Multiple positive solutions for singular problems with mixed boundary data and derivative dependence

The existence of at least two positive solutions is presented for the singular second-order boundary value problems { 1 p ( t ) ( p ( t ) x ′ ( t ) ) ′ + Φ ( t ) f ( t , x ( t ) , p ( t ) x ′ ( t ) ) = 0 , 0 < t < 1 lim t → 0 p ( t ) x ′ ( t ) = 0 , x ( 1 ) = 0 using the fixed point index, where ∫ 0 1 1 p ( r ) d r = + ∞ and f may be singular at x = 0 and p x ′ = 0 . The solutions we construct may be unbounded.