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Periodic problem involving quasilinear differential operator and weak singularity. (English) Zbl 1146.34016

The paper deals with the periodic problem \[ (\phi_p(u'))'=f(t,u),\;\;u(0)=u(T),\;u'(0)=u'(T), \] where \(T\in(0,\infty),\;\phi_p(y)=| y| ^{p-2}y,\;p>1,\;f(t,x)\) satisfies the Carathéodory conditions on \([0,T]\times(0,\infty)\) and may be singular at \(x=0\). It is assumed that there are constants \(A,B,r\) and measurable and Lebesgue integrable on \([0,T]\) functions \(\mu,\beta:[0,T]\to\mathbb R\) such that \(B>A\geq r>0\), \(\mu(t)\geq0\) a.e. on \([0,T]\),
\[ \overline\mu=\tfrac1T \int_0^T\mu(s)\, ds>0, \quad \overline\beta\leq0\text{ (or }\beta<0\text{ if }p\in(1,2)), \]
\[ f(t,x)\leq\beta(t)\;\text{for a.e.}\;t\in[0,T]\;\text{and}\;x\in[A,B], \]
\[ f(t,x)+\mu(t)\phi_p(x-r)\geq0\;\text{for a.e.}\;t\in[0,T]\;\text{and}\;x\in[r,B], \]
and \(B-A\geq {T\over{2}}\phi_p^{-1}(| | m| | _1),\) where \(m(t)>\max\{\sup\{f(t,x):x\in[r,A]\},\beta(t),0\}\) for a.e. \(t\in[0,T].\) It is proved under these assumptions that the considered problem has a solution \(u:[0,T]\to\mathbb R\) such that \(r\leq u\leq B\) on \([0,T]\) and \(| | u'| | _\infty<\phi_p^{-1}(| | m| | _1).\) Lower and upper solutions technique and a new antimaximum principle are used.

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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