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In this paper, we consider the fractional Langevin equation with finite delay. Our aim is to establish new conditions ensuring the existence, uniqueness and stability of solutions. This is realized...

In this work, we investigate the existence of solutions for the system of two singular fractional q-differential equations under integral boundary conditions via the concept of Caputo fractional q-derivative and fractional Riemann-Liouville type q-integral. Some new existence results are obtained by applying Krattowski measure of non-compactness. Also, the Darbo?s fixed point theorem and the Lebesgue dominated convergence theorem are the main tools in deriving our proofs. Lastly, we present an example illustrating the primary effects.

There are many forms of solutions that occur obviously, and in various situations one cannot just switch to one of them and establish solutions in that sense. It is valuable to study solutions with strong differentiability properties, but it might be difficult to confirm their existence. Therefore, one essentially considers solutions in a weaker sense. In this note, we establish the existence of monotone solutions of the dynamic system of the bacterial growth of iterative fractional differential equations, found by a new coupled and triple fixed point theorem in the sense of Darbo processing. We propose these theorems (Darbo type) for a new contraction by using a measure of non-compactness in Banach spaces. The new outcomes show some special well-known recent results.

In this paper, we prove the existence of decay integral solutions to a class of fractional differential inclusions with finite delays and estimate their decay rate. For these purposes, we have to construct a suitable regular measure of noncompactness on the space of solutions and then deploy the fixed point theory for condensing multivalued maps. An application to a class of fractional PDE with almost sectorial operator is also given.

We prove the global solvability and weakly asymptotic stability for a semilinear fractional differential inclusion subject to impulsive effects by analyzing behavior of its solutions on the half-line. Our analysis is based on a fixed point principle for condensing multi-valued maps, which is employed for solution operator acting on the space of piecewise continuous functions. The obtained results will be applied to a lattice fractional differential system.

We deal with the Cauchy problem associated with integro-differential inclusions of diffusion-wave type involving infinite delays. Based on the behavior of resolvent operator associated with the linear part, an explicit estimate for solutions will be established. As a consequence, the weak stability of zero solution is proved in case the resolvent operator is asymptotically stable.

We study spectral properties of operators on logarithmic perturbations of the real interpolation spaces with θ = 0 or 1. We also establish estimates for the measure of non-compactness of operators interpolated by those methods.

In this paper, firstly, we introduce Z µ -contraction and ? µ -contraction via stimulation functions. Secondly, we prove some new fixed point theorems for Z µ -contractive mappings and ? µ -contractive mappings. Our results generalize and extend some existing results. Moreover, some examples and an application to functional integral equations are given to support the obtained results.

This paper studies the existence of solutions for nonlocal semi-linear fractional differential equations of Hilfer type in Banach space by using the non-compact measure method in the weighted space of continuous functions. The main result is illustrated with the aid of an example.

By using the partial order method and some new comparison results, the maximal or minimal solution of the initial value problem for nonlinear second order integro-differential equations with delayed arguments in Banach spaces are investigated. In this paper, we require only a lower solution or an upper solution and some weaker conditions presented here, and we extend and improve some recent results (see [1-11]).

The aim of this paper is to establish the existence of decay integral solutions to a class of retarded fractional differential equations involving impulsive effects. The results are obtained by using the fixed point approach and fractional calculus tools in Banach spaces. Applications to both ordinary and partial differential equations are presented.

In this paper, we prove the existence of solutions of some nonlinear functional-integral equation by using a fixed point theorem which satisfy the Darbo condition. The results extend the corresponding results of many authors. In the sequel, we give an example of our main result to highlight the realized improvements.

First we establish some necessary and sufficient conditions for the existence of exponential attractors by using ω -limit compactness and a measure of non-compactness. Then we provide a new method for proving the existence of exponential attractors. We prove the existence of exponential attractors for reaction–diffusion equations and 2D Navier–Stokes equations as simple applications.

This paper concerns with the IVP for set differential equations involving causal operators with memory. By using the techniques of the measure of non-compactness, the existence of solutions has been established. The maximal and minimal solutions are obtained by means of Arzela-Ascoli Theorem.

We study the control systems governed by abstract Volterra equations without uniqueness in a Banach space. By using the technique of the theory of condensing maps and multivalued analysis tools, we obtain the existence result, investigate the topological structure of the solution set, and prove the invariance of a reachability set of the control system under nonlinear perturbations. Examples concerning fractional order differential equations and first order evolution equations with multiple delays are proposed to demonstrate the applications to approximate controllability results.

In this paper we investigate some topological properties of solution sets of a differential equation of fractional order in Banach spaces. Our assumptions and proofs are expressed in terms of measures of noncompactness.

In this paper, the Mönch fixed point theorem is used to investigate the existence of positive solutions for the second-order boundary value problem with integral boundary conditions of nonlinear impulsive differential equations on an infinite interval in a Banach space.

In this paper, by using the cone theory and monotone iterative technique, we investigate the existence of extremal solutions and unique solution of the integral boundary value problem for a class of first-order impulsive integro-differential equations in a real Banach space. An explicit iterative scheme for the unique solution and an error estimate of the approximation sequence are also derived.

