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1

Stability by Krasnoselskii's theorem in totally nonlinear neutral differential equations

Derrardjia I., Ardjouni A., Djoudi A.

Opuscula Mathematica

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2013

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Vol. 33, no. 2

255--272

EN

In this paper we use fixed point methods to prove asymptotic stability results of the zero solution of a class of totally nonlinear neutral differential equations with functional delay. The study concerns x'(t)= -a(t)x3(t) + c(t)x'(t-r(t)) + b(t)x3(t-r(t)). The equation has proved very challenging in the theory of Liapunov’s direct method. The stability results are obtained by means of Krasnoselskii-Burton’s theorem and they improve on the work of T.A. Burton (see Theorem 4 in [Liapunov functionals, fixed points, and stability by Krasnoselskii’s theorem, Nonlinear Studies 9 (2001), 181–190]) in which he takes c=0 in the above equation

2

Fixed points and stability in neutral nonlinear differential equations with variable delays

Ardjouni A., Djoudi A.

Opuscula Mathematica

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2012

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Vol. 32, no. 1

5-19

EN

By means of Krasnoselskii's fixed point theorem we obtain boundedness and stability results of a neutral nonlinear differential equation with variable delays. A stability theorem with a necessary and sufficient condition is given. The results obtained here extend and improve the work of C.H. Jin and J.W. Luo [Nonlinear Anal. 68 (2008), 3307-3315], and also those of T.A. Burton [Fixed Point Theory 4 (2003), 15-32; Dynam. Systems Appl. 11 (2002), 499-519] and B. Zhang [Nonlinear Anal. 63 (2005), e233-e242]. In the end we provide an example to illustrate our claim.

3

General fixed point theorems for weakly compatible maps

Djoudi A.

Demonstratio Mathematica

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2005

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Vol. 38, nr 1

197--205

EN

The purpose of this note is to use general expanssive conditions and minimal type commutativity without continuity requirements to prove some fixed point theorems. The theorems extend known results from the class of compatible continuous expansive maps to a wider class of mappings.

4

Fixed points for set and single valued maps without continuity

Djoudi A., Khemis R.

Demonstratio Mathematica

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2005

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Vol. 38, nr 3

739-751

EN

In this paper we prove some common fixed point theorems for single valued and set valued mappings on a metric space. The theorems use minimal type commutativity and strict contractivity conditions with no continuity requirements. The theorems extend previous results on delta-compatible and continuous maps of [1] and [11] to a wider class of weakly compatible mappings not necessarly continuous. Also, a common fixed point theorem which applies to a sequence of set valued mappings is given.

5

A unique common fixed point for compatible mappings of type (B) satisfying an implicit relation

Djoudi A.

Demonstratio Mathematica

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2003

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Vol. 36, nr 3

763--770

6

A common fixed point theorem for compatible mappings of type (B) in complete metric spaces

Djoudi A.

Demonstratio Mathematica

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2003

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Vol. 36, nr 2

463--470

EN

In the present note we give a common fixed point theorem. The theorem extend the results of Popa [8] to the case of four self mappings which are compatible of type (B) and satisfying a contractive relation.

7

A simple proof of Kasparov's technical theorem

Djoudi A.

Demonstratio Mathematica

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2001

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Vol. 34, nr 3

651-657

EN

In this note we use the technic of quasi-central approximate units to give anew simple proof of Kasparov's technical theorem [5, §3. th. 4]. An extended version of the theorem to a finite number of C*-algebras is given and proved. In the end we show that the theorem has implications for the problem of lifting derivations.