Functional analysis and nonlinear boundary value problems: the legacy of Andrzej Lasota
Abstract
2012 Annual Lecture dedicated to the memory of Professor Andrzej Lasota.
Keywords
multipoint boundary value problems; periodic solutions; nonlinear integral equations; functional analysis
References
1 Bailey P.B., Shampine L., Waltman P.E., Nonlinear Two-Point Boundary Value Problems, Academic Press, New York, 1968.
2. Banach S., Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133–181.
3. Barbalat I., Halanay A., Solutions périodiques des systèmes d’équations différentielles non-linéaires, Revue Math. Pures Appl. 3 (1958), 395–411.
4. Bernfeld S., Lakshmikantham V., An Introduction to Nonlinear Boundary Value Problems, Academic Press, New York, 1974.
5. Biernacki M., Sur un problème d’interpolation relatif aux équations différentielles linéaires, Ann. Soc. Polon. Math. 20 (1947), 169–214.
6. Birkhoff G.D., Kellogg O.D., Invariant points in function space, Trans. Amer. Math. Soc. 23 (1922), 96–115.
7. Borg G., Über die Stabilität gewisser Klassen von linearen Differentialgleichungen, Ark. Mat. Astr. Fyz. 31 A (1944).
8. Browder F.E., Nonlinear Operator Equations of Evolution in Banach Spaces, Amer. Math. Soc., Providence, 1976.
9. Caccioppoli R., Un teorema generale sull’esistenza di elementi uniti in una trasformazione funzionale, Rend. Accad. Naz. Lincei (6) 11 (1930), 794–799.
10. Caccioppoli R., Sugli elementi uniti delle transformazioni funzionali; un osservazione sui problemi di valori ai limiti, Rend. Acad. Naz. Lincei, Cl. Sci. Mat. Fiz. Nat. (1931), 498–502.
11. Cañada A., Villegas S., An applied mathematical excursion through Lyapunov inequalities, classical analysis and differential equations, SeMA J. 57 (2012), 69–106.
12. Cinquini S., Problemi di valori al contorno per equazioni differenziali di ordine n, Ann. Scuola Norm. Sup. Pisa (2) 9 (1940), 61–67.
13. Chow S.N., Lasota A., On boundary value problems for ordinary differential equations, J. Differential Equations 14 (1973), 326–337.
14. Chow S.N., Lasota A., Dunninger D.R., A maximum principle for fourth order ordinary differential equations, J. Differential Equations 14 (1973), 101–105.
15. de La Vallée Poussin Ch.J., Sur l’équation différentielle linéaire du second ordre. Détermination d’une intégrale par deux valeurs assignées. Extension aux équations d’ordre n, J. Math. Pures Appl. 8 (1929), 125–144.
16. Denkowski Z., On the boundary value problems for the ordinary differential equation of second order, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 12 (1968), 11–16.
17. Denkowski Z., On the existence of solution of two-point boundary-value problem for second order differential equation, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 13 (1969), 27–33.
18. Denkowski Z., The boundary value problems for ordinary non-linear differential and difference equations of the fourth order, Ann. Polon. Math. 24 (1971), 87–102.
19. Gaines R.E., Mawhin J., Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math. Vol. 568, Springer, Berlin, 1977.
20. Hartman P., On boundary value problems for systems of ordinary nonlinear second-order differential equations, Trans. Amer. Math. Soc. 96 (1960), 493–509.
21. Henderson J., Boundary value problems for n th order Lipschitz equations, J. Math. Anal. Appl. 134 (1988), 196–210.
22. Henderson J., McGwier Jr R.W., Uniqueness, existence, and optimality for fourth-order Lipschitz equations, J. Differential Equations 67 (1987), 414–440.
23. Henrard M., Optimal integral criterion of nonresonance for asymptotically positively homogeneous equations with Sturm-Liouville boundary conditions, Acad. Roy. Belg. Cl. Sci. Mém. in-8^0 (3) 18 (2000), 51 pp.
24. Jackson L.K., Existence and uniqueness of solutions of boundary value problems for Lipschitz equations, J. Differential Equations 32 (1979), 76–90.
