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On theorems for weak solutions of nonlinear differential equations with and without delay in Banach spaces

Adel Mahmoud Gomaa — 2007

Commentationes Mathematicae

In the present work we give an existence theorem for bounded weak solution of the differential equation x ˙ ( t ) = A ( t ) x ( t ) + f ( t , x ( t ) ) , t 0 where { A ( t ) : t I + } is a family of linear operators from a Banach space E into itself, B r = { x E : x r } and f : + × B r E is weakly-weakly continuous. Furthermore, we give existence theorem for the differential equation with delay x ˙ ( t ) = A ^ ( t ) x ( t ) + f d ( t , θ t x ) if t [ 0 , T ] , where T , d 0 , C B r ( [ - d , 0 ] ) is the Banach space of continuous functions from [ - d , 0 ] into B r , f d : [ 0 , T ] × C B r ( [ - d , 0 ] ) E weakly-weakly continuous function, A ^ ( t ) : [ 0 , T ] L ( E ) is strongly measurable and Bochner integrable operator on [ 0 , T ] and θ t x ( s ) = x ( t + s ) for all s [ - d , 0 ] .

Existence and Topological Properties of Solution Sets for Differential Inclusions with Delay

Adel Mahmoud Gomaa — 2008

Commentationes Mathematicae

We consider the problem x ˙ ( t ) A ( t ) x ( t ) + F ( t , θ t x ) ) a.e. on [ 0 , b ] , x = κ on [ - d , 0 ] in a Banach space E , where κ belongs to the Banach space, C E ( [ - d , 0 ] ) , of all continuous functions from [ - d , 0 ] into E . A multifunction F from [ 0 , b ] × C E ( [ - d , 0 ] ) into the set, P f c ( E ) , of all nonempty closed convex subsets of E is weakly sequentially hemi-continuous, θ t x ( s ) = x ( t + s ) for all s [ - d , 0 ] and { A ( t ) : 0 t b } is a family of densely defined closed linear operators generating a continuous evolution operator S ( t , s ) . Under a generalization of the compactness assumptions, we prove an existence result and give some topological...

Existence theorems for nonlinear differential equations having trichotomy in Banach spaces

Adel Mahmoud Gomaa — 2017

Czechoslovak Mathematical Journal

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation x ˙ ( t ) = ( t ) x ( t ) + f ( t , x ( t ) ) , t ( P ) where { ( t ) : t } is a family of linear operators from a Banach space E into itself and f : × E E . By L ( E ) we denote the space of linear operators from E into itself. Furthermore, for a < b and d > 0 , we let C ( [ - d , 0 ] , E ) be the Banach space of continuous functions from [ - d , 0 ] into E and f d : [ a , b ] × C ( [ - d , 0 ] , E ) E . Let ^ : [ a , b ] L ( E ) be a strongly measurable and Bochner integrable operator on [ a , b ] and for t [ a , b ] define τ t x ( s ) = x ( t + s ) for each s [ - d , 0 ] . We prove that, under certain conditions,...

On four-point boundary value problems for differential inclusions and differential equations with and without multivalued moving constraints

Adel Mahmoud Gomaa — 2012

Czechoslovak Mathematical Journal

We deal with the problems of four boundary points conditions for both differential inclusions and differential equations with and without moving constraints. Using a very recent result we prove existence of generalized solutions for some differential inclusions and some differential equations with moving constraints. The results obtained improve the recent results obtained by Papageorgiou and Ibrahim-Gomaa. Also by means of a rather different approach based on an existence theorem due to O. N. Ricceri...

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