0+ u(t) = f (t, u(t), D β+1 0+ u(t), D β 0+ u(t)), 0 < t < 1, u(0) = u (0) = 0, u(1) = 1 0 u(t) dA(t), where C D α 1-is the left Caputo fractional derivative of order α ∈ (1, 2], and D β 0+ is the right Riemann-Liouville fractional derivative of order β ∈ (0, 1]. The coincidence degree theory is the main theoretical basis to prove the existence of solutions of such problems.

No abstract

This paper deals with the existence and multiplicity of solutions for the integral boundary value problem of fractional differential systems: D0+α1u1t=f1t,u1t,u2t,D0+α2u2t=f2t,u1t,u2t,u10=0, D0+β1u10=0, D0+γ1u11=∫01D0+γ1u1ηdA1η,u20=0, D0+β2u20=0, D0+γ2u21=∫01D0+γ2u2ηdA2η,, where fi:0,1×0,∞×0,∞⟶0,∞ is continuous and αi−2<βi≤2,αi−γi≥1,2<αi≤3,γi≥1i=1,2.D0+α is the standard Riemann–Liouville’s fractional derivative of order α. Our result is based on an extension of the Krasnosel’skiĭ’s fixed-point theorem due to Radu Precup and Jorge Rodriguez-Lopez in 2019. The main results are explained by the help of an example in the end of the article.

In this paper, the existence of extremal solutions for fractional differential equations with integral boundary conditions is obtained by using the monotone iteration technique and the method of upper and lower solutions. The main equations studied are as follows: −D0+αut=ft,ut, t∈0,1,u0=0, u1=∫01utdAt, where D0+α is the standard Riemann–Liouville fractional derivative of order α∈1,2 and At is a positive measure function. Moreover, an example is given to illustrate the main results.

Under a resonance condition involving integral boundary value problems for a second-order nonlinear differential equation in $\mathbb{R}^{n}$ R n , we show its solvability by using the coincidence degree theory of Mawhin and the theory of matrix diagonalization in linear algebra.