25. Jackson L.K., Boundary value problems for Lipschitz equations, in: Differential equations (Proc. Eighth Fall Conf., Oklahoma State Univ., Stillwater, Okla., 1979), Academic Press, New York, 1980, pp. 31–50.
26. Jackson L.K., Schrader K., Existence and uniqueness of solutions of boundary value problems for third order differential equations, J. Differential Equations 9 (1971), 46–54.
27. Kamenskii M., Obukhovskii V., Zecca P., Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach spaces, De Gruyter, Berlin, 2001.
28. Kaplan J.L., Lasota A., Yorke J.A., An application of the Wazewski retract method to boundary value problems, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 16 (1974), 7–14.
29. Kasprzyk S., Myjak J., On the existence of solutions of the Floquet problem for ordinary differential equations, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 13 (1969), 35–39.
30. Krakowiak A., An application of the homotopy method to linear problems for nonlinear differential systems, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 16 (1974), 71–75.
31. Krakowiak A., On the existence of solution of homogeneous boundary value problem for ordinary differential equations, Univ. Iagel. Acta Math. 26 (1987), 179–181.
32. Kubicová M., Periodic solutions to a nonlinear differential equation of the order 2n+1, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 83 (1989), 133–137.
33. Lasota A., A note on the relationship between the initial and the boundary value problems for an ordinary differential equation of the n th order, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 5 (1959), 59–65 (in Polish).
34. Lasota A., Sur l’existence des solutions d’un problème d’interpolation pour l’équation différentielle ordinaire d’ordre n, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 10 (1962), 523–528.
35. Lasota A., Sur les problèmes linéaires aux limites pour un système d’équations différentielles ordinaires, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 10 (1962), 565–570.
36. Lasota A., Sur la distance entre les zéros de l’équation différentielle linéaire du troisième ordre, Ann. Polon. Math. 13 (1963), 129–132.
37. Lasota A., Sur une généralisation du premier théorème de Fredholm, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 11 (1963), 89–94.
38. Lasota A., Sur l’existence de solutions des problèmes aux limites de Neumann et de Dirichlet pour l’équation différentielle elliptique de second ordre, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 11 (1963), 441–446.
39. Lasota A., Un problème aux limites pour l’équation différentielle du second ordre, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 9 (1963), 49–54.
40. Lasota A., Sur l’existence de solutions des problèmes linéaires aux limites pour les équations différentielles ordinaires, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 10 (1965), 45–60.
41. Lasota A., Une généralisation du premier théorème de Fredholm et ses applications à la théorie des équations différentielles ordinaires, Ann. Polon. Math. 18 (1966), 65–77.
42. Lasota A., Sur l’existence et l’unicité des solutions du problème aux limites de Nicoletti pour un système d’équations differentielles ordinaires, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 11 (1966), 41–48.
43. Lasota A., A discrete boundary value problem, Ann. Polon. Math. 20 (1968), 183–190.
44. Lasota A., Contingent equations and boundary value problems, in: Numerical Analysis of Partial Differential Equations (C.I.M.E. 2 Ciclo, Ispra, 1967), Edizioni Cremonese, Roma, 1968, pp. 255–266.
45. Lasota A., Boundary value problems for second order differential equations, in: Seminar on Differential Equations and Dynamical Systems, II (Univ. Maryland, College Park, Md., 1969), Lecture Notes in Math. Vol. 144, Springer, Berlin, 1970, pp. 140–152.
46. Lasota A., Équations intégrales au contingent, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 15 (1971), 91–97.
47. Lasota A., On two-point boundary value problems for systems of ordinary non-linear, first-order differential equations, Collection of articles dedicated to the memory of Tadeusz Wazewski, IV. Ann. Polon. Math. 29 (1975), 391–396.
48. Lasota A., Open mappings in the proofs of theorems on the existence of solutions of differential equations, Wiadom. Mat. (2) 20 (1976), 76–80 (in Polish).
49. Lasota A., On the existence and uniqueness of solutions of a multipoint boundary value problem, Ann. Polon. Math. 38 (1980), 305–310.
50. Lasota A., Łuczyński M., A note on the uniqueness of two point boundary value problems. I, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 12 (1968), 27–29.
51. Lasota A., Łuczyński M., A note on the uniqueness of two point boundary value problems. II, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 13 (1969), 45–48.
