In this paper, we use fixed‐point index to study the existence of positive solutions for a system of Hadamard fractional integral boundary value problems involving nonnegative nonlinearities. By virtue of integral‐type Jensen inequalities, some appropriate concave and convex functions are used to depict the coupling behaviors for our nonlinearities f_i(i = 1, 2).

In this paper, we introduce new sequential fractional differential equations with mixed-type boundary conditionsCDq+kCDq−1ut=ft,ut,CDq−1ut,t∈0,1,α1u0+β1u1+γ1Iruη=ε1,η∈0,1,α2u′0+β2u′1+γ2Iru′η=ε2,whereq∈1,2is a real number,k,r>0,αi,βi,γi,εi∈ℝ,i=1,2,CDqis the Caputo fractional derivative, and the boundary conditions include antiperiodic and Riemann-Liouville fractional integral boundary value cases. Our approach to treat the above problem is based upon standard tools of fixed point theory and some new inequalities of norm form. Some existence results are obtained and well illustrated through the aid of examples.

In this paper, we present some theorems on impulsive periodic boundary value problems with fractional derivative dependence. In particular, we discuss the existence of solutions of a class of fractional-order impulsive periodic boundary values with nonlinear terms and impulsive terms satisfying certain growth conditions. Three examples are provided to illustrate our results.

We study the existence of multiple positive solutions for nth-order multipoint boundary value problem. u n t a t f u t 0, t ∈ 0, 1 , u j-1 0 0 j 1, 2,. .. , n-1 , u 1 m i 1 α i u η i , where n ≥ 2, 0 < η 1 < η 2 < • • • < η m < 1, α i > 0, i 1, 2,. .. , m. We obtained the existence of multiple positive solutions by applying the fixed point theorems of cone expansion and compression of norm type and Leggett-Williams fixed-point theorem. The results obtained in this paper are different from those in the literature.