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

2012 ◽  
Vol 32 (1) ◽  
pp. 5 ◽  
Author(s):  
Abdelouaheb Ardjouni ◽  
Ahcene Djoudi
Author(s):  
Hocine Gabsi ◽  
Abdelouaheb Ardjouni ◽  
Ahcene Djoudi

In this work we study a class of second order nonlinear neutral integro-differential equations    x(t)+f(t,x(t),x(t))x(t)+∑_{j=1}^{N}∫_{t-τ_{j}(t)}^{t}a_{j}(t,s)g_{j}(s,x(s))ds    +∑_{j=1}^{N}b_{j}(t)x′(t-τ_{j}(t))=0,with variable delays and give some new conditions ensuring that the zero solution is asymptotically stable by means of the fixed point theory. Our work extends and improves previous results in the literature such as, D. Pi <cite>pi2,pi3</cite> and T. A. Burton <cite>b12</cite>. An example is given to illustrate our claim.


Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3157-3172
Author(s):  
Mujahid Abbas ◽  
Bahru Leyew ◽  
Safeer Khan

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.


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