Point Theorems For Multivalued Contractions Research Articles

Nonlocal Boundary Value Problems for (k,ψ)-Hilfer Fractional Differential Equations and Inclusions

In the present research, single and multi-valued (k,ψ)-Hilfer type fractional boundary value problems of order in (1,2] involving nonlocal integral boundary conditions were studied. In the single-valued case, the Banach and Krasnosel’skiĭ fixed point theorems as well as the Leray–Schauder nonlinear alternative were used to establish the existence and uniqueness results. In the multi-valued case, when the right-hand side of the inclusion has convex values, we established an existence result via the Leray–Schauder nonlinear alternative method for multi-valued maps, while the second existence result, dealing with the non-convex valued right-hand side of the inclusion, was obtained by applying Covitz-Nadler fixed point theorem for multi-valued contractions. The obtained theoretical results are well illustrated by the numerical examples provided.

On (k,ψ)-Hilfer Fractional Differential Equations and Inclusions with Mixed (k,ψ)-Derivative and Integral Boundary Conditions

In this paper we study single-valued and multi-valued (k,ψ)-Hilfer-type boundary value problems of fractional order in (1,2], subject to nonlocal boundary conditions involving (k,ψ)-Hilfer-type derivative and integral operators. The results for single-valued case are established by using Banach and Krasnosel’skiĭ fixed point theorems as well as Leray–Schauder nonlinear alternative. In the multi-valued case, we establish an existence result for the convex valued right-hand side of the inclusion via Leray–Schauder nonlinear alternative for multi-valued maps, while the second one when the right-hand side has non-convex values is obtained by applying Covitz–Nadler fixed point theorem for multi-valued contractions. Numerical examples illustrating the obtained theoretical results are also presented.

(k,ψ)-Hilfer Nonlocal Integro-Multi-Point Boundary Value Problems for Fractional Differential Equations and Inclusions

In this paper, we establish existence and uniqueness results for single-valued as well as multi-valued (k,ψ)-Hilfer boundary value problems of order in (1,2], subject to nonlocal integro-multi-point boundary conditions. In the single-valued case, we use Banach and Krasnosel’skiĭ fixed point theorems as well as a Leray–Schauder nonlinear alternative to derive the existence and uniqueness results. For the multi-valued problem, we prove two existence results for the convex and non-convex nature of the multi-valued map involved in a problem by applying a Leray–Schauder nonlinear alternative for multi-valued maps, and a Covitz–Nadler fixed point theorem for multi-valued contractions, respectively. Numerical examples are presented for illustration of all the obtained results.

Nonlocal Integro-Multi-Point (k, ψ)-Hilfer Type Fractional Boundary Value Problems

In this paper we investigate the criteria for the existence of solutions for single-valued as well as multi-valued boundary value problems involving (k,ψ)-Hilfer fractional derivative operator of order in (1,2], equipped with nonlocal integral multi-point boundary conditions. For the single-valued case, we rely on fixed point theorems due to Banach and Krasnosel’skiĭ, and Leray–Schauder alternative to establish the desired results. The existence results for the multi-valued problem are obtained by applying the Leray–Schauder nonlinear alternative for multi-valued maps for convex-valued case, while the nonconvex-valued case is studied with the aid of Covit–Nadler’s fixed point theorem for multi-valued contractions. Numerical examples are presented for the illustration of the obtained results.

Existence for Impulsive Semilinear Functional Differential Inclusions

In this paper we investigate the existence of solutions for first-order impulsive semilinear functional differential inclusions in Banach spaces. Sufficient condition for the existence is obtained with the well-known Covitz and Nadler’s fixed point theorem for multivalued contractions.

Existence results for a nonlinear fractional differential inclusion

In this paper, we establish some existence results for higher-order nonlinear fractional differential inclusions with multi-strip conditions, when the right-hand side is convex-compact as well as nonconvexcompact values. First, we use the nonlinear alternative of Leray-Schauder type for multivalued maps. We investigate the next result by using the well-known Covitz and Nadler?s fixed point theorem for multivalued contractions. The results are illustrated by two examples.

