Hybrid Fixed Point Theorem Research Articles

Approximating positive solutions of PBVPs of nonlinear first order ordinary quadratic differential equations

In this paper the authors prove the existence as well as approximations of the positive solutions for a periodic boundary value problem of first order ordinary nonlinear quadratic differential equations. An algorithm for the solutions is developed and it is shown that the sequence of successive approximations converges monotonically to the positive solution of related quadratic differential equations under some suitable mixed hybrid conditions. Our results rely on the Dhage iteration principle embodied in a recent hybrid fixed point theorem of Dhage (2014) in partially ordered normed linear spaces. A numerical example is also provided to illustrate the abstract theory developed in the paper.

A hybrid nonlinear alternative theorem and some hybrid fixed point theorems for multimaps

We present a Krasnosel’skii–Schaefer-type nonlinear alternative theorem and some fixed point theorems of Krasnosel’skii–Sadowskii (or Daher) type for multimaps. Our results improve or extend in a broad sense recent ones obtained in literature.

Nonlinear D-set contraction mappings in partially ordered normed linear spaces and applications to functional hybrid integral equations

In this paper the author introduces the notion of partially nonlinear D-set-contraction mappings in a partially ordered normed linear space and prove some hybrid fixed point theorems under certain mixed conditions from algebra, analysis and topology. The applications of abstract results presented here are given to perturbed nonlinear hybrid functional integral equations for proving the existence as well as global attractivity of the comparable solutions under certain monotonic conditions. The abstract theory developed in this paper is also useful to develop the algorithms for the solutions of some nonlinear problems of analysis and allied areas of mathematics.

Partially condensing mappings in partially ordered normed linar spaces and applications to functional integral equations

In this paper, the author introduces a notion of partially condensing mappings in a partially ordered normed linear space and proves some hybrid fixed point theorems under certain mixed conditions of algebra, analysis and topology. The applications of abstract results presented here are given to some nonlinear functional integral equations for proving the existence as well as global attractivity of the comparable solutions under certain monotonicity conditions. The abstract theory presented here is very much useful to develop the algorithms for the solutions of some nonlinear problems of analysis and allied areas of mathematics. A realization of of our hypotheses is also indicated by a numerical example.

Hybrid Fixed Point Theorems in Symmetric Spaces via Common Limit Range Property

Abstract In this paper, we point out that some recent results of Vijaywar et al. (Coincidence and common fixed point theorems for hybrid contractions in symmetric spaces, Demonstratio Math. 45 (2012), 611-620) are not true in their present form. With a view to prove corrected and improved versions of such results, we introduce the notion of common limit range property for a hybrid pair of mappings and utilize the same to obtain some coincidence and fixed point results for mappings defined on an arbitrary set with values in symmetric (semi-metric) spaces. Our results improve, generalize and extend some results of the existing literature especially due to Imdad et al., Javid and Imdad, Vijaywar et al. and some others. Some illustrative examples to highlight the realized improvements are also furnished.

Approximating solutions of nonlinear hybrid differential equations

Abstract We prove the existence as well as approximations of the solutions of initial value problems of first order ordinary nonlinear hybrid differential equations. We rely our results on a recent hybrid fixed point theorem of Dhage (2014) in partially ordered normed linear spaces. A realization is also indicated to illustrate the abstract theory developed in the paper.

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

Some hybrid fixed point theorems of Krasnosel’skii type, which involve product of two operators, are proved in partially ordered normed linear spaces. These hybrid fixed point theorems are then applied to fractional integral equations for proving the existence of solutions under certain monotonicity conditions blending with the existence of the upper or lower solution.

On the existence of positive periodic solutions for totally nonlinear neutral differential equations of the second-order with functional delay

We prove that the totally nonlinear second-order neutral differential equation \[\frac{d^2}{dt^2}x(t)+p(t)\frac{d}{dt}x(t)+q(t)h(x(t))\] \[=\frac{d}{dt}c(t,x(t-\tau(t)))+f(t,\rho(x(t)),g(x(t-\tau(t))))\] has positive periodic solutions by employing the Krasnoselskii-Burton hybrid fixed point theorem.

