# Results on Fixed Point Theory, obtained at the Institute

## Fixed point theorems in Nonlinear Analysis.

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

- M.-C. Alicu (Anisiu), O. Mark,
*Some properties of the fixed point set for multifunctions*, Studia Univ. “Babeş-Bolyai”, Math. XXV (4) (1980), 77-79.

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

- M.-C. 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 - M.-C. 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.

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

- M.-C. 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.

In the book

- M.-C. Anisiu,
*Nonlinear analysis methods applied in Celestial Nechanics*, Presa Universitară Clujeană, 1998. (in Romanian)

a chapter is dedicated to fixed point theorems.

## 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 Ş. M. Soltuz,
*On the equivalence of Mann and Ishikawa iteration methods*, Internat. J. Math. Math. Sci. 2003 (7), 451-459. - B. E. Rhoades, Ş. M . Şoltuz,
*The equivalence between T-stabilities of Mann and Ishikawa iterations*, J. Math. Anal. Appl. 318 (2006), 472-475. - B. E. Rhoades and Ş. M . Şoltuz,
*The equivalence between Mann-Ishikawa iterations and multistep iteration*, Nonlinear Analysis 58 (2004), 219-228.

## 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 are given in:

- E. Cătinaş,
*On the superlinear convergence of the successive approximations method*, J. Optim. Theory Appl., v. 113 (2002) no. 3, pp. 473-485. - E. Cătinaş,
*Estimating the radius of an attraction ball*, Applied Mathematics Letters, v. 22 (2009) no. 5, pp. 712-714. - 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 we obtained existence, uniqueness results, data dependence and Ulam stability 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. - D. Otrocol, I.A. Rus,
*Functional-differential equations with maxima of mixed type*, Fixed Point Theory, 9 (2008) no. 1, 207-220. - D. Otrocol, V.A. Ilea, Ulam stability for a delay differential equation, Central European Journal of Mathematics, 11 (2013) no. 7, 1296-1303.

The convergence of the sequence of the successive approximation by using contraction principle and step method with a weaker Lipschitz condition and a new algorithm of successive approximation sequence generated by the step method were obtained in

- V. Ilea, D. Otrocol, M.-A. Şerban and D. Trif, Integro-differential equation with two time lags, Fixed Point Theory, 13 (2012), no. 1, pp. 85-97.