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Nonlinear superlinear singular and nonsingular second order boundary value problems. (English) Zbl 0902.34015
In the study of nonlinear phenomena many mathematical models give rise to the differential equation $${1\over p} (py')'+ q f(t,y,py')= 0,\quad 0< t<1\tag 1$$ subject to the boundary conditions $$y(0)= y(1)= 0,\tag 2$$ $$\lim_{t\to 0^+} p(t) y'(t)= y(1)= 0\tag 3$$ with $p\in C[0,1]\cap C^1(0,1)$ and $p> 0$ on $(0,1)$, $q\in C(0,1)$ with $q>0$ on $(0,1)$ and $$\int^1_0 p(x)q(x)dx< \infty,\quad\int^1_0{1\over p(s)} \int^s_0 p(x)q(x)dx ds< \infty,$$ and $f: [0,1]\times (0,\infty)\times(- \infty,0]\to \bbfR$ is continuous. The authors prove the existence of a solution $y(t)\in C[0,1]\cap C^2(0,1)$ with $y> 0$ on $(0,1)$ to the problems (1), (2) and (1), (3) if there are some supplementary assumptions on $p(t)$, $q(t)$, and $f(t,y,z)$.

34B15Nonlinear boundary value problems for ODE
Full Text: DOI
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