52. Lasota A., Olech C., An optimal solution of Nicoletti’s boundary value problem, Ann. Polon. Math. 18 (1966), 131–139.
53. Lasota A., Opial Z., Sur un problème d’interpolation pour l’équation différentielle ordinaire d’ordre n, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 9 (1961), 667–671.
54. Lasota A., Opial Z., L’application du principe de Pontryagin à l’évaluation de l’intervalle d’existence et d’unicité des solutions d’un problème aux limites, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 11 (1963), 41–46.
55. Lasota A., Opial Z., L’existence et l’unicité des solutions du problème d’interpolation pour l’équation différentielle ordinaire d’ordre n, Ann. Polon. Math. 15 (1964), 253–271.
56. Lasota A., Opial Z., Sur les solutions périodiques des équations différentielles ordinaires, Ann. Polon. Math. 16 (1964), 69–94.
57. Lasota A., Opial Z., Nouvelles méthodes d’analyse fonctionelle dans la théorie des solutions périodiques des équations différentielles ordinaires, Abh. Deutsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech. 1965 No. 1 (1965), 186–189.
58. Lasota A., Opial Z., Linear problems for ordinary non-linear differential equations and integral equations of Hammerstein’s type, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 13 (1965), 715–718.
59. Lasota A., Opial Z., An application of the Kakutani–Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 13 (1965), 781–786.
60. Lasota A., Opial Z., On the existence of solutions of linear problems for ordinary differential equations, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 14 (1966), 371–376.
61. Lasota A., Opial Z., Sur la dépendance continue des solutions des équations différentielles ordinaires de leurs seconds membres et des conditions aux limites, Ann. Polon. Math. 19 (1967), 13–36.
62. Lasota A., Opial Z., On the existence and uniqueness of solutions of nonlinear functional equations, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 15 (1967), 797–800.
63. Lasota A., Opial Z., On the existence and uniqueness of solutions of a boundary value problem for an ordinary second-order differential equation, Colloq. Math. 18 (1967), 1–5.
64. Lasota A., Opial Z., Fixed-point theorems for multi-valued mappings and optimal control problems, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 16 (1968), 645–649.
65. Lasota A., Szafraniec F.H., Sur les solutions périodiques d’une équation différentielle ordinaire d’ordre n, Ann. Polon. Math. 18 (1966), 339–344.
66. Lasota A., Szafraniec F.H., On Hammerstein integral equations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 59 (1975), 65–67.
67. Lasota A., Traple J., Nicoletti boundary value problems for systems of linear differential equations with distributional perturbations, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 15 (1971), 103–108.
68. Lasota A., Yorke J.A., Existence of solutions of two-point boundary value problems for nonlinear systems, J. Differential Equations 11 (1972), 509–518.
69. Leray J., Schauder J., Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. 51 (1934), 45–78.
70. Levin A.Yu., Sur un problème polylocal, Nauk. Dokl. Vusch. Chk. 5 (1958), 34–37 (in Russian).
71. Levin A.Yu., Sur certaines estimations d’une fonction différentiable, Dokl. Akad. Nauk SSSR 138 (1961), 37–38 (in Russian).
72. Lyapunov A., The general problem of stability of motion, Kharkov Math. Soc., Kharkov, 1892 (in Russian).
73. Mawhin J., Tshinanga S.B., Extension des théorèmes d’existence de Schmitt et Smith aux perturbations d’opérateurs de Fredholm d’indice zéro, Acad. Roy. Belg. Bull. Cl. Sci. (5) 75 (1989), 322–340.
74. Mawhin J., Ward Jr J.R., Nonresonance and existence for nonlinear elliptic boundary value problems, Nonlinear Anal. 5 (1981), 677–684.
75. Mawhin J., Ward Jr J.R., Nonuniform nonresonance conditions at the two first eigenvalues for periodic solutions of forced Liénard and Duffing equations, Rocky Mountain J. Math. 12 (1982), 643–654.
76. Melentsova Yu.A., Best constants in integral inequalities for functions with n zeros, Ural. Gos. Univ. Mat. Zap. 10, Issled. Sovremen. Mat. Anal. 217 (1977), 93–107 (in Russian).
