fixed-point theorems

The existence problem is studied for the impulsive nonlinear Sturm–Liouville equation on the whole line. Existence and uniqueness results are obtained.

This study uses a classic fixed point theorem of cone type in a Banach space to identify the eigenvalue intervals of parameters for which an iterative system of a Hadamard fractional boundary value problem has at least one positive solution. To the best of our knowledge, no attempt has been made to obtain such results for Hadamard-type problems in the literature. We provided an example to illustrate the feasibility of our findings in order to show how effective they are.

In this work, we investigate the existence of weak solutions for the following semi-linear elliptic system\begin{equation*}\left\{\begin{array}{c}-\Delta u+p(x)u=\alpha u+\phi \left( x,v\right) \ \ \ \ \text{in }\Omega ,\\-\Delta v+q(x)v=\beta v+\psi \left( x,u\right)  \ \ \ \  \text{in }\Omega ,%\end{array}\right.\end{equation*}with Dirichlet boundary condition, where $\Omega $ is a bounded open set of $\mathbb{R}^{N}$ $\left( N\geq 2\right) ,$ $\alpha ,\beta $ two real parameters, $\left( p(x),q(x)\right) \in \left( L^{\infty }\left( \Omega \right) \right) ^{2}$ and $p(x),q(x)\geq 0.$ using the Leray-Schauder's topological degree and under some suitable conditions for the non linearities $\phi $ and $\psi$, we show the existence of nontrivial solutions.

In this research article, we study the existence, uniqueness and Ulam-Hyers stability of solutions in connection to the generalized Caputo Langevin fractional differential equations in Banach Space. The existence, uniqueness, and stability in the sense of Ulam are established for the proposed system. Our approach is based on the technique of measure of noncompactness combined with the Monch fixed point theorem, the implementation Banach contraction principle fixed point theorem.  Moreover,  the Ulam--Hyers stability is discussed by utilizing the Urs's. Lastly, we deliver an example to check the efficiency and accuracy of the proposed methods.

The present paper is mainly concerned with the existence, uniqueness, interval of existence and continuous dependence of solutions on initial conditions for nonlinear implicit Caputo-Fabrizio fractional hybrid differential equations. The results are obtained by using fractional calculus, contraction principle theorem and the Gronwall's inequality to show the estimate of the solutions. An example is given to illustrate the effectiveness of our main results.

In this paper, contractive mappings of \CM{} type in fuzzy metric spaces are studied. A class $\Psi_1$ of gauge functions $\psi:(0,1]\to(0,1]$ such that, for any $r\in(0,1)$, there exists $\rho\in(r,1)$ such that $1-r> \tau >1-\rho$ implies $\psi(\tau)\geq 1-r$, is introduced, and it is shown that fuzzy $\psi$-contractive mappings are fuzzy contractive mappings of \CM{} type. A characterization of Cauchy sequences in fuzzy metric spaces is presented, and it is utilized to establish fixed point theorems. Examples are given to support the results. Our results cover those of Mihet (Fuzzy $\psi$-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets Syst. 159(2008) 739-744), Wardowski (Fuzzy contractive mappings and fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 222(2013) 108-114) and others.

In this article, a class of first order neutral delay differential equations with iterative terms is investigated. The proofs of the existence of positive periodic solutions rely on the Krasnoselskii's fixed point theorem together with the Green's functions method. Furthermore, by the aid of the Banach fixed point theorem and under an extra condition, we establish the existence, uniqueness and stability results. We provide an example to show the accuracy of the conditions of the obtained findings which extend and generalize earlier ones in the literature.

In this paper, we investigate the existence and uniqueness of solutions for nonlinear implicit Hilfer-Hadamard fractional differential equations involving both retarded and advanced arguments and nonlocal mixed boundary conditions. We also use the Banach contraction mapping principle and Schaefer’s fixed point theorem to show the existence and uniqueness of solutions. The results obtained here extend the work of Benchohra et al. [10] and Haoues et al. [18]. An example is also
given to illustrate the results.

In this paper, we review some research works on exploring image processing in digital spaces using fixed point theorems. The basic concepts of digital images are mentioned. Moreover, we prove some theorems on digital metric spaces by replacing the conditions in the previously established theorem with a suitable condition.

In this study, we present the existence of solutions for Urysohn integral equations. By using the techniques of noncompactness measures, we employ the basic fixed point theorems such as Petryshyn's fixed point theorem to obtain the mentioned aim in Banach algebra. Then this paper presents a numerical approach based on Haar wavelets to solve the equation. This numerical method does not lead to a nonlinear algebraic equations system. Conducting numerical experiments confirm the theoretical results of the applied method and endorse the accuracy of the method.

This paper investigates the existence of single and multiple positive positive solutions of fractional differential equations with p-Laplacian by means of the Green's function properties, the Guo-Krasnosel'skii fixed point theorem, the monotone iterative technique accompanied by established sufficient conditions and the Leggett-Williams fixed point theorem. Additionally, the main results are illustrated by some examples to show their validity.

Nonlocal functional fractional differential inclusions with impulses effect in Banach spaces are studied. This paper deals with the case when the multivalued function is lower semicontinuous and nonconvex as well as the linear term generates a semigroup which is not,in general, compact. Our results are obtained by using NCHM (noncompactness Hausdorff measure), multivalued properties and theorems of fixed point. We finally present an example to lighten our results.

‎We investigate the existence and uniqueness of solutions for multi-point nonlocal boundary value problems of higher-order nonlinear fractional differential equations by using some well known fixed point theorems‎.

‎In this paper‎, ‎Avery-Henderson (Double) fixed point theorem and Ren fixed point theorem are used to investigate the existence of positive solutions for fractional-order nonlinear boundary value problems on infinite interval‎. ‎As applications‎, ‎some examples are given to illustrate the main results‎.

This paper gives existence results for impulsive fractional semilinear differential inclusions involving Caputo derivative in Banach spaces. We are concerned with the case when the linear part generates a semigroup not necessarily compact, and the multivalued function is upper semicontinuous and compact. The methods used throughout the paper range over applications of Hausdorff measure of noncompactness, and multivalued fixed point theorems. Finally, we provide an example to clarify our results.

This paper is devoted to investigate some existence and uniqueness results of integrable solutions for nonlinear fractional order initial value differential equations involving Caputo operator. We develop the existence of integral solution using Schauder's fixed point theorem. In addition, by applying the Banach contraction principle, we establish uniqueness result. To illustrate the applicability of main results, two examples are presented.

In this paper, we concerned with positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions. We establish the criteria for the existence of at least one, two and three positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions by using a result from the theory of fixed point index, Avery-Henderson fixed point theorem and the Legget-Williams fixed point theorem, respectively.

In this paper, we establish some new critical fixed point theorems for the sum AB in a Banach algebra relative to the weak topology, where I−CA allows to be noninvertible. In addition, a special class of Banach algebras will be considered.

In this paper, we provide two different kinds of fixed point theorems in fuzzy metric spaces. The first kind is for the fuzzy $varepsilon$-contractive type mappings and the second kind is for the fuzzy order $psi$-contractive type mappings. They improve the corresponding conclusions in the literature.