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.     

A Semilocal Convergence of a Secant–Type Method for Solving Generalized Equations

In this paper we present a study of the existence and the convergence of a secant–type method for solving abstract generalized equations in Banach spaces. With different assumptions for divided differences, we obtain a procedure that have superlinear convergence. This study follows the recent results of semilocal convergence related to the resolution of nonlinear equations (see [11])     

Fixed point theorems for a class of nonlinear operators in Banach spaces and applications

In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results.     

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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Applications of Hilbert’s projective metric to a class of positive nonlinear operators

The purpose of this paper is to study the eigenvalue problems for a class of positive nonlinear operators. Using projective metric techniques and the contraction mapping principle, we establish existence, uniqueness and continuity results for positive eigensolutions of a particular type of positive nonlinear operator. In addition, we prove the existence of a unique fixed point of the operator with explicit norm-estimates. Applications to nonlinear systems of equations and to matrix equations are considered.     

Existence and uniqueness of fixed points for mixed monotone operators with applications

In this paper, we study the uniqueness and existence of fixed points of mixed monotone operators in the partially ordered Banach space. Our conclusions essentially improve the relevant results obtained by Liang and others. Moreover, as an application of our results, we prove the existence and uniqueness of a positive solution for a class of integral equations which cannot be solved by using previously available methods.     

Common fixed points for Banach operator pairs with applications

The existence of common fixed point results for a Banach operator pair under certain generalized contractions is established. The invariant best approximation results are proved as applications and the existence of solutions of variational inequalities is obtained. We also study the solution of functional equations arising from dynamic programming.     

A generalization of Browder’s fixed point theorem with applications

In this paper, the concept of c$c$-compact mapping is introduced. A generalization of Browder’s fixed point theorem and some equivalence forms are given. As applications, the existence of solutions for some variational inequalities and monotone operator equations is discussed.     

Fixed point theorems for generalized set-contraction maps and their applications

A new generalized set-valued contraction on topological spaces with respect to a measure of noncompactness is introduced. Two fixed point theorems for the KKM type maps which are either generalized set-contraction or condensing ones are given. Furthermore, applications of these results for existence of coincidence points and maximal elements are deduced.     

Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations

In this paper, coincidence and common fixed point results are established in a partially ordered cone metric space. An application of our results obtained to prove the existence of a common solution to integral equations is presented.     

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.     

Multiple positive solutions of non-local initial value problems for first order differential systems

The paper gives a new and natural method for the existence of multiple positive solutions for first order differential systems with non-local initial value conditions involving linear functionals. The case of higher order differential equations is also considered. The results are accompanied by numerical examples confirming the theory and proving for practice the importance of the bounds in solution localization.     

A note on the solution of a class of functional equations

On the basis of fixed point theorems for linear topological spaces the authors prove existence theorems for the solutions of functional equations that arise in Dynamic Programming.     

Shooting with degree theory: Analysis of some weighted poly-harmonic systems

In this paper, the author establishes the existence of positive entire solutions to a general class of semilinear poly-harmonic systems, which includes equations and systems of the weighted Hardy–Littlewood–Sobolev type. The novel method used implements the classical shooting method enhanced by topological degree theory. The key steps of the method are to first construct a target map which aims the shooting method and the non-degeneracy conditions guarantee the continuity of this map. With the continuity of the target map, a topological argument is used to show the existence of zeros of the target map. The existence of zeros of the map along with a non-existence theorem for the corresponding Navier boundary value problem imply the existence of positive solutions for the class of poly-harmonic systems.     

Global attractors for von Karman evolutions with a nonlinear boundary dissipation

Dynamic von Karman equations with a nonlinear boundary dissipation are considered. Questions related to long time behaviour, existence and structure of global attractors are studied. It is shown that a nonlinear boundary dissipation with a large damping parameter leads to an existence of global (compact) attractor for all weak (finite energy) solutions. This result has been known in the case of full interior dissipation, but it is new in the case when the boundary damping is the main dissipative mechanism in the system. In addition, we prove that fractal dimension of the attractor is finite. The proofs depend critically on the infinite speed of propagation associated with the von Karman model considered.     

Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator

This paper is devoted to proving the existence and uniqueness of solutions to Cauchy type problems for fractional differential equations with composite fractional derivative operator on a finite interval of the real axis in spaces of summable functions. An approach based on the equivalence of the nonlinear Cauchy type problem to a nonlinear Volterra integral equation of the second kind and applying a variant of the Banach’s fixed point theorem to prove uniqueness and existence of the solution is presented. The Cauchy type problems for integro-differential equations of Volterra type with composite fractional derivative operator, which contain the generalized Mittag-Leffler function in the kernel, are considered. Using the method of successive approximation, and the Laplace transform method, explicit solutions of the open problem proposed by Srivastava and Tomovski (2009) [11] are established in terms of the multinomial Mittag-Leffler function.     

Continuity, compactness, and degree theory for operators in systems involving p-Laplacians and inclusions

The study of weak solutions for systems of nonlinear partial differential equations of elliptic type with inclusions leads to a multivalued operator of superposition type in Sobolev spaces. We show that, under natural assumptions, this operator has the properties which allow to apply degree theory (fixed point index) for multivalued maps. More precisely, this operator is upper semicontinuous and compact with nonempty convex compact values. For the particular case of systems involving p-Laplacians, we show that there is a homeomorphism transforming the whole system to a situation for which a fixed point index is available.     

A class of function spaces and its application in solving second-order nonlinear differential equations

In this paper, we prove an existence theorem for time global monotone positive solutions of nonlinear second-order ordinary differential equations by applying the Schauder-Tikhonov fixed point theorem. This result generalizes the result of existence on a half-line given in Yin (2003) [8].     

Nonvariational problems with critical growth

In this paper, we develop new topological methods for handling nonvariational elliptic problems of critical growth. Our primary goal is to demonstrate how concentration compactness can be applied to achieve topological existence theorems in the nonvariational setting. Our methods apply to both semilinear single equations and systems whose nonlinearity is of critical type.     

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.