(Differencialnie Uravnenia i Protsesy Upravlenia)

About
History
Editorial Page
Addresses
Scope
Editorial Staff
Submission Review
For Authors
Publication Ethics
Issues
Русская версия

**A.M.A. El-Sayed**

Faculty of Science,

Alexandria University,

Alexandria,

Egypt

**W.G. El-Sayed**

Faculty of Science,

Alexandria University,

Alexandria,

Egypt

**A. A. H. Abd El-Mowla**

Faculty of Science,

Omar Al-Mukhtar University,

Derna, Libya

The existence of solution of the functional equations and functional integral and differential equations, in different spaces of functions, have been s tudied by some authors. Here, we are concerned with an initial value problem of the nonlinear functional differential equation and present two theorems for the existence of at least one weak solution for this functional differential equation in the reflexive and nonreflexive Banach spaces relative to the weak topology. For this aim we study, firstly, the existence of weakly continuous solutions for a nonlinear functional equation in the reflexive and nonreflexive Banach spaces

- fixed point theorem
- functional equation
- measure of weak noncompactness
- Weak solution

- J. M. Ball,
*Weak continuity properties of mapping and semi-groups*, Proc. Royal Soc. Edinbourgh Sect. A,**72**(1973-1974), 275-280 - M. Cichon,
*Weak solutions of ordinary differential equations in Banach spaces*, Discuss. Math. Differ. Incl. Control Optim.,**15**(1995), 5-14 - M. Cichon, I. Kubiaczyk,
*Existence theorem for the Hammerstein integral equation*, Discuss. Math. Differ. Incl. Control Optim.,**16**(2) (1996), 171-177 - F. S. De Blasi,
*On a property of the unit sphere in Banach spaces*, Bull. Math. Sot. Sci. Math. Roum.**21**(1977)259-262 - A. M. A. El-Sayed, H. H. G. Hashem,
*Weak maximal and minimal solutions for Hammerstein andUrysohn integral equations in reflexive Banach spaces*, Differential Equation and Control Processes, No.**4**(2008), 50-62 - I. Kubiaczyk,
*On a fixed point theorem for weakly sequentially continuous mapping*. Discuss. Math., Differ. Incl.**15**(1995)15-20 - W. J. Knight,
*Solutions of differential equations in B-spaces*, Duke Math. J.**41**(1974), 437-442 - D. O'Regan,
*Integral equations in reflexive Banach spaces and weak topologies*, Amer. Math. Soc., Vol.**124**, No. 2, (1996), 607-614 - D. O'Regan,
*Fixed point theory for weakly sequentially continuous mapping*, Math. Comput. Modeling,**27**(1998), 1-14 - D. O'Regan,
*Weak solutions of ordinary differential equations in Banach spaces*, Appl. Math. Lett.**12**(1999), 101-105 - W. Rubin,
*Functional Analysis*, McGraw-Hill, Inc. New York, Harlow, (1991. ( - H. A. H. Salem, A. M. A. El-Sayed,
*Fractional order integral equations in reflexive Banach spaces*, Math. Slovaca,**55**, No. 2 (2005), 169-181