Recently, the study of set-differential equations (SDEs) was initiated in a metric space and some basic results of interest were obtained

Existence and monotone iterative technique for some boundary value problems

V.Lakshmikantham, J.V.Devi

Florida Institute of Technology

Department of Mathematical Sciences

Melbourne, FL 32901, USA

Recently, the study of set-differential equations (SDEs) was initiated in a metric space and some basic results of interest were obtained. The investigation of set-differential equations as an independent subject has some advantages. For example, when the set is a single-valued mapping, it is easy to see that the Hukuhara derivative and the integral utilized in formulating the SDEs reduce to the ordinary vector derivative and the integral and therefore, the results obtained in this new framework become the corresponding results in ordinary differential systems. Also, we have only a semilinear complete metric space to work with in the present setup, compared to the normed linear space that one employs in the usual study of ordinary differential systems.

The monotone iterative technique is an effective and flexible mechanism that offers theoretical, as well as constructive existence results in a closed set, namely, the sector. The upper and lower solutions that generate the sector serve as upper and lower bounds for the solutions which can be improved by monotone procedures.

In this paper, an attempt is made to prove existence results using contraction mapping and Schauder fixed point theorem and to develop the monotone iterative technique for second order Boundary Value Problems (BVPs) in the context of SDEs in a metric space.

We provide first the preliminaries needed for setting up set-differential equations. We then discuss the existence results for the boundary value problem of the second order set differential equations utilizing the contraction mapping theorem and the Schauder's fixed point result. Finally, we develop the monotone iterative technique for the considered boundary value problem. The results obtained include, in a natural way, second order boundary value problems for systems of ordinary differential equations.