In this paper we investigate linear operators between arbitrary BK spaces $X$ and spaces $Y$ of sequences that are $(\bar{N},q)$ summable or bounded. We give necessary and sufficient conditions for infinite matrices $A$ to map $X$ into $Y$. Further, the Hausdorff measure of noncompactness is applied to give necessary and sufficient conditions for $A$ to be a compact operator.
We consider a nonlinear fractional differential inclusion with nonlocal fractional integro-differential boundary conditions in a Banach space. The existence of at least one solution is proved by using the set-valued analog of Mönch fixed point theorem associated with the technique of measures of noncompactness.
We present an existence theorem for monotonic solutions of a quadratic integral equation of Abel type in C[0, 1]. The famous Chandrasekhar’s integral equation is considered as a special case. The concept of measure of noncompactness and a fixed point theorem due to Darbo are the main tools in carrying out our proof.
We give sufficient conditions for the existence of at least one integrable solution of equation x(t) = f(t) + ∫ t 0 K(t, s)g(s, x(s)) ds. Our assumptions and proofs are expressed in terms of measures of noncompactness.
Let $(\Omega ,\Sigma )$ be a measurable space, $X$ a Banach space whose characteristic of noncompact convexity is less than 1, $C$ a bounded closed convex subset of $X$, $KC(C)$ the family of all compact convex subsets of $C.$ We prove that a set-valued nonexpansive mapping $T\: C\rightarrow KC(C)$ has a fixed point. Furthermore, if $X$ is separable then we also prove that a set-valued nonexpansive operator $T\: \Omega \times C\rightarrow KC(C)$ has a random fixed point.