This paper is mainly concerned with the exact controllability for a class of impulsive ψ-Caputo fractional evolution equations with nonlocal conditions. First, by generalized Laplace transforms, a mild solution for considered problems is introduced. Next, by the Mönch fixed point theorem, the exact controllability result for the considered systems is obtained under some suitable assumptions. Finally, an example is given to support the validity of the main results.

This paper is concerned with a class of implicit-type coupled system with integral boundary conditions involving Caputo fractional derivatives. First, the existence result of solutions for the considered system is obtained by means of topological degree theory. Next, Ulam–Hyers stability and generalized Ulam–Hyers stability are studied under some suitable assumptions. Finally, one example is worked out to illustrate the main results.

This paper is concerned with multiple solutions for a class of nonlinear fourth-order boundary value problems with parameters. By constructing a special cone and applying fixed point index theory, the multiple solutions for the considered systems are obtained under some suitable assumptions. The main feature of obtained solutions (u(t),v(t)) is that the solution u(t) is positive, and the other solution v(t) may change sign. Finally, two examples with continuous function f1 being positive and f2 being semipositone are worked out to illustrate the main results.

The aim of this paper is to establish some results about the existence of multiple solutions for the following singular semipositone boundary value problem of fourth-order differential systems with parameters: $$ \textstyle\begin{cases} u^{(4)}(t)+\beta _{1}u''(t)-\alpha _{1}u(t)=f_{1}(t,u(t),v(t)),\quad 0< t< 1; \\ v^{(4)}(t)+\beta _{2}v''(t)-\alpha _{2}v(t)=f_{2}(t,u(t),v(t)),\quad 0< t< 1; \\ u(0)=u(1)=u''(0)=u''(1)=0; \\ v(0)=v(1)=v''(0)=v''(1)=0, \end{cases} $$ { u ( 4 ) ( t ) + β 1 u ″ ( t ) − α 1 u ( t ) = f 1 ( t , u ( t ) , v ( t ) ) , 0 < t < 1 ; v ( 4 ) ( t ) + β 2 v ″ ( t ) − α 2 v ( t ) = f 2 ( t , u ( t ) , v ( t ) ) , 0 < t < 1 ; u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 ; v ( 0 ) = v ( 1 ) = v ″ ( 0 ) = v ″ ( 1 ) = 0 , where $f_{1},f_{2}\in C[(0,1)\times \mathbb{R}^{+}_{0}\times \mathbb{R}, \mathbb{R}]$ f 1 , f 2 ∈ C [ ( 0 , 1 ) × R 0 + × R , R ] , $\mathbb{R}_{0}^{+}=(0,+\infty )$ R 0 + = ( 0 , + ∞ ) . By constructing a special cone and applying fixed point index theory, some new existence results of multiple solutions for the considered system are obtained under some suitable assumptions. Finally, an example is worked out to illustrate the main results.