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We study an approximate coincidence point and a common fixed point problem for a hybrid pair of mappings with constraints in Menger PM-spaces, and obtain some new results. We derive interesting consequences of the main results by using the properties of a Menger–Hausdorff metric, and analogous results based on graphs instead of partial orders can be similarly formulated. Moreover, we construct two examples to reveal that the main results are valid, and show that the main results can be used to explore the existence of solutions to a system of nonlinear integral equations.

In this paper, we introduce the concepts of cyclic $$\varphi $$ -contractions in Menger PM-spaces and cyclic weak $$\phi $$ -contractions in generalized Menger PM-spaces. Based on these concepts, some fixed point results for cyclic $$\varphi $$ -contractions in Menger PM-spaces are obtained. Furthermore, some fixed point theorems for cyclic weak $$\phi $$ -contractions and cyclic $$\varphi $$ -contractions in generalized Menger PM-spaces are obtained by assuming that the number of cyclic sets is odd. An example is also given to show the validity of our main results.

In this paper, the new concepts of \(\alpha \)-admissible mappings with respect to \(\eta \) in single-valued case and \(\alpha _{*}\)-admissible mappings with respect to \(\eta _{*}\) in set-valued case in Menger PM-spaces are introduced, and some new fixed point results for both single-valued and set-valued mappings under certain contractive conditions are studied, which extend and generalize some results in corresponding literatures. Some examples are provided to illustrate the validity of our main results, and an application is also given to discuss the existence of solution for Volterra type integral equations.

Some new coupled coincidence point and coupled fixed point theorems are established in partially ordered metric-like spaces, which generalize many results in corresponding literatures. An example is given to support our main results. As an application, we discuss the existence of the solutions for a class of nonlinear integral equations.

When A ∈ B ( H ) and B ∈ B ( K ) are given, we denote by M C the operator acting on the infinite dimensional separable Hilbert space H ⊕ K of the form M C = ( A C 0 B ) . In this paper, we first give some necessary and sufficient conditions for M C to be a left invertible operator (an upper semi-Weyl, upper semi-Fredholm) operator for some C ∈ B ( K , H ) , which extend the corresponding results in Cao et al. (2006) [4] , Cao and Meng (2005) [5] , Hwang and Lee (2001) [12] and Li and Du (2006) [15] . Then we present some counter-examples.

In this paper, some Banach spaces are introduced. Based on these spaces and the coincidence degree theory, a 2m-point boundary value problem for a coupled system of impulsive fractional differential equations at resonance is considered, and the new criterion on existence is obtained. Finally, an example is also given to illustrate the availability of our main results. MSC:34A08, 34B10, 34B37.