Aftabizadeh, A. R. Existence and uniqueness theorems for fourth-order boundary value problems. (English) Zbl 0634.34009 J. Math. Anal. Appl. 116, 415-426 (1986). The differential equation (1) \(y^{(IV)}=f(x,y,y'')\) is considered under the following types of boundary conditions: (2) \(y(0)=y_ 0\), \(y(1)=y_ 1\), \(y''(0)=\bar y_ 0\), \(y''(1)=\bar y_ 1\); (3) \(y(0)=y_ 0\), \(y(1)=y_ 1\), \(y''(0)=\bar y_ 0\), \(y'''(1)=\bar y_ 1\); (4) \(y(0)=y(1)=y''(0)=y''(1)=0\) or (5) \(y(0)=y(1)=0\); \(y'''(0)-hy''(0)=0\), \(y'''(1)+ky''(1)=0\), with \(h, k\geq 0\), \(h+k>0\). Existence theorems for all of the boundary value problems (1-2)–(1-5) are obtained by application of Schauder’s fixed point theorem under continuity and boundedness hypotheses on \(f\) and its partial derivative \(f_ 3\). A uniqueness theorem is obtained for the problem (1-4) under the additional assumption of a bound on the partial derivative \(f_ 2\). Further uniqueness results for the problem (1-2)–(1-5) are reduced to uniqueness questions for solutions of corresponding second order boundary value problems for the integrodifferential equation \[ u''=f(x,y_ 0+x(y_ 1- y_ 0)+\int^{1}_{0}G(x,t)u(t)\,dt,u) \] obtained by setting \(y''=u\), with \(G(x,t)\) the Green’s function for the problem \[ u''=0,\quad u(0)=u(1)=0. \] Boundary conditions of the form (2)–(5) have been less extensively studied than the familiar conjugate or focal-type problems. R. A. Usmani [Proc. Am. Math. Soc. 77, 329–335 (1979; Zbl 0424.34019)] has studied a problem of the form (1-2) in which the equation is linear and independent of \(y''\). Reviewer: L.J.Grimm Cited in 1 ReviewCited in 151 Documents MSC: 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:two-point boundary value problems; Schauder’s fixed point theorem; Green’s function Citations:Zbl 0424.34019 PDF BibTeX XML Cite \textit{A. R. Aftabizadeh}, J. Math. Anal. Appl. 116, 415--426 (1986; Zbl 0634.34009) Full Text: DOI OpenURL References: [1] Bebernes, J.W.; Gaines, R., Dependence on boundary data and a generalized boundary-value problem, J. differential equations, 4, 359-368, (1968) · Zbl 0169.10602 [2] Corduneanu, C., Sopra I problemi ai limiti per alcuni sistemi di equazioni differenziali non lineari, Rend. acad. napoli, 4, 98-106, (1958) · Zbl 0091.26201 [3] Reiss, E.L.; Callegari, A.J.; Ahluwalia, D.S., Ordinary differential equations with applications, (1976), Holt, Rinehart & Winston Berlin/New York/Heidelberg · Zbl 0334.34002 [4] Usmani, R.A., A uniqueness theorem for a boundary value problem, (), 329-335, 3 · Zbl 0424.34019 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.