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boundary value problem; $p$-Laplacian; half-linear equation; positive solution; uniqueness; decaying solution; principal solution
We investigate two boundary value problems for the second order differential equation with $p$-Laplacian \[ (a(t)\Phi _{p}(x'))'=b(t)F(x), \quad t\in I=[0,\infty ), \] where $a$, $b$ are continuous positive functions on $I$. We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: \[ {\rm i)}\ x(0)=c>0, \ \lim _{t\rightarrow \infty }x(t)=0; \quad {\rm ii)}\ x'(0)=d<0, \ \lim _{t\rightarrow \infty }x(t)=0. \]
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