A construction of monotonically convergent sequences from successive approximations in certain Banach spaces (Q1824989)

Let A: \(X\to X\) (where \(X=C^ q[a,b]\) or \(L^ p[a,b])\) be an \(\alpha\)- contraction \((0<\alpha <1)\) having the fixed point f. Denoting by \(y_ n(x)=A^ n(f_ 0)(x)\) the successive approximations, the main result is the following: ``For any \(f_ 0\in C^ q[a,b]\) (respectively \(f_ 0\in L^ p[a,b])\), any \(\epsilon >0\) and any \(s_ i=\pm 1\), the new sequence defined by \(r_ n(x):=y_ n(x)+M(x)\cdot \alpha^ n\) (where M(x) is a polynomial, respectively a constant when \(X=L^ p[a,b]\), depending on \(A(f_ 0)\), \(f_ 0\), a, b, \(\epsilon\) and \(s_ i)\), satisfies the properties: \(r_ n\to f\) in \(C^ q[a,b]\) (respectively in \(L^ p(a,b])\) and \(s_ i\cdot r^{(i)}_{n+1}(x)<s_ i\cdot r_ n^{(i)}(x)\), for all \(x\in [a,b]\), \(i=0,...,q\), \(n\in N\) (respectively \(\int^{b}_{a}r_{n+1}(x)dx<\int^{b}_{a}r_ n(x)dx\), \(\forall n\in N\), when \(X=L^ p[a,b])''\). Finally, the paper contains an application to the study of an integro-differential equation of the Fredholm type.