Fixed Point Theory

(in Romanian )

Fixed point theorems in Nonlinear Analysis.

Fixed point theory is applied to prove the existence and uniqueness of solutions of the Darboux problem with deviating argument. The properties of the fixed point set for special multivalued mappings were studied in

• Mira-Cristiana Alicu, O. Mark, Some properties of the fixed point set for multifunctions, Studia Univ. "Babeş-Bolyai", Math. XXV (4) (1980), 77-79 (citations in ISI journals).

The approximation of the fixed points in Caristi theorem and the connection with Ekeland theorem were considered in

• Mira-Cristiana Anisiu, On Caristi's theorem and successive approximations, Seminar on Functional Analysis and Numerical Methods, 1-10, Preprint, 86-1, Univ. "Babeş-Bolyai" Cluj-Napoca, 1986
• Mira-Cristiana Anisiu, On maximality principles related to Ekeland's theorem, Seminar on Functional Analysis and Numerical Methods, 1-8, Preprint, 87-1, Univ. "Babeş-Bolyai" Cluj-Napoca, 1987 (citations in ISI journals).

It was proved that the convex sets with nonvoid interior (in a Banach space) for which every contraction has a fixed point are necessarily closed in

• Mira-Cristiana Anisiu, V. Anisiu, On the closedness of sets with the fixed point property for contractions, Rev. Anal. Numer. Theor. Approx. 26 (1-2) (1997), 13-17 (citations).

In the book

• Mira-Cristiana Anisiu, Metode ale analizei neliniare cu aplicaţii în mecanica cerească, Presa Universitară Clujeană, 1998.

Convergence of the Mann and Ishikawa type iterations.

Results on the equivalence of the convergence of certain iterations have been obtained.

• B. E. Rhoades and Stefan M. Soltuz, On the equivalence of Mann and Ishikawa iteration methods, Internat. J. Math. Math. Sci. 2003 (7), 451-459 (impact factor 2009: 0.996).
(citations in journals).
• B. E. Rhoades, Ş. M . Şoltuz, The equivalence between T-stabilities of Mann and Ishikawa iterations, J. Math. Anal. Appl. 318 (2006), 472-475. (impact factor 2009: 1.225)
(citations in ISI journals, citations in other journals).
• B. E. Rhoades and Ş. M . Şoltuz, The equivalence between Mann-Ishikawa iterations and multistep iteration, Nonlinear Analysis 58 (2004), 219-228. (impact factor 2009: 1.487)
(citations in ISI journals, citations in other journals).

Local convergence of the successive approximations.

The high convergence orders of the successive approximations were characterized, an estimation of the attraction balls were obtained, and some results on the acceleration of the convergence of the successive approximations.

• E. Cătinaş, On the superlinear convergence of the successive approximations method, J. Optim. Theory Appl., v. 113 (2002) no. 3, pp. 473-485 (impact factor 2009: 0.996).
(citations in ISI journals, citations in other journals).
• E. Cătinaş, Estimating the radius of an attraction ball, Applied Mathematics Letters, v. 22 (2009) no. 5, pp. 712-714. (impact factor 2009: 0.978)
(citations in ISI journals, citations in other journals).
• E. Cătinaş, On accelerating the convergence of the successive approximations method, Rev. Anal. Numer. Theor. Approx., 30 (2001) no. 1, pp. 3-8.

Solutions of differential equations as fixed points.

In this direction were obtained existence and uniqueness results, data dependence for the solution of functional-differential equations, differential equations with delays and mixed type argument

• D. Otrocol, I.A. Rus, Functional-differential equations with "maxima" via weakly Picard operators theory, Bull. Math. Soc. Sci. Math. Roumanie, 51(99) 2008, no. 3, 253-261, (impact factor: 0.554).
• D. Otrocol, Ioan A. Rus, Functional-differential equations with maxima of mixed type, Fixed Point Theory, 9 (2008) no. 1, 207-220. (CNCSIS jurnal ranked B+, ISI journal since 2009)