Condensing operators and topological degree theory in standard fuzzy normed spaces

On cyclic (ψ, �)-contractions in Kaleva-Seikkala's type fuzzy metric spaces

“…Applying Theorem 4.1., we deduce the existence of a unique fixed point of T , which is the unique solution to (26).…”

“…Under the assumptions (H1) and (H2), the integral equation(26) has one and only one solution x * ∈ C(I).Proof. Consider the operator T : Y → Y defined byTx(t) = T 0 k(t, s)f (s, x(s)) ds, t ∈ I, x ∈ Y.Then x ∈ Y is a solution to(26) if and only if x ∈ Y is a fixed point of T .…”

“…Consider the operator T : Y → Y defined byTx(t) = T 0 k(t, s)f (s, x(s)) ds, t ∈ I, x ∈ Y.Then x ∈ Y is a solution to(26) if and only if x ∈ Y is a fixed point of T . Using (H1), for all x, y ∈ Y , for all t ∈ I, we have|Tx(t) − Ty(t)| ≤ T 0 |k(t, s)||f (s, x(s)) − f (s, y(s))|ds ≤ α T 0 |k(t, s)| ln((x(s) − y(s)) d(x, y) 2 + 1),which implies that d(Tx, Ty) 2 ≤ α 2 T max t∈I T 0 k 2 (t, s) ds ln(d(x, y) 2 + 1).…”