (1995), no. 4, 407–421. MR 1386758
J. Andres, Ważewski-type results without transversality, Proceed. of Equadiff 95 (Lisbon, July 24–29), World Scientific, Singapore, 1998, 233–238.
Jan Andres, Lech Górniewicz, and Marta Lewicka, Partially dissipative periodic processes, Topology in nonlinear analysis (Warsaw, 1994) Banach Center Publ., vol. 35, Polish Acad. Sci. Inst. Math., Warsaw, 1996, pp. 109–118. MR 1448430
J. Andres, J. Mikolajski und J. Palát, Über die Trichotomie von Lösungen einer nichtlinear Vektordifferentialgleichnung zweiter Ordnung, Acta UPO 91, Math. 27 (1988), 211–224.
Jan Andres and Tomáš Turský, Asymptotic estimates of solutions and their derivatives for $n$-th order nonhomogeneous ordinary differential equations with constant coefficients, Discuss. Math. Differential Incl. 16 (1996), no. 1, 75–89. MR 1429037
Ján Andres and Vladimír Vlček, Asymptotic behaviour of solutions to the $n$th order nonlinear differential equation under forcing, Rend. Istit. Mat. Univ. Trieste 21 (1989), no. 1, 128–143 (1991) (English, with Italian summary). MR 1142529
Giuseppe Anichini, Giuseppe Conti, and Pietro Zecca, Using solution sets for solving boundary value problems for ordinary differential equations, Nonlinear Anal. 17 (1991), no. 5, 465–472. MR 1124119, DOI 10.1016/0362-546X(91)90141-M
R. Conti, G. Sestini, and G. Villari (eds.), Equazioni differenziali ordinarie ed equazioni funzionali, Consiglio Nazionale delle Ricerche (CNR), Rome, 1978. MR 690589
G. Anichini and P. Zecca, Problemi ai limiti per equazioni differenziali multivoche su intervalli non compatti, Riv. Mat. Univ. Parma (4) 1 (1975), 199–212 (Italian, with English summary). MR 447728
Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR 755330, DOI 10.1007/978-3-642-69512-4
C. Avramescu, Sur l’existence des solutions convergentes des systèmes d’équations différentielles non linéaires, Ann. Mat. Pura Appl. (4) 81 (1969), 147–168 (French). MR 249738, DOI 10.1007/BF02413501
John V. Baxley, Existence and uniqueness for nonlinear boundary value problems on infinite intervals, J. Math. Anal. Appl. 147 (1990), no. 1, 122–133. MR 1044690, DOI 10.1016/0022-247X(90)90388-V
J. W. Bebernes and L. K. Jackson, Infinite interval boundary value problems for $y^{\prime \prime }=f(x,\,y)$, Duke Math. J. 34 (1967), 39–47. MR 206386
M. M. Belova, On bounded solutions of non-linear second-order differential equations, Mat. Sb. (N.S.) 56 (98) (1962), 469–503 (Russian). MR 0141824
H. Ben-El-Mechaiekh, Continuous approximations of multifunctions, fixed points and coincidences, Approximation and optimization in the Caribbean, II (Havana, 1993) Approx. Optim., vol. 8, Peter Lang, Frankfurt am Main, 1995, pp. 69–97. MR 1368166
H. Ben-El-Mechaiekh and P. Deguire, Approachability and fixed points for nonconvex set-valued maps, J. Math. Anal. Appl. 170 (1992), no. 2, 477–500. MR 1188567, DOI 10.1016/0022-247X(92)90032-9
Czesław Bessaga and Aleksander Pełczyński, Selected topics in infinite-dimensional topology, Monografie Matematyczne, Tom 58. [Mathematical Monographs, Vol. 58], PWN—Polish Scientific Publishers, Warsaw, 1975. MR 0478168
Ugo Bessi, A variational proof of a Sitnikov-like theorem, Nonlinear Anal. 20 (1993), no. 11, 1303–1318. MR 1220837, DOI 10.1016/0362-546X(93)90133-D
R. Bielawski and L. Górniewicz, Some applications of the Leray-Schauder alternative to differential equations, Nonlinear functional analysis and its applications (Maratea, 1985) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 173, Reidel, Dordrecht, 1986, pp. 187–194. MR 852578
R. Bielawski and L. Górniewicz, A fixed point index approach to some differential equations, Topological fixed point theory and applications (Tianjin, 1988) Lecture Notes in Math., vol. 1411, Springer, Berlin, 1989, pp. 9–14. MR 1031778, DOI 10.1007/BFb0086436
Karol Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. MR 0216473
