Positive Solutions for a Second-Order Nonlinear Coupled System with Derivative Dependence Subject to Coupled Stieltjes Integral Boundary Conditions

Abstract

In this paper, we discuss the following second-order coupled differential system with coupled integral boundary value conditions and nonlinearities depending on the first derivatives:

$$\begin{aligned} \left\{ \begin{array}{l}{-u^{\prime \prime }(t)=f_{1}\left( t, u(t), v(t), u^{\prime }(t), v^{\prime }(t)\right) , t \in [0,1]}, \\ {-v^{\prime \prime }(t)=f_{2}\left( t, u(t), v(t), u^{\prime }(t), v^{\prime }(t)\right) , t \in [0,1]}, \\ {u(0)=\alpha [v], \quad u^{\prime }(1)=\beta [u]}, \\ {v(0)=\beta [u], \quad v^{\prime }(1)=\alpha [v]},\end{array}\right. \end{aligned}$$

where $\alpha$ and $\beta$ denote linear functionals given by

$$\begin{aligned} \alpha [u]=\int _{0}^{1}u(t){\text {d}}A(t),\ \ \beta [u]=\int _{0}^{1}u(t){\text {d}}B(t) \end{aligned}$$

involving Stieltjes integrals with suitable functions A, B of bounded variation. By the theory of fixed point index on a special cone in $C^1[0,1]\times C^1[0,1]$, the existence of the positive solutions of the system is obtained through posing some inequality conditions and the spectral radius conditions on the nonlinearities.