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Second and higher order systems of boundary value problems. (English) Zbl 0744.34024
The author studies some linear homogeneous boundary conditions for second and higher order systems of ordinary differential equations. The main proofs are carried out by means of topological techniques. The main result concerns a Dirichlet boundary value problem for a second order differential equation which can be singular at the end points. A careful use of some inequalities yields the possibility a Nonlinear Alternative result due to A. Granas. The results are then extended to a boundary value problem of the type $$y(0)=y'(1)=0$$ and then to higher order equations with similar boundary conditions.
Reviewer: P.Zezza (Firenze)

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations
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##### References:
 [1] Aftabizadeh, A.R, Existence and uniqueness theorems for fourth order boundary value problems, J. math. anal. appl., 116, 415-426, (1986) · Zbl 0634.34009 [2] Agarwal, R.P, Some new results on two point boundary value problems for higher order differential equations, Funkcial. ekvac., 29, 197-212, (1986) · Zbl 0623.34019 [3] Agarwal, R.P, Existence, uniqueness and iteration methods for third order boundary value problems, J. comput. appl. math., 17, 271-289, (1987) · Zbl 0617.34008 [4] Bobisud, L.E; O’Regan, D; Royalty, W.D, Singular boundary value problems, Appl. anal., 23, 233-243, (1986) · Zbl 0584.34012 [5] Eloe, P.W; Henderson, J, Nonlinear boundary value problems and a priori bounds on solutions, SIAM J. math. anal., 15, 642-647, (1984) · Zbl 0547.34015 [6] Fabry, Ch; Habels, P, The Picard boundary value problem for nonlinear second order vector differential equations, J. differential equations, 42, 186-198, (1981) · Zbl 0439.34018 [7] Granas, A; Guenther, R.B; Lee, J.W, Nonlinear boundary value problems for ordinary differential equations, Dissertations math. (rozprawy mat.), 244, (1985) · Zbl 0476.34017 [8] Granas, A; Guenther, R.B; Lee, J.W, Existence principles for classical and caratheodory solutions of nonlinear systems and applications, (), 353-364 · Zbl 0717.34022 [9] Granas, A; Guenther, R.B; Lee, J.W, Nonlinear boundary value problems for some classes of ordinary differential equations, Rocky mountain J. math., 10, 35-58, (1980) · Zbl 0476.34017 [10] Gupta, C.P, Existence and uniqueness theorems for the bending of an elastic beam equation, Appl. anal., 26, 289-304, (1988) · Zbl 0611.34015 [11] Hart, V.G; Holland, F, Existence and uniqueness problems for an elastic annular plate at large transverse deflections, (), 95-106, No. 2 · Zbl 0592.73073 [12] Hartman, P, On boundary value problems for systems of ordinary nonlinear second order differential equations, Trans. amer. math. soc., 96, 493-509, (1960) · Zbl 0098.06101 [13] Hartman, P, On two point boundary value problems for nonlinear second order systems, SIAM J. math. anal., 5, 172-177, (1974) · Zbl 0297.34015 [14] Henderson, J, Best interval lengths for boundary value problems for third order Lipschitz equations, SIAM J. math. anal., 18, 293-305, (1987) · Zbl 0668.34017 [15] Jackson, L.K, Existence and uniqueness of solutions of boundary value problems for Lipschitz equations, J. differential equations, 32, 76-90, (1979) · Zbl 0407.34018 [16] Lee, J.W; O’Regan, D, Existence of solutions to some initial value, two point and multipoint boundary value problems with discontinuous nonlinearities, Appl. anal., 33, 57-77, (1989) · Zbl 0643.34018 [17] Lee, J.W; O’Regan, D, Nonlinear boundary value problems in Hilbert spaces, J. math. anal. appl., 137, 59-69, (1989) · Zbl 0672.34056 [18] {\scJ. W. Lee and D. O’Regan}, Boundary value problems for nonlinear fourth order equations with applications to nonlinear beams, to appear. [19] O’Regan, D, Topological transversality, applications to third order boundary value problems, SIAM J. math. anal., 18, 630-641, (1987) · Zbl 0628.34017 [20] Thompson, B.H, Existence and uniqueness problem for an elastic annular plate, (), 61-70, No. 1 · Zbl 0683.73026 [21] Dym, H; McKean, H.P, Fourier series and integrals, (1972), Academic Press New York · Zbl 0242.42001 [22] Hardy, G.H; Littlewood, J.E; Polya, G, Inequalities, (1952), Cambridge University Press London/New York · Zbl 0047.05302 [23] Dugundji, J; Granas, A, Fixed point theory, () · Zbl 1025.47002
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