Previous |  Up |  Next


functional boundary conditions; functional differential equation; existence; multiplicity; Bihari lemma; homotopy; Leray Schauder degree; Borsuk theorem
A class of functional boundary conditions for the second order functional differential equation $x''(t)=(Fx)(t)$ is introduced. Here $F:C^1(J) \rightarrow L_1(J)$ is a nonlinear continuous unbounded operator. Sufficient conditions for the existence of at least four solutions are given. The proofs are based on the Bihari lemma, the topological method of homotopy, the Leray-Schauder degree and the Borsuk theorem.
[1] Ambrosetti A., Prodi G.: On the inversion of some differentiable mappings with singularities between Banach spaces. Ann. Mat. Pura Appl. 93, 1972, 231–247. MR 0320844 | Zbl 0288.35020
[2] Azbelev N. V., Maksimov V. P., Rakhmatullina L. F.: Introduction to the Theory of Functional Differential Equations. Moscow, Nauka, 1991 (in Russian). MR 1144998 | Zbl 0725.34071
[3] Bihari I.: A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math. Sci. Hungar. 7, 1956, 71–94. MR 0079154 | Zbl 0070.08201
[4] Brüll L., Mawhin J.: Finiteness of the set of solutions of some boundary- value problems for ordinary differential equations. Arch. Math. (Brno) 24, 1988, 163–172. MR 0983234 | Zbl 0678.34023
[5] Brykalov S. A.: Solvability of a nonlinear boundary value problem in a fixed set of functions. Diff. Urav. 27, 1991, 2027–2033 (in Russian). MR 1155041 | Zbl 0788.34070
[6] Brykalov S. A.: Solutions with given maximum and minimum. Diff. Urav. 29, 1993, 938–942 (in Russian). MR 1254551
[7] Brykalov S. A.: A second-order nonlinear problem with two-point and integral boundary conditions. Proceedings of the Georgian Academy of Science, Math. 1, 1993, 273–279. MR 1262564 | Zbl 0798.34021
[8] Deimling K.: Nonlinear Functional Analysis. Springer, Berlin Heidelberg 1985. MR 0787404 | Zbl 0559.47040
[9] Ermens B., Mawhin J.: Higher order nonlinear boundary value problems with finitely many solutions. Séminaire Mathématique, Université de Louvain, No. 139, 1988, 1–14, (preprint). MR 1065638
[10] Filatov A. N., Sharova L. V.: Integral Inequalities and the Theory of Nonlinear Oscillations. Nauka, Moscow 1976 (in Russian). MR 0492576 | Zbl 0463.34001
[11] Mawhin J.: Topological Degree Method in Nonlinear Boundary Value Problems. CMBS Reg. Conf. in Math., No. 40, AMS, Providence, 1979. MR 0525202
[12] Mawhin J., Willem M.: Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations. J. Differential Equations 52, 1984, 264–287. MR 0741271 | Zbl 0557.34036
[13] Nkashama M. N., Santanilla J.: Existence of multiple solutions for some nonlinear boundary value problems. J. Differential Equations 84, 1990, 148–164. MR 1042663 | Zbl 0693.34011
[14] Rachůnková I., Staněk S.: Topological degree method in functional boundary value problems. Nonlinear Analysis 27, 1996, 153–166.
[15] Rachůnková I.: On the existence of two solutions of the periodic problem for the ordinary second-order differential equation. Nonlinear Analysis 22, 1994, 1315–1322. Zbl 0808.34023
[16] Staněk S.: Existence of multiple solutions for some functional boundary value problems. Arch. Math. (Brno) 28, 1992, 57–65. MR 1201866 | Zbl 0782.34074
[17] Staněk S.: Multiple solutions for some functional boundary value problems. Nonlinear Analysis, to appear. MR 1610598 | Zbl 0945.34049
[18] Staněk S.: Multiplicity results for second order nonlinear problems with maximum and minimum. Math. Nachr., to appear. MR 1626344 | Zbl 0920.34058
[19] Šeda V.: Fredholm mappings and the generalized boundary value problem. Differential and Integral Equations 8, 1995, 19–40. MR 1296108
Partner of
EuDML logo