On some inverse problems for partial differential equations
In this paper, some new fixed point theorems concerning the nonlinear alternative of Leray-Schauder type are proved in a Banach algebra. Applications are given to nonlinear functional integral equations in Banach algebras for proving the existence results. Our results of this paper complement the results that appear in Granas et. al. (Granas, A., Guenther, R. B. and Lee, J. W., Some existence principles in the Caratherodony theory of nonlinear differential system, J. Math. Pures Appl. 70 (1991),...
The aim of this paper is to study problems of the form: with where V is a set of admissible controls and yu is the solution of the Cauchy problem: , and each is a nonnegative measure with support in [0,t]. After studying the Cauchy problem, we establish existence of minimizers, optimality conditions (in particular in the form of a nonlocal version of the Pontryagin principle) and prove some regularity results. We also consider the more general case where the control also enters the dynamics...
In the paper it is shown that each solution ot the initial value problem (2), (3) has a finite limit for , and an asymptotic formula for the nontrivial solution tending to 0 is given. Further, the existence of such a solutions is established by examining the number of zeros of two different solutions , .