## Abstract

In this paper we obtain nonlinear alternatives of Leray-Schauder and Monch type for nonself vector-valued operators, under hybrid conditions of Perov contraction and compactness.

Thus, we give vector versions of the theorems of Krasnosel’skii, Avramescu, Burton-Kirk and Gao-Li Zhang. An application is given to a boundary value problem for a system of second order differential equations in which some of the equations are implicit.

## Authors

**Veronica Ilea
**Babes–Bolyai University, Faculty of Mathematics and Computer Science, Cluj-Napoca, Romania

**Adela Novac
**Technical University of Cluj-Napoca, Department of Mathematics, Cluj-Napoca, Romania

**Diana Otrocol
**Technical University of Cluj-Napoca, Department of Mathematics, Cluj-Napoca, Romania

Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

**Radu Precup**

Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania

Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

## Keywords

Nonlinear operator; nonself map; fixed point; Perov contraction; nonlinear boundary value problem.

## Paper coordinates

V. Ilea, A. Novac, D. Otrocol, R. Precup, *Nonlinear alternatives of hybrid type for nonself vector-valued maps and application, *Fixed Point Theory, **24** (2023) no. 1, 221-232, http://doi.org/10.24193/fpt-ro.2023.1.11

## About this paper

##### Journal

Fixed Point Theory

##### Publisher Name

Casa Cărţii de Ştiinţă Cluj-Napoca

(House of the Book of Science Cluj-Napoca)

##### Print ISSN

1583-5022

##### Online ISSN

2066-9208

google scholar link

[1] C. Avramescu, On a fixed point theorem (in Romanian), St. Cerc. Mat., 22(1970), no. 2, 215–221.

[2] C.S. Barroso, E.V. Teixeira, A topological and geometric approach to fixed points results for sum of operators and applications, Nonlinear Anal., 60(2005), 625–650.

[3] I. Basoc, T. Cardinali, A hybrid nonlinear alternative theorem and some hybrid fixed point theorems for multimaps, J. Fixed Point Theory Appl., 17(2015), 413–424.

[4] I. Benedetti, T. Cardinali, R. Precup, Fixed point–critical point hybrid theorems and application to systems with partial variational structure, J. Fixed Point Theory Appl., 23(2021), 63, 1–19.

[5] T.A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii-Schaefer type, Math. Nachr., 189(1998), 23–31.

[6] T. Cardinali, R. Precup, P. Rubbioni, Heterogeneous vectorial fixed point theorems, Mediterr. J. Math., 14(2017), 83, 1–12.

[7] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985

[8] B.C. Dhage, Local fixed point theory for the sum of two operators in Banach spaces, Fixed Point Theory, 4(2003), 49–60.

[9] H. Gao, Y. Li, B. Zhang, A fixed point theorem of Krasnoselskii-Schaefer type and its applications in control and periodicity of integral equations, Fixed Point Theory, 12(2011), 91–112.

[10] L. Gorniewicz, A. Ouahab, Some fixed point theorems of a Krasnosel’skii type and application to differential inclusions, Fixed Point Theory, 17(2016), 85–92.

[11] G.L. Karakostas, An extension of Krasnoselsk’s fixed point theorem for contractions and compact mappings, Topol. Methods Nonlinear Anal., 22(2003), 181–191.

[12] M.A. Krasnosel’skii, Some problems of nonlinear analysis, Amer. Math. Soc. Transl. Ser. 2, 10(1958), 345–409.

[13] D. O’Regan, Fixed-point theory for the sum of two operators, Appl. Math. Lett., 9(1996), 1–8.

[14] A. Ouahab, Some Perov’s and Krasnosel’skii type fixed point results and application, Comm. Appl. Anal., 19(2015), 623–642.

[15] S. Park, Generalizations of the Krasnoselskii fixed point theorem, Nonlinear Anal., 67(2007), 3401–3410.

[16] I.-R. Petre, A. Petru¸sel, Krasnoselskii’s theorem in generalized Banach spaces and applications, Electron. J. Qual. Theory Differ. Equ., 2012, 85, 1–20.

[17] R. Precup, Methods in Nonlinear Integral Equations, Springer Science + Business Media, Dordrecht, 2011.

[18] I.A. Rus, On the fixed points of mappings defined on a Cartesian product, III (in Romanian), Studia Univ. Babes–Bolyai Math., 24(1979), no. 2, 55–56.

[19] T. Xiang, R. Yuan, Critical type of Krasnosel’skii fixed point theorem, Proc. Amer. Math. Soc., 139(2007), 1033–1044.