Theory of Difference Equations Numerical Methods and by V. Lakshmikantham

By V. Lakshmikantham

Offers a transparent and complete review of the basic theories, numerical equipment, and iterative approaches encountered in distinction calculus.

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6 CHAPTER 1. DISCRETE CALCULUS Comparison Principle One of the most efficient methods of obtaining information on the behavior of solutions of difference equations, even when they cannot be solved explicitly, is the comparison principle. In general, the comparison principle is concerned with estimating a function satisfying a difference inequality by the solution of the corresponding difference equation. In this section, we shall present various forms of this principle. 1 Let n e N+0,r > 0 andg(n,r) be a nondecreasing function with respect to r for any fixed n.

Be a positive sequence satisfying ( n-l n-2 \ 2/n'S^'Z]^ ' j=0 j=0 / where g ( y , z, w) is a nondecreasing function with respect to its arguments. If 7/0 < U then, for all n > 0 'i/n < '"n- Proof. The proof is by induction. The claim is true for n = 0. Suppose it is true for n — k. l j=0 / Copyright © 2002 Marcel Dekker, Inc. 7. 6 Let yi i = 0, 1, . . be a positive sequence satisfying inequality the yn+i < g(yrnyn-\, • • • ,yn-k}, where g is nondecreasing with respect to its argument. Un-lT-iUn-k}, Uj>y^ Proof.

Then, Pn < un < r n , where Pn and rn are the solutions of the difference - 9i(n,rn). Proof. 1 twice, we obtain the needed estimate. 4 (Discrete Bihari inequality). Suppose that hn is a nonnegative function defined on -/V+0, M > 0. and W is a positive strictly increasing function defined on IR + . If for n > no, yn< Vn where n-l Vn=y0 + M Copyright © 2002 Marcel Dekker, Inc. 28 CHAPTER!. DISCRETE CALCULUS then for n l ( ~ 1 n G N, = in e 7V+|Af ^ ha < G(oo) - G(x 0 ) , I s=no J n-l yn < G-1 ( G(y 0 ) + M X!

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