Moses A. Boudourides, On bounded solutions of nonlinear ordinary differential equations, Comment. Math. Univ. Carolin. 22 (1981), no. 1, 15–26. MR 609933
Alberto Bressan, Directionally continuous selections and differential inclusions, Funkcial. Ekvac. 31 (1988), no. 3, 459–470. MR 987798
Alberto Bressan, On the qualitative theory of lower semicontinuous differential inclusions, J. Differential Equations 77 (1989), no. 2, 379–391. MR 983301, DOI 10.1016/0022-0396(89)90150-2
Felix E. Browder and Chaitan P. Gupta, Topological degree and nonlinear mappings of analytic type in Banach spaces, J. Math. Anal. Appl. 26 (1969), 390–402. MR 257826, DOI 10.1016/0022-247X(69)90162-0
M. Cecchi, M. Furi, and M. Marini, On continuity and compactness of some nonlinear operators associated with differential equations in noncompact intervals, Nonlinear Anal. 9 (1985), no. 2, 171–180. MR 777986, DOI 10.1016/0362-546X(85)90070-7
M. Cecchi, M. Furi, and M. Marini, About the solvability of ordinary differential equations with asymptotic boundary conditions, Boll. Un. Mat. Ital. C (6) 4 (1985), no. 1, 329–345 (English, with Italian summary). MR 805224
M. Cecchi, M. Marini, and P. Zecca, Existence of bounded solutions for multivalued differential systems, Nonlinear Anal. 9 (1985), no. 8, 775–786. MR 799883, DOI 10.1016/0362-546X(85)90017-3
Olga Taussky, An algebraic property of Laplace’s differential equation, Quart. J. Math. Oxford Ser. 10 (1939), 99–103. MR 83, DOI 10.1093/qmath/os-10.1.99
M. Cecchi, M. Marini, and P. L. Zezza, Asymptotic properties of the solutions of nonlinear equations with dichotomies and applications, Boll. Un. Mat. Ital. C (6) 1 (1982), no. 1, 209–234 (English, with Italian summary). MR 696272
M. Cecchi, M. Marini, and P. L. Zezza, Boundary value problems on $[a,b)$ and singular perturbations, Ann. Polon. Math. 44 (1984), no. 1, 73–80. MR 764806, DOI 10.4064/ap-44-1-73-80
Mieczysław Cichoń, Trichotomy and bounded solutions of nonlinear differential equations, Math. Bohem. 119 (1994), no. 3, 275–284. MR 1305530
Roberto Conti, Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. Un. Mat. Ital. (3) 22 (1967), 135–178. MR 0218650
C. Corduneanu, Quelques problèmes globaux concernant les équations différentielles non-linéaires du second ordre, Acad. R. P. Romîne. Fil. Iaşi. Stud. Cerc. Şti. Mat. 7 (1956), no. 1, 1–7 (Romanian, with Russian and French summaries). MR 0096863
C. Corduneanu, L’existence des solutions bornées pour certaines équations différentielles du second ordre, Acad. R. P. Romîne. Fil. Iaşi. Stud. Cerc. Şti. Mat. 8 (1957), no. 2, 127–134 (Romanian, with Russian and French summaries). MR 0096866
Krzysztof Czarnowski and Tadeusz Pruszko, On the structure of fixed point sets of compact maps in $B_0$ spaces with applications to integral and differential equations in unbounded domain, J. Math. Anal. Appl. 154 (1991), no. 1, 151–163. MR 1087965, DOI 10.1016/0022-247X(91)90077-D
Marian Dawidowski and Bogdan Rzepecki, On bounded solutions of nonlinear differential equations in Banach spaces, Demonstratio Math. 18 (1985), no. 1, 91–102. MR 816022
A. Dold, Lectures on algebraic topology, Die Grundlehren der mathematischen Wissenschaften, Band 200, Springer-Verlag, New York-Berlin, 1972 (German). MR 0415602
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
J. Dugundji, Modified Vietoris theorems for homotopy, Fund. Math. 66 (1969/70), 223–235. (errata insert). MR 254844, DOI 10.4064/fm-66-2-223-235
A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
A. F. Filippov, Differentsial′nye uravneniya s razryvnoĭ pravoĭ chast′yu, “Nauka”, Moscow, 1985 (Russian). MR 790682
Leonard Fountain and Lloyd Jackson, A generalized solution of the boundary value problem for $y^{\prime \prime }=f(x,\,y,\,y^{\prime } )$, Pacific J. Math. 12 (1962), 1251–1272. MR 163002
M. Frigon, Theoreme d’existence de solutions faibles pour un probleme aux limites du second ordre dans l’intervale $(0,\infty )$, Preprint. no. 85-18 (Juin 1985). Univ. de Montreal. Dépt. de. Math. et de Statistique.
