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2 Bulletin of the Polish Academy of Sciences. Mathematics

2 2003

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On functionals with the Cauchy difference bounded by a homogeneous functional

Baron K., Kominek Z.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 3

301-307

Assume V is a real vector space. We consider the functional inequality f(x+y)-f(x)-f(y) is more than or equal to Phi(x,y) (x,y is an element of V) for f : V --> R and Phi : V x V --> R such that Phi(x, .) is homogeneous for every x is an element of V and we provide conditions on f which force the representation f(x)=L(x)+B(x,x) (x is an element of V) with a linear L : V --> R and a bilinear and symmetric B : V x V --> R.

General solutions of two functional inequalities and converses to contraction theorems

Jachymski J.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 2

147-156

We give necessary and sufficient conditions for the existence of some general solutions of the following two functional inequalities with an unknown real function phi: phi(Fx) is less than or equal to eta(phi(x)) and gamma(phi(Fx)) is less than or equal to phi(x). As an application we establish reciprocals to fixed point theorems of Matkowski and Wong. This extends an earlier result of Bessaga on a converse to the Banach contraction principle.