Browsing by Author "Pillay, Paranjothi."

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Aspects of connectedness in metric frames.
(2019) Rathilal, Cerene.; Pillay, Paranjothi.; Baboolal, Dharmanand.
Abstract available in PDF file.
Degree theory in nonlinear functional analysis.
(1989) Pillay, Paranjothi.; Hill, C. K.
The objective of this dissertation is to expand on the proofs and concepts of Degree Theory, dealt with in chapters 1 and 2 of Deimling [28], to make it more readable and accessible to anyone who is interested in the field. Chapter 1 is an introduction and contains the basic requirements for the subsequent chapters. The remaining chapters aim at defining a ll-valued map D (the degree) on the set M = {(F, Ω, y) / Ω C X open, F : Ὠ → X, y ɇ F(∂Ω)} (each time, the elements of M satisfying extra conditions) that satisfies : (D1) D(I, Ω, y) = 1 if y Є Ω. (D2) D(F, Ω, y) = D(F, Ω1 , y) + D(F, Ω2, y) if Ω1 and Ω2 are disjoint open subsets of Ω o such that y ɇ F(Ὠ \ Ω1 U Ω2 ). (D3) D(I - H(t, .), Ω, y(t)) is independent of t if H : J x Ὠ →X and y : J → X. An important property that follows from these three properties is (D4) F-1(y) ≠ Ø if D(F, Ω, y) ≠ 0. This property ensures that equations of the form Fx = y have solutions if D(F, Ω, y) ≠ 0. Another property that features in these chapters is the Borsuk property which gives us conditions under which the degree is odd and hence nonzero.
Interative approaches to convex feasibility problems.
(2001) Pillay, Paranjothi.; Xu, Hongjun.; O'Hara, John Gerard.
Solutions to convex feasibility problems are generally found by iteratively constructing sequences that converge strongly or weakly to it. In this study, four types of iteration schemes are considered in an attempt to find a point in the intersection of some closed and convex sets. The iteration scheme Xn+l = (1 - λn+1)y + λn+1Tn+lxn is first considered for infinitely many nonexpansive maps Tl , T2 , T3 , ... in a Hilbert space. A result of Shimizu and Takahashi [33] is generalized, and it is shown that the sequence of iterates converge to Py, where P is some projection. This is further generalized to a uniformly smooth Banach space having a weakly continuous duality map. Here the iterates converge to Qy, where Q is a sunny nonexpansive retraction. For this same iteration scheme, with finitely many maps Tl , T2, ... , TN , a complementary result to a result of Bauschke [2] is proved by introducing a new condition on the sequence of parameters (λn). The iterates converge to Py, where P is the projection onto the intersection of the fixed point sets of the Tis. Both this result and Bauschke's result [2] are then generalized to a uniformly smooth Banach space, and to a reflexive Banach space having a weakly continuous duality map and having Reich's property. Now the iterates converge to Qy, where Q is the unique sunny nonexpansive retraction onto the intersection of the fixed point sets of the Tis. For a random map r : N  {I, 2, ... ,N}, the iteration scheme xn+l = Tr(n+l)xn is considered. In a finite dimensional Hilbert space with Tr(n) = Pr(n) , the iterates converge to a point in the intersection of the fixed point sets of the PiS. In an arbitrary Banach space, under certain conditions on the mappings, the iterates converge to a point in the intersection of the fixed point sets of the Tis. For the scheme xn+l = (1- λn+l)xn+λn+lTr(n+l)xn, in a finite dimensional Hilbert space the iterates converge to a point in the intersection of the fixed point sets of the Tis, and in an infinite dimensional Hilbert space with the added assumption that the random map r is quasi-cyclic, then the iterates converge weakly to a point in the intersection of the fixed point sets of the Tis. Lastly, the minimization of a convex function θ is considered over some closed and convex subset of a Hilbert space. For both the case where θ is a quadratic function and for the general case, first the unique fixed points of some maps Tλ are shown to converge to the unique minimizer of θ and then an algorithm is proposed that converges to this unique minimizer.
Iterative algorithms for solutions of nonlinear equations in Banach spaces.
(2019) Aibinu, Mathew Olajiire.; Pillay, Paranjothi.
Perfect compactifications of frames.
(2018) Mthethwa, Simo Sisize.; Baboolal, Dharmanand.; Pillay, Paranjothi.
We study the compacti cations of frames. In particular, we study the compacti cations of frames which are perfect. That is, those compacti cations for which the right adjoint of the compacti cation mapping preserves disjoint binary joins. The Stone-C ech compacti cation of a completely regular frame and the Freudenthal compacti cation of a rim-compact frame are known to be examples of such compacti cations. We study the Freudenthal compacti cation of a rim-compact frame with an aim of providing more properties and characterizations of this compacti cation in the context of frames since this is less studied in the literature compared to the Stone-C ech compacti cation of frames. One of the main results that we obtain about the Freudenthal compacti cation of a rim-compact frame is that it is the minimal perfect compacti cation for this class of frames and the maximal -compacti cation. The notion of a full -compact basis is known in the context of spaces. We de ne an analogous concept in the context of frames and show that the Freudenthal compacti cation of a rim-compact frame arises from such a basis. We also establish the one-to-one correspondence between such bases and the -compacti cations of a rim-compact frame. The fact that the compacti cations arriving from such basis are zero-dimensional is also established.It is well known that a frame has the least compacti cation if and only if it is regular continuous. Some conditions under which the least compacti - cation of a regular continuous frame is perfect have been studied by Baboolal in [1] and the study is furthered herein. An N-point compacti cation of a space is any compacti cation whose remainder consists of N points. The N-star compacti cations of frames are known to be the frame analogue of the N-point compacti cations for spaces. It has been shown that the least compacti cation of a regular continuous frame is an example of an N-star compacti cation. We study the conditions under which a 2-star compacti- cation of a regular continuous frame is perfect and we conjecture that the results can be generalized to any N > 1: We prove that, under perfectness, the 2-star compacti cation of a regular continuous frame is the only N-star compacti cation. We also show some results related to the connectedness of the remainder of the 1-star compacti cation. Some contribution to the theory of compacti cations of frames not relating to perfectness has also been made. The concept and the construction of a freely generated frame is well known. We have shown that any compacti cation of a frame L can be realized as a frame freely generated by L subject to certain relations.