We obtain a new Suzuki type coupled fixed point theorem for a multivalued mapping $T$ from $X\times X$ into $\text{C}\text{B}(X)$, satisfying a generalized contraction condition in a complete metric space. Our result unifies and generalizes various known comparable results in the literature. We also give an application to certain functional equations arising in dynamic programming.

Some fixed point theorems for ρ-expansive mappings in modular spaces are presented. As an application, two nonlinear integral equations are considered and the existence of their solutions is proved.

We prove a generalized result on the existence of equilibria for a monotone set-valued map defined on noncompact domain and take its values in an order of topological vector space. As consequence, we give a new variational inequality.

Using the concepts of $G$-metric, partial metric, and $b$-metric spaces, we define a new concept of generalized partial $b$-metric space. Topological and structural properties of the new space are investigated and certain fixed point theorems for contractive mappings in such spaces are obtained. Some examples are provided here to illustrate the usability of the obtained results.

The quasidifferential of a quasidifferentiable function in the sense of Demyanov and Rubinov is not uniquely defined. Xia proposed the notion of the kernelled quasidifferential, which is expected to be a representative for the equivalence class of quasidifferentials. Although the kernelled quasidifferential is known to have good algebraic properties and geometric structure, it is still not very convenient for calculating the kernelled quasidifferentials of $-f$ and $\text{min}\left\{f_i\mid i\in \text{a finite index set}\hspace{0.17em}I\right\}$, where $f$ and $f_i$ are kernelled quasidifferentiable functions. In this paper, the notion of adjoint kernelled quasidifferential, which is well-defined for $-f$ and $\text{min}\left\{f_i\mid i\in I\right\}$, is employed as a representative of the equivalence class of quasidifferentials. Some algebraic properties of the adjoint kernelled quasidifferential are given and the existence of the adjoint kernelled quasidifferential is explored by means of the minimal quasidifferential and the Demyanov difference of convex sets. Under some condition, a formula of the adjoint kernelled quasidifferential is presented.

The purpose of this paper is to introduce and analyze hybrid viscosity methods for a general system of variational inequalities (GSVI) with hierarchical fixed point problem constraint in the setting of real uniformly convex and 2-uniformly smooth Banach spaces. Here, the hybrid viscosity methods are based on Korpelevich’s extragradient method, viscosity approximation method, and hybrid steepest-descent method. We propose and consider hybrid implicit and explicit viscosity iterative algorithms for solving the GSVI with hierarchical fixed point problem constraint not only for a nonexpansive mapping but also for a countable family of nonexpansive mappings in X, respectively. We derive some strong convergence theorems under appropriate conditions. Our results extend, improve, supplement, and develop the recent results announced by many authors.

Some common fixed point theorems for $𝒥\mathscr{H}$-operator pairs are proved. As an application, the existence and uniqueness of the common solution for systems of functional equations arising in dynamic programming are discussed. Also, an example to validate all the conditions of the main result is presented.

We introduce the new type of nonself mapping and study sufficient conditions for the existence of past-present-future (for short PPF) dependent fixed point for such mapping in the Razumikhin class. Also, we apply our result to prove the PPF dependent coincidence point theorems. Finally, we use PPF dependence techniques to obtain solution for a nonlinear integral problem with delay.