Functional fractional boundary value problems with singular ϕϕ-Laplacian

This paper discusses the existence of solutions of the fractional differential equations cDμ(ϕ(cDαu))=Fu,cDμ(ϕ(cDαu))=f(t,u,cDνu) satisfying the boundary conditions u(0)=A(u),u(T)=B(u). Here μ,α∈(0,1],ν∈(0,α],cD is the Caputo fractional derivative, ϕ∈C(-a,a)ϕ∈C(-a,a)(a>0(a>0), F   is a continuous operator, A,B are bounded and continuous functionals and f∈C([0,T]×R2)f∈C([0,T]×R2). The existence results are proved by the Leray–Schauder degree theory.