Multiple solutions of problems with nonlinear first-order differential operators. (English) Zbl 1334.34052
The paper gives existence and multiplicity results for first order differential equations of the form
\[
(\phi(u(t)))'=f(t,u(t))\text{ a.e. }t\in[0,T]
\]
subject to the periodic condition \(u\left( 0\right) =u\left( T\right) \) or to the initial value condition \(u\left( 0\right) =r\). For these equations, \(\phi :]a,b[\rightarrow \mathbb{R}\) is an increasing homeomorphism and \(f:\left[ 0,T\right] \times \mathbb{R} \rightarrow \mathbb{R}\) is a Carathéodory function. The results are based on techniques of upper and lower solutions and fixed point index theory.
Reviewer: Radu Precup (Cluj-Napoca)
MSC:
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
Keywords:
first-order differential equation; nonlinear differential operator; multiple solutions; upper and lower solutions; fixed point index; initial value problem; periodic boundary value problemReferences:
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