Proceedings of the American Mathematical Society

Let E 1 , E 2 {E_1},{E_2} be two Banach spaces, and let f : E 1 → E 2 f:{E_1} \to {E_2} be a mapping, that is “approximately linear". S. M. Ulam posed the problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist". The purpose of this paper is to give an answer to Ulam’s problem.

Let K be a subset of a Banach space X. A mapping F : K → K F:K \to K is said to be asymptotically nonexpansive if there exists a sequence { k i } \{ {k_i}\} of real numbers with k i → 1 {k_i} \to 1 as i → ∞ i \to \infty such that ‖ F i x − F i y ‖ ≦ k i ‖ x − y ‖ , x , y ∈ K \left \| {{F^i}x - {F^i}y} \right \| \leqq {k_i}\left \| {x - y} \right \|,x,y \in K . It is proved that if K is a non-empty, closed, convex, and bounded subset of a uniformly convex Banach space, and if F : K → K F:K \to K is asymptotically nonexpansive, then F has a fixed point. This result generalizes a fixed point theorem for nonexpansive mappings proved independently by F. E. Browder, D. Göhde, and W. A. Kirk.

Let I I be a norm-continuous functional on the space B B of bounded Σ \Sigma -measurable real valued functions on a set S S , where Σ \Sigma is an algebra of subsets of S S . Define a set function v v on Σ \Sigma by: v ( E ) v (E) equals the value of I I at the indicator function of E E . For each a a in B B let \[ J ( a ) = ∫ − ∞ 0 ( v ( a ≥ α ) − v ( S ) ) d α + ∫ 0 ∞ v ( a ≥ α ) d α . J(a) = \int _{ - \infty }^0 {(v (a \geq \alpha ) - v (S))d\alpha + \int _0^\infty {v (a \geq \alpha )d\alpha .} } \] Then I = J I = J on B B if and only if I ( b + c ) = I ( b ) + I ( c ) I(b + c) = I(b) + I(c) whenever ( b ( s ) − b ( t ) ) ( c ( s ) − c ( t ) ) ⩾ 0 (b(s) - b(t))(c(s) - c(t)) \geqslant 0 for all s s and t t in S S .

We consider a nonlocal analogue of the Fisher-KPP equation \[ u t = J ∗ u − u + f ( u ) ,   x ∈ R ,   f ( 0 ) = f ( 1 ) = 0 ,   f > 0   on   ( 0 , 1 ) , u_t =J*u-u+f(u),~x\in R,~f(0)=f(1)=0,~f>0 ~\textrm {on}~(0,1), \] and its discrete counterpart u ˙ n = ( J ∗ u ) n − u n + f ( u n ) {\dot u}_n =(J*u)_n -u_n +f(u_n ) , n ∈ Z n\in Z , and show that travelling wave solutions of these equations that are bounded between 0 0 and 1 1 are unique up to translation. Our proof requires finding exact a priori asymptotics of a travelling wave. This we accomplish with the help of Ikehara’s Theorem (which is a Tauberian theorem for Laplace transforms).

Let X be a uniformly convex Banach space which satisfies Opial’s condition or has a Fréchet differentiable norm, C a bounded closed convex subset of X, and T : C → C T:C \to C an asymptotically nonexpansive mapping. It is then shown that the modified Mann and Ishikawa iteration processes defined by x n + 1 = t n T n x n + ( 1 − t n ) x n {x_{n + 1}} = {t_n}{T^n}{x_n} + (1 - {t_n}){x_n} and x n + 1 = t n T n ( s n T n x n + ( 1 − s n ) x n ) + ( 1 − t n ) x n {x_{n + 1}} = {t_n}{T^n}({s_n}{T^n}{x_n} + (1 - {s_n}){x_n}) + (1 - {t_n}){x_n} , respectively, converge weakly to a fixed point of T.