Massimo Furi and Patrizia Pera, A continuation method on locally convex spaces and applications to ordinary differential equations on noncompact intervals, Ann. Polon. Math. 47 (1987), no. 3, 331–346. MR 927581, DOI 10.4064/ap-47-3-331-346
M. Furi and M. P. Pera, On the fixed point index in locally convex spaces, Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), no. 1-2, 161–168. MR 899950, DOI 10.1017/S0308210500018291
J. C. Oxtoby and S. M. Ulam, On the existence of a measure invariant under a transformation, Ann. of Math. (2) 40 (1939), 560–566. MR 97, DOI 10.2307/1968940
Jack Girolo, Approximating compact sets in normed linear spaces, Pacific J. Math. 98 (1982), no. 1, 81–89. MR 644940
Lech Górniewicz, Homological methods in fixed-point theory of multi-valued maps, Dissertationes Math. (Rozprawy Mat.) 129 (1976), 71. MR 394637
Lech Górniewicz, Topological approach to differential inclusions, Topological methods in differential equations and inclusions (Montreal, PQ, 1994) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 472, Kluwer Acad. Publ., Dordrecht, 1995, pp. 129–190. MR 1368672, DOI 10.1007/978-94-011-0339-8_{4}
Lech Górniewicz, Andrzej Granas, and Wojciech Kryszewski, On the homotopy method in the fixed point index theory of multi-valued mappings of compact absolute neighborhood retracts, J. Math. Anal. Appl. 161 (1991), no. 2, 457–473. MR 1132121, DOI 10.1016/0022-247X(91)90345-Z
Lech Górniewicz and Sławomir Plaskacz, Periodic solutions of differential inclusions in $\textbf {R}^n$, Boll. Un. Mat. Ital. A (7) 7 (1993), no. 3, 409–420 (English, with Italian summary). MR 1249117
A. Granas, R. B. Guenther, J. W. Lee, and D. O’Regan, Boundary value problems on infinite intervals and semiconductor devices, J. Math. Anal. Appl. 116 (1986), no. 2, 335–348. MR 842804, DOI 10.1016/S0022-247X(86)80002-6
O. A. Gross, The boundary value problem on an infinite interval: Existence, uniqueness, and asymptotic behavior of bounded solutions to a class of nonlinear second order differential equations, J. Math. Anal. Appl. 7 (1963), 100–109. MR 153901, DOI 10.1016/0022-247X(63)90080-5
V. V. Gudkov, Boundary value problems on finite and infinite intervals, Differencial′nye Uravnenija 12 (1976), no. 3, 555–557, 575 (Russian). MR 0473301
Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
D. Bainov and V. Covachev (eds.), Proceedings of the Third International Colloquium on Differential Equations, VSP, Utrecht, 1993. Held in Plovdiv, August 18–22, 1992. MR 1460525
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Masuo Hukuhara, Sur l’application semi-continue dont la valeur est un compact convexe, Funkcial. Ekvac. 10 (1967), 43–66 (French). MR 222856
D. M. Hyman, On decreasing sequences of compact absolute retracts, Fund. Math. 64 (1969), 91–97. MR 253303, DOI 10.4064/fm-64-1-91-97
D. V. Izyumova, Positive bounded solutions of second-order nonlinear ordinary differential equations, Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 22 (1987), 100–109, 234 (Russian, with English and Georgian summaries). MR 946663
Jiří Jarník and Jaroslav Kurzweil, On conditions on right hand sides of differential relations, Časopis Pěst. Mat. 102 (1977), no. 4, 334–349, 426 (English, with Czech and loose Russian summaries). MR 0466702
Athanassios G. Kartsatos, The Leray-Schauder theorem and the existence of solutions to boundary value problems on infinite intervals, Indiana Univ. Math. J. 23 (1973/74), 1021–1029. MR 340697, DOI 10.1512/iumj.1974.23.23083
Athanassios G. Kartsatos, A stability property of the solution to a boundary value problem on an infinite interval, Math. Japon. 19 (1974), no. 3, 187–194. MR 399598
Athanassios G. Kartsatos, A boundary value problem on an infinite interval, Proc. Edinburgh Math. Soc. (2) 19 (1974/75), 245–252. MR 377165, DOI 10.1017/S0013091500015510
