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We present some sufficient conditions for the existence of positive solutions to a third order differential equation subject to nonlocal boundary conditions. Our approach is based on the Krasnosel'skiı-Guo fixed point theorem in cones and the properties of the Green's function corresponding to the BVP under study. The main results are illustrated by suitable examples. [ABSTRACT FROM AUTHOR]

We establish existence results for differential equations of the form: (φ(u′))′=f(t,u,u′),$$ {\left(\varphi \left({u}^{\prime}\right)\right)}^{\prime }=f\left(t,u,{u}^{\prime}\right), $$where f$$ f $$ is continuous and φ$$ \varphi $$ is a homeomorphism, with nonlinear boundary conditions: g(u(0),G(u′))=h(u′(1),H(u))=0;g(u′(0),G(u))=h(u(1),H(u′))=0;g(u(0),G(u))=h(u′(1),H(u′))=0org(u′(0),G(u′))=h(u(1),H(u))=0,$$ {\displaystyle \begin{array}{cc}\hfill g\left(u(0),G\left({u}^{\prime}\right)\right)=h\left({u}^{\prime }(1),H(u)\right)=& 0;\kern6.05pt g\left({u}^{\prime }(0),G(u)\right)=h\left(u(1),H\left({u}^{\prime}\right)\right)=0;\hfill \\ {}\hfill g\left(u(0),G(u)\right)=h\left({u}^{\prime }(1),H\left({u}^{\prime}\right)\right)=& 0\kern6.05pt \mathrm{or}\kern6.05pt g\left({u}^{\prime }(0),G\left({u}^{\prime}\right)\right)=h\left(u(1),H(u)\right)=0,\hfill \end{array}} $$where g,h$$ g,h $$ are continuous and G,H$$ G,H $$ are continuous functionals. Our methods of proofs are based on the extension of Mawhin's continuation theorem for quasi‐linear operators. [ABSTRACT FROM AUTHOR]

In this paper, we deal with two point boundary value problem (BVP) for the functional-differential equation of second order where the function takes values in a cone of a Banach space . For and we obtain the BVP with reflection of the argument. Applying fixed point theorem on strict set-contraction from G. Li, Proc. Amer. Math. Soc. 97 (1986), 277–280, we prove the existence of positive solution in the space . Some inequalities involving and the respective Green's function are used. We also give the application of our existence results to the infinite system of functional–differential equations in the case .

We study the existence and stability of periodic solutions of two kinds of regular equations by means of classical topological techniques like the Kolmogorov-Arnold-Moser (KAM) theory, the Moser twist theorem, the averaging method and the method of upper and lower solutions in the reversed order. As an application, we present some results on the existence and stability of T-periodic solutions of a Liebau-type equation. [ABSTRACT FROM AUTHOR]