           Available online through www.rjpa.info   Research Journal of Pure Algebra- 1 (5), August – 2011 111 A NOTE ON SYSTEMS OF SUMMATION INEQUALITIES Dr. K. L. Bondar* P. G. Dept. of Mathematics, N. E. S. Science College, Nanded - 431 605 (M.S.) India E-mail: klbondar_75@rediffmail.com (Received on: 13-08-11; Accepted on: 23-08-11) ------------------------------------------------------------------------------------------------------------------------------------------------ ABSTRACT In this paper we discuss some systems of summation inequalities. We also discuss the under and over functions of systems of summation equations. Keywords: Difference Equation, Summation Equation, Summation Inequality, Under and Over Function. ------------------------------------------------------------------------------------------------------------------------------------------------ 1. INTRODUCTION: Agarwal [1], Kelley and Peterson [9] developed the theory of difference equations and difference inequalities. Some difference inequalities and comparison results are obtained by K. L. Bondar [2, 3]. Some summation and difference inequalities are obtained in K. L. Bondar [4, 5]. K. L. Bondar, V. C. Borkar, S. T. Patil [6, 7] and Dang H., Oppenheimer S.F.[8] obtained the existence and uniqueness results for difference equations. Some differential and integral inequalities are given in [10]. In this paper we discuss about systems of summation inequalities. We also discuss the under and over functions of systems of summation equations. 2. PRELIMINARY NOTES Let J = {t 0 , t 0 + 1… t 0 + a}, t 0  0, t 0 ∈ R, and E be an open subset of R n , consider the difference equations with an initial condition, u(t) = g(t, u(t)), u(t 0 ) = u 0 (1) where u 0 ∈ E, u: J  E, g : J × E  R n . The function φ : J  R n is said to be a solution of initial value problem (1), if it satisfies  φ (t) = g (t, φ (t)); φ (t 0 ) = u 0 . The initial value problem is equivalent to the problem  - = + = 1 0 0 )). ( , ( ) ( t t s s u s g u t u By summation convention  - = = 1 0 0 0 )) ( , ( t t s s u s g and so u(t) given above is the solution of (1). 3. MAIN RESULTS: Theorem: 3.1 Assume that (i) K : J × J × R n  R n and K (t, s, x) is nondecreasing in x for each fixed (t, s) and one of the inequalities  - = + ≤ 1 0 )) ( , , ( ) ( ) ( t t s s x s t K t h t x , (2)  - = + ≥ 1 0 )) ( , , ( ) ( ) ( t t s s y s t K t h t y (3) is strict where x,y : J  R n ; ------------------------------------------------------------------------------------------------------------------------------------------------   Dr. K. L. Bondar*  klbondar_75@rediffmail.com