Uniqueness criterion for solutions of nonlocal problems on a finite interval for abstract singular equations. (English. Russian original) Zbl 1500.34053
Math. Notes 111, No. 1, 20-32 (2022); translation from Mat. Zametki 111, No. 1, 24-39 (2022).
Let \(E\) be a complex Banach space and let \(A,B\) be linear closed operators on \(E\) whose domains are not necessary dense. Consider the equation
\[
B(u''(t)+\frac{k}{t}u'(t))=Au(t),\quad 0<t<1.
\]
The following cases are considered
Applications to the equation \(t^\gamma v''+bt^{\gamma-1}v'=Av\) are given.
- ●
- \(k\geq0\) with Neumann boundary condition \(u'(0)=0\) and non-local condition
\(\int_0^1t^k(1-t^2)^{\alpha-1}u(t)dt=0\) (\(\alpha>0\)) - ●
- \(k<1\) with Dirichlet boundary condition \(u(0)=0\) and non-local condition
\(\int_0^1t(1-t^2)^{\beta-1}u(t)dt=0\) (\(\beta>0\)) - ●
- \(k\geq0\) with Neumann boundary condition \(u'(0)=0\) and non-local condition
\(a\int_0^1t^ku(t)dt+bu'(1)=0\) (\(a,b\ne0\)) - ●
- \(k<1\) with Dirichlet boundary condition \(u(0)=0\) and non-local condition
\(a\int_0^1t^ku(t)dt+b\lim_{t\to1}(t^{k-1}u(t))'=0\)
Applications to the equation \(t^\gamma v''+bt^{\gamma-1}v'=Av\) are given.
Reviewer: Nikita V. Artamonov (Moskva)
MSC:
| 34G10 | Linear differential equations in abstract spaces |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 34B16 | Singular nonlinear boundary value problems for ordinary differential equations |
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