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Solvability of some fourth (and higher) order singular boundary value problems. (English) Zbl 0795.34018

This paper deals primarily with solutions of (1) ${y}^{\left(4\right)}=f\left(t,y,{y}^{\text{'}\text{'}}\right)$, $0, satisfying boundary conditions of the type (2) $y\left(0\right)=a\ge 0$, ${y}^{\text{'}\text{'}}\left(0\right)=c\le 0$, ${y}^{\text{'}}\left(1\right)=b\ge 0$, ${y}^{\text{'}\text{'}\text{'}}\left(1\right)=0$, (3) $y\left(0\right)=a\ge 0$, ${y}^{\text{'}}\left(0\right)=b\ge 0$, ${y}^{\text{'}\text{'}}\left(1\right)=0$, ${y}^{\text{'}\text{'}\text{'}}\left(1\right)=0$, (4) $y\left(0\right)=a\ge 0$, ${y}^{\text{'}}\left(0\right)=0$, ${y}^{\text{'}}\left(1\right)=0$, ${y}^{\text{'}\text{'}\text{'}}\left(1\right)=0$, or (5) $y\left(0\right)=a\ge 0$, ${y}^{\text{'}\text{'}}\left(0\right)=0$, ${y}^{\text{'}}\left(1\right)=b\ge 0$, ${y}^{\text{'}\text{'}}\left(1\right)=0$, where $f\left(t,{y}_{1},{y}_{2}\right)$ may have singularities at $t=0,1$ and at ${y}_{1}=0$, ${y}_{2}=0$.

The first results concern problem (1), (2), where $f\left(t,{y}_{1},{y}_{2}\right)$ has singularities at ${y}_{1}=0$, not at ${y}_{2}=0$. Growth assumptions are made on $f$ such that one can conclude the existence of a priori bounds (independent of $\lambda$), on solutions of ${y}^{\left(4\right)}=\lambda f\left(t,y,{y}^{\text{'}\text{'}}\right)$, $0, $0\le \lambda \le 1$, satisfying $y\left(0\right)=1/n$, ${y}^{\text{'}\text{'}}\left(0\right)=c\le 0$, ${y}^{\text{'}}\left(1\right)=b\ge 0$, ${y}^{\text{'}\text{'}\text{'}}\left(1\right)=0$. By applying the topological transversality theorem of Granas, one obtains a solution $y\in C\left[0,1\right]\cap {C}^{3}\left(0,1\right)\cap {C}^{4}\left(0,1\right)$ of (1), (2). Later, boundary value problems are treated similarly for ${y}^{\left(4\right)}=\psi \left(t\right)f\left(t,y,{y}^{\text{'}\text{'}}\right)$, where $f\left(t,{y}_{1},{y}_{2}\right)$ has singularities at ${y}_{1}=0$, and $\psi \left(t\right)$ is positive and improper integrable over (0,1).

Boundary value problems for equation (1) with any of the conditions (3), (4), or (5) are dealt with similarly, and then in the last section, results are given for boundary problems for ${y}^{\left(n\right)}=f\left(t,y,{y}^{\text{'}\text{'}}\right)$, $0, where, as above, $f\left(t,{y}_{1},{y}_{2}\right)$ may have singularities at $t=0,1$ and at ${y}_{1}=0$, ${y}_{2}=0$.

MSC:
 34B15 Nonlinear boundary value problems for ODE