Singular two point boundary value problems for second order differential systems
The authors consider singular two-point boundary value problems \[ A(t,x,x') x''= f(t,x,x'),\quad x(a)- Px'(a)= 0,\quad x(b)+ Qx'(b)= 0,\tag{1} \] with \(x: [a,b]\to \mathbb{R}^n\), \(A\in C([a,b]\times \mathbb{R}^n\times \mathbb{R}^n,\mathbb{R}^{n\times n})\) is symmetric and positive definite on \([a,b)\times \mathbb{R}^n\times \mathbb{R}^n\), with \(\text{rank }A(b,-Qy, y)< n\) for \(y\in \mathbb{R}^n\), \(P\) and \(Q\) are constant \(n\times n\)-matrices, \(f\in C([a, b]\times \mathbb{R}^n\times \mathbb{R}^n,\mathbb{R}^n)\). The authors give some existence results for (1) provided \(f\) fulfils a growth condition and the eigenvalues of \(A\) are bounded below by a continuous function \(\lambda_0\), positive on \([a,b)\), with \(\int^b_a (ds/\lambda_0(s))< +\infty\). In the proofs, the method of a priori estimates, guiding functions and Leray-Schauder topological degree arguments are used. An application of these existence results to some examples of semilinear boundary value problems is shown.
singular two-point boundary value problems
scientific article; zbMATH DE number 1295261
scientific article