On a quadratic integral equation with supremum involving Erdélyi‐Kober fractional order

We introduce Erdélyi‐Kober fractional quadratic integral equation with supremum, namely . This equation contains as special cases numerous integral equations studied by other authors. We show that there exists at least one monotonic solution belonging to C[0, 1] of our equation. The main tools in our analysis are Darbo fixed point theorem and the measure of noncompactness related to monotonicity which was introduced by Bana? and Olszowy. Finally, we present an example for illustrating the natural realizations of our abstract results.     

Solving nonlinear integral equations of Fredholm type with high order iterative methods

The application of high order iterative methods for solving nonlinear integral equations is not usual in mathematics. But, in this paper, we show that high order iterative methods can be used to solve a special case of nonlinear integral equations of Fredholm type and second kind. In particular, those that have the property of the second derivative of the corresponding operator have associated with them a vector of diagonal matrices once a process of discretization has been done.     

The Hukuhara–Kneser property for a nonlinear integral equation

Applying a structure theorem of Krasnosel’skii and Perov, we show that the solution set of a nonlinear integral equation satisfies the classical Hukuhara–Kneser property.     

Applying a fixed point theorem of Krasnosel’skii type to the existence of asymptotically stable solutions for a Volterra-Hammerstein integral equation

Using a fixed point theorem of Krasnosel’skii type, the paper proves the existence of asymptotically stable solutions for a Volterra-Hammerstein integral equation.     

On a nonlinear Hammerstein integral equation with a parameter

This paper, following theories for asymptotically linear operators, the Schaefer fixed point theorem, decomposition of operators, and critical point theory, is mainly concerned with the existence and multiplicity of solutions to a nonlinear Hammerstein integral equation with a parameter. The results show that when the nonlinearity satisfies certain conditions, different parametric intervals lead to different existence results; however, in some cases only the sign of the parameter makes a contribution to the existence of solutions for the problem. Our results can be applied to some well known boundary value problems, and some examples are given.     

ON THE TRANSPORT EQUATION WITH INTEGRALBOUNDARY CONDITIONS AND ITS CONSERVATIVE LAW

Abstract. The well-posedness for the time-dependent neutron transport equation with integral boundary conditions is established in LI space. Some spectral properties of the transport operator are discussed, the dorninant dgenvalue is proved existhlg,and furthermore, the conservative law of migrating particle numbers is established.     

On the existence of asymptotically stable solutions of certain integral equations

In this paper two existence results of asymptotically stable solutions of certain integral equations are presented.     

Conditions for blow-up of solutions of some nonlinear Volterra integral equations

In this paper we give necessary and sufficient conditions for blow-up of solutions for a particular class of nonlinear Volterra equations. We also give some examples.     

Solvability of an implicit fractional integral equation via a measure of noncompactness argument

In this paper, we study the existence of solutions to an implicit functional equation involving a fractional integral with respect to a certain function, which generalizes the Riemann-Liouville fractional integral and the Hadamard fractional integral. We establish an existence result to such kind of equations using a generalized version of Darbo's theorem associated to a certain measure of noncompactness. Some examples are presented.     

Semilinear ordinary differential equation coupled with distributed order fractional differential equation

The system , where D^γ,γ∈[0,2] are operators of fractional differentiation, is investigated and the existence of a mild and classical solution is proven. Also, a necessary and sufficient condition for the existence and uniqueness of a solution to a general linear fractional differential equation , in is given.     

Existence, uniqueness and analyticity for periodic solutions of a non-linear convolution equation

We study existence, uniqueness and analyticity for periodic solutions ofu(x)=( _IR J(y)u(x–y)dy) forxIR.     

On the convergence of Newton's method for a class of nonsmooth operators

We provide an analog of the Newton–Kantorovich theorem for a certain class of nonsmooth operators. This class includes smooth operators as well as nonsmooth reformulations of variational inequalities. It turns out that under weaker hypotheses we can provide under the same computational cost over earlier works [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305] a semilocal convergence analysis with the following advantages: finer error bounds on the distances involved and a more precise information on the location of the solution. In the local case not examined in [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305] we can show how to enlarge the radius of convergence and also obtain finer error estimates. Numerical examples are also provided to show that in the semilocal case our results can apply where others [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305] fail, whereas in the local case we can obtain a larger radius of convergence than before [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305].     

On the convergence of modified Newton methods for solving equations containing a non-differentiable term

We provide a semilocal convergence analysis for certain modified Newton methods for solving equations containing a non-differentiable term. The sufficient convergence conditions of the corresponding Newton methods are often taken as the sufficient conditions for the modified Newton methods. That is why the latter methods are not usually treated separately from the former. However, here we show that weaker conditions, as well as a finer error analysis than before can be obtained for the convergence of modified Newton methods. Numerical examples are also provided.     

The existence of asymptotically stable solutions for a nonlinear functional integral equation with values in a general Banach space

Motivated by the recent known results about the solvability and existence of asymptotically stable solutions for nonlinear functional integral equations in spaces of functions defined on unbounded intervals with values in the n-dimensional real space, we establish asymptotically stable solutions for a nonlinear functional integral equation in the space of all continuous functions on R₊ with values in a general Banach space, via a fixed point theorem of Krasnosel’skii type. In order to illustrate the result obtained here, an example is given.     

Integrable solutions of a nonlinear functional integral equation on an unbounded interval

In this paper we prove the existence of integrable solutions of a generalized functional-integral equation, which includes many key integral and functional equations that arise in nonlinear analysis and its applications. This is achieved by means of an improvement of a Krasnosel’skii type fixed point theorem recently proved by K. Latrach and the author. The result presented in this paper extends the corresponding result of [J. Banas, A. Chlebowicz, On existence of integrable solutions of a functional integral equation under Carathéodory condition, Nonlinear Anal. (2008) doi:10.1016/j.na.2008.04.020]. An example which shows the importance and the applicability of our result is also included.     

Multiple positive solutions for a class of integral inclusions

This paper deals with sufficient conditions for the existence of at least two positive solutions for a class of integral inclusions arising in the traffic theory. To show our main results, we apply a norm-type expansion and compression fixed point theorem for multivalued map due to Agarwal and O’Regan [A note on the existence of multiple fixed points for multivalued maps with applications, J. Differential Equation 160 (2000) 389–403].