Functional analysis

Research team

Ioan Rasa, Dorian Popa, Daniela Inoan, Bogdan Gavrea, Mircea Rus, Daniela Marian 

 

Research fields

• Semigroups of operators
• Positive operators
• Invariant classes of functions
• Evolution equations
• Multifunctions

 

Recent research project

“Qualitative aspects in studying some generalized convex mappings. Analytic and geometric methods”, research grant funded by CNCSIS A- 865, 2004-2006,

Rasa Ioan, project manager, Popa Dorian, Inoan Daniela, Rus Mircea, Lung Nicolaie, Vornicescu Neculae, Pop Vasile, Ciupa Alexandra, Ioan Radu Peter, Novac Adela

 

The project is devoted to the study of some classes of generalized convex functions and multifunctions by using both analytic and geometric methods. We investigate such classes of functions defined on a convex subset of a locally convex space), which are invariant with respect to some linear operators. This is important at least for two reasons. On the one hand, in numerical analysis a function is often approximated by its images under linear operators. The convexity properties of the function should be preserved by the operators. If the operators are given, one studies the properties of the function which are preserved. On the other hand some iterates of the operators converge to some semigroup; the classes of functions invariant under the operators are also invariant under the semigroup. The evolution equation associated to the semigroup can be related to a stochastic equation. The information about a problem can be translated to the other one.

We characterize the continuity and the semicontinuity of some multifunctions from classes defined by generalized convexity. The study of the equality case in the definition of some convex multifunctions leads to specific functional equations. For such functional equations we characterize the solutions and generalize some known results.

We are interested in the impact of different kinds of curvatures on the convexity of Finsler submanifolds, and on the smooth mappings defined on them. We are also interested in the applications of stochastic equations on Finsler manifolds, a method which is already successfully used in applications in Engineering and Biology.

 

“Shape sensitivity analysis for the solutions of some variational problems”, research grant funded by CNCSIS AT-33702, 2004-2005,

Daniela Inoan, project manager, Adela Chis, Daniela Rosca

 

The theory of shape optimization studies problems in which the optimization variable is a geometric domain. The aspects concerning these problems are multiple, from the theoretical ones to the implementation of the results in concrete situations. An important aspect is shape sensitivity analysis, both for the cost functional and for the variational problem which has the role of the state equation from optimal control.

In the project different variational problems are studied, especially with nonlinear operators, from the point of view of the behavior of their solutions when the geometric domain is perturbed. We analyse both abstract problems (searching for conditions on the operators as general as possible), and examples (with possible applications in technics and physics).We use different topologies known in the literature for classes of geometric domains defined by the transformation (mapping) method, speed method or the Hausdorff complementary topology. One of the objectives of the project is to study the continuity and stability with respect to the domain of the solutions of some variational or hemivariational inequalities. In addition, the differentiability with respect to the domain is also investigated, studying the existence of different types of derivatives (Gateaux, Hadamard, Frechet). Formulas for these derivatives are searched.

Some shape optimizaton problems do not have solutions in the a-priori considered addmissible class. In such cases one uses generalized solutions- for example by homogenization method.

In the numerical analysis the method of finite elements is used most frequently. In some situations, wavelets proved to be more efficient then the finite elements. These functions are studied in the project.

Publications

B. Gavrea, J. Jaksetic, J. Pecaric, “On a global upper bound for Jessen’s inequality”, ANZIAM Journal, 50 (2), 2009, 246-257.

 

D. Marian, “Some types of convex functions on network”, Revue d'Analyse Numerique et de Theorie de l'Approximation, 38 (1), 2009, 51-63.

 

D. Marian, S. Tigan, E. Iacob, “Some remarks on the high order convexity of Tberiu Popoviciu type for functions of several variables”, Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity, 7, 2009, 119-145.