Fixed point and extended coupled fixed point theorems for multi-valued contractions with respect to the Pompeiu functional

Fixed point and extended coupled fixed point theorems for multi-valued contractions with respect to the Pompeiu functional

Fixed point theorems for multivalued contractions on b-metric spaces with graph

In this paper, we consider multivalued maps defined on a complete b-metric space endowed with direct graph. We discuss some fixed point results for maps, called multivalued Gb-contraction, multivalued weak Gb-contraction and multivalued (Gb,ϕ)-contraction. Our results extend/generalize many pre-existing results in the literature.

Fixed Point Theorems for Multi-Valued Contractions in $b$ -Metric Spaces With Applications to Fractional Differential and Integral Equations

The aim of this manuscript is to establish common fixed points results for multi-valued mappings via generalized rational type contractions in complete b-metric spaces. Using the derived results, existence of solutions to certain integral equations and fractional differential equations in the frame of Caputo fractional derivative are studied. Examples are provided for the authenticity of the presented work.

A note on the equivalence of some metric and H-cone metric fixed point theorems for multivalued contractions

AbstractIn this paper, by using Minkowski functional introduced by Kadelburg et al. (Appl. Math. Lett. 24:370-374, 2011) or nonlinear scalarization function introduced by Du (Nonlinear Anal. 72:2259-2261, 2010), we prove some equivalences between vectorial versions of fixed point theorems for H-cone metrics in the sense of Arshad and Ahmad and scalar versions of fixed point theorems for (general) Hausdorff-Pompeiu metrics (in usual sense).

Fixed point theorems for multivalued contractions in Kasahara spaces

In this paper we present some fixed point results for multivalued α-contractions. Our results are obtained in a more general setting, the so called Kasahara space. Some of them are generalizations of Maia’s type fixed point result for multivalued α-contractions. As application, a fixed point theorem for integral inclusions is given.

Fixed point theorems for multivalued generalized contractions of rational type in complete metric spaces

The purpose of this paper is to present some fixed point theorems for multivalued contractions of rational type. We extend the results of I. Cabrera, J. Harjani and K. Sadarangan, A fixed point theorem for contractions of rational type in partially ordered metric spaces, Annali dellUniversita di Ferrara, DOI 10.1007/s11565-013-0176-x, to the case of multivalued operators.

Existence of solutions for fractional differential inclusions with nonlocal Riemann-Liouville integral boundary conditions

In this paper, we discuss the existence of solutions for a boundary value problem of fractional differential inclusions with nonlocal Riemann-Liouville integral boundary conditions. Our results include the cases when the multivalued map involved in the problem is (i) convex valued, (ii) lower semicontinuous with nonempty closed and decomposable values and (iii) nonconvex valued. In case (i) we apply a nonlinear alternative of Leray-Schauder type, in the second case we combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo, while in the third case we use a fixed point theorem for multivalued contractions due to Covitz and Nadler.

Fixed point theorems for multi-valued contractions with w-distance

Some fixed point theorems for a few nonlinear multi-valued contraction mappings in complete metric spaces with w-distance are proved. The results presented in this paper generalize the recent results due to Ćirić (2009) [3], Feng and Liu (2006) [4], Klim and Wardowski (2007) [8], Latif and Abdou (2011) [12] and Liu et al. (2010) [14]. Seven examples are given to explain our results.

Fixed point theorems for multi-valued contractions in distance spaces

In this paper, we first introduce a distance space and then give a new fixed point theorem for multi-valued contractions in such spaces. Even in the case of metric spaces, our main theorem unifies and generalizes some recent results in the literature. Some examples are given to show that the fixed point result given here is a genuine generalization.

On some generalizations of Ekeland’s principle and inward contractions in gauge spaces

We give generalizations in complete gauge spaces of the following results: Bishop-Phelps’ theorem, Ekeland’s variational principle, Caristi’s fixed point theorem, the drop theorem and the flower petal theorem. We show that our generalizations are equivalent. We apply those results to obtain fixed point theorems for multivalued contractions defined on a closed subset of a complete gauge space and satisfying a generalized inwardness condition.