77. Melentsova Yu.A., A non-improvable estimate of the interval of disconjugacy for a linear differential equation with coefficients that are bounded in $L^r$, Differentsial’nye Uravnenija 13 (1977), 1776–1786, 1906–1907 (in Russian). English translation: Differential equations 13 (1977), 1236–1244.
78. Melentsova Yu.A., Mil’shtein G.N., Optimal estimation of the interval of solvability of a multipoint boundary value problem, Differentsial’nye Uravnenija 10 (1974), 1630–1641, 1732 (in Russian). English translation: Differential Equations 10 (1974), 1257–1265.
79. Melentsova Yu.A., Mil’shtein G.N., On the optimal estimate of the interval of nonoscillation for linear differential equations with bounded coefficients, Differentsial’nye Uravneniya 17 (1981), 2160–2175, 2299 (in Russian). English translation: Differential Equations 17 (1981), 1368–1379.
80. Mikusiński J.G., Sur un problème d’interpolation pour les intégrales des équations différentielles linéaires, Ann. Soc. Polon. Math. 19 (1946), 165–205.
81. Myjak J., Boundary value problems for nonlinear differential and difference equations of the second order, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 15 (1971), 113–123.
82. Niccoletti O., Sulle condizioni initiali che determinano gli integrali delle equazioni differenziali ordinarie, Atti R. Accad. Sci. Torino (1898), 746–759.
83. Picard E., Sur l’application des méthodes d’approximations successives à l’étude de certaines équations différentielles ordinaires, J. Math. Pures Appl. 9 (1893), 217–271.
84. Picard E., Traité d’analyse, Gauthier-Villars, Paris, 1896.
85. Piccinini L., Stampacchia G., Vidossich G. Equazioni differenziali ordinarie in $\mathbb{R}^n$, Liguori Editore, Napoli, 1978. English translation: Ordinary Differential Equations in ℝ^n. Problems and Methods, Springer, New York, 1984.
86. Reissig R., Sansone G., Conti R., Non-linear Differential Equations of Higher Order, Noordhoff, Leyden, 1974.
87. Rouche N., Mawhin J., Équations différentielles ordinaires, vol. 2, Masson, Paris, 1973. English translation: Ordinary Differential Equations. Stability and Periodic Solutions, Pitman, Boston, 1980.
88. Schauder J., Zur Theorie stetiger Abbildungen in Funktionalräumen, Math. Z. 26 (1927), 47–65.
89. Schauder J., Bemerkungen zu meiner Arbeit ‘Zur Theorie stetiger Abbildungen in Funktionalräumen’, Math. Z. 26 (1927), 417–431.
90. Schauder J., Der Fixpunktsatz in Funktionalräumen, Studia Math. 2 (1930), 171–180.
91. Schmitt K., Smith H.L., On eigenvalue problems for nondifferentiable mappings, J. Differential Equations 33 (1979), 294–319.
92. Schmitt K., Smith H.L., Fredholm alternatives for nonlinear differential equations, Rocky Mountain J. Math. 12 (1982), 817–841.
93. Švec M., On the uniqueness problem for integrals of a system of linear differential equations, Matem.-fiz. Sbornik Slov. Akad. Ved. 2 (1952), 3–22 (in Slovakian).
94. Szafraniec F.H., Note on the existence of periodic solutions of a second order differential equation, Ann. Polon. Math. 21 (1969), 305–307.
95. Traple J., On a boundary value problem for systems of ordinary differential equations of second order, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 15 (1971), 159–168.
96. Traple J., A continuous and a discrete boundary value problem, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 19 (1977), 39–53.
97. Volpato M., Sopra un problema di valori al contorno per l’equazione differenziale $y^{( n )} = f( x,y,...,y^{(n-1)}$, Rend. Sem. Mat. Univ. Padova 22 (1953), 334–349.
98. Volpato M., Sull’ esistenza di soluzioni periodiche per equazioni differenziali ordinarie del secondo ordine, Rend. Sem. Mat. Univ. Padova 25 (1956), 371–385.
99. Wypich M., On the existence of a solution of nonlinear periodic boundary value problems, Univ. Iagel. Acta Math. 28 (1991), 289–295.
100. Zwirner G., Su un problema di valori al contorno per equazioni differenziali ordinarie dell ordine n, Rend. Sem. Mat. Univ. Padova 12 (1941), 114–122.
