The Green's function for \((k,n-k)\) conjugate boundary value problems and its applications (Q5934243)

The authors prove that the Green function of the \((k,n-k)\) conjugate boundary value problem NEWLINE\[NEWLINE(-1)^{n-k}y^{(n)}(x)=f(x); \quad 0<x<1, n\geq 2, 0<k<n,NEWLINE\]NEWLINE NEWLINE\[NEWLINEy^{(i)}(0)=y^{(j)}(1)=0; \quad 0\leq i\leq k-1, \;0\leq j\leq n-k-1,NEWLINE\]NEWLINE is given by the expression NEWLINE\[NEWLINEG(x,s)= \frac{1}{(k-1)!(n-k-1)!}\begin{cases}\int_0^{s(1-x)}t^{n-k-1}(t+x-s)^{k-1} dt; \quad 0\leq s\leq x\leq 1,\\ \int_0^{x(1-s)}t^{k-1}(t+s-x)^{n-k-1} dt; \quad 0\leq x\leq s\leq 1.\end{cases}NEWLINE\]NEWLINE Furthermore, using this expression and under suitable conditions on the function \(p \in L_{loc}^1(0,1)\), \(p \geq 0\), and on the continuous and nonnegative function on \([0, +\infty)\), \(f\), they prove some existence results on the problem NEWLINE\[NEWLINE(-1)^{n-k}y^{(n)}(x)=p(x)f(y); \quad 0\leq x \leq 1,NEWLINE\]NEWLINE NEWLINE\[NEWLINEy^{(i)}(0)=y^{(j)}(1)=0; \quad 0\leq i\leq k-1, \;0\leq j\leq n-k-1.NEWLINE\]