Eigenvalues of a general class of boundary value problem with derivative-dependent nonlinearity

We consider a general class of boundary value problem (BVP) comprising the differential equation y ( m ) ( t ) = λ F t , y ( t ) , y ' ( t ) , y ? ( t ) , ? , y ( q ) ( t ) , t ? ( 0 , 1 ) where 1 ≤ q ≤ m - 1 and λ 0 , together with multi-point boundary conditions y ( 0 ) = y ' ( 0 ) = y ? ( 0 ) = ? = y ( q - 1 ) ( 0 ) = 0 , A i y ( q ) ( t j ) , y ( q + 1 ) ( t j ) , ? , y ( m - 1 ) ( t j ) ; 0 ≤ j ≤ r = 0 , 1 ≤ i ≤ m - q where 0 = t 0 < t 1 < ? < t r = 1 . Particular cases of this general BVP include the well known Abel-Gontscharoff, focal, ( m , p ) , Sturm-Liouville and complementary Lidstone BVPs. It is noted that BVPs with derivative-dependent nonlinear terms are less investigated in the literature due to technical difficulty. In this paper, a new technique is developed to characterize the eigenvalues λ so that the BVP has a positive solution. Explicit eigenvalue intervals are also established. We include several well known examples in the literature to illustrate the results obtained.