Partially condensing mappings in partially ordered normed linar spaces and applications to functional integral equations

Abstract

In this paper, the author introduces a notion of partially condensing mappings in a partially ordered normed linear space and proves some hybrid fixed point theorems under certain mixed conditions of algebra, analysis and topology. The applications of abstract results presented here are given to some nonlinear functional integral equations for proving the existence as well as global attractivity of the comparable solutions under certain monotonicity conditions. The abstract theory presented here is very much useful to develop the algorithms for the solutions of some nonlinear problems of analysis and allied areas of mathematics. A realization of of our hypotheses is also indicated by a numerical example.

Dhage iteration method in the theory of ordinary nonlinear PBVPs of first order functional differential equations

“…The proof of this lemma is well-known and appears in the papers of Dhage [13] via the Arzelá-Ascoli theorem for compactness. Here we give the proof of the lemma using somewhat different arguments via cones in a Banach space C(J, R).…”

“…the above partially order relation ≤. It is known that the partially ordered Banach space C(J, R) is regular and lattice so that every pair of elements of E has a lower and an upper bound; see [6,11,13] and the references therein. The following useful lemma concerning the Janhavi subsets of C(J, R) follows immediately from the Arzelá-Ascoli theorem for compactness or the cones in a Banach space C(J, R).…”

“…The details of other hybrid fixed point theorems involving the Dhage iteration principle and method are given in [1,2,6,11,12] and the references therein.…”

“…The Dhage iteration method embodied in the following applicable hybrid fixed point principle of Dhage [11] in a partially ordered normed linear space is used as a key tool for our work contained in this paper. The details of other hybrid fixed point theorems involving the Dhage iteration principle and method are given in [1,2,6,11,12] and the references therein.…”

“…(Dhage [11,13]) Let E, , · be a regular partially ordered complete normed linear space such that every compact chain C in E is Janhavi. Let T : E → E be a partially continuous, nondecreasing and partially compact operator.…”