Integrable solutions of a mixed type operator equation

In this paper we prove the existence of integrable solutions for a generalized mixed type operator equation, which contains many key integral and functional equations appearing frequently in Mathematical literature. Our main tool is a Krasnosel’skii type fixed point theorem recently proved by Latrach and Taoudi, the first author. An existence theory for a class of nonlinear transport equations is also developed.     

Local asymptotic attractivity for nonlinear quadratic functional integral equations

In this paper, an existence result for local asymptotic attractivity of the solutions is proved for a nonlinear quadratic functional integral equation under certain growth conditions which in turn gives the existence as well as asymptotic stability of solutions. A couple of examples are provided for indicating the natural realizations of abstract theory presented in the paper.     

Measures of weak noncompactness,nonlinear leray-schauder alternatives in banach algebras satisfying condition (p) and an application

In this paper, we establish some new nonlinear Leray-Schauder alternatives for the sum and the product of weakly sequentially continuous operators in Banach algebras satisfying certain sequential condition (P). The main condition in our results is formulated in terms of axiomatic measures of weak noncompactness. As an application, our results are used to prove the existence of solutions for a nonlinear integral equation.     

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.     

Monotone convergence theorems for Henstock-Kurzweil integrable functions and applications

In this paper we prove monotone convergence theorems for Henstock-Kurzweil integrable functions from a compact real interval to an ordered Banach space. These theorems are then applied to prove existence results for solutions of a discontinuous functional integral equation containing Henstock-Kurzweil integrable functions.     

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.     

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.     

Existence of asymptotically stable solutions for a mixed functional nonlinear integral equation in N variables

Motivated by recent known results about the solvability of nonlinear functional integral equations in one, two or N variables, this paper proves the existence of asymptotically stable solutions for a mixed functional integral equation in N variables with values in a general Banach space via a fixed point theorem of Krasnosels'ki?  type. In order to illustrate the results obtained here, an example is given.     

On global existence and attractivity results for nonlinear functional integral equations

In this paper two existence results concerning the global attractivity and global asymptotic attractivity for a certain functional nonlinear integral equation are proved. Our existence results include several existence as well as attractivity results obtained earlier by Banas and Dhage (2008) [1], Hu and Yan (2006) [3], Dhage (2009) [15] and Banas and Rzepka (2003) [7] as special cases under some weaker Lipschitz conditions. A measure theoretic fixed point theorem of Dhage (2008) [6] is used in formulating our main results and the characterizations of solutions are obtained in the space of functions defined, continuous and bounded on unbounded intervals.     

Existence and continuous dependence results for nonlinear differential inclusions with infinite delay

This paper is concerned with nonlinear functional differential inclusions with infinite delay in Banach spaces. Using tools involving the measure of noncompactness and multi-valued fixed point theory, existence and continuous dependence results are obtained, for integral solutions, without the assumption of compactness on the associated nonlinear semigroup.     

Existence and uniqueness of fixed points for some mixed monotone operators

In this paper, we introduce the e-concave-convex operator. Without any compact or continuous assumptions, we prove the existence and uniqueness of fixed points, giving monotone iterative sequences for the unique fixed point for the operator. Finally, we apply the results to an integral equation of polynomial type which possesses items of measurable functions.     

On existence of integrable solutions of a functional integral equation under Carathéodory conditions

We study the solvability of a functional integral equation in the space of Lebesgue integrable functions on an unbounded interval. Using the conjunction of the technique of measures of weak noncompactness with the classical Schauder fixed point principle we show that the equation in question is solvable in the mentioned function space. Our existence result is obtained under the assumption that functions involved in the investigated functional integral equation satisfy Carathéodory conditions. Moreover, that result generalizes several ones obtained earlier in many research papers and monographs.     

Existence of positive periodic solutions for a functional differential equation with a parameter

In this paper, we study the existence, multiplicity and nonexistence of positive ω$\omega$-periodic solutions for a kind of nonautonomous functional differential equation by employing the fixed point theorem in cones. Some new results are obtained. As an application of our results, we discuss the existence of a positive periodic solution for a class of models of blood cell production with a parameter, which has not been investigated by using previous methods.     

Equations containing locally Henstock-Kurzweil integrable functions

A fixed point theorem in ordered spaces and a recently proved monotone convergence theorem are applied to derive existence and comparison results for solutions of a functional integral equation of Volterra type and a functional impulsive Cauchy problem in an ordered Banach space. A novel feature is that equations contain locally Henstock-Kurzweil integrable functions.     

Existence of solutions for generalized variational inequality problems in Banach spaces

In this paper, we prove the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over compact convex subsets in a reflexive Banach space with a Fréchet differentiable norm. Moreover, we give some conditions that guarantee the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over unbounded closed convex subsets. The result obtained in this paper improves and extends the recent ones announced by Yu and Yang [J. Yu, H. Yang, Existence of solutions for generalized variational inequality problems, Nonlinear Anal., 71 (2009) e2327-e2330] and many others.     

Existence and continuity with respect to parameter of solutions to stochastic Volterra equations in a plane

In this paper we study stochastic Volterra equations in a plane. These equations contain integrals with respect to fields of locally bounded variation and square-integrable strong martingales. We prove the existence and the uniqueness of solutions of such equations with locally integrable (in some measure) trajectories, assuming that the coefficients of equations possess the Lipschitz property with respect to the functional argument. We prove that a solution of a stochastic Volterra integral equation in a plane is continuous with respect to parameter.     

ON THE EXISTENCE AND UNIQUENESS THEOREMS OF SOLUTIONS FOR A CLASS OF THE SYSTEMS OF MIXED MONOTONE OPERATOR EQUATIONS WITH APPLICATION

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On the existence of integrable solutions for a nonlinear quadratic integral equation

In this paper, we investigate the existence of integrable solution to a nonlinear integral equation, which includes many important integral and functional equations that arise in nonlinear analysis and its applications. Our results are obtained under rather general assumptions. In particular, the solvability of the well known Chandrasekhar equation is discussed under appropriate assumptions. Our analysis uses the technique of measures of weak noncompactness and rely on a variant of Schauder’s fixed point theorem.