2. Banach S., Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133–181.
3. Barbalat I., Halanay A., Solutions périodiques des systèmes d’équations différentielles non-linéaires, Revue Math. Pures Appl. 3 (1958), 395–411.
4. Bernfeld S., Lakshmikantham V., An Introduction to Nonlinear Boundary Value Problems, Academic Press, New York, 1974.
5. Biernacki M., Sur un problème d’interpolation relatif aux équations différentielles linéaires, Ann. Soc. Polon. Math. 20 (1947), 169–214.
6. Birkhoff G.D., Kellogg O.D., Invariant points in function space, Trans. Amer. Math. Soc. 23 (1922), 96–115.
7. Borg G., Über die Stabilität gewisser Klassen von linearen Differentialgleichungen, Ark. Mat. Astr. Fyz. 31 A (1944).
8. Browder F.E., Nonlinear Operator Equations of Evolution in Banach Spaces, Amer. Math. Soc., Providence, 1976.
9. Caccioppoli R., Un teorema generale sull’esistenza di elementi uniti in una trasformazione funzionale, Rend. Accad. Naz. Lincei (6) 11 (1930), 794–799.
10. Caccioppoli R., Sugli elementi uniti delle transformazioni funzionali; un osservazione sui problemi di valori ai limiti, Rend. Acad. Naz. Lincei, Cl. Sci. Mat. Fiz. Nat. (1931), 498–502.
11. Cañada A., Villegas S., An applied mathematical excursion through Lyapunov inequalities, classical analysis and differential equations, SeMA J. 57 (2012), 69–106.
12. Cinquini S., Problemi di valori al contorno per equazioni differenziali di ordine n, Ann. Scuola Norm. Sup. Pisa (2) 9 (1940), 61–67.
13. Chow S.N., Lasota A., On boundary value problems for ordinary differential equations, J. Differential Equations 14 (1973), 326–337.
14. Chow S.N., Lasota A., Dunninger D.R., A maximum principle for fourth order ordinary differential equations, J. Differential Equations 14 (1973), 101–105.
15. de La Vallée Poussin Ch.J., Sur l’équation différentielle linéaire du second ordre. Détermination d’une intégrale par deux valeurs assignées. Extension aux équations d’ordre n, J. Math. Pures Appl. 8 (1929), 125–144.
16. Denkowski Z., On the boundary value problems for the ordinary differential equation of second order, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 12 (1968), 11–16.
17. Denkowski Z., On the existence of solution of two-point boundary-value problem for second order differential equation, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 13 (1969), 27–33.
18. Denkowski Z., The boundary value problems for ordinary non-linear differential and difference equations of the fourth order, Ann. Polon. Math. 24 (1971), 87–102.
19. Gaines R.E., Mawhin J., Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math. Vol. 568, Springer, Berlin, 1977.
20. Hartman P., On boundary value problems for systems of ordinary nonlinear second-order differential equations, Trans. Amer. Math. Soc. 96 (1960), 493–509.
21. Henderson J., Boundary value problems for n th order Lipschitz equations, J. Math. Anal. Appl. 134 (1988), 196–210.
22. Henderson J., McGwier Jr R.W., Uniqueness, existence, and optimality for fourth-order Lipschitz equations, J. Differential Equations 67 (1987), 414–440.
23. Henrard M., Optimal integral criterion of nonresonance for asymptotically positively homogeneous equations with Sturm-Liouville boundary conditions, Acad. Roy. Belg. Cl. Sci. Mém. in-8^0 (3) 18 (2000), 51 pp.
24. Jackson L.K., Existence and uniqueness of solutions of boundary value problems for Lipschitz equations, J. Differential Equations 32 (1979), 76–90.
25. Jackson L.K., Boundary value problems for Lipschitz equations, in: Differential equations (Proc. Eighth Fall Conf., Oklahoma State Univ., Stillwater, Okla., 1979), Academic Press, New York, 1980, pp. 31–50.
26. Jackson L.K., Schrader K., Existence and uniqueness of solutions of boundary value problems for third order differential equations, J. Differential Equations 9 (1971), 46–54.
27. Kamenskii M., Obukhovskii V., Zecca P., Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach spaces, De Gruyter, Berlin, 2001.
