Existence and Uniqueness Theorem for Voltera Equation First Order

• Abd El-Salam Bo-Geldain

  Dep. of Mach, Faculty of Science, Omar Al-Mukhtar University, Libya.

  Author

In this paper I introduced two theorems for the local existence of a unique solution, one for Nonlinear Volterra Integro-Differential Equation in the additive form , and the other for the general form ; where , with the i.c.  and .

The proof is done by proving that the following operator:

, with , is a contraction mapping in the metric space:

 where  and ; noting that  is a subset of the Banach space  given by:

, each of u and w is a continuous function in its arguments, and  is equipped with the following weighted norm which is known as bielecki's type norm:

 for any finite numbers  and  such that  is the Lipschitz's coefficient of  and  is the Lipschitz's coefficient of  By using the above mentioned norm, I concluded that  is contractive on the following form:

Which reveals that the contraction coefficient  is in (0, 1) ; whence the existence of a unique solution is guaranteed globally if we can guarantee that . Indeed we got this result for all the examples.

A. A. Bojeldain, On the numerical solving of Nonlinear Volterra Integro-Differential Equations, Annales Univ. Sci. Budapest Sect. Compo XI (1991), pp. 105-125.

Bielescki A., Ramarks on the applications of the Banach- Kantorowich- Tichonoff method for the equation S= f (x,y,z,p,q), Acad. Polon. Bull. Sci. IV No.5, (1956) , pp. 259-262.

Janko B., The solving of the nonlinear operational equations in Banach spaces, Monograph in Romanian, Publishing house of the the Romanian academy, (1969).

Pierre Pouzet, Method d'Integration Numerique des Equations Integrales et Integro-Differentielles du Type Volterra de Seconde Espece formulas de Runge-Kutta , Symposium on The Numerical Treatment of ODE's, Integral, and Integro-Differential Equations, Rome (1960), pp. 362-368.

Peter Linz , Analytical and Numerical methods for Volterra Equations, Siam Studies in Applied Mathematics, (1985).

Hutson V. and J. S. Pym, Application of Functional Analysis and Operator Theory, Academic press, (1980).

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