Abstract
This paper deals with the existence of extremal solutions for the third-order nonlinear boundary value problem - [ φ ( u ″ ( t ) ) ] ′ = f ( t , u ( t ) ) , t ∈ [ a , b ] , u ( a ) = A , u ″ ( a ) = B , u ″ ( b ) = C , in the presence of a pair of lower and upper solutions in reversed order. Here φ : R → R is an increasing homeomorphism, f : [ a , b ] × R → R is a Carathédory function and A , B , C ∈ R . The proof follows from monotone iterative techniques which are based on suitable anti-maximum principles for adequate operators. To deduce such results, we study some related problems coupled with boundary value conditions of the form p 0 u ( a ) - q 0 u ′ ( a ) = A , p 1 u ( b ) + q 1 u ′ ( b ) = B , u ″ ( a ) = C , and u ( a ) = A , u ′ ( a ) = B , u ″ ( b ) = C .
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