28. Kaplan J.L., Lasota A., Yorke J.A., An application of the Wazewski retract method to boundary value problems, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 16 (1974), 7–14.
29. Kasprzyk S., Myjak J., On the existence of solutions of the Floquet problem for ordinary differential equations, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 13 (1969), 35–39.
30. Krakowiak A., An application of the homotopy method to linear problems for nonlinear differential systems, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 16 (1974), 71–75.
31. Krakowiak A., On the existence of solution of homogeneous boundary value problem for ordinary differential equations, Univ. Iagel. Acta Math. 26 (1987), 179–181.
32. Kubicová M., Periodic solutions to a nonlinear differential equation of the order 2n+1, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 83 (1989), 133–137.
33. Lasota A., A note on the relationship between the initial and the boundary value problems for an ordinary differential equation of the n th order, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 5 (1959), 59–65 (in Polish).
34. Lasota A., Sur l’existence des solutions d’un problème d’interpolation pour l’équation différentielle ordinaire d’ordre n, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 10 (1962), 523–528.
35. Lasota A., Sur les problèmes linéaires aux limites pour un système d’équations différentielles ordinaires, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 10 (1962), 565–570.
36. Lasota A., Sur la distance entre les zéros de l’équation différentielle linéaire du troisième ordre, Ann. Polon. Math. 13 (1963), 129–132.
37. Lasota A., Sur une généralisation du premier théorème de Fredholm, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 11 (1963), 89–94.
38. Lasota A., Sur l’existence de solutions des problèmes aux limites de Neumann et de Dirichlet pour l’équation différentielle elliptique de second ordre, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 11 (1963), 441–446.
39. Lasota A., Un problème aux limites pour l’équation différentielle du second ordre, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 9 (1963), 49–54.
40. Lasota A., Sur l’existence de solutions des problèmes linéaires aux limites pour les équations différentielles ordinaires, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 10 (1965), 45–60.
41. Lasota A., Une généralisation du premier théorème de Fredholm et ses applications à la théorie des équations différentielles ordinaires, Ann. Polon. Math. 18 (1966), 65–77.
42. Lasota A., Sur l’existence et l’unicité des solutions du problème aux limites de Nicoletti pour un système d’équations differentielles ordinaires, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 11 (1966), 41–48.
43. Lasota A., A discrete boundary value problem, Ann. Polon. Math. 20 (1968), 183–190.
44. Lasota A., Contingent equations and boundary value problems, in: Numerical Analysis of Partial Differential Equations (C.I.M.E. 2 Ciclo, Ispra, 1967), Edizioni Cremonese, Roma, 1968, pp. 255–266.
45. Lasota A., Boundary value problems for second order differential equations, in: Seminar on Differential Equations and Dynamical Systems, II (Univ. Maryland, College Park, Md., 1969), Lecture Notes in Math. Vol. 144, Springer, Berlin, 1970, pp. 140–152.
46. Lasota A., Équations intégrales au contingent, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 15 (1971), 91–97.
47. Lasota A., On two-point boundary value problems for systems of ordinary non-linear, first-order differential equations, Collection of articles dedicated to the memory of Tadeusz Wazewski, IV. Ann. Polon. Math. 29 (1975), 391–396.
48. Lasota A., Open mappings in the proofs of theorems on the existence of solutions of differential equations, Wiadom. Mat. (2) 20 (1976), 76–80 (in Polish).
49. Lasota A., On the existence and uniqueness of solutions of a multipoint boundary value problem, Ann. Polon. Math. 38 (1980), 305–310.
50. Lasota A., Łuczyński M., A note on the uniqueness of two point boundary value problems. I, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 12 (1968), 27–29.
51. Lasota A., Łuczyński M., A note on the uniqueness of two point boundary value problems. II, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 13 (1969), 45–48.
52. Lasota A., Olech C., An optimal solution of Nicoletti’s boundary value problem, Ann. Polon. Math. 18 (1966), 131–139.
53. Lasota A., Opial Z., Sur un problème d’interpolation pour l’équation différentielle ordinaire d’ordre n, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 9 (1961), 667–671.
54. Lasota A., Opial Z., L’application du principe de Pontryagin à l’évaluation de l’intervalle d’existence et d’unicité des solutions d’un problème aux limites, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 11 (1963), 41–46.