 

A. I. Mitrea, P. Mitrea, “Inequalities involving the superdense unbounded divergence of some approximation processes”, International Series of Numerical Mathematics, Birkhauser Verlag AG Basel-Boston-Berlin, vol. 157, 2009, 193-200.

 

D. Popa, K. Nikodem, “On single valuedness of the set-valued maps satisfiyng linear inclusions”, Banach Journal of Mathematical Analysis, 3 (1), 2009, 44-51.

 

D. Popa, K. Nikodem, “On selection of general linear inclusions”, Publicationes Mathematicae Debrecen, 75 (2), 2009, 239-249.

 

D. Popa, M. S. Moslehian, K. Nikodem, “Asymptotic aspect of the quadratic equations in multinormed spaces”, Journal of Mathematical Analysis and Applications, 355 (2), 2009, 717-724.

 

 D. Popa, “A property of a functional inclusion connected with Hyers-Ulam stability”, Journal of Mathematical Inequalities, 3 (4), 2009, 591-598.

 

D. Popa, “On the stability of recurrences”, Nonlinear Functional Analysis and Applications, 2009.

 

I. Raşa, H. Gonska, M. Heilmann, “Convergence of iterates of genuine and ultraspherical Durrmeyer operators to the limiting semigroup: C2-estimates”, Journal of Approximation Theory, 160 (1-2), 2009, 243-255.

 

I. Raşa, H. Gonska, “On the composition and decomposition of positive linear operators (II)”, Studia Scientiarum Mathematicarum Hungarica, 2009, DOI:10.1556/SScMath.2009.1144

 

I. Raşa, “Asymptotic behaviour of certain semigroups generated by differential operators”, Jaen Journal on Approximation, 1 (1), 2009, 27-36.

 

I. Raşa, “Positive linear operators and the asociated semigroup: asymptotic behaviour”, Jaen Journal on Approximation, 1 (2), 2009, 195-204.

 

D. Marian, “E-rough analysis on undirected networks”, Annals of the Tiberiu Popoviciu Seminar of Functionals Equations, Approximation and Convexity, 6, 2008, 67-74.

 

D. Marian, “Some properties of semi_E-d-convex functions”, Automation Computers and Applied Mathematics, 17 (3), 2008, 485-495.

 

A. I. Mitrea, “Superdense unbounded divergence of numerical differentiation on Sturm-Liouville node matrix”, Pure Mathematics and Applications, 17 (3), 2008, 359-365.

 

A. I. Mitrea, “Double condensation of singularities for interpolating operators”, Automation Computers and Applied Mathematics, 17 (2), 2008, 147-153.

 

I. Rasa, F. Altomare, V. Leonessa, “On Bernstein-Schnabl operators on the unit interval”, Zeitschrift fur Analysis und ihre Anwendungen (Journal for Analysis and its Applications), 27 (3), 2008, 353-379.

 

I. Rasa, A. Attalienti, “Overiterated linear operators and asymptotic behaviour of semigroups”, Mediterranean Journal of Mathematics, 5 (3), 2008, 315-324.

 

I. Rasa, A. Attalienti, “Shape-preserving properties and asymptotic behaviour of the semigroup generated by the Black-Scholes operator”, Czechoslovak Mathematical Journal, 58 (2), 2008, 457-467.

 

I. Rasa, M. Campiti, C. Tacelli, “Steklov operators and semigroups in weighted spaces of continuous real functions”, Acta Mathematica Hungarica, 120 (1), 2008, 103-125.

 

I. Rasa, M. Campiti, C. Tacelli, Steklov operators and their associated semigroups, Acta Scientiarum Mathematicarum (Szeged), 74 (1), 2008, 171-189.

 

E. Mangino, I. Rasa, “A quantitative version of Trotter’s approximation theorem”, Journal of Approximation Theory, 146(2), 2007, 149-156.

