Staněk, Svatoslav An application of the Leray-Schauder degree theory to boundary value problems of third and fourth order differential equations depending on a parameter. (English) Zbl 0858.34020 Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 34, 155-166 (1995). Sufficient conditions for the existence and uniqueness of solutions of the boundary value problem (1) \(x'''=f(t,x,x',x'',\lambda)\), (2) \(x(0)=x(1)=0\), \(x'(0)-x'(1)=0\), \(x''(0)-x''(1)=0\) are established. The proof of the existence theorem is based on the Leray-Schauder degree theory. The results are applied to fourth-order one-parameter functional differential equations using the quasilinearization technique and the Schauder fixed point theorem. Reviewer: W.Šeda (Bratislava) MSC: 34B15 Nonlinear boundary value problems for ordinary differential equations 34K10 Boundary value problems for functional-differential equations Keywords:existence; uniqueness; boundary value problem; Leray-Schauder degree theory; fourth-order one-parameter functional differential equations; quasilinearization technique; Schauder fixed point theorem × Cite Format Result Cite Review PDF References: [1] Fabry, Ch., Habets P.: The Picard boundary value problem for nonlinear second order vector differential equations. J. Differential Equations 42 (1981), 186-198. · Zbl 0439.34018 · doi:10.1016/0022-0396(81)90025-5 [2] Hartman P.: Ordinary Differential Equations. Wiley-Interscience, New York, 1964. · Zbl 0125.32102 [3] Pachpatte B. G.: On certain boundary value problem for third order differential equations. An. st. Univ. Iasi, f. 1, s. Ia, Mat. (1986), 61-74. · Zbl 0619.34024 [4] Staněk S.: Three-point boundary value problem for nonlinear third-order differential equations with parameter. Acta Univ. Palacki. Olomuc., Fac. rer. nat. 100, Math. 30 (1991), 61-74. · Zbl 0752.34019 [5] Staněk S.: On a class of functional boundary value problems for nonlinear third-order functional differential equations depending on the parameter. Arch. Math. 62 (1994), 462-469. · Zbl 0801.34065 · doi:10.1007/BF01196438 [6] Staněk S.: Leray-Schauder degree method in functional boundary value problems depending on the parameter. Math. Nach. 164 (1993), 333-344. · Zbl 0805.34053 · doi:10.1002/mana.19931640124 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.