The first goal of this article is to discuss the existence of solutions of nonlinear quadratic integral equations. These equations are considered in the Banach space L p (ℝ+). The arguments used in the existence proofs are based on Schauder's and Darbo's fixed point theorems. In particular, to apply Schauder's fixed point theorem based method, a special care is devoted to the proof of the L p -compactness of the operators associated with our nonlinear quadratic integral equations. The second goal of this work is to study a numerical method for solving nonlinear Volterra integral equations of a fairly general type. Finally, we provide the reader with some examples that illustrate the different results of this work.

We first present a necessary and sufficient condition for the existence of a pullback -attractor for non-autonomous infinite-dimensional dynamical systems. Then, as an application, we apply our method to non-autonomous weakly dissipative wave equations and prove the existence of a strong pullback -attractor in .

We establish a formula for the measure of non-compactness of an operator interpolated by the general real method generated by a sequence lattice Γ. The formula is given in terms of the norms of the shift operators in Γ.

In this paper, the minimal and maximal solutions of initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces are investigated by using new comparison results and the partial order method. The results presented here essentially improve, generalize and unify many well-known results.

Let Ω be an open subset of ℝn and let p ∈ [1, n). We prove that the measure of non–compactness of the Sobolev embedding Wk,p0(Ω) Lp*(Ω) is equal to its norm. This means that the entropy numbers of this embedding are constant and equal to the norm. The same is true, when λn(Ω) is small enough, for the embedding of W1,n0(Ω) into the Orlicz space with Young function exp(tn/(n−1) − 1. The position is different for the embedding of W1,p0(J) in C0,1−1/p(), J equals; (0, 1), when p ∈ (1,∞): in this case the measure of non–compactness is less than the norm. (© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We investigate a class of operator-integral equations of Volterra-Stieltjes type and we study the solvability of those equations in the space of continuous functions. Equations in question create a generalization of numerous integral equations considered in nonlinear analysis. The main tool used in our considerations is the technique associated with measures of noncompactness. We show the applicability of our existence result in the study of a few integral equations of Volterra type.

We establish an estimate for the measure of non-compactness of an interpolated operator acting from a J-space into a K-space. Our result refers to general Banach N-tuples. We also derive estimates for entropy numbers if some of the N-tuples reduce to a single Banach space.

We establish formulae for the measure of non-compactness of operators interpolated by limiting methods that come up by the choice θ = 0 and θ = 1 in the definition of the real method.

We consider the Edmunds--Triebel logarithmic spaces $A_\theta (logA)_{b,q} $ produced by a Banach couple $\overline{A}=(A_0, \; A_1),$ as special cases of extrapolation spaces and get estimates of a measure of weak noncompactness of the unit balls of these spaces in terms of the measures of weak noncompactness of the unit balls of the spaces $A_0$ and $A_1.$ We obtain also estimates of the $n$-th Jordan--von Neumann constant $C^n_{NJ}$ and the $n$-th James constant $J_n$ of the spaces $A_\theta (logA)_{b,q} $ in terms of the corresponding constants of the spaces $A_0$ and $A_1.$

This paper deals with an impulsive Cauchy problem governed by the semilinear evolution differential inclusion x ′ ( t ) ∈ A ( t ) x ( t ) + F ( t , x ( t ) ) , where { A ( t ) } t ∈ [ 0 , b ] is a family of linear operators (not necessarily bounded) in a Banach space E generating an evolution operator and F is a Caratheodory type multifunction. First a theorem on the compactness of the set of all mild solutions for the problem is given. Then this result is applied to obtain the existence of mild solutions for the impulsive Cauchy problem defined on non-compact domains.

We study the behavior of the ball measure of non-compactness under several interpolation methods. First we deal with methods that interpolate couples of spaces, and then we proceed to extend the results to methods that interpolate finite families of spaces. We will need an approximation hypothesis on the target family of spaces.

It is proved that there exists no weight pair(v, w)for which a singular integral operator is compact from the weighted Lebesgue spaceLwp(Rn)toLvp(Rn). Moreover, a measure of non-compatness for this operator is estimated from below. Analogous problems for Cauchy singular integrals defined on Jordan smooth curves are studied.

We characterize upper semicontinuity of multifunctions in terms of upper Hausdorff semicontinuity, measure of non compactness and active boundary. The results are applicable in optimization, theory of best approximation and in metrizability theory.

In this paper, the concept of contraction via the measure of non-compactness on a Banach space is investigated by generalizing some results which have been previously discussed in literatures. Furthermore, to validity of the theorems and homotopy perturbation method (HPM), as a technical solution, they are applied on some nonlinear singular integral equations.

We study the embeddings E : W(X(Ω), Y(Ω)) ↪ Z(Ω), where X(Ω), Y(Ω) and Z(Ω) are rearrangement–invariant Banach function spaces (BFS) defined on a generalized ridged domain Ω, and W denotes a first–order Sobolev–type space. We obtain two–sided estimates for the measure of non–compactness of E when Z(Ω) = X(Ω) and, in turn, necessary and sufficient conditions for a Poincare–type inequality to be valid and also for E to be compact. The results are used to analyse the example of a trumpet–shaped domain Ω in Lorentz spaces. We consider the problem of determining the range of possible target spaces Z(Ω), in which case we prove that the problem is equivalent to an analogue on the generalized ridge Γ of Ω. The range of target spaces Z(Ω) is determined amongst a scale of (weighted) Lebesgue spaces for “rooms and passages” and trumpet–shaped domains.