 

H. Gonska, P. Pitul, I. Rasa, “Over-iterates of Bernstein-Stancu operators”, Calcolo, 44(2), 2007, 117-125.

 

J. Brzdek, D. Popa, XuBing, “The Hyers-Ulam Stability of Nonlinear Recurrences”, Journal of Mathematical Analysis and Applications, 335(1), 2007, 443-449.

 

H. Gonska, I. Rasa, “The limiting semigroup of the Bernstein iterates: degree of convergence”, Acta Mathematica Hungarica, 111(1-2), 2006, 119-130.

 

H. Gonska, P. Pitul, I. Rasa, “On differences of positive operators”, Carpathian Journal of Mathematics, 22(1-2), 2006, 65-78.

 

I. Rasa, “Iterated Boolean sums of Bernstein and related operators”, Revue d’Anayse Numerique et de Theorie de l’Approximation, 35(1), 2006, 111-115.

 

H. Gonska, P. Pitul, I. Rasa, ”On Peano's form of the Taylor remainder, Voronovskaja's theorem and the commutator of positive linear operators”, Proceedings of the Interational Conference on Numerical Analysis and Approximation Theory, Cluj-Napoca, Romania, 2006, 55-80.

 

A. Attalienti, I. Rasa, “Asymptotic behaviour of C_0 semigroups”, Proceedings of the Interational Conference on Numerical Analysis and Approximation Theory, Cluj-Napoca, Romania, 2006, 127-130.

 

J. Brzdek, D. Popa, XuBing, “Note on the nonstability of the linear recurrence”, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 76, 2006, 183-189.

 

D. Inoan, J. Kolumban, “Two existence results for variational inequalities”, in Studia Universitatis, Seria Mathematica LI(3), 2006, 85-95.

 

F. Altomare, I. Rasa, “On some classes of diffusion equations and related approximation problems”, Proceedings of the International Conference Trends and Applications in Constructive Approximation, Birkhäuser Verlag Basel, 2005, 13-26.

 

I. Rasa, “Semigroups associated to Mache operators (II)”, Proceedings of the International Conference Trends and Applications in Constructive Approximation, Birkhäuser Verlag Basel, 2005, 225-228.

 

F. Altomare, I. Rasa, “On a class of exponential-type operators and their limit semigroups”, Journal Approximation Theory, 135(1), 2005, 258-275.

 

I. Rasa, “One-dimensional diffusions and approximation”, Mediterranean Journal of Mathematics, 2(2), 2005, 153-169.

 

D. Popa, “Hyers-Ulam-Rassias stability of a linear recurrence”, Journal of Mathematical Anaysis and Applications,309(2), 2005, 591-597.

 

D. Popa, N. Lungu, “On an operatorial inequality”, Demonstratio Mathematica 38(3), 2005, 667-674.

 

D. Popa, ”On the Hyers-Ulam stability of the first order linear recurrence”, Pure Mathematics and Applications, PU.M.A. 15(2-3), 2005, 285-293.

 

D. Popa, “Hyers-Ulam stability of the linear recurrence with constant coefficients”, Advances in Differential Equations, No. 2, 2005, 101-107.

 

D. Inoan, “Stability of a variational inequality with respect to domain perturbations”, in Acta Universitatis Apulensis, no. 10, 2005, 123-130

 

M. D. Rus, “A note on the existence of positive solutions of Fredholm integral equations”, Fixed Point Theory, 5(2), 2004, 369-377.

 

D. Inoan, “Continuity and derivatives with respect to the perturbations of a domain”, Automation Computers Applied Mathematics. Scientific Journal, 13(1), 2004, 101-106.

 

D. Inoan, “Derivatives with respect to the perturbations of the domain”, Annals of University of Craiova, seria Mathematica & Computer Science, 31, 2004, 35-42

 

V. Campian, M. D. Rus, “A note on the Cauchy-Riemann conditions”, Proceedings of the 11th Conference on Applied and Industrial Mathematics, Oradea, Romania, 2003, 54-59.