Let $A$ and $B$ be nonempty subsets of a metric space with the distance function d, and $T:A\to B$ is a given non-self-mapping. The purpose of this paper is to solve the nonlinear programming problem that consists in minimizing the real-valued function $x\mapsto d(x,Tx)$, where $T$ belongs to a new class of contractive mappings. We provide also an iterative algorithm to find a solution of such optimization problems.

We study the optimal control problem of a controlled time-symmetric forward-backward doubly stochastic differential equation with initial-terminal state constraints. Applying the terminal perturbation method and Ekeland’s variation principle, a necessary condition of the stochastic optimal control, that is, stochastic maximum principle, is derived. Applications to backward doubly stochastic linear-quadratic control models are investigated.

We prove the equivalence of the convergence of the Mann and multistep iterations with errors for uniformly continuous generalized weak $\mathrm{\Phi }$-pseduocontractive mappings in Banach spaces. We also obtain the convergence results of Mann and multistep iterations with errors. Our results extend and improve the corresponding results.

We consider and study the modified extragradient methods for finding a common element of the solution set $\mathrm{\Gamma }$ of a split feasibility problem (SFP) and the fixed point set $\text{Fix}(S)$ of a strictly pseudocontractive mapping $S$ in the setting of infinite-dimensional Hilbert spaces. We propose an extragradient algorithm for finding an element of $\text{Fix}(S)\cap \mathrm{\Gamma }$ where $S$ is strictly pseudocontractive. It is proven that the sequences generated by the proposed algorithm converge weakly to an element of $\text{Fix}(S)\cap \mathrm{\Gamma }$. We also propose another extragradient-like algorithm for finding an element of $\text{Fix}(S)\cap \mathrm{\Gamma }$ where $S:C\to C$ is nonexpansive. It is shown that the sequences generated by the proposed algorithm converge strongly to an element of $\text{Fix}(S)\cap \mathrm{\Gamma }$.