scholarly journals A New Approach to the Solution of Non-Linear Integral Equations via Various FBe-Contractions

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 206 ◽  
Author(s):  
Sumati Panda ◽  
Asifa Tassaddiq ◽  
Ravi Agarwal
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Metric Space ◽  
Efficient Solution ◽  
New Approach ◽  
Non Linear ◽  
Fixed Point Technique ◽  
Linear Integral ◽  
Linear Integral Equations

In this article, we introduce and establish various approaches related to the F-contraction using new sorts of contractions, namely the extended F B e -contraction, the extended F B e -expanding contraction, and the extended generalized F B e -contraction. Thereafter, we propose a simple and efficient solution for non-linear integral equations using the fixed point technique in the setting of a B e -metric space. Moreover, to address conceptual depth within this approach, we supply illustrative examples where necessary.

scholarly journals An extension of Karapinar and Sadaranganis’ result through the C-class functions by using α-admissible mapping with applications

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 973-993
Author(s):  
Sudipta Ghosh ◽  
C. Nahak
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Ordered Structure ◽  
Linear Contraction ◽  
Admissible Mapping ◽  
New Findings ◽  
Non Linear ◽  
New Type ◽  
Linear Integral ◽  
Linear Integral Equations

The main objective of this work is to introduce a new type of non-linear contraction via C-class functions by using ?-admissible mapping. Our new results extend and generalize the very recent results of Karapinar and Sadarangani (2015. RACSAM. [37]). Illustrative examples are given to support our new findings. We have shown that our results satisfy the periodic fixed point results after modifying the contraction. Next, we extend our main findings from a self-mapping T to two self-mappings T; S. Also, an example is provided to justify the effectiveness of our new result on two self mappings, where the partially ordered structure fails. Finally, we apply our new findings to solve ordinary differential and non-linear integral equations.

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 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.

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