Ireneusz Kubiaczyk

On bounded pseudo and weak solutions of a nonlinear differential equation in Banach spaces

In this paper, we prove an existence theorem for bounded pseudo and weak solution of the differential equation x(t) = A(t)x(t) + f{t,x(t)) where /(•, x(-)) is Pettis-integrable for each strongly absolutely continuous function x and / ( t , •) is weakly-weakly sequentially continuous. We also assume some condition expressed in terms of De Blasis measure of weak noncompactness.

EXISTENCE OF SOLUTIONS OF THE DYNAMIC CAUCHY PROBLEM IN BANACH SPACES

Abstract In this paper we obtain the existence of solutions and Carathéodory type solutions of the dynamic Cauchy problem in Banach spaces for functions defined on time scales xΔ(t)=f(t,x(t)),x(0)=x0,t∈Ia,

COMMON FIXED POINT THEOREMS IN METRIC SPACES

\matrix{{x^\Delta (t) = f(t,x(t)),} \hfill & {} \hfill \cr {x(0) = x_0 ,} \hfill & {t \in I_a ,} \hfill }

FIXED POINTS FOR CONTRACTIVE CORRESPONDENCES

where f is continuous or f satisfies Carathéodory conditions and some conditions expressed in terms of measures of noncompactness. The Mönch fixed point theorem is used to prove the main result, which extends these obtained for real valued functions.

Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces

Let (X,d) be a metric space and l e t S, T be two cor re spondences ( i . e . mappings from poin ts to s e t s ) from X to CB(X), which are ne i the r necessa r i ly continuous nor commuting. We s h a l l denots by CB(X) the se t of a l l non-empty closed and bounded subsets of X, by CL(X) the se t of a l l closed subsets of X, by H(A,B) the Hausdorff d i s tance of A, BeCB(X). We s h a l l a lso wr i t e D(A,B) = in f{d(a ,b ) : a e A , be b ) , i U , B ) = = sup j d ( a , b ) : a e A , b e b} . A point x e X w i l l be ca l led a f ixed point of S or T i f x e Sx or x e Ix r e s p e c t i v e l y . Suppose t h a t t he re a re nonnegative r e a l numbers a^ , a2» a^, a^ such t h a t

A FUNCTIONAL DIFFERENTIAL EQUATION IN BANACH SPACES

I n r ecen t papers by Boyd and Wong [ 1 ] , Kannan [ 2 ] , Nadler [ 3 ] , Reich [ 4 , 5 ] , Sehgal [6] and o the r a u t h o r s , t he problem of ex i s t ence of f i x e d po in t has been i n v e s t i g a t e d . I n the p re sen t paper we g e n e r a l i z e t h e s e r e s u l t s . Throughout t h i s pape r , (X,d) denotes a me t r i c space. I f A,B are any non-empty subse t s of X we put D(A,B) = i n f { d ( a , b ) i a £ A, b e B ] , i(A,B) = sup{d(a ,b ) : a e A, b e B ] , H(A,B) = max [sup^D (a , B) s a e A), supfD(b,A) i b e B } ] . Let BN(X) = {C t C i s a non-empty bounded subse t of X], CB(X) = {0 i C i s a non-empty closed and bounded subset of X}, C(X) = {C i C i s a non-empty compact subset of X , and l e t Q = ( d ( x , y ) i x , y e X j , P = Q * Q * Q . I n t h i s paper l e t x X — CB(X) ( i = 1 , 2 ) , xQ £ X and f o r k < 1 , l e t t h e sequence (xQ) be de f ined by

Weak solutions for the dynamic Cauchy problem in Banach spaces

In this paper we prove a common coincidence point theorem for singlevalued and multivalued mappings satisfying an implicit relation under the condition of R-weak commutativity on metric spaces.

Kneser's theorems for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces

In the p r e sen t paper v/e prove some f i x e d and common f i x e d po in t theorems f o r c o n t r a c t i v e type mappings and mul t iva lued mappings i n the met r i c space . These theorems extend theorems of Achari [1] , Ray [6 ] , Reich [7 ] , Singh and W h i t f i e l d [ s ] . Let (X,d) be a met r ic space and l e t S, T be two c o r r e s pondences ( i . e . mappings from p o i n t s t o s e t s ) from X t o CB(X), which a re n e i t h e r n e c e s s a r i l y con t inuous nor commuting. We s h a l l denote by CB(X) the s e t of a l l non-empty c losed and bounded s u b s e t s of X, by CL(X) the s e t of a l l c losed s u b s e t s of X, by H(A,B) the Hausdorf f d i s t a n c e of A,B eCB(X). We s h a l l a l s o w r i t e D(A,B) = i n f { d ( a , b ) : a e A, b eB} , 5(A,B) = = sup ( d ( a , b ) : a e A , b e B } . A poin t x e X w i l l be c a l l e d a f i x e d po in t of S or T i s x e S x or x eTx r e s p e c t i v e l y . The theorems proved i n t h i s paper g e n e r a l i z e some r e s u l t s from [1-10] f o r mappings and c o r r e s p o n d e n c e s . T h e o r e m 1. Let (X,d) be a compact me t r i c space and S f T s X CL(X) and l e t S or T be con t inuous and s a t i s f y c o n d i t i o n s t h e r e e x i s t s a r e a l valued f u n c t i o n