Bounded Solutions for some Gradient Type Systems

Abstract

**Please note that the full text is embargoed** ABSTRACT: C. Corduneanu in [3,TH.1] investigated the existence of a unique bounded solution of the system [see pdf for notation] (1.1) with [see pdf for notation] continuous and of class c(2) in u. In this paper we shall prove the existence of a unique bounded solution of (1.1), stated in [3,TR.1], under weaker conditions, and also we shall discuss the asymptotic behavior of bounded solution of (1.1) on a half-line. We wish to mention that M. M. Belova [1] had similar ideas, but she did consider only a special case (space L2), and without concern for existence. Before we proceed further, we present some results, without proofs, which help simplify the proofs of our main results. Lemma 1.1 is analogous to Lemma 2 of [2, P. 102], Theorem 1.1 is analogous to Theorem 1 of [3], and Lemma 1.2 is a special case of Theorem 1.1. We also need the following definitions for M and M0. [see pdf for notation] where the norm in M0 is the same as in M, also it is easy to see that M0 C M. The author is grateful to professor C. Corduneanu for all his suggestions and comments made during the preparation of this paper.