In this article we introduce a new definition of impulsive conditions for boundary value problems of first order impulsive integro-differential equations with multi-point boundary conditions. By using the method of lower and upper solutions in reversed order coupled with the monotone iterative technique, we obtain the extremal solutions of the boundary value problem. An example is also discussed to illustrate our results. Mathematics Subject Classification 2010: 34B15; 34B37.Keywords: impulsive integro-differential equations, multi-point boundary value problem, lower and upper solutions, monotone iterative technique.

This paper is concerned with the existence of extremal solutions of periodic boundary value problems for second-order impulsive integro-differential equations with integral jump conditions. We introduce a new definition of lower and upper solutions with integral jump conditions and prove some new maximum principles. The method of lower and upper solutions and the monotone iterative technique are used.

Square Cycle, C 2 n is a graph that has n vertices and two vertices u and v are adjacent if and only if distance between u and v not greater than 2. In this paper, we show that the determinant of adjacency matrix of square cycle C 2 n are as follows 0, n ≡ 0, 2, 4 mod 6, 16, n ≡ 3 mod 6, 4, n ≡ 1, 5 mod 6.

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