On fixed point index theory for the sum of operators and applications to a class of ODEs and PDEs





positive solution, fixed point index, cone, sum of operators, ODEs, PDEs


The aim of this work is two fold: first  we  extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a $k$-set contraction  obtained in \cite{DjebaMeb, Svet-Meb}, to  the case of the sum $T+F$, where $T$ is a mapping such that $(I-T)$ is Lipschitz invertible and $F$ is a $k$-set contraction.  Secondly, as  illustration of some our theoretical results,  we study  the existence of positive solutions  for two classes of differential equations, covering a class of first-order ordinary differential equations (ODEs for short) posed on the positive half-line as well as  a class of  partial differential equations (PDEs for short).


Download data is not yet available.

Author Biographies

Svetlin Georgiev Georgiev, University of Sofia

Department of Differential Equations, Faculty of Mathematics and Informatics

Karima Mebarki, University of Bejaia

Laboratory of Applied Mathematics, Faculty of Exact Sciences.


S. Benslimane, S. G. Georgiev and K. Mebarki, Expansion-compression fixed point theorem of Leggett-Williams type for the sum of two operators and application in three-point BVPs, Studia UBB Math, to appear.

G. Cain and M. Nashed, Fixed points and stability for a sum of two operators in locally convex spaces, Pacific J. Math. 39 (1971), 581-592. https://doi.org/10.2140/pjm.1971.39.581

S. Djebali and K. Mebarki, Fixed point index theory for perturbation of expansive mappings by k-set contraction, Topol. Meth. in Nonlinear Anal. 54, no. 2 (2019), 613-640. https://doi.org/10.12775/TMNA.2019.055

S. Djebali and K. Mebarki, Fixed point index on translates of cones and applications, Nonlinear Studies 21, no. 4 (2014), 579-589.

D. Edmunds, Remarks on nonlinear functional equations, Math. Ann. 174 (1967), 233-239. https://doi.org/10.1007/BF01360721

S. G. Georgiev and K. Mebarki, Existence of positive solutions for a class ODEs, FDEs and PDEs via fixed point index theory for the sum of operators, Commun. on Appl. Nonlinear Anal. 26, no. 4 (2019), 16-40.

S. G. Georgiev and K. Mebarki, Existence of solutions for a class of IBVP for nonlinear parabolic equations via the fixed point index theory for the sum of two operators, New Trends in Nonlinear Analysis and Applications, to appear.

D. Guo, Y. J. Cho and J. Zhu, Partial Ordering Methods in Nonlinear Problems, Shangdon Science and Technology Publishing Press, Shangdon, 1985.

M. Nashed and J. Wong, Some variants of a fixed point theorem Krasnoselskii and applications to nonlinear integral equations, J. Math. Mech. 18 (1969), 767-777.

A. Polyanin and A. Manzhirov, Handbook of integral equations, CRC Press, 1998. https://doi.org/10.1201/9781420050066

V. Sehgal and S. Singh, A fixed point theorem for the sum of two mappings, Math. Japonica 23 (1978), 71-75.

T. Xiang and R. Yuan, A class of expansive-type Krasnosel'skii fixed point theorems, Nonlinear Anal. 71, no. 7-8 (2009), 3229-3239. https://doi.org/10.1016/j.na.2009.01.197

T. Xiang and S. G. Georgiev, Noncompact-type Krasnoselskii fixed-point theorems and their applications, Math. Methods Appl. Sci. 39, no. 4 (2016), 833-863. https://doi.org/10.1002/mma.3525




How to Cite

S. Georgiev Georgiev and K. Mebarki, “On fixed point index theory for the sum of operators and applications to a class of ODEs and PDEs”, Appl. Gen. Topol., vol. 22, no. 2, pp. 259–294, Oct. 2021.



Regular Articles