Positive solutions for a second-order nonlinear coupled system with derivative dependence subject to coupled Stieltjes integral boundary conditions
The paper is concerned with the second-order differential system with nonlinearities depending on the first derivatives, subject to coupled Riemann-Stieltjes integral boundary conditions \[\left\{\begin{array}{l} -u''(t)=f_1(t,u(t),v(t),u'(t),v'(t)),\,\,\,t\in [0,1],\\ -v''(t)=f_2(t,u(t),v(t),u'(t),v'(t)),\,\,\,t\in [0,1],\\ u(0)=\alpha[v],\,\,\,u'(1)=\beta[u],\\ v(0)=\beta[u],\,\,\,v'(1)=\alpha[v], \end{array}\right.\leqno(1)\] where \(\alpha\) and \(\beta\) are linear functionals given by \(\alpha[u]=\int_0^1u(t)dA(t)\), \(\beta[u]=\int_0^1u(t)dB(t)\), and \(A,\,B\) are bounded variation functions. Under some assumptions on the functions \(f_1\) and \(f_2\), and on the spectral radius of an associated linear operator, the authors prove the existence of positive solutions for problem \((1)\), by using the fixed point index theory.
scientific article; zbMATH DE number 7488606
scientific article