Existence of solution for third order m-point boundary value problem.
In this paper we deal with the boundary value problem in the Hilbert space. Existence of a solutions is proved by using the method of lower and upper solutions. It is not necessary to suppose that the homogeneous problem has only the trivial solution. We use some results from functional analysis, especially the fixed-point theorem in the Banach space with a cone (Theorem 4.1, [5]).
In this paper we examine nonlinear periodic systems driven by the vectorial -Laplacian and with a nondifferentiable, locally Lipschitz nonlinearity. Our approach is based on the nonsmooth critical point theory and uses the subdifferential theory for locally Lipschitz functions. We prove existence and multiplicity results for the “sublinear” problem. For the semilinear problem (i.e. ) using a nonsmooth multidimensional version of the Ambrosetti-Rabinowitz condition, we prove an existence theorem...
In this work, we are interested in the existence of solutions for a class of first order boundary value problems (BVPs for short). We give new sufficient conditions under which the considered problems have at least one solution, one nonnegative solution and two non trivial nonnegative solutions, respectively. To prove our main results we propose a new approach based upon recent theoretical results. The results complement some recent ones.
We establish the existence of solutions for evolution equations in Hilbert spaces with anti-periodic boundary conditions. The energies associated to these evolution equations are quadratic forms. Our approach is based on application of the Schaefer fixed-point theorem combined with the continuity method.
The Cauchy problem for an infinite system of parabolic type equations is studied. General operators of parabolic type of second order with variable coefficients are considered and the system is weakly coupled. We prove the existence and uniqueness of a bounded solution under Carathéodory type conditions and its differentiability, as well as the existence and uniqueness in the class of functions satisfying a natural growth condition. Both results are obtained by the fixed point method.