Extremal solutions of Cauchy problems for abstract fractional differential equations

Abstract In this paper, we study the extremal solutions of Cauchy problems for abstract fractional differential equations. Some definitions such as L 1-Lipschitz-like, L 1-Carathéodory-like and L 1-Chandrabhan-like are introduced. By virtue of the singular integral inequalities with several nonlinearities due to Medved’, the properties of solutions are given. By using a hybrid fixed point theorem due to Dhage, existence results for extremal solutions are established. Finally, we present an example to illustrate our main results.

Hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations

In this paper, some basic hybrid fixed point theorems of Banach and Schauder type and some hybrid fixed point theorems of Krasnoselskii type involving the sum of two operators are proved in a partially ordered normed linear spaces which are further applied to nonlinear Volterra fractional integral equations for proving the existence of solutions under certain mono- tonic conditions blending with the existence of either a lower or an upper solution type function. This research is dedicated in the loving memory of my late father and mother who imbibed in me the honesty, hard-work and services for all.

Existence of extremal solutions for discontinuous fractional functional differential equations

In this paper, we apply the hybrid fixed point theorem to prove the existence of extremal solutions for certain discontinuous fractional neutral functional differential equations. We do not require the continuity of the nonlinearities involved in the equations.

Existence theorems for perturbed abstract measure differential equations

In this paper, an existence result for perturbed abstract measure differential equations is proved via hybrid fixed point theorems of Dhage [B.C. Dhage, On some nonlinear alternatives of Leray–Schauder type and functional integral equations, Arch. Math. (Brno) 42 (2006) 11–23] under the mixed generalized Lipschitz and Carathéodory conditions. The existence of the extremal solutions is also proved under certain monotonicity conditions and using a hybrid fixed point theorem of Dhage given in the above-mentioned reference, on ordered Banach spaces. Our existence results include the existence results of Sharma [R.R. Sharma, An abstract measure differential equation, Proc. Amer. Math. Soc. 32 (1972) 503–510], Joshi [S.R. Joshi, A system of abstract measure delay differential equations, J. Math. Phy. Sci. 13 (1979) 497–506] and Shendge and Joshi [G.R. Shendge, S.R. Joshi, Abstract measure differential inequalities and applications, Acta Math. Hung. 41 (1983) 53–54] as special cases under weaker continuity condition.

HYBRID FIXED POINT THEORY AND EXISTENCE OF EXTREMAL SOLUTIONS FOR PERTURBED NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, some hybrid fixed point theorems are proved which are further applied to first and second order neutral functional differential equations for proving the existence results for the extremal solutions under the mixed Lipschitz, compactness and monotonic conditions.

Hybrid fixed point theory for strictly monotone increasing multi-valued mappings with applications

In this paper, a general hybrid fixed point theorem for the strict monotone increasing multi-valued mappings in ordered Banach spaces is proved via measure of noncompactness and it is further applied to perturbed functional nonconvex differential inclusions for proving the existence results for the extremal solutions under mixed Lipschitz, compactness and strict monotonic conditions.

MULTI-VALUED OPERATORS AND FIXED POINT THEOREMS IN BANACH ALGEBRAS I

In this paper the multi-valued versions of some well-known hybrid fixed point theorems of Dhage [6, 7] in Banach algebras are proved. As an application, an existence theorem for a certain integral inclusion in Banach algebras is proved.

A general multi-valued hybrid fixed point theorem and perturbed differential inclusions

Combining three basic multi-valued versions of Banach, Schauder and Tarski fixed point theorems, a general hybrid fixed point theorem for multi-valued mappings in Banach spaces is proved via measure of noncompactness and it is further applied to perturbed differential inclusions for proving the existence results under mixed Lipschitz, compactness and monotone conditions.

Caristi type hybrid fixed point theorems in menger probabilistic metric space

This paper brings forward the concept of Caristi type hybrid fixed point in MPM-space, by giving two hybrid fixed point theorems and two common hybrid fixed point theorems of sequences of set-valued mappings, the theorems improve and generalize the Caristi's fixed point and correspond to recent important results.

Common fuzzy hybrid fixed point theorems for a sequence of fuzzy mappings

In this paper, we discuss the concepts of fuzzy hybrid fixed points, of g‐Φ‐contractive type fuzzy mappings and common fuzzy hybrid fixed point theorems of a sequence of fuzzy mappings. Our theorems improve and generalize the corresponding recent important results.

Some hybrid fixed point theorems related to optimization

Some hybrid fixed point theorems related to optimization