... upper solution of the problem (3. 30)– (3. 34) , ,1 , (3. 36) F M Atici and D C Biles 131 Theorem 3. 3 assures that there exists a solution y(t) of the problem (3. 30)– (3. 34) such that y ∈ [α,β] We note ... I(x, y) = x − (3. 29) Next we consider the following boundary value problem: y (t) = f t, y σ (t) , t ∈ [0,1]κ \ , (3. 30) y(0) = 1, y(1) = 0, y y∆ + (3. 31) (3. 32) −y − = −1, (3. 33) 1+ 1− 1− − y∆ ... (3. 8) Now we introduce the concept of lower and upper solutions of problem (3. 1)– (3. 5) as follows F M Atici and D C Biles 127 Definition 3. 1 The functions α and β are, respectively, a lower and...