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On some nonlinear alternatives of Leray-Schauder type and functional integral equations

Bapurao Chandra Dhage (2006)

Archivum Mathematicum

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),...

On some optimal control problems governed by a state equation with memory

Guillaume Carlier, Rabah Tahraoui (2008)

ESAIM: Control, Optimisation and Calculus of Variations

The aim of this paper is to study problems of the form: i n f ( u V ) J ( u ) with J ( u ) : = 0 1 L ( s , y u ( s ) , u ( s ) ) d s + g ( y u ( 1 ) ) where V is a set of admissible controls and yu is the solution of the Cauchy problem: x ˙ ( t ) = f ( . , x ( . ) ) , ν t + u ( t ) , t ( 0 , 1 ) , x ( 0 ) = x 0 and each ν t 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...

On some properties of the solution of the differential equation u ' ' + 2 u ' r = u - u 3

Valter Šeda, Ján Pekár (1990)

Aplikace matematiky

In the paper it is shown that each solution u ( r , α ) ot the initial value problem (2), (3) has a finite limit for r , and an asymptotic formula for the nontrivial solution u ( r , α ) tending to 0 is given. Further, the existence of such a solutions is established by examining the number of zeros of two different solutions u ( r , α ¯ ) , u ( r , α ^ ) .

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