ABSTRACTS

Historical Account of the creation of Wavelets and suggestion connection with Approximation of Fixed point. Geraldo De Souza

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In this short course, we will try to have a historical account of several results on the (D) the Hardy Spaces, introduced by Frigyes Riesz in 1923, and named after G. H. Hardy (1915). These spaces are interesting in that they are connected to the Lebesgue spaces ([0,2π]), for 0< p ≤ ꚙ, and are useful as the spaces of solutions to numerous  problems in mathematics and physics. For longtime then, finding dual space of (D) was an open problem until C. Fefferman identified it as the space of Bounded Means Oscillation (BMO), space introduced by F. John and L. Nirenberg in the sixties. R.R. Coifman then used this result to propose another characterization of the dual space of (D), using an Atomic Decomposition. Antony Zigmund commented at the time that this characterization of the dual space of (D)  via an atomic decomposition was more important than the duality of (D). Also motivated by this atomic decomposition, Geraldo De Souza in 1980, introduced “the special atomic decomposition” of a subspace of (D), which was called The Special Atom Space. Yves Meyer careful analyzed this special atomic decomposition and connected it with the Haar functions, and and proposed a general framework for Wavelets.

In this presentation, we will go through the main results on special atomic decomposition, Haar functions and wavelets on [0,1] and on real line. We, also generalize these to higher dimension (joint work with Eddy Kwesi et al), in a very natural way and give some applications. We will mention the Multi-Resolution Analysis, that I believe relate to approximations in fixed-point theory. In my last hour, I will pay a Tribute to Prof. Charles Chidume, whom I knew since 1976 and with whom I collaborated on several projects. I will also share some unique stories, perspectives, and conversations we had together through out these years.

Recent trends in vector optimization. Sorin-Mihai Grad

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In this minicourse we discuss about some current developments in Vector Optimization.  Emphasis will be placed on recent contributions on algorithmic methods for solving vector optimization problems. In particular we will talk about the existing extensions of the proximal point methods towards Vector Optimization and about some open questions in this direction. 

Asymptotic behaviours of solutions to elliptic and parabolic monotone operators depending on vector fields. Alberto Maione

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During this mini-course I will present recent results obtained in collaboration with Fabio Paronetto (University of Padova) and Eugenio Vecchi (University of Bologna) concerning variational convergences for differential operators depending on a family of Lipschitz continuous vector fields X.

I will focus, in particular, to H-compactness (or G-compactness) results for a class of monotone operators both elliptic and parabolic.

This theory, known as "compensated compactness", was initiated by François Murat and Luc Tartar in the Euclidean case and to date it still finds numerous applications. Here we try to extend the Murat-Tartar H-compactness theorem for monotone operators to a class of operators depending on the family X, under suitable assumptions on X.

I will provide many examples of relevant families of suitable vector fields and at least an example in which the standard theory seems not to be appliable.

New Simple Projection-Type Methods for Variational Inequalities. Yekini Shehu

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In this talk, we present new projection type methods which involve one projection and one functional evaluation at each iteration for variational inequality problems in Hilbert spaces. The proposed methods are variants of the projected reflected gradient type methods (PRGMs) and forward-reflected-backward splitting method (FRBM) of Malitsky-Tam. Inertial versions of these methods are also presented alongside weak and linear convergence results.

Effect of spatial heterogeneity, population’s size and movement on the dynamics of an infectious disease. Salako Rachidi

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We investigate how population’s size and movement affect the dynamics of an infectious disease by studying the asymptotic profiles of endemic equilibrium solutions of a diffusive epidemic model in spatially heterogeneous environment. Our results suggest that when the diffusion rates dS of the susceptible and dI of the infected groups approach zero, the size of dI/dS plays a crucial dS role on the persistence of the disease in the sense that: (i) if dI/dS is small, the dS disease may persist and the total size of the infected group will be maximized; (ii) if dI/dS is large, then the total size of the susceptible group is maximized while dS the total size of the infected group is minimized. Our results also indicate a concentration phenomenon of the infected population on the most vulnerable regions when the disease persists and the infected group movement is consider- ably lowered.

Global boundedness and pointwise persistence in full chemotaxis models in heterogeneous environments. Tahir Bachar Issa

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In this talk, we will discuss global boundedness and pointwise persistence in full chemotaxis models with local as well as nonlocal time and space-dependent logistic sources in bounded domains. We first prove the global existence and boundedness of nonnegative classical solutions under some conditions on the coefficients in the models. Next, under the same conditions on the coefficients, we show that pointwise persistence occurs, that is, any globally defined positive solution is bounded below by a positive constant independent of its initial condition when the time is large enough. It should be pointed out that Tao and Winkler in 2015, have established the persistence of mass for globally defined positive solutions, which indicates that any extinction phenomenon, if occurring at all, necessarily must be spatially local in nature, whereas the population as a whole always persists. The pointwise persistence proved in this work implies that not only the population as a whole persists, but also it persists at any location eventually. It also implies the existence of strictly positive entire solutions.

Well-posedness and stability for a fractional di􏰀erential problem arising in porous media. Jamilu Hashim Hassan

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In this talk, we shall discuss the passage from integer-order di􏰀erential equations to fractional-order di􏰀erential equations. Its advantages and disadvantages will be high- lighted. Moreover, we will explain how to derive well-posedness of a fractional di􏰀erential problem arising in porous media from the corresponding integer-order problem. In addi- tion, some stability results will be presented.

Beyond the abstraction of nonlinear operator theory: Examples of inverse problems in cancer therapies. Lois C. Okereke

 

The beauty and elegance of the inherent abstraction in nonlinear operator theory lie in the use of simple principles to execute complex tasks. This provides a powerful balance of generalization and specialization, thus paving the way for applications in related areas such as convex optimization and differential equations. Most importantly, beyond the abstraction lies its natural ability to provide a rich set of tools for solving challenging real-life problems. This talk highlights this important ability through an expository discussion of inverse problems in radiation and combined immune-radiation treatments of cancer.