55. Lasota A., Opial Z., L’existence et l’unicité des solutions du problème d’interpolation pour l’équation différentielle ordinaire d’ordre n, Ann. Polon. Math. 15 (1964), 253–271.
56. Lasota A., Opial Z., Sur les solutions périodiques des équations différentielles ordinaires, Ann. Polon. Math. 16 (1964), 69–94.
57. Lasota A., Opial Z., Nouvelles méthodes d’analyse fonctionelle dans la théorie des solutions périodiques des équations différentielles ordinaires, Abh. Deutsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech. 1965 No. 1 (1965), 186–189.
58. Lasota A., Opial Z., Linear problems for ordinary non-linear differential equations and integral equations of Hammerstein’s type, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 13 (1965), 715–718.
59. Lasota A., Opial Z., An application of the Kakutani–Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 13 (1965), 781–786.
60. Lasota A., Opial Z., On the existence of solutions of linear problems for ordinary differential equations, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 14 (1966), 371–376.
61. Lasota A., Opial Z., Sur la dépendance continue des solutions des équations différentielles ordinaires de leurs seconds membres et des conditions aux limites, Ann. Polon. Math. 19 (1967), 13–36.
62. Lasota A., Opial Z., On the existence and uniqueness of solutions of nonlinear functional equations, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 15 (1967), 797–800.
63. Lasota A., Opial Z., On the existence and uniqueness of solutions of a boundary value problem for an ordinary second-order differential equation, Colloq. Math. 18 (1967), 1–5.
64. Lasota A., Opial Z., Fixed-point theorems for multi-valued mappings and optimal control problems, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 16 (1968), 645–649.
65. Lasota A., Szafraniec F.H., Sur les solutions périodiques d’une équation différentielle ordinaire d’ordre n, Ann. Polon. Math. 18 (1966), 339–344.
66. Lasota A., Szafraniec F.H., On Hammerstein integral equations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 59 (1975), 65–67.
67. Lasota A., Traple J., Nicoletti boundary value problems for systems of linear differential equations with distributional perturbations, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 15 (1971), 103–108.
68. Lasota A., Yorke J.A., Existence of solutions of two-point boundary value problems for nonlinear systems, J. Differential Equations 11 (1972), 509–518.
69. Leray J., Schauder J., Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. 51 (1934), 45–78.
70. Levin A.Yu., Sur un problème polylocal, Nauk. Dokl. Vusch. Chk. 5 (1958), 34–37 (in Russian).
71. Levin A.Yu., Sur certaines estimations d’une fonction différentiable, Dokl. Akad. Nauk SSSR 138 (1961), 37–38 (in Russian).
72. Lyapunov A., The general problem of stability of motion, Kharkov Math. Soc., Kharkov, 1892 (in Russian).
73. Mawhin J., Tshinanga S.B., Extension des théorèmes d’existence de Schmitt et Smith aux perturbations d’opérateurs de Fredholm d’indice zéro, Acad. Roy. Belg. Bull. Cl. Sci. (5) 75 (1989), 322–340.
74. Mawhin J., Ward Jr J.R., Nonresonance and existence for nonlinear elliptic boundary value problems, Nonlinear Anal. 5 (1981), 677–684.
75. Mawhin J., Ward Jr J.R., Nonuniform nonresonance conditions at the two first eigenvalues for periodic solutions of forced Liénard and Duffing equations, Rocky Mountain J. Math. 12 (1982), 643–654.
76. Melentsova Yu.A., Best constants in integral inequalities for functions with n zeros, Ural. Gos. Univ. Mat. Zap. 10, Issled. Sovremen. Mat. Anal. 217 (1977), 93–107 (in Russian).
77. Melentsova Yu.A., A non-improvable estimate of the interval of disconjugacy for a linear differential equation with coefficients that are bounded in $L^r$, Differentsial’nye Uravnenija 13 (1977), 1776–1786, 1906–1907 (in Russian). English translation: Differential equations 13 (1977), 1236–1244.
78. Melentsova Yu.A., Mil’shtein G.N., Optimal estimation of the interval of solvability of a multipoint boundary value problem, Differentsial’nye Uravnenija 10 (1974), 1630–1641, 1732 (in Russian). English translation: Differential Equations 10 (1974), 1257–1265.
