K. Jha

Note on common fixed points under strict contractive conditions

The first error occurs in lines 4–7 on page 329 which claim that the inequality d(f u,ff u) < max { d(gu,gf u), k[d(f u,gu)+ d(ff u,gf u)]/2, k[d(ff u,gu)+ d(f u,gf u)]/2= kd(f u,ff u) leads to a contradiction. However, this inequality does not lead to a contradiction unless condition (ii) is slightly modified. To overcome this problem we replace the condition (ii) of Theorem 2.1 by d(f x,fy) < max { d(gx,gy), k[d(f x,gx)+ d(fy,gy)]/2, [d(fy,gx)+ d(f x,gy)]/2, 1 k < 2. (ii′) With the above modification the theorem can be proved along the similar lines as given in the original one with minor changes in accordance with the replaced condition (ii′).

Minimization of the Principal Eigenvalue Under Neumann Boundary Conditions

We investigate the minimization of the principal eigenvalue of the Laplacian under Neumann boundary conditions in case the weight has indefinite sign and varies in a class of rearrangements. Biologically, this minimization problem is motivated by the question of determining the most convenient spatial arrangement of favorable and unfavorable resources for a species to survive. The question may have practical importance in the context of reserve design.