Type Fixed Point Theorem Research Articles

Stability results on mild solution of impulsive neutral fractional stochastic integro-differential equations involving Poisson jumps

This work studies the existence and the p-th moment asymptotic stability of the mild solution of some neutral fractional stochastic integro-differential equations involving non-instantaneous impulses and Poisson jumps. Sufficient conditions proving existence and asymptotic stability of solutions are obtained utilizing stochastic analysis, resolvent operator and Krasnoselskii-Schaefer type fixed point theorem.

Some fixed point results on generalized metric spaces

In this paper, generalized metric spaces are introduced as a common generalization of tvs-cone metric spaces, partial metric spaces and b-metric spaces, and a unified approach is proposed to some fixed point results by using generalized metric spaces. Specifically, Banach's contraction principle and Kannan type fixed point theorem, as well as other types fixed point results on generalized metric spaces are given, respectively.

On coupled impulsive fractional integro-differential equations with Riemann-Liouville derivatives

In this paper, we investigate the existence, uniqueness and stability of coupled impulsive fractional integro-differential equations with Riemann-Liouville derivatives. To prove the existence and uniqueness results for afore mentioned system, we use the techniques of Kransnoselskiios type fixed point theorem. Furthermore, different kinds of Ulam stabilities are discussed along with examples, to demonstrate the validity of main results.

Existence, stability and controllability results of fractional dynamic system on time scales with application to population dynamics

Abstract In this manuscript, we investigate the existence, uniqueness, Hyer-Ulam stability and controllability analysis for a fractional dynamic system on time scales. Mainly, this manuscript has three segments: In the first segment, we give the existence of solutions. The second segment is devoted to the study of stability analysis while in the last segment, we establish the controllability results. We use the Banach and nonlinear alternative Lery-Schauder–type fixed point theorem to establish these results. Also, we give some numerical examples for different time scales. Moreover, we give two applications to outline the effectiveness of these obtained results.

Unified Feng-Liu type fixed point theorems solving control problems

The propose of this work is to study unified Feng-Liu type fixed point theorems using $$(\alpha ,\eta )$$ -muti-valued admissible mappings with more general contraction condition in complete metric spaces. The obtained results generalize and improve several existing theorems in the literature. We use these results in metric spaces endowed with binary relations and partially ordered sets. Some non-trivial example have been presented to illustrate facts and show genuineness of our work. At the end, the established results will be used to obtain existence solutions for a fractional-type integral inclusion.

Set-valued Meir–Keeler, Geraghty and Edelstein type fixed point results in b-metric spaces

In the current paper, some new set-valued Meir–Keeler, Geraghty and Edelstein type fixed point theorems are presented in a b-metric space. Some new contractive conditions are introduced and the results are proved using different techniques than their single-valued analogues. Applicability of these newly established theorems is demonstrated with some examples.

Positive solutions to classes of infinite semipositone (p,q)-Laplace problems with nonlinear boundary conditions

We consider one-dimensional (p,q)-Laplace problems:{−(φ(u′))′=λh(t)f(u),t∈(0,1),u(0)=0=au′(1)+g(λ,u(1))u(1), where λ>0, a≥0, φ(s):=|s|p−2s+|s|q−2s, 1<p<q<∞, h∈C((0,1),(0,∞)), f∈C((0,∞),R) with lims→0+⁡f(s)∈(−∞,0)∪{−∞}, and g∈C((0,∞)×[0,∞),(0,∞)) such that g(r,s)s is nondecreasing with respect to s∈[0,∞). Classifying the behaviors of f near infinity, we establish the existence, multiplicity and nonexistence of positive solutions. In particular, we provide a sufficient condition on f to obtain a multiplicity result for the case when lims→∞⁡f(s)sr−1∈(0,∞), 1<r<q, which is new even in semilinear problems (p=q=2). The proofs are based on a Krasnoselskii type fixed point theorem which is fit to overcome a lack of homogeneity.

New Monch–Krasnosel’skii type fixed point theorems applied to solve neutral partial integrodifferential equations without compactness

In this paper, we show the existence of mild solutions for a class of neutral partial integrodifferential equations with lack of compactness. The results are obtained using noncompact resolvent operators and a new fixed point theorem of Monch-Krasnosel’skii type. Our results are applied to a large variety of partial differential equations in which memory effects are considered. An example is provided at the end of the paper to illustrate the main results of this work.

Browder type fixed point theorems and Nash equilibria in generalized games

In this paper, we present two generalizations of the well-known Browder fixed point theorem, one of which is equivalent to the well-known Fan–Knaster–Kuratowski–Mazurkiewicz theorem. As applications, we apply these fixed point theorems to derive existence theorems for Nash equilibria in generalized games which generalize some existing existence theorems in the literature, including the well-known equilibrium existence theorem by Arrow and Debreu (Econometrica 22:265–290, 1954) and the existence theorem by Cubiotti (Int J Game Theory 26:267–273, 1997).

Existence of periodic solutions for hybrid evolution equation with delay in partially ordered Banach spaces

In this paper, we are devoted to consider the periodic problem for the hybrid delayed evolution equation in partially ordered Banach spaces. By means of positive $$C_{0}$$ -semigroup and Krasnoselkii type fixed point theorem, we obtain some new existence theorems of periodic mild solutions under some mixed conditions. The results obtained in this paper generalize and improve the recent achievements on this subject. Finally, an example shows the feasibility of the abstract results.