79. Melentsova Yu.A., Mil’shtein G.N., On the optimal estimate of the interval of nonoscillation for linear differential equations with bounded coefficients, Differentsial’nye Uravneniya 17 (1981), 2160–2175, 2299 (in Russian). English translation: Differential Equations 17 (1981), 1368–1379.
80. Mikusiński J.G., Sur un problème d’interpolation pour les intégrales des équations différentielles linéaires, Ann. Soc. Polon. Math. 19 (1946), 165–205.
81. Myjak J., Boundary value problems for nonlinear differential and difference equations of the second order, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 15 (1971), 113–123.
82. Niccoletti O., Sulle condizioni initiali che determinano gli integrali delle equazioni differenziali ordinarie, Atti R. Accad. Sci. Torino (1898), 746–759.
83. Picard E., Sur l’application des méthodes d’approximations successives à l’étude de certaines équations différentielles ordinaires, J. Math. Pures Appl. 9 (1893), 217–271.
84. Picard E., Traité d’analyse, Gauthier-Villars, Paris, 1896.
85. Piccinini L., Stampacchia G., Vidossich G. Equazioni differenziali ordinarie in $\mathbb{R}^n$, Liguori Editore, Napoli, 1978. English translation: Ordinary Differential Equations in ℝ^n. Problems and Methods, Springer, New York, 1984.
86. Reissig R., Sansone G., Conti R., Non-linear Differential Equations of Higher Order, Noordhoff, Leyden, 1974.
87. Rouche N., Mawhin J., Équations différentielles ordinaires, vol. 2, Masson, Paris, 1973. English translation: Ordinary Differential Equations. Stability and Periodic Solutions, Pitman, Boston, 1980.
88. Schauder J., Zur Theorie stetiger Abbildungen in Funktionalräumen, Math. Z. 26 (1927), 47–65.
89. Schauder J., Bemerkungen zu meiner Arbeit ‘Zur Theorie stetiger Abbildungen in Funktionalräumen’, Math. Z. 26 (1927), 417–431.
90. Schauder J., Der Fixpunktsatz in Funktionalräumen, Studia Math. 2 (1930), 171–180.
91. Schmitt K., Smith H.L., On eigenvalue problems for nondifferentiable mappings, J. Differential Equations 33 (1979), 294–319.
92. Schmitt K., Smith H.L., Fredholm alternatives for nonlinear differential equations, Rocky Mountain J. Math. 12 (1982), 817–841.
93. Švec M., On the uniqueness problem for integrals of a system of linear differential equations, Matem.-fiz. Sbornik Slov. Akad. Ved. 2 (1952), 3–22 (in Slovakian).
94. Szafraniec F.H., Note on the existence of periodic solutions of a second order differential equation, Ann. Polon. Math. 21 (1969), 305–307.
95. Traple J., On a boundary value problem for systems of ordinary differential equations of second order, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 15 (1971), 159–168.
96. Traple J., A continuous and a discrete boundary value problem, Zeszyty Nauk. Uniw. Jagiello. Prace Mat. 19 (1977), 39–53.
97. Volpato M., Sopra un problema di valori al contorno per l’equazione differenziale $y^{( n )} = f( x,y,...,y^{(n-1)}$, Rend. Sem. Mat. Univ. Padova 22 (1953), 334–349.
98. Volpato M., Sull’ esistenza di soluzioni periodiche per equazioni differenziali ordinarie del secondo ordine, Rend. Sem. Mat. Univ. Padova 25 (1956), 371–385.
99. Wypich M., On the existence of a solution of nonlinear periodic boundary value problems, Univ. Iagel. Acta Math. 28 (1991), 289–295.
100. Zwirner G., Su un problema di valori al contorno per equazioni differenziali ordinarie dell ordine n, Rend. Sem. Mat. Univ. Padova 12 (1941), 114–122.
MawhinJ. (2013). Functional analysis and nonlinear boundary value problems: the legacy of Andrzej Lasota. Annales Mathematicae Silesianae, 27, 7-38. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13998
Jean Mawhin
jean.mawhin@uclouvain.be
Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Belgium Belgium
Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Belgium Belgium
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