First extremal point comparison for a fractional boundary value problem with a fractional boundary condition

Abstract

Let n ≥ 2 n \geq 2 be a natural number, and let n − 1 > α ≤ n n-1>\alpha \le n and 0 > γ ≤ α − 1 0>\gamma \le \alpha -1 be real numbers. Let β > 0 \beta >0 and b ∈ ( 0 , β ] b\in (0,\beta ] . We compare first extremal points of the differential equations D 0 + α u + p ( t ) u = 0 D_{0+}^\alpha u+p(t)u=0 , D 0 + α u + q ( t ) u = 0 D_{0+}^\alpha u+q(t)u=0 , t ∈ ( 0 , β ) t\in (0,\beta ) , each of which satisfies the boundary conditions u ( i ) ( 0 ) = 0 u^{(i)}(0)=0 , i = 0 , 1 , … , n − 2 i=0,1,\dots ,n-2 , D 0 + γ u ( b ) = 0 \quad D_{0^+}^\gamma u(b)=0 . While it is assumed that q q is nonnegative, no sign restrictions are put on p p . The fact that the associated Green’s function G ( b ; t , s ) G(b;t,s) is nonnegative and increasing with respect to b b plays an important role in the analysis.