If (alpha=0), there are a great deal of papers devoted to equations (1.2), many of them are important and interesting.
In the literature devoted to equations with finite delay, the phase space is much of time the space of all continuous functions on,, endowed with the uniform norm topology.
Further, if f ∈ C ( ( 0, b ] ) ∩ L loc 1 ( a, b ) such that D α f ∈ C ( ( 0, b ] ) ∩ L loc 1 ( 0, b ), then I α D α f ( t ) = f ( t ) + C t α − 1. for t > 0 and some constant C ∈ R. In the literature devoted to equations with infinite delay, the selection of the state space ℬ plays an important role in the study of both qualitative and quantitative theory.
Many papers have been devoted to CH equation and FCCH equation, see, for example, [10] [15].
In Section 3, we study the existence and stability of solitary waves of equation (1.1) with some special nonlinearities f ( u ) = 1 p u p. Section 4 is devoted to studying equation (1.1) with general nonlinearities f ( u ) satisfying the assumption (A).
An application of this can be found for instance the uncertainty model devoted to Paris Erdogan equation discussed in [58,59].
Things changed with the arrival of SEMNET [ 3], a free internet listserv hosted at the University of Alabama and devoted to structural equation modeling, including factor modeling.
Therefore, considerable attention was paid to study the field devoted to differential equations of fractional order.
However, up to now, almost all studies on anti-periodic fractional boundary value problems have been devoted to fractional equations with the Caputo fractional derivative.
In spite of the variety of physical applications, the analysis of differential and difference equations with mixed singularities is at this moment very incomplete, and few references can be cited (see [12, 18 20]) if compared with the large number of references devoted to singular equations either of attractive or repulsive type (see the review [21 23] and the references therein).