Some Caristi type fixed point theorems

In 1976, Caristi fixed point theorem was proved for ordinary metric spaces. After that, it was shown that this result is equivalent to Ekeland variational principle which has a great number of applications in many branches of mathematics. In this paper, we first introduce a new concept so called strong $$M_{b}$$ -metric to remedy lackness of continuity of $$M_{b}$$ -metric. Then, we investigate whether Caristi fixed point theorem can be extended to this space. Next, we obtain Caristi type fixed point theorem and some generalizations on strong $$M_{b}$$ -metric spaces. Also, we provide some illustrative and interesting examples showing that our theorems extend the results existing in the literature. Finally, we present some applications of our results to ordinary metric spaces.

Some fixed point theorems via measure of noncompactness with applications to differential equations

In this article, we prove some Darbo’s type fixed point theorems associated with measure of noncompactness via the concept of operators A(f; .) and weakly JS-contractive condition in Banach space. Moreover, as an application of our results, we study the existence of solutions for a system of differential equations. Finally, we present an example to support the effectiveness of our results.

New results on discontinuity at fixed point

We obtain a Meir–Keeler type fixed-point theorem which gives a new solution to the Rhoades’ problem on the existence of contractive mappings that admit discontinuity at the fixed point. Meir–Keeler type solutions of the Rhoades’ problem have not been reported in literature before this. Presenting a new approach, we give another solution to this problem using the set of simulation functions. To emphasize the importance of our theoretical results, we obtain two applications of the main results with some illustrative examples.

Optimal control of Sobolev‐type stochastic Hilfer fractional non‐instantaneous impulsive differential inclusion involving Poisson jumps and Clarke subdifferential

This article is concerned with the optimal control of Sobolev-type Hilfer fractional non-instantaneous impulsive differential inclusion driven by Poisson jumps and Clarke subdifferential. Initially, the existence of a mild solution is established for the proposed Hilfer type fractional problem with novel ideas of non-instantaneous impulses. The non-linear alternative of Leray-Schauder type fixed point theorem, stochastic analysis, the measure of non-compactness and the multivalued analysis are applied to prove the mild solution. Further, the existence of optimal control is derived by employing Balder's theorem. Finally, the application as a stochastic dam pollution model is provided to illustrate the developed theoretical results.

Solvability, Completeness and Computational Analysis of A Perturbed Control Problem with Delays

As a first step, we provide a precise mathematical framework for the class of control problems with delays (which we refer to as the control problem) under investigation in a Banach space setting, followed by careful definitions of the key properties to be analyzed such as solvability and complete controllability. Then, we recast the control problem in a reduced form that is especially amenable to the innovative analytical approach that we employ. We then study in depth the solvability and completeness of the (reduced) nonlinearly perturbed linear control problem with delay parameters. The main tool in our approach is the use of a Borsuk–Ulam type fixed point theorem to analyze the topological structure of a suitably reduced control problem solution, with a focus on estimating the dimension of the corresponding solution set, and proving its completeness. Next, we investigate its analytical solvability under some special, mildly restrictive, conditions imposed on the linear control and nonlinear functional perturbation. Then, we describe a novel computational projection-based discretization scheme of our own devising for obtaining accurate approximate solutions of the control problem along with useful error estimates. The scheme effectively reduces the infinite-dimensional problem to a sequence of solvable finite-dimensional matrix valued tasks. Finally, we include an application of the scheme to a special degenerate case of the problem wherein the Banach–Steinhaus theorem is brought to bear in the estimation process.

Darbo type best proximity point results via simulation function with application

Primarily this work intends to prove the best proximity point (pair) results using the concept of measure of noncompactness and simulation functions. The obtained results generalize and extend some present state of the art on Darbo type fixed point theorems. The main results are applied to actualize the optimum solutions of a system of nonlinear mixed Fredholm–Volterra functional integro-differential equations with local initial conditions.

New fixed point theorem for generalized contractions

In this paper, we give a Ciric type fixed point theorem in a complete metric space; this theorem extends other well-known fixed point theorems ([7], [8], [9]). Two examples are given to demonstrate the importance of our work.

A probabilistic Meir-Keeler type fixed point theorem whichcharacterizes metric completeness

A probabilistic version of the Meir-Keeler type fixed point theorem, which characterizes completeness of themetric space is established. In addition to it, a fixed point theorem for non-expansive mappings satisfying(−δ)type condition in Menger probabilistic metric space (Menger PM-space) is proved. As a byproduct we find anaffirmative answer to the open question on the existence of contractive mappings which admit discontinuity atthe fixed point (see Rhoades, B. E.,Contractive definitions and continuity, Contemporary Mathematics72(1988),233–245, p. 242) in the setting of Menger PM-space.

Ciric type fixed point theorems under c-distance on non-normal cone metric spaces over Banach algebras

In this paper, we obtain Ciric type fixed point theorems for continuous or non-continuous mappings under c-distance on mapping-orbitally complete cone metric spaces over Banach algebras without normalities.

Boyd and Wong Type Fixed Point Theorems in Partial Metric Spaces

Abstract In this paper, we present fixed point results for Boyd and Wong type [3] generalized contractive condition in partial metric spaces. In particular, we generalize the fixed point results due to Akkouchi [1] in complete partial metric spaces in which the continuity requirement for a mapping is relaxed to obtain the results. In addition to that we present a common fixed point theorem for a pair of maps. An illustrative example is also constructed to exhibit the results.