On a class of functional boundary value problems for nonlinear third- order functional differential equations depending on the parameter. (English) Zbl 0801.34065

Staněk, Svatoslav

Arch. Math. 62, No. 5, 462-469 (1994).

Under some conditions the functional differential equation \(x'''= f(t,x_ t,x',x'',\lambda)\) depending on the parameter \(\lambda\) has a solution \(x\in C([-h, 2])\) with \(x\bigl|_{[0,2]}\in C^ 3([0,2])\) satisfying the boundary conditions \(\alpha(x)= A\), \(x''(0)= B\), \(x'(1)= C\), \(x'(2)= D\) and the initial condition \(x_ 0= \varphi+ x(0)\) for \(\lambda= \lambda_ 0\), where \(\alpha: C([0,2])\to \mathbb{R}\) is a continuous increasing functional, \(\varphi\in C([-h, 0])\), \(\varphi(0)= 0\), \(h>0\), \(A\), \(B\), \(C\), \(D\) are reals. The result is based on the Schauder nonlinearization technique, the Schauder fixed point theorem and a surjectivity result in \(\mathbb{R}^ n\).

Reviewer: W.Šeda (Bratislava)

MSC:

34K10

Boundary value problems for functional-differential equations

Keywords:

boundary value problem; functional differential equation; parameter; Schauder nonlinearization technique; Schauder fixed point theorem; surjectivity

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