Positive solutions of a derivative dependent second-order problem subject to Stieltjes integral boundary conditions

摘要

In this paper, we investigate the derivative dependent second-order problem subject to Stieltjes integral boundary conditions {?u′′(t)=f(t,u(t),u′(t)),t∈[0,1],au(0)?bu′(0)=α[u], cu(1)+du′(1)=β[u], {?u″(t)=f(t,u(t),u′(t)),t∈[0,1],au(0)?bu′(0)=α[u], cu(1)+du′(1)=β[u], where ff: [0,1]×R+×R→R+[0,1]×R+×R→R+ is continuous, α[u]α[u] and β[u]β[u] are linear functionals involving Stieltjes integrals. Some inequality conditions on nonlinearity ff and the spectral radius condition of linear operator are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index. Not only is the general case considered but a large range of coefficients can be chosen to weaken the conditions in previous work for some special cases. The conditions allow that f(t,x1,x2)f(t,x1,x2) has superlinear or sublinear growth in x1,x2x1,x2. Two examples are provided to illustrate the theorems under multi-point and integral boundary conditions with sign-changing coefficients.