scholarly journals On Monotonic and Nonnegative Solutions of a Nonlinear Volterra-Stieltjes Integral Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Tomasz Zając
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Bounded Variation ◽  
Fractional Integral ◽  
Functions Of Bounded Variation ◽  
Stieltjes Integral ◽  
Measures Of Noncompactness ◽  
Bounded Interval ◽  
Nonnegative Solutions ◽  
Real Functions

We study the existence of monotonic and nonnegative solutions of a nonlinear quadratic Volterra-Stieltjes integral equation in the space of real functions being continuous on a bounded interval. The main tools used in our considerations are the technique of measures of noncompactness in connection with the theory of functions of bounded variation and the theory of Riemann-Stieltjes integral. The obtained results can be easily applied to the class of fractional integral equations and Volterra-Chandrasekhar integral equations, among others.

Applications of Henstock-Kurzweil integrals on an unbounded interval to differential and integral equations

2018 ◽  
Vol 68 (1) ◽  
pp. 77-88
Author(s):  
Marcin Borkowski ◽  
Daria Bugajewska
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Integral Equations ◽  
Bounded Variation ◽  
Volterra Integral Equation ◽  
Functions Of Bounded Variation ◽  
Hammerstein Integral Equation ◽  
Linear Differential ◽  
Unbounded Intervals ◽  
Differential And Integral Equations

Abstract In this paper we are going to apply the Henstock-Kurzweil integrals defined on an unbounded intervals to differential and integral equations defined on such intervals. To deal with linear differential equations we examine convolution involving functions integrable in Henstock-Kurzweil sense. In the case of nonlinear Hammerstein integral equation as well as Volterra integral equation we look for solutions in the space of functions of bounded variation in the sense of Jordan.

scholarly journals Some properties of Urysohn-Stieltjes integral operators

1998 ◽  
Vol 21 (1) ◽  
pp. 79-88 ◽  
Author(s):  
Józef Banaś
Keyword(s):  
Integral Equation ◽  
Bounded Variation ◽  
Integral Operators ◽  
Functions Of Bounded Variation ◽  
Stieltjes Integral ◽  
Stieltjes Type

We investigate some properties of Urysohn-Stieltjes integral operators. The assumptions will be formulated under which the operators in question transform the space of coatinuous functions into itself or in the space of functions of bounded variation, are continuous or are compact. The solvability of an integral equation of Urysohn-Stieltjes type will be also discussed

scholarly journals From fractional order equations to integer order equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Daniel Cao Labora ◽  
Rosana Rodríguez-López
Keyword(s):  
Integral Equation ◽  
Integral Operator ◽  
Vector Space ◽  
Integral Equations ◽  
Fractional Order ◽  
Fractional Integral ◽  
State Of The Art ◽  
Constant Coefficients ◽  
Fractional Order Integral ◽  
Ordinary Integral

AbstractThe main goal of this article is to show a new method to solve some Fractional Order Integral Equations (FOIE), more precisely the ones which are linear, have constant coefficients and all the integration orders involved are rational. The method essentially turns a FOIE into an Ordinary Integral Equation (OIE) by applying a suitable fractional integral operator.After discussing the state of the art, we present the idea of our construction in a particular case (Abel integral equation). After that, we propose our method in a general case, showing that it does work when dealing with a family of “additive” operators over a vector space. Later, we show that our construction is always possible when dealing with any FOIE under the above-mentioned hypotheses. Furthermore, it is shown that our construction is “optimal” in the sense that the OIE that we obtain has the least possible order.

scholarly journals Existence and Characterization of Solutions of Nonlinear Volterra-Stieltjes Integral Equations in Two Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Mohamed Abdalla Darwish ◽  
Józef Banaś
Keyword(s):  
Integral Equations ◽  
Nonlinear Integral Equation ◽  
Existence Of Solutions ◽  
Stieltjes Integral ◽  
Basic Tool ◽  
Measures Of Noncompactness ◽  
Characterization Of Solutions ◽  
Fractional Orders ◽  
Stieltjes Type

The paper is devoted mainly to the study of the existence of solutions depending on two variables of a nonlinear integral equation of Volterra-Stieltjes type. The basic tool used in investigations is the technique of measures of noncompactness and Darbo’s fixed point theorem. The results obtained in the paper are applicable, in a particular case, to the nonlinear partial integral equations of fractional orders.

scholarly journals Some Nonlinear Gronwall-Bellman-Gamidov Integral Inequalities and Their Weakly Singular Analogues with Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Kelong Cheng ◽  
Chunxiang Guo ◽  
Min Tang
Keyword(s):  
Integral Equation ◽  
Qualitative Study ◽  
Integral Equations ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Operator ◽  
Power Nonlinearity ◽  
Weakly Singular ◽  
Type Integral ◽  
Fractional Integral Equation

Some Gronwall-Bellman-Gamidov type integral inequalities with power nonlinearity and their weakly singular analogues are established, which can give the explicit bound on solution of a class of nonlinear fractional integral equations. An example is presented to show the application for the qualitative study of solutions of a fractional integral equation with the Riemann-Liouville fractional operator.

scholarly journals Existence and Stability of Solutions for Hadamard-Stieltjes Fractional Integral Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Saïd Abbas ◽  
Eman Alaidarous ◽  
Mouffak Benchohra ◽  
Juan J. Nieto
Keyword(s):  
Integral Equations ◽  
Fractional Integral ◽  
Existence Result ◽  
Existence Results ◽  
Stieltjes Integral ◽  
Ulam Stability ◽  
Stability Of Solutions ◽  
Existence And Stability ◽  
Stability Results ◽  
Rassias Stability

We give some existence results and Ulam stability results for a class of Hadamard-Stieltjes integral equations. We present two results: the first one is an existence result based on Schauder’s fixed point theorem and the second one is about the generalized Ulam-Hyers-Rassias stability.

scholarly journals An Extension of Darbo’s Theorem and Its Application

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
A. Samadi ◽  
M. B. Ghaemi
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Existence Of Solutions ◽  
Measures Of Noncompactness ◽  
Darbo Fixed Point Theorem ◽  
Real Functions

Here, some extensions of Darbo fixed point theorem associated with measures of noncompactness are proved. Then, as an application, our attention is focused on the existence of solutions of the integral equationx(t)=F(t,f(t,x(α1(t)),  x(α2(t))),((Tx)(t)/Γ(α))×∫0t‍(u(t,s,max⁡[0,r(s)]⁡|x(γ1(τ))|,  max⁡[0,r(s)]⁡|x(γ2(τ))|)/(t-s)1-α)ds,  ∫0∞v(t,s,x(t))ds),    0<α≤1,t∈[0,1]in the space of real functions defined and continuous on the interval[0,1].

scholarly journals On existence theorems for some generalized nonlinear functional-integral equations with applications

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 2081-2091 ◽  
Author(s):  
Mishra Narayan ◽  
Mausumi Sen ◽  
Ram Mohapatra
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Banach Algebra ◽  
Integral Equations ◽  
Fixed Point Theorems ◽  
Existence Theorems ◽  
Functional Integral ◽  
Measures Of Noncompactness ◽  
Nonlinear Functional ◽  
Functional Integral Equations

In the present paper, utilizing the techniques of suitable measures of noncompactness in Banach algebra, we prove an existence theorem for nonlinear functional-integral equation which contains as particular cases several integral and functional-integral equations that appear in many branches of nonlinear analysis and its applications. We employ the fixed point theorems such as Darbo?s theorem in Banach algebra concerning the estimate on the solutions. We also provide a nontrivial example that explains the generalizations and applications of our main result.

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