Athanassios G. Kartsatos, The Hildebrandt-Graves theorem and the existence of solutions of boundary value problems on infinite intervals, Math. Nachr. 67 (1975), 91–100. MR 377166, DOI 10.1002/mana.19750670603
Athanassios G. Kartsatos, Locally invertible operators and existence problems in differential systems, Tohoku Math. J. (2) 28 (1976), no. 2, 167–176. MR 430385, DOI 10.2748/tmj/1178240831
J. Horn, Über eine hypergeometrische Funktion zweier Veränderlichen, Monatsh. Math. Phys. 47 (1939), 359–379 (German). MR 91, DOI 10.1007/BF01695508
M. A. Krasnosel′skiĭ, Zh. Moven, and A. V. Pokrovskiĭ, New theorems on forced periodic oscillations and bounded solutions, Dokl. Akad. Nauk SSSR 321 (1991), no. 3, 491–495 (Russian); English transl., Soviet Phys. Dokl. 36 (1991), no. 11, 743–745 (1992). MR 1151282
A. M. Krasnosel’skii, J. Mawhin, M. A. Krasnosel’skii and A. V. Pokrovskii, Generalized guiding functions in a problem of high frequency forced oscillations, Rapp. no. 222 (Feb. 1993), Sem. Math. (N.S.), Inst. Math. Pura Appl. Univ. Cath. de Louvain).
Mária Kečkemétyová, On the existence of a solution for nonlinear operator equations in Fréchet spaces, Math. Slovaca 42 (1992), no. 1, 43–54. MR 1159490
Mária Kečkemétyová, Continuous solutions of nonlinear boundary value problems for ODEs on unbounded intervals, Math. Slovaca 42 (1992), no. 3, 279–297. MR 1182959
I. T. Kiguradze, Bounded and periodic solutions of higher-order linear differential equations, Mat. Zametki 37 (1985), no. 1, 48–62, 138 (Russian). MR 792232
A. Kneser, Untersuchung und asymptotische Darstellung der Integrale gewisser Differentialgleichungen bei grossen Werthen des Arguments, J. Reine Angen. Math 1, 116 (1896), 178–212.
M. A. Krasnosel′skiĭ, Operator sdviga po traktoriyam differentsial′nykh uravneniĭ, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0203159
W. Kryszewski, An application of $A$-mapping theory to boundary value problems for ordinary differential equations, Nonlinear Anal. 15 (1990), no. 8, 697–717. MR 1074949, DOI 10.1016/0362-546X(90)90087-W
W. Kryszewski, Graph-approximation of set-valued maps on noncompact domains, Topology Appl. 83 (1998), 1–21.
I. T. Kiguradze and I. Rach nková, On a certain nonlinear problem for two-dimensional differential systems, Arch. Math. (Brno) 16 (1980), no. 1, 15–37. MR 594664
I. T. Kiguradze and B. L. Shekhter, Singular boundary value problems for second-order ordinary differential equations, Current problems in mathematics. Newest results, Vol. 30 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987, pp. 105–201, 204 (Russian). Translated in J. Soviet Math. 43 (1988), no. 2, 2340–2417. MR 925830
Zbyněk Kubáček, On the structure of fixed point sets of some compact maps in the Fréchet space, Math. Bohem. 118 (1993), no. 4, 343–358. MR 1251881
Zhong Chao Liang, Limit boundary value problem for nonlinear differential equation of the second order, Acta Math. Sinica (N.S.) 1 (1985), no. 2, 119–125. MR 858560, DOI 10.1007/BF02560026
S. G. Lobanov, Peano’s theorem is invalid for any infinite-dimensional Fréchet space, Mat. Sb. 184 (1993), no. 2, 83–86 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 78 (1994), no. 1, 211–214. MR 1214945, DOI 10.1070/SM1994v078n01ABEH003465
David Lowell Lovelady, Bounded solutions of whole-line differential equations, Bull. Amer. Math. Soc. 79 (1973), 752–753. MR 322267, DOI 10.1090/S0002-9904-1973-13298-7
A. Margheri and P. Nistri, An existence result for a class of asymptotic boundary value problems, Differential Integral Equations 6 (1993), no. 6, 1337–1347. MR 1235197
A. Mambriani, Su un teoreme relativo alle equazioni differenziali ordinarie del $2^0$ ordine, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. 9 (1929), 620–622.
