On a class of Volterra and Fredholm non-linear integral equations

1976 ◽  
Vol 79 (2) ◽  
pp. 329-335 ◽  
Author(s):  
P. J. Bushell
Keyword(s):  
Integral Equations ◽  
Kernel Function ◽  
Existence And Uniqueness ◽  
Volterra Integral Equations ◽  
Fredholm Equations ◽  
Non Linear ◽  
Linear Volterra Integral Equations ◽  
Linear Integral ◽  
Image Position ◽  
Linear Integral Equations

This paper concerns the existence and uniqueness of non-negative solutions of non-linear Volterra integral equations of the typeandwhere the kernel function k(.,.) is non-negative and sufficiently smooth, and either 0 < p < 1 or – 1 < p < 1. We will consider also the corresponding Fredholm equationsand

scholarly journals Non-linear integral equations for Heun functions

1969 ◽  
Vol 16 (4) ◽  
pp. 281-289 ◽  
Author(s):  
B. D. Sleeman
Keyword(s):  
Integral Equations ◽  
Heun Equation ◽  
Mathieu Functions ◽  
Special Cases ◽  
Mathieu’S Equation ◽  
Heun Functions ◽  
Non Linear ◽  
Linear Integral ◽  
Image Position ◽  
Linear Integral Equations

Some years ago Lambe and Ward (1) and Erdélyi (2) obtained integral equations for Heun polynomials and Heun functions. The integral equations discussed by these authors were of the formFurther, as is well known, the Heun equation includes, among its special cases, Lamé's equation and Mathieu's equation and so (1.1) may be considered a generalisation of the integral equations satisfied by Lamé polynomials and Mathieu functions. However, integral equations of the type (1.1) are not the only ones satisfied by Lamé polynomials; Arscott (3) discussed a class of non- linear integral equations associated with these functions. This paper then is concerned with discussing the existence of non-linear integral equations satisfied by solutions of Heun's equation.

Fixed points of set-valued F-contractions and its application to non-linear integral equations

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3377-3390 ◽  
Author(s):  
Satish Shukla ◽  
Dhananjay Gopal ◽  
Juan Martínez-Moreno
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Uniqueness Of Solutions ◽  
Complete Metric Spaces ◽  
Non Linear ◽  
Linear Integral ◽  
Linear Integral Equations

We observe that the assumption of set-valued F-contractions (Sgroi and Vetro [13]) is actually very strong for the existence of fixed point and can be weakened. In this connection, we introduce the notion of set-valued ?-F-contractions and prove a corresponding fixed point theorem in complete metric spaces. Consequently, we derive several fixed point theorems in metric spaces. Some examples are given to illustrate the new theory. Then we apply our results to establishing the existence and uniqueness of solutions for a certain type of non-linear integral equations.

scholarly journals On Some Classes of Linear Volterra Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Anatoly S. Apartsyn
Keyword(s):  
Integral Equations ◽  
Existence And Uniqueness ◽  
Volterra Equation ◽  
Continuous Solution ◽  
Sufficient Conditions ◽  
Volterra Integral Equations ◽  
Piecewise Smooth ◽  
Smooth Kernels ◽  
Linear Volterra Integral Equations

The sufficient conditions are obtained for the existence and uniqueness of continuous solution to the linear nonclassical Volterra equation that appears in the integral models of developing systems. The Volterra integral equations of the first kind with piecewise smooth kernels are considered. Illustrative examples are presented.

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