Extremal solutions for third-order nonlinear problems with upper and lower solutions in reversed order Original Research Article

This paper deals with the existence of extremal solutions for the third-order nonlinear boundary value problem View the MathML source-[φ(u″(t))]′=f(t,u(t)),t∈[a,b], Turn MathJax on View the MathML sourceu(a)=A,u″(a)=B,u″(b)=C, Turn MathJax on in the presence of a pair of lower and upper solutions in reversed order. Here φ:R→Rφ:R→R is an increasing homeomorphism, f:[a,b]×R→Rf:[a,b]×R→R is a Carathédory function and A,B,C∈R.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 View the MathML sourcep0u(a)-q0u′(a)=A,p1u(b)+q1u′(b)=B,u″(a)=C, Turn MathJax on and View the MathML source