The Extended Cone b-Metric-like Spaces over Banach Algebra and Some Applications

In this paper, we introduce the structure of extended cone b-metric-like spaces over Banach algebra as a generalization of cone b-metric-like spaces over Banach algebra. In this generalized space we define the notion of generalized Lipschitz mappings in the setup of extended cone b-metric-like spaces over Banach algebra and investigated some fixed point results. We also provide examples to illustrate the results presented herein. Finally, as an application of our main result, we examine the existence and uniqueness of solution for a Fredholm integral equation.

Solution of system of the Fredholm integral equation of the first kind

The article is devoted to the study of a system of two inhomogeneous Fredholm integral equations of the first kind with two required functions depending on one variable. Integral equations describe the restoration of a blurred image, production costs, etc. Fredholm integral equations with one desired function have been considered in many works, but relatively few works have been devoted to systems of such equations. The questions of stability for the solution of systems and the construction of a regularizing system of equations were investigated, but the solution was not constructed in an explicit form. In this paper, the kernels depend on two variables. The case is considered: in the kernels and inhomogeneities, the variables are separated in the equations; these functions are decomposed on the basis of two functions on the interval of integration. Examples of basic functions are given. A condition is determined under which the system has a unique solution in the chosen basis, formulated as a theorem. The solution is found in the form of an expansion in this basis. To illustrate the results obtained, an example is considered

Infinite Geraghty type extensions and their applications on integral equations

In this article, we discuss about a series of infinite dimensional extensions of some theorems given in (Shumrani et al. in SER Math. Inform. 33(2):197–202, 2018), (Fisher in Math. Mag. 48(4):223–225, 1975), and (Fogh, Behnamian and Pashaie in Int. J. Maps in Mathematics 2(41):1–13, 2019). We also prove a similar Geraghty type construction for Fisher (Math. Mag. 48(4):223–225, 1975) in an infinite dimension using similar techniques as in (Shumrani et al. in SER Math. Inform. 33(2):197–202, 2018) and (Fogh, Behnamian and Pashaie in Int. J. Maps in Mathematics 2(41):1–13, 2019). As an application, we ensure the existence of solutions for infinite dimensional Fredholm integral equation and Uryshon type integral equation.

A new Model for Solving Three Mixed Integral Equations with Continuous and Discontinuous Kernels

In this paper, we discuss a new model to obtain the answer to the following question: how can we establish the different types of mixed integral equations from the Fredholm integral equation? For this, we consider three types of mixed integral equations (MIEs), under certain conditions. The existence of a unique solution of such equations is guaranteed. Using analytic and numerical methods, the three MIEs formulas yield the same Fredholm integral equation (FIE) formula of the second kind. For continuous kernel, the solution of these three MIEs, via the FIEs, is discussed analytically. In addition, for a discontinuous kernel, the Toeplitz matrix method (TMM) and Product Nyström method (PNM) are used to obtain, in each method, a linear algebraic system (LAS). Then, the numerical results are obtained, the error is computed in each case, and compared as well.

Unique Fixed-Point Results in Fuzzy Metric Spaces with an Application to Fredholm Integral Equations

This paper aims at proving some unique fixed-point results for different contractive-type self-mappings in fuzzy metric spaces by using the “triangular property of the fuzzy metric”. Some illustrative examples are presented to support our results. Moreover, we present an application by resolving a particular case of a Fredholm integral equation of the second kind.

Solving a Nonlinear Fredholm Integral Equation via an Orthogonal Metric

In this paper, we prove fixed point theorems using orthogonal triangular α -admissibility on orthogonal complete metric spaces. Some of the well-known outcomes in the literature are generalized and expanded by the obtained results. An instance to help our outcome is being presented.

A Fixed Point Technique for Set-Valued Contractions with Supportive Applications

In this manuscript, exciting fixed point results for a pair of multivalued mappings justifying rational Gupta-Saxena type Ω -contractions in the setting of extended b -metric-like spaces are established. The theoretical results have also been strengthened by some nontrivial examples. Finally, the theoretical results are used to study the existence of the solution of Fredholm integral equation which arises from the damped harmonic oscillator, to study initial value problem which arises from Newton’s law of cooling and to study infinite systems of fractional ordinary differential equations (ODEs).

On Controlled Rectangular Metric Spaces and an Application

In this paper, we introduce the notion of controlled rectangular metric spaces as a generalization of rectangular metric spaces and rectangular b -metric spaces. Further, we establish some related fixed point results. Our main results extend many existing ones in the literature. The obtained results are also illustrated with the help of an example. In the last section, we apply our results to a common real-life problem in a general form by getting a solution for the Fredholm integral equation in the setting of controlled rectangular metric spaces.