Well-posedness And Uniqueness For Some Optimization And Approximation Problems

The existence and well-posedness results for a class of perturbed optimization problemsin a Banach space are given, and the relationships between the strict Kolmogorov Criterionand the uniqueness elements of a sun in vector-valued function spaces are established. Themain work done in this dissertation is organized as follows:In Chapter 1, we shall study the existence and nonexistence of solutions for the perturbedoptimization problems (x, Z, J)-min and (x, Z, J)-max in a Banach space X. Under theassumptions that X is Kadec and Z is a closed, boundedly relatively weakly compact andnon-empty subset (but unnecessarily bounded), the existence result for the problem(x, Z, J)-min is established. Furthermore, if additionally assume that X is strictly convex and p＞1,the well-posedness result for the problem (x, Z, J)-min is given. Under the assumptionsthat X is compactly fully 2-convex and compactly locally uniformly convex and p=1,the porosity result for the problem (x, Z, J)-min is given. Under the assumptions that Xis kadec and J satisfies the condition BC_p, the existence result for (x, Z, J)-max is given.Furthermore, if additionally assume that X is strictly convex and p＞1, the well-posednessresult for the problem (x, Z, J)-max is given. Under the assumptions that X is uniformlyconvex and J is upper semicontinuous and bounded and p=1, the porosity result for theproblem (x, Z, J)-max is given. At the same time, we show that if p=1 then the uniquenessresults for the problem (x, Z, J)-min and (x, Z, J)-max do not hold.In Chapter 2, we will investigate the relationships between the strict Kolmogorov Cri-terion and the uniqueness element of a sun in terms of property SK in the vector-valuedfunctions spaces Y_A(p,q) and Y_B(p,q). The results we obtain depending on A, B, p, q. In criticalcases, based on the related results established for the space of real-valued continuous func-tions and the space L₁, we will prove that each sun of Y_A(∞,1), CY_{A(∞,∞)}, CY_{B(∞,∞)}, Y_A(1,1)and Y_B(1,1) has Property SK. We also establish similar results for CY_B(∞,1) in the case whenm=2 and for Y_B(1,∞) in the case of the simultaneous approximation. While in noncriticalcases, a series of counterexamples will be given to show that we cannot characterize theuniqueness element in terms of the strict Kolmogorov Criterion.