Abstract. Let U be an open bounded subset of $R^d$. We aim to investigate the multiplication operator on Hölder space $C^{1,1}(U)$, specifically focusing on FREDHOLM operators. Our goal is to provide an interpretation of these operators in these spaces and to discuss and prove their properties. In the case where U is a subset of R, we present and prove results regarding these operators. Furthermore, we establish a necessary condition for the FREDHOLM property of these operators.

The collocation methods based on cubic B-spline, are developed to approximate solution of the second and first kind FREDHOLM integral equations. First we collocate the solution by B-spline and the Newton-Cotes formula is used to approximate integral. Convergence analysis has been investigated and proved that the quadratur rule is fourth order convergent. The presented methods are tested to the problem, and the absolute error in the solution are compared with existing methods [1, 2, 7, 10] to verify the accuracy and convergent nature of proposed methods.

A new six order method developed for the approximation FREDHOLM integral equation of the second kind. This method is based on the quintic spline functions (QSF). In our approach, we , rst formulate the Quintic polynomial spline then the solution of integral equation approximated by this spline. But we need to develop the end conditions which can be associated with the quntic spline. To avoid the reduction accuracy, we formulate the end condition in such a way to obtain the band matrix and also to obtain the same order of accuracy. The convergence of the method is discussed by using matrix algebra. Finally, four test problems have been used for numerical illustration to demonstrate the practical ability of the new method.

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In this study, a new application of Taylor expansion is considered to estimate the solution of Volterra-FREDHOLM integral equations (VFIEs) and systems of Volterra-FREDHOLM integral equations (SVFIEs). Our proposed method is based upon utilizing the nth-order Taylor polynomial of unknown function at an arbitrary point and employing integration method to convert VFIEs into a system of linear equations with respect to unknown function and its derivatives. An approximate solution can be easily determined by solving the obtained system. Furthermore, this method leads always to the exact solution if the exact solution is a polynomial function of degree up to n. Also, an error analysis is given. In addition, some problems are provided to demonstrate the validity and applicability of the proposed method.

In this paper we characterize FREDHOLM and invertible composition operators on harmonic Bloch function spaces. Indeed, we provide some necessary and sufficient conditions for FREDHOLMness of composition operators. Also, we investigate the relation between the FREDHOLM and invertible composition operators on harmonic Bloch function spaces.

This paper suggests a simple analytical method for Volterra-FREDHOLM integral equations, the solution process is similar to that by variational-based analytical method, e. g., Ritz method, however, the method requires no establishment of the variational principle for the discussed problem, making the method much attractive for practical applications. The examples show the method is straightforward and effective, and the method can also be extended to other nonlinear problems.