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Sturm-Liouville and focal higher order BVPs with singularities in phase variables. (English) Zbl 1069.34029

This paper is devoted to the existence of solutions for two types of singular boundary value problems. The Sturm-Liouville boundary value problem \[ -x^n(t)=f\bigl(t,x(t),\dots ,x^{(n-1)}(t)\bigr),\quad n\geq2, \]
\[ x^{(i)}(0)=0,\quad 0\leq i\leq n-3,\quad \alpha x^{(n-2)}(0)-\beta x^{(n-1)}(0)=0,\quad \gamma x^{(n-2)}(T)+\delta x^{(n-1)}(T)=0, \] where \(T\) is a positive constant, \(\alpha,\gamma>0,\;\beta,\delta\geq0\) and \(f\) satisfies the local Caratheodory conditions on \([0,T]\times D\) with \(D=\mathbb{R}_+^{n-1}\times \mathbb{R}_0\), \(\mathbb{R}_+=(0,\infty)\) and \(\mathbb{R}_0=\mathbb{R}\setminus\{0\}\), and the \((p,n-p)\) right focal BVP \[ (-1)^{n-p}x^{(n)}(t)=f\bigl(t,x(t),\dots .x^{(n-1)}(t)\bigr),\quad n\geq2, \]
\[ x^{(i)}(0)=0,\;0\leq i\leq p-1,\quad x^{(i)}(T)=0,\quad p\leq i\leq n-1, \] where \(p\in \mathbb{N}\) is fixed, \(1\leq p\leq n-1\) and \(f\) satisfies the Carathéodory conditions on \([0,T]\times X,\;\) where \(X\) is a proper subset of \(\mathbb{R}^n\). In both problems the function \(f(t,x_0,\dots ,x_{n-1})\) may be singular at the points \(x_i=0,\;0\leq i\leq n-1,\) of all its phase variables \(x_0,\dots ,x_{n-1}.\) To prove the existence of solutions for each of the above problems, the authors construct a suitable sequence of regular problems. The construction of regular BVPs is based on a priori bounds. The proofs of the existence results for the auxiliary regular BVPs are based on the nonlinear Fredholm alternative.

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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