Hybrid Fixed Point Theorem Research Articles

Fractional hybrid differential equations with three-point boundary hybrid conditions

In this paper, we study the existence of solutions for hybrid fractional differential equations involving fractional Caputo derivative of order 1 < alphaleq 2. Our results rely on a hybrid fixed point theorem for a sum of three operators due to Dhage. An example is provided to illustrate the theory.

A coupled hybrid fixed point theorem involving the sum of two coupled operators in a partially ordered Banach space with applications

In this paper we prove a coupled hybrid fixed point theorem involving the sum of two coupled operators in a partially ordered Banach space and apply to a pair of nonlinear second order coupled linearly perturbed hybrid differential equations with the periodic boundary conditions for proving the existence and approximation of coupled solutions under certain mixed hybrid conditions. The abstract existence result of the coupled periodic boundary value problems is also illustrated by furnishing a numerical example.

EXISTENCE RESULTS TO A CLASS OF HYBRID FRACTIONAL DIFFERENTIAL EQUATIONS

This article is devoted to the study of existence results to a class of boundary value problems for hybrid fractional differential equations. A couple of hybrid fixed point theorems for the sum of three operators are used for proving the main results. Examples illustrating the results are also presented.

Existence Results of Initial Value Problems for Hybrid Fractional Sum-Difference Equations

We consider a hybrid fractional sum-difference initial value problem and a hybrid fractional sequential sum-difference initial value problem. The existence results of these two problems are proved by using the hybrid fixed point theorem for three operators in a Banach algebra and the generalized Krasnoselskii’s fixed point theorem, respectively.

Existence results for hybrid fractional neutral differential equations

We discuss the existence of solutions of initial value problems for a class of hybrid fractional neutral differential equations. To prove the main results, we use a hybrid fixed point theorem for the sum of three operators. We also derive the dependence of a solution on the initial data and present an example to illustrate the results.

Coupled hybrid fixed point theory involving the sum and product of three coupled operators in a partially ordered Banach algebra with applications

In this paper we prove a couple of coupled hybrid fixed point theorems involving the sum and the product of three coupled operators in a partially ordered Banach algebra and derive two applicable fixed point corollaries on the lines of Dhage (Math Stud 61:81–88, 1992, Panam Math J 9(4):93–102, 1999). An application of our new abstract coupled fixed point result is given to nonlinear coupled hybrid quadratic integral equations on a unbounded interval of real line for proving the existence, approximation and attractivity of the coupled solutions under certain mixed partial Lipschitz and partial compactness type hybrid conditions. The obtained abstract existence and attractivity results of the coupled quadratic integral equations are also illustrated by furnishing a numerical example.

Dhage iteration method for PBVPs of nonlinear first order hybrid integro-differential equations

In this paper, author proves the algorithms for the existence as well as the approximation of solutions to a couple of periodic boundary value problems of nonlinear first order ordinary integro-differential equations using operator theoretic techniques in a partially ordered metric space. The main results rely on the Dhage iteration method embodied in the recent hybrid fixed point theorems of Dhage in a partially ordered normed linear space. The approximation of the solutions are obtained under weaker mixed partial continuity and partial Lipschitz conditions. Our hypotheses and abstract results are also illustrated by some numerical examples.

Hybrid fixed point theorems with application to fractional evolution equations

In this paper, we first prove some hybrid fixed point theorems of Krasnoselskii type via the measure of noncompactness in Banach spaces. Then some coupled fixed point theorems are established using the obtained hybrid fixed point theorems. As an application, we consider the existence of PC-mild solutions for the fractional evolution equations involving sectorial operators and impulsive effects.

Coupled hybrid fixed point theory in a partially ordered metric space and attractivity of nonlinear hybrid fractional integral equations

In this paper, we present some coupled hybrid fixed point theorems for partially condensing mixed monotone mappings in a partially ordered metric space which include among others the coupled hybrid fixed point theorems of Bhaskar and Lakshmikantham (Nonlinear Anal TMA 65:1379–1393, 2006), Berinde (Nonlinear Anal 74:7347–7355, 2011), Dhage (Differ Equ Appl 8:77–97, 2016) and Dhage and Dhage (Nonlinear Stud 21(4):675–656, 2014) as special cases with a different method. An application of the main hybrid fixed point principle is given to coupled nonlinear hybrid fractional functional integral equations for proving an algorithm for the existence and attractivity of the approximate solutions on a unbounded interval under certain mixed algebraic and analytical conditions.

