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 \(f: [0,1]\times (0,\infty)\times(- \infty,0]\to \mathbb{R}\) 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 \(f(t,y,z)\).
\[ \lim_{t\to 0^+} p(t) y'(t)= y(1)= 0\tag{3} \] 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 \(f: [0,1]\times (0,\infty)\times(- \infty,0]\to \mathbb{R}\) 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 \(f(t,y,z)\).
Reviewer: A.I.Kolosov (Khar’kov)
MSC:
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
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