Weighted pseudo asymptotically Bloch periodic solutions to nonlocal Cauchy problems of integrodifferential equations in Banach spaces

This paper is mainly concerned with some new asymptotic properties on mild solutions to a nonlocal Cauchy problem of integrodifferential equation in Banach spaces. Under some well-imposed conditions on the nonlocal Cauchy, the neutral and forced terms, respectively, we establish some existence results for weighted pseudo S-asymptotically (ω, k)-Bloch periodic mild solutions to the referenced equation on R + ${\mathbb{R}}_{+}$ by suitable superposition theorems. The results show that the strict contraction of the nonlocal Cauchy and the neutral terms with the state variable has an appreciable effect on the existence and uniqueness of such a solution compared with the forced term. As an auxiliary result, the existence of weighted pseudo S-asymptotically (ω, k)-Bloch periodic mild solutions is deduced under the sublinear growth condition on the force term with its state variable. The existence of weighted pseudo S-asymptotically ω-antiperiodic mild solution is also obtained as a special example.

A Meir-Keeler Type Common Fixed Point Result in Dislocated Metric Space

Meir and E. Keeler [11] generalized the Banach Contraction Principle [1] with the notion of weakly uniformly strict contraction which is famous as a (ε - δ) contraction. In this article, we establish a Meir- Keeler type common fixed point result in dislocated metric space which generalize and extend similar fixed point results in the literature.

Proper contractions and invariant subspaces

LetTbe a contraction andAthe strong limit of{T∗nTn}n≥1. We prove the following theorem: if a hyponormal contractionTdoes not have a nontrivial invariant subspace, thenTis either a proper contraction of class𝒞00or a nonstrict proper contraction of class𝒞10for whichAis a completely nonprojective nonstrict proper contraction. Moreover, its self-commutator[T*,T]is a strict contraction.

INTERVAL MAPS WITH STRICTLY CONTRACTING PERRON–FROBENIUS OPERATORS

In this paper we give a computable criterion for a piecewise-expanding interval map T to be mixing, which at the same time not only establishes explicit bounds on the spectral gap of the associated Perron–Frobenius operator acting on the space of functions of bounded variation, but also establishes strict contraction rates for this operator. Of course such a result cannot be completely general, but our procedure covers a number of examples with inf |T′|>2 and the bounds derived for them compare favorably with other estimates found in the literature.

A fixed point theorem for contraction mappings

LetSbe a closed subset of a Banach spaceEandf:S→Ebe a strict contraction mapping. Suppose there exists a mappingh:S→(0,1]such that(1−h(x))x+h(x)f(x)∈Sfor eachx∈S. Then for anyx0∈S, the sequence{xn}inSdefined byxn+1=(1−h(xn))xn+h(xn)f(xn),n≥0, converges to au∈S. Further, if∑h(xn)=∞, thenf(u)=u.

A Fixed Point Theorem for Weakly Uniformly Strict Contractions(1)

In Meir and Keeler [3], the authors proved a fixed point theorem in a complete metric space (X, d) for a mapping f that satisfies the following condition of weakly uniformly strict contraction:Given ∊ > 0, there exists δ > 0 such that(A)