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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.

In this article it is shown that some of the hypotheses of a fixed point theorem of the present author [B.C. Dhage, On some variants of Schauder's fixed point principle and applications to nonlinear integral equations, J. Math. Phys. Sci. 25 (1988) 603-611] involving two operators in a Banach algebra are redundant. Our claim is also illustrated with the applications to some nonlinear functional integral equations for proving the existence results.

In this paper, some basic hybrid fixed point theorems of Banach and Schauder type and some hybrid fixed point theorems of Krasnoselskii type involving the sum of two operators are proved in a partially ordered normed linear spaces which are further applied to nonlinear Volterra fractional integral equations for proving the existence of solutions under certain monotonic conditions blending with the existence of either a lower or an upper solution type function.

Abstract. In this paper, some basic results concerning the strict and nonstrict differential inequalities and existence of the maximal and minimal solutions are proved for a hybrid differential equation with linear perturbations of second type.