Two positive solutions of third-order BVP with integral boundary condition and sign-changing Green's function
Summary: We are concerned with the following third-order boundary value problem with integral boundary condition: \[ u'''(t)= f(t,u(t)),\;t\in [0,1], \] \[ u'(0)= u(1)= 0,\;u''(\eta)+ \int^\beta_\alpha u(t)\,dt= 0, \] where \(1/2< \alpha\leq\beta\leq 1\), \(\alpha+\beta\leq 4/3\), and \(\eta\in(1/2, \alpha]\). Although the corresponding Greens function is sign-changing, we still obtain the existence of at least two positive and decreasing solutions under some suitable conditions on \(f\) by using the two-fixed-point theorem due to Avery and Henderson. An example is also included to illustrate the main results obtained.
scientific article; zbMATH DE number 6474217
scientific article