Mario Martelli, Continuation principles and boundary value problems, Topological methods for ordinary differential equations (Montecatini Terme, 1991) Lecture Notes in Math., vol. 1537, Springer, Berlin, 1993, pp. 32–73. MR 1226929, DOI 10.1007/BFb0085075
Margaret C. Memory and James R. Ward Jr., The Conley index and the method of averaging, J. Math. Anal. Appl. 158 (1991), no. 2, 509–518. MR 1117579, DOI 10.1016/0022-247X(91)90253-V
Marian Mrozek, A cohomological index of Conley type for multi-valued admissible flows, J. Differential Equations 84 (1990), no. 1, 15–51. MR 1042657, DOI 10.1016/0022-0396(90)90125-9
J. Musielak, Introduction to Functional Analysis, (in Polish). PWN, Warszawa 1976.
Rafael Ortega, A boundedness result of Landesman-Lazer type, Differential Integral Equations 8 (1995), no. 4, 729–734. MR 1306589
A. I. Povolockiĭ and E. A. Gango, Periodic solutions of differential equations with a multivalued right hand side, Leningrad. Gos. Ped. Inst. Učen. Zap. 541 (1972), 145–154 (Russian). MR 0310350
B. Przeradzki, On a two-point boundary value problem for differential equations on the half-line, Ann. Polon. Math. 50 (1989), no. 1, 53–61. MR 1016620, DOI 10.4064/ap-50-1-53-61
D. O’Regan, Existence of positive solutions to some singular and nonsingular second order boundary value problems, J. Differential Equations 84 (1990), no. 2, 228–251. MR 1047568, DOI 10.1016/0022-0396(90)90077-3
B. Przeradzki, The existence of bounded solutions for differential equations in Hilbert spaces, Ann. Polon. Math. 56 (1992), no. 2, 103–121. MR 1159982, DOI 10.4064/ap-56-2-103-121
Irena Rach nková, On a Kneser problem for a system of nonlinear ordinary differential equations, Czechoslovak Math. J. 31(106) (1981), no. 1, 114–126. With a loose Russian summary. MR 604119
Irena Rachůnková, On Kneser problem for differential equations of the 3rd order, Časopis Pěst. Mat. 115 (1990), no. 1, 18–27 (English, with Russian and Czech summaries). MR 1044010
Irena Rachůnková, Nonnegative nonincreasing solutions of differential equations of the 3rd order, Czechoslovak Math. J. 40(115) (1990), no. 2, 213–221. MR 1046289
A. P. Robertson and W. J. Robertson, Topological vector spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 53, Cambridge University Press, New York, 1964. MR 0162118
Bogdan Rzepecki, An existence theorem for bounded solutions of differential equations in Banach spaces, Rend. Sem. Mat. Univ. Padova 73 (1985), 89–94. MR 799899
S. K. Sachdev, Positive solutions of a nonlinear boundary value problem on half-line, Panamer. Math. J. 1 (1991), no. 2, 27–40. MR 1116241
Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
Keith W. Schrader, Existence theorems for second order boundary value problems, J. Differential Equations 5 (1969), 572–584. MR 239175, DOI 10.1016/0022-0396(69)90094-1
Keith Schrader, Second and third order boundary value problems, Proc. Amer. Math. Soc. 32 (1972), 247–252. MR 291548, DOI 10.1090/S0002-9939-1972-0291548-4
J. D. Schuur, The existence of proper solutions of a second order ordinary differential equation, Proc. Amer. Math. Soc. 17 (1966), 595–597. MR 192097, DOI 10.1090/S0002-9939-1966-0192097-1
Jerry D. Schuur, A class of nonlinear ordinary differential equations which inherit linear-like asymptotic behavior, Nonlinear Anal. 3 (1979), no. 1, 81–86 (1978). MR 520475, DOI 10.1016/0362-546X(79)90038-5
Sek Wui Seah, Bounded solutions of multivalued differential systems, Houston J. Math. 8 (1982), no. 4, 587–598. MR 688258
Valter Šeda, On an application of the Stone theorem in the theory of differential equations, Časopis Pěst. Mat. 97 (1972), 183–189, 206 (English, with Czech summary). MR 0348188
Valter Šeda, On a generalization of the Thomas-Fermi equation, Acta Math. Univ. Comenian. 39 (1980), 97–114 (English, with Russian and Slovak summaries). MR 619267
Valter Šeda and Zbyněk Kubáček, On the connectedness of the set of fixed points of a compact operator in the Fréchet space $C^m(\langle b,\infty ),\textbf {R}^n)$, Czechoslovak Math. J. 42(117) (1992), no. 4, 577–588. MR 1182189
