NON-HOMOGENEOUS CONTINUOUS AND DISCRETE GRADIENT SYSTEMS: THE QUASI-CONVEX CASE

‎In this paper‎, ‎first we study the weak and strong convergence of solutions to the‎ ‎following first order nonhomogeneous gradient system‎ ‎ {−x′(t)=∇ϕ(x(t))+f(t), a.e. on (0,∞) {‎‎x(0)=x0∈H to a critical point of ϕ‎, ‎where‎ ‎ϕ is a C1 quasi-convex function on a real Hilbert space‎ ‎H with Argminϕ≠∅ and f∈L1(0,+∞;H)‎. ‎These results extend the‎ ‎results in the literature to non-homogeneous case‎. ‎Then the‎ ‎discrete version of the above system by backward Euler‎ ‎discretization has been studied‎. ‎Beside of the proof of the‎ ‎existence of the sequence given by the discrete system‎, ‎some‎‎results on‎ ‎the weak and strong convergence to the critical point of ϕ are also proved‎. ‎These results when ϕ is pseudo-convex (therefore the critical points‎ ‎are the same minimum points) may be applied in optimization for approximation of a‎ ‎minimum point of ϕ‎.