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The main aim of this work is to introduce the new concept of λ−Υ,χ-contraction self-mappings and prove the existence of χ-fixed points for such mappings in metric spaces. Our results generalize and improve some results in existing literature. Moreover, some fixed point results in partial metric spaces can be derived from our χ-fixed points results. Finally, the existence of solutions of nonlinear integral equations is investigated via the theoretical results in this work.

The main goal of this article is to study the existence of a unique positive definite common solution to a pair of matrix equations of the form \begin{eqnarray*} X^r=Q_1 + \displaystyle \sum_{i=1}^{m} {A_i}^*F(X)A_i \mbox{ and } X^s=Q_2 + \displaystyle \sum_{i=1}^{m} {A_i}^*G(X)A_i \end{eqnarray*} where $Q_1,Q_2\in P(n)$, $A_i\in M(n)$ and $F,G:P(n)\to P(n)$ are certain functions and $r,s>1$. In order to achieve our target, we take the help of elegant properties of Thompson metric on the set of all $n \times n$ Hermitian positive definite matrices. To proceed this, we first derive a common fixed point result for a pair of mappings utilizing a certain class of control functions in a metric space. Then, we obtain some sufficient conditions to assure a unique positive definite common solution to the said equations. Finally, to validate our results, we provide a couple of numerical examples with diagrammatic representations of the convergence behaviour of iterative sequences.

In this paper, we study the generalized Hyers-Ulam stability of Euler-Lagrange type cubic functional equation of the form \begin{align*} 2mf(x + my) + 2f(mx - y) = (m^3 + m)[f(x+ y) + f(x - y)] + 2(m^4 - 1)f(y) \end{align*} for all $x,y \in X$, where $m$ is a fixed scalar such that $m \neq 0,1$, and $f$ is a map from a quasi-normed space $X$ to a quasi-Banach space $Y$ over the same field with $X$ by applying the alternative fixed point theorem.

In this paper, we give some further comments to the counterexample and the results of R.~K. Bisht in [R.~K. Bisht. \newblock {Comment on: A new fixed point theorem in the fractal space}. \newblock {\em Indag. Math. (N.S.)}, 29(2):819--823, 2018.]., Comment: 5 pages

We prove some common fixed point results for two mappings satisfying generalized contractive condition in b-metric space. Note that b-metric of main results in this work are not necessarily continuous. So our results extend and improve several previous works. We also present one example that shows the applicability and usefulness of our results.

The aim of this paper is to obtain some new integral type fixed point theorems for nonself weakly compatible mappings in symmetric spaces satisfying generalized $(\psi ,\phi )$ -contractive conditions employing the common limit range property. We furnish some interesting examples which support our main theorems. Our results generalize and extend some recent results contained in Imdad et al. (2013) to symmetric spaces. Consequently, a host of metrical common fixed theorems are generalized and improved. In the process, we also derive a fixed point theorem for four finite families of mappings which can be utilized to derive common fixed point theorems involving any number of finite mappings.

We introduce two new concepts of weakly relaxed $\eta \text{-}\alpha $ monotone mappings and weakly relaxed $\eta \text{-}\alpha $ semimonotone mappings. Using the KKM technique, the existence of solutions for variational-like problems with weakly relaxed $\eta \text{-}\alpha $ monotone mapping in reflexive Banach spaces is established. Also, we obtain the existence of solution for variational-like problems with weakly relaxed $\eta \text{-}\alpha $ semimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem.

We introduce the notion of generalized ${\alpha }_{\ast}$ -admissible mappings. By using this notion, we prove a fixed point theorem. Our result generalizes Mizoguchi-Takahashi’s fixed point theorem. We also provide some examples to show the generality of our work.

In this article, we introduce two new different classes of noncommuting selfmaps. The first class is more general than J H -operator class of Hussain et al. (Common fixed points for J H -operators and occasionally weakly biased pairs under relaxed conditions. Nonlinear Anal. 74(6), 2133-2140, 2011) and occasionally weakly compatible class. We establish the existence of common fixed point theorems for these classes. Several invariant approximation results are obtained as applications. Our results unify, extend, and complement several well-known results. 2000 Mathematical Subject Classification: 47H09; 47H10.

In 2007, Kamran [T. Kamran, Multivalued f-weakly Picard mappings, Nonlinear Anal. 67 (2007) 2289–2296] extended the notion of multi-valued mapping from weak contraction and generalized (α,L)-weak contraction to f-weak contraction and generalized multi-valued f-weak contraction. He also obtained some common fixed point theorems with the notion of T-weakly commuting at a coincidence point of a hybrid pair. In this paper, we can drop the condition of T-weakly commuting in Theorems 2.9 and 3.5 in [5]. We further extend the notion of generalized multi-valued f-weak contraction and introduce the notion of generalized multi-valued (f,α,β)-weak contraction. We also establish some coincidence and common fixed point theorems with generalized multi-valued (f,α,β)-weak contraction mappings. Our results extend and generalize several common fixed point theorems of many authors.