Adalberto Spezamiglio, Bounded solutions of ordinary differential equations, and stability, Rev. Mat. Estatíst. 8 (1990), 1–9 (Portuguese, with English summary). MR 1133135
Roman Srzednicki, Periodic and constant solutions via topological principle of Ważewski, Univ. Iagel. Acta Math. 26 (1987), 183–190. MR 943337
Roman Srzednicki, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations, Nonlinear Anal. 22 (1994), no. 6, 707–737. MR 1270166, DOI 10.1016/0362-546X(94)90223-2
Svatoslav Staněk, Bounded solutions of second order functional-differential equations, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 30 (1991), 97–105. MR 1166429
Svatoslav Staněk, On the boundedness and periodicity of solutions of second-order functional-differential equations with a parameter, Czechoslovak Math. J. 42(117) (1992), no. 2, 257–270. MR 1179497
Marko Švec, Fixpunktsatz und monotone Lösungen der Differentialgleichung$y^{(n)}+B(x,y,y^{\prime }, \cdots ,y^{(n-1)})y=0$, Arch. Math. (Brno) 2 (1966), 43–55. MR 206368
Marko Švec, Les propriétés asymptotiques des solutions d’une équations différentielle nonlinéaire d’ordre $n$, Czechoslovak Math. J. 17(92) (1967), 550–557 (French, with Russian summary). MR 218677
Stanisław Szufla, On the existence of bounded solutions of nonlinear differential equations in Banach spaces, Funct. Approx. Comment. Math. 15 (1986), 117–123. MR 880140
S. Umamaheswaram, Boundary value problems for higher order differential equations, J. Differential Equations 18 (1975), 188–201. MR 372309, DOI 10.1016/0022-0396(75)90089-3
S. Umamaheswaram and M. Venkata Rama, Multipoint focal boundary value problems on infinite intervals, J. Appl. Math. Stochastic Anal. 5 (1992), no. 3, 283–289. MR 1183601, DOI 10.1155/S1048953392000236
John von Neumann, Collected works. Vol. VI: Theory of games, astrophysics, hydrodynamics and meteorology, A Pergamon Press Book, The Macmillan Company, New York, 1963. General editor: A. H. Taub. MR 0157876
Paul Waltman, On the asymptotic bahavior of solutions of an $n\textrm {th}$ order equation, Monatsh. Math. 69 (1965), 427–430. MR 185217, DOI 10.1007/BF01299949
James Robert Ward Jr., Averaging, homotopy, and bounded solutions of ordinary differential equations, Differential Integral Equations 3 (1990), no. 6, 1093–1100. MR 1073059
James R. Ward Jr., A topological method for bounded solutions of nonautonomous ordinary differential equations, Trans. Amer. Math. Soc. 333 (1992), no. 2, 709–720. MR 1066450, DOI 10.1090/S0002-9947-1992-1066450-8
J. R. Ward Jr., Global continuation for bounded solutions of ordinary differential equations, Topol. Methods Nonlinear Anal. 2 (1993), no. 1, 75–90. MR 1245479, DOI 10.12775/TMNA.1993.031
James R. Ward Jr., Homotopy and bounded solutions of ordinary differential equations, J. Differential Equations 107 (1994), no. 2, 428–445. MR 1264531, DOI 10.1006/jdeq.1994.1021
James R. Ward Jr., A global continuation theorem and bifurcation from infinity for infinite-dimensional dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 4, 725–738. MR 1405753, DOI 10.1017/S0308210500023039
James R. Ward Jr., Bifurcating continua in infinite-dimensional dynamical systems and applications to differential equations, J. Differential Equations 125 (1996), no. 1, 117–132. MR 1376062, DOI 10.1006/jdeq.1996.0026
Pui-kei Wong, Existence and asymptotic behavior of proper solutions of a class of second-order nonlinear differential equations, Pacific J. Math. 13 (1963), 737–760. MR 153912
T. Yoshizawa, Stability theory and the existence of periodic solutions and almost periodic solutions, Applied Mathematical Sciences, Vol. 14, Springer-Verlag, New York-Heidelberg, 1975. MR 0466797
Pietro Zecca and Pier Luigi Zezza, Nonlinear boundary value problems in Banach spaces for multivalue differential equations on a noncompact interval, Nonlinear Anal. 3 (1979), no. 3, 347–352. MR 532895, DOI 10.1016/0362-546X(79)90024-5