On Lipschitz Perturbations of a Self-Adjoint Strongly Positive Operator

In this paper we study semilinear equations of the formAu+λF(u)=f, whereAis a linear self-adjoint operator, satisfying a strong positivity condition, andFis a nonlinear Lipschitz operator. As applications we develop Krasnoselskii and Ky Fan type approximation results for certain pair of maps and to illustrate the usability of the obtained results, the existence of solution of an integral equation is provided.

A General Iterative Algorithm with Strongly Positive Operators for Strict Pseudo-Contractions

This paper deals with a new iterative algorithm{xn}with a strongly positive operatorAfor ak-strict pseudo-contractionTand a non-self-Lipschitzian mappingSin Hilbert spaces. Under certain appropriate conditions, the sequence{xn}converges strongly to a fixed point ofT, which solves some variational inequality. The results here improve and extend some recent related results.

A Note on the Parabolic Differential and Difference Equations

The differential equationu'(t)+Au(t)=f(t)(−∞<t<∞)in a general Banach spaceEwith the strongly positive operatorAis ill-posed in the Banach spaceC(E)=C(ℝ,E)with norm‖ϕ‖C(E)=sup−∞<t<∞‖ϕ(t)‖E. In the present paper, the well-posedness of this equation in the Hölder spaceCα(E)=Cα(ℝ,E)with norm‖ϕ‖Cα(E)=sup−∞<t<∞‖ϕ(t)‖E+sup−∞<t<t+s<∞(‖ϕ(t+s)−ϕ(t)‖E/sα),0<α<1, is established. The almost coercivity inequality for solutions of the Rothe difference scheme inC(ℝτ,E)spaces is proved. The well-posedness of this difference scheme inCα(ℝτ,E)spaces is obtained.