Singular Antiperiodic Boundary Value Problem with Given Maximal Values for Solutions

Singular Antiperiodic Boundary Value Problem with Given Maximal Values for Solutions

Peer-reviewed article not indexed in WoS or Scopus

The singular boundary value problem $(\phi(x'))' + \mu f(t,x,x')=0$, $x(0)+x(T)=0$, $x'(0)+x'(T)=0$, $\max\{x(t): 0 \le t \le T\}=A$ depending on the parameter $\mu$ is considered. Here the function $f$ satisfies local Carath\'eodory conditions on $[0,T] \times (\R\setminus \{0\})^2$ and $f$ may be singular at the zero value of its phase variables. The paper presents conditions which guarantee that for any $A>0$ there exists $\mu_A >0$ such that the above problem with $\mu=\mu_A$ has a solution. The proofs are based on regularization and sequential techniques and use the Leray-Schauder degree.

The singular boundary value problem $(\phi(x'))' + \mu f(t,x,x')=0$, $x(0)+x(T)=0$, $x'(0)+x'(T)=0$, $\max\{x(t): 0 \le t \le T\}=A$ depending on the parameter $\mu$ is considered. Here the function $f$ satisfies local Carath\'eodory conditions on $[0,T] \times (\R\setminus \{0\})^2$ and $f$ may be singular at the zero value of its phase variables. The paper presents conditions which guarantee that for any $A>0$ there exists $\mu_A >0$ such that the above problem with $\mu=\mu_A$ has a solution. The proofs are based on regularization and sequential techniques and use the Leray-Schauder degree.

Singular boundary value problem, antiperiodic boundary conditions, dependence on a parameter, $\phi$-Laplacian, existence, Leray-Schauder degree.

Singular boundary value problem, antiperiodic boundary conditions, dependence on a parameter, $\phi$-Laplacian, existence, Leray-Schauder degree.

PŘIBYL, O.; STANĚK, S.

01.06.2007

Functional Differential Equations

0793-1786

Functional Differential Equations

14

2/4

State of Israel

103

114

12

http://hdl.handle.net/