A hybrid fixed-point theorem for set-valued maps

In 1955, M. A. Krasnosel’skii proved a fixed-point theorem for a single-valued map which is a completely continuous contraction (a hybrid theorem). Subsequently, his work was continued in various directions. In particular, it has stimulated the development of the theory of condensing maps (both single-valued and set-valued); the images of such maps are always compact. Various versions of hybrid theorems for set-valued maps with noncompact images have also been proved. The set-valued contraction in these versions was assumed to have closed images and the completely continuous perturbation, to be lower semicontinuous (in a certain sense). In this paper, a new hybrid fixed-point theorem is proved for any set-valued map which is the sum of a set-valued contraction and a compact set-valued map in the case where the compact set-valued perturbation is upper semicontinuous and pseudoacyclic. In conclusion, this hybrid theorem is used to study the solvability of operator inclusions for a new class of operators containing all surjective operators. The obtained result is applied to solve the solvability problem for a certain class of control systems determined by a singular differential equation with feedback.

A coupled hybrid fixed point theorem involving the sum of two coupled operators in a partially ordered Banach space with applications

In this paper we prove a coupled hybrid fixed point theorem involving the sum of two coupled operators in a partially ordered Banach space and apply to a pair of nonlinear second order coupled linearly perturbed hybrid differential equations with the periodic boundary conditions for proving the existence and approximation of coupled solutions under certain mixed hybrid conditions. The abstract existence result of the coupled periodic boundary value problems is also illustrated by furnishing a numerical example.

A nonlocal hybrid boundary value problem of caputo fractional integro-differential equations

In this paper, we discuss the existence of solutions for a nonlocal hybrid boundary value problem of Caputo fractional integro-differential equations. Our main result is based on a hybrid fixed point theorem for a sum of three operators due to Dhage, and is well illustrated with the aid of an example.

Approximating solutions of nonlinear periodic boundary value problems with maxima

In this paper we study a periodic boundary value problem of first order nonlinear differential equations with maxima and discuss the existence and approximation of the solutions. The main result relies on the Dhage iteration method embodied in a recent hybrid fixed point theorem of Dhage (2014) in a partially ordered normed linear space. At the end, we give an example to illustrate the applicability of the abstract results to some concrete periodic boundary value problems of nonlinear differential equations.

Hybrid fixed point theorems for multivalued mappings in Banach algebras under a weak topology setting

In this paper, we introduce a new class of multivalued mappings of the form \({\frac{I-C}{A}}\), where \({A}\) and \({C}\) are multivalued mappings acting on Banach algebras. Using this concept, we present some fixed point theorems in Banach algebras satisfying a certain sequential condition (\({P}\)) in the weak topology setting. Our main tool is the axiomatic measure of weak noncompactness. Our results generalize, extend and improve wellknown results on the subject. An application to illustrate our theory is included.

Existence of Solutions for a Nonlinear Integral Equation via a Hybrid Fixed Point Theorem

This paper is devoted to prove the existence of continuous solutions for a nonlinear integral equation. Our existence theorem extends in a broad sense the analogous one obtained by L.T.P. Ngoc and N.T. Long. To this aim we first prove new variants of the Krasnosel’skii–Sadowskii and of the Krasnosel’skii–Daher fixed point theorems in Hausdorff locally convex topological vector spaces. These hybrid theorems improve some recently obtained results. Moreover a fixed point theorem for a multimap defined on a cartesian product subset of a Banach space is stated.

Approximating solutions of nonlinear second order ordinary differential equations via Dhage iteration principle

In this paper the authors prove algorithms for the existence as well as approximation of the solutions for an initial and a periodic boundary value problem of nonlinear second order ordinary differential equations. The main results rely on the Dhage iteration principle embodied in a recent hybrid fixed point theorem of Dhage (2013) in the partially ordered normed linear spaces and the numerical solution of the considered equations is obtained under weaker mixed partial continuity and partial Lipschitz conditions. Our hypotheses and results are also illustrated by some numerical examples.

Some generalizations of a hybrid fixed point theorem in partially ordered metric spaces and nonlinear functional integral equations

Some generalizations of a hybrid fixed point theorem in partially ordered metric spaces and nonlinear functional integral equations

Dhage iteration method for initial value problems for nonlinear first order hybrid integrodifferential equations

In this paper, we prove algorithms for the existence as well as the approximation of solutions to initial value problems (IVPs) for nonlinear first order ordinary integrodifferential equations using operator theoretic techniques in a partially ordered metric space. The main results rely on the Dhage iteration method embodied in the recent hybrid fixed point theorems of Dhage in a partially ordered normed linear space. The approximations to the solutions are obtained under weaker mixed partial continuity and Lipschitz conditions. The hypotheses and results are also illustrated by some examples.

Existence results for hybrid fractional integro-differential equations

In this paper we study existence results for initial value problems for hybrid fractional integro-differential equations. A couple of hybrid fixed point theorems for the sum of three operators are used for proving the main results. Examples illustrating the results are also presented.

Corrigendum to “Krasnosel’skii Type Hybrid Fixed Point Theorems and Their Applications to Fractional Integral Equations”

Corrigendum to “Krasnosel’skii Type Hybrid Fixed Point Theorems and Their Applications to Fractional Integral Equations”