A Filippov’s Theorem, Some Existence Results and the Compactness of Solution Sets of Impulsive Fractional Order Differential Inclusions

In this paper, we first present an impulsive version of Filippov's Theorem for fractional differential inclusions of the form, $$\begin{array}{lll} \quad \qquad D^{\alpha}_{*}y(t) & \in & F(t, y(t)), \quad\; {\rm a.e.}\ t\, \in \, J{\backslash} \{t_{1}, \ldots, t_{m}\}, \ \alpha\, \in \, (0,1], \\ y(t^{+}_{k}) - y(t^{-}_{k}) & = & I_{k}(y(t^{-}_{k})), \quad k = 1, \ldots, m, \\ \qquad \qquad y(0) & = & a,\end{array}$$ where J = [0, b], \({D^{\alpha}_{*}}\) denotes the Caputo fractional derivative and F is a set-valued map. The functions Ikcharacterize the jump of the solutions at impulse points tk (\({k = 1, \ldots , m}\)). In addition, several existence results are established, under both convexity and nonconvexity conditions on the multivalued right-hand side. The proofs rely on a nonlinear alternative of Leray-Schauder type and on Covitz and Nadler’s fixed point theorem for multivalued contractions. The compactness of solution sets is also investigated.

Towards Lim

The paper contains an elegant extension of the Nadler fixed point theorem for multivalued contractions (see Theorem 21). It is based on a new idea of the α-step mappings (see Definition 17) being more efficient than α-contractions. In the present paper this theorem is a tool in proving some fixed point theorems for “nonexpansive” mappings in the bead spaces (metric spaces that, roughly speaking, are modelled after convex sets in uniformly convex spaces). More precisely the mappings are nonexpansive on a set with respect to only one point – the centre of this set (see condition (4)). The results are pretty general. At first we assume that the value of the mapping under consideration at this central point looks “sharp” (see Definition 6). This idea leads to a group of theorems (based on Theorem 7). Their proofs are compact and the theorems, in particular, are natural extensions of the classical results for (usual) nonexpansive mappings. In the second part we apply the idea of Lim to investigate the regular sequences and here the proofs are based on our extension of Nadler's Theorem. In consequence we obtain some fixed point theorems that generalise the classical Lim Theorem for multivalued nonexpansive mappings (see e.g. Theorem 26).

On Fixed Point Theorems for Multivalued Contractions

Three concepts of multivalued contractions in complete metric spaces are introduced, and the conditions guaranteeing the existence of fixed points for the multivalued contractions are established. The results obtained in this paper extend genuinely a few fixed point theorems due to Ćirić (2009) Feng and Liu (2006) and Klim and Wardowski (2007). Five examples are given to explain our results.

Impulsive differential inclusions with fractional order

In this paper, we first present an impulsive version of the Filippov–Ważewski theorem and a continuous version of the Filippov theorem for fractional differential inclusions of the form D∗αy(t)∈F(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=I¯k(y(tk−)),k=1,…,m,y(0)=a,y′(0)=c, where J=[0,b],D∗α denotes the Caputo fractional derivative, and F is a set-valued map. The functions Ik,I¯k characterize the jump of the solutions at impulse points tk(k=1,…,m). Additional existence results are obtained under both convexity and nonconvexity conditions on the multivalued right-hand side. The proofs rely on the nonlinear alternative of Leray–Schauder type, a Bressan–Colombo selection theorem, and Covitz and Nadler’s fixed point theorem for multivalued contractions. The compactness of the solution set is also investigated. Finally, some geometric properties of solution sets, Rδ sets, acyclicity and contractibility, corresponding to Aronszajn–Browder–Gupta type results, are obtained. We also consider the impulsive fractional differential equations D∗αy(t)=f(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=Īk(y(tk−)),k=1,…,m,y(0)=a,y′(0)=c, and D∗αy(t)=f(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(0,1],y(tk+)=Ik(y(tk−)),k=1,…,m,y(0)=a, where f:J×R→R is a single map. Finally, we extend the existence result for impulsive fractional differential inclusions with periodic conditions, D∗αy(t)∈φ(t,y(t)),a.e. t∈J∖{t1,…,tm},α∈(1,2],y(tk+)=Ik(y(tk−)),k=1,…,m,y′(tk+)=I¯k(y(tk−)),k=1,…,m,y(0)=y(b),y′(0)=y′(b), where φ:J×R→P(R) is a multivalued map. The study of the above problems use an approach based on the topological degree combined with a Poincaré operator.