In this paper, we introduce a new contraction condition that combines the framework of Singh's extension with the classical Chatterjea contraction. This generalized form, called the Singh-Chatterjea contraction, is defined on the p-th iterate of a mapping. We establish fixed point theorems for such mappings in complete metric spaces and show that our results extend and unify both Singh's and Chatterjea's classical fixed point theorems. Illustrative examples and a simple numerical implementation are provided to demonstrate the applicability of the obtained results.

In this paper, we study the existence of solution for the fourth-order three-point boundary value problem having the following form u((4)) (t) + f(t, u(t)) = 0, 0 < t < 1, u(0) = alpha u(eta), u' (0) = u '' (0) = 0, u(1) = beta u(eta), where eta is an element of (0, 1), alpha, beta is an element of R, f is an element of C([0, 1] x R, R), and f (t, 0) not equal 0. We give sufficient conditions that allow us to obtain the existence of solution. And by using the Leray-Schauder nonlinear alternative we prove the existence of at least one solution of the posed problem. As an application, we also given some examples to illustrate the results obtained.

This paper presents a reformulation of classical existence and uniqueness results for second-order boundary value problems (BVPs) using the Kannan fixed point theorem, extending beyond the Banach contraction principle. We shift focus from the nonlinearity j to the solution operator T defined via Green’s function and establish a sufficient condition under which T satisfies the Kannan contraction criterion. Specifically, if the derivative of j is bounded by K and $K·{(\eta -\zeta )}^2/8<1/3$, then T is a Kannan contraction, ensuring a unique solution. This condition applies even when the Banach contraction principle fails. We also explore the plausibility of applying the Chatterjea contraction, though rigorous verification remains open. Examples illustrate the applicability of the results. This work highlights the utility of generalized contractions in differential equations.

Introduction/purpose: The aim of the paper is to prove the existence of solutions for a special case of the sixth-order boundary value problem. Methods: The Leray-Schauder fixed point theorem is used in order to determine lower and upper bound solutions. Results: Lower and upper bound solutions have been found. Conclusions: The sixth-order boundary value problem admits solutions.

This paper develops new unified fixed point theorems by extending the classical Ćirić contractions through the iterative framework introduced by Singh in his generalization of Kannan-type mappings. We first revisit the two well-known definitions of Ćirić contractions in their standard forms and then introduce the concepts of $ \rho $-Ćirić contraction and generalized $ \rho $-Ćirić contraction, in which the contractive condition is imposed on higher iterates of the mapping. These extended classes retain the intrinsic structure of Círic-type mappings while significantly enlarging the family of operators that admit a unique fixed points. For each class, we establish comprehensive fixed point theorems ensuring existence, uniqueness, and global convergence of the associated Picard iteration. Our results unify and generalize several classical fixed-point theorems, including those of Banach, Kannan, Chatterjea, and the original Ćirić contractions, which arise as special cases within this broader framework. Illustrative examples are provided to demonstrate the applicability and sharpness of the proposed generalizations.