... minr∈Ipr.4 Journal of Inequalities and ApplicationsOur main results are as the following theorem.Theorem 1.1. If f is Caratheodory and satisfies (H1) and (H2), α and β satisfy αT1 ≤ c ≤ βT1,αT2 ... for any t ∈ T1,T2. The argument of the case ofαt ≤ ut for any t ∈ T1,T2 is similar.Assume that ut >βt for some t ∈ T1,T2, then there exist a t0∈ T1,T2 and ... βT1,αT2 ≤ d ≤ βT2,thenP with 1.1 possesses a solution.Theorem 1.2. If f is Caratheodory and satisfies (H1) and (H2), α and β satisfy αT2 ≤ d ≤ βT2, and gαT1,wT11/pT1−1αT1≥...