Non-homogeneous continuous and discrete gradient systems‎: ‎the quasi-convex case

ýIn this paperý, ýfirst we study the weak and strong convergence of solutions to theý ýfollowing first order nonhomogeneous gradient systemý ý{−x′(t)=∇ϕ(x(t))(t), a.e.\ \ on (0,∞)ýýx(0)=x0∈H to a critical point of ϕý, ýwhereý ýϕ is a C1 quasi-convex function on a real Hilbert spaceý ýH with Argminϕ≠∅ and f∈L1(0,∞;H)ý. ýThese results extend theý ýresults in the literature to non-homogeneous caseý. ýThen theý ýdiscrete version of the above system by backward Eulerý ýdiscretization has been studiedý. ýBeside of the proof of theý ýexistence of the sequence given by the discrete systemý, ýsomeýýresults oný ýthe weak and strong convergence to the critical point of ϕ are also provedý. ýThese results when ϕ is pseudo-convex (therefore the critical pointsý ýare the same minimum points) may be applied in optimization for approximation of aý ýminimum point of ϕý.