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The Cauchy problem for the Kawahara equation with data in analytic Gevrey spaces on the circle is considered and its local well-posedness in these spaces is proved. Using Bourgain-Gevrey type analytic spaces and appropriate bilinear estimates, it is shown that local in time wellposedness holds when the initial data belong to an analytic Gevrey spaces of order σ. Moreover, the solution is not necessarily Gσ in time. However, it belongs to G5σ near zero for every x on the circle. We study a Cauchy problem for Kawahara equation . With data in analytic Gevrey spaces on the circal, we prove that the problem is well defined. We also treat the regularity in time which belongs to $G^{5\sigma}$ .

The task of our work is to consider the initial value problem based on the model of the generalized Kadomtsev-Petviashvili I equation and prove the local well-posedness in an anisotropic Gevrey spaces and then global well-posedness which improves the recent results on the well-posedness of this model in anisotropic Sobolev spaces [ 17 ]. Also, wide information about the regularity of the solution in the time variable is provided.

In this paper, we study the existence of solutions for hybrid fractional integro-differential equations involving $\psi$-Caputo derivative. We use an hybrid fixed point theorem for a sum of three operators due to Dhage for proving the main results. An example is provided to illustrate main results.

This paper investigates the existence results and uniqueness of solutions for a class of boundary value problems for fractional differential equations with the Hilfer fractional derivative. The reasoning is mainly based upon Mönch’s fixed point theorem associated with the technique of measure of noncompactness. We illustrate our main findings, with a particular case example, included to show the applicability of our outcomes. The boundary conditions introduced in this work are of quite general nature and reduce to many special cases by fixing the parameters involved in the conditions.

This paper studies a Cauchy problems for fifth-order shallow water equations with nonlinear terms in (1). With data in analytic Gevrey spaces on the line, we prove that the problem is well defined. We also treat the regularity in time which belongs to $$G^{5\sigma }$$ near zero for every x on the line. The proof is based mainly on bilinear and trilinear estimates in the analytic Gevrey–Bourgain spaces, relies on the contraction mapping theorem to improve the results in Jia and Huo (J Differ Equ 246:2448–2467, 2009).

The paper is devoted to study a Cauchy problem for fifth-order Kadomtsev–Petviashvili I equation. With data in analytic Bourgain spaces on the line and the circle, we prove that the problem is well posed. We also treat the regularity in t and x, y, where the solution is analytic in x, y and belongs to $$G^{5 }$$ in t.

We study the Cauchy problem of the Ostrovsky equation (Ost) $$\partial _{t}u +\partial _{x}^{3}u -\partial _{x}^{-1}u+u\partial _{x}u = 0$$ , where the data in analytic Gevrey spaces on the line and the circle is considered and its local well-posedness in these spaces is proved. The proof is based on bilinear estimates in Bourgain type spaces. Also, Gevrey regularity of the solution in time variable is provided.

This work is concerned with the solvability of multi-point boundary value problems for fractional differential equations with nonlinear growth at the resonance. Existence results are obtained with the use of the coincidence degree theory. As an application, we discuss an example to illustrate the obtained results.

Studies of modified Korteweg-de Vries-type equations are of considerable mathematical interest due to the importance of their applications in various branches of mechanics and physics. In this article, using trilinear estimate in Bourgain spaces, we show the local well-posedness of the initial value problem associated with a coupled system consisting of modified Korteweg-de Vries equations for given data. Furthermore, we prove that the unique solution belongs to Gevrey space G &sigma, &times, G &sigma, in x and G 3 &sigma, G 3 &sigma, in t. This article is a continuation of recent studies reflected.

This manuscript aims to study the existence and uniqueness of solutions for a new class of boundary value problems of nonlinear fractional differential equations with Katugampola fractional integral conditions. Ours results are given by applying some standard fixed point theorems, the Boyd-Wong nonlinear contraction principle, and Leray-Schauder nonlinear alternative. Examples illustrating the obtained results are also presented.

We show that the fifth-order Kadomtsev-Petviashvili II equation is globally well-posed in an anisotropic Gevrey space, which complements earlier results on the well-posedness of this equation in anisotropic Sobolev spaces.

In this paper, we discuss the existence of solutions for a hybrid boundary value problem of Caputo fractional differential equations. The main tool used in our study is associated with the technique of measures of noncompactness. As an application, we give an example to illustrate our results.

In this paper, we study the existence of mild solutions of impulsive evolution fractional functional differential equation of order [Formula: see text] involving a Lipschitz condition on term [Formula: see text]. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators due to Burton and Kirk.

Resume Le but de notre travail est l'etude du probleme de Cauchy pour une large classe d'operateurs lineaires, non necessairement kowalevskiens, avec argument absorbant de la variable spatiale. On montre l'existence et l'unicite d'une solution du probleme dans l'espace des fonctions analytiques par rapport au temp et de classe Gevrey par rapport a la variable d'espace. Les outils utilises sont inspires des travaux [Cl. Wagschal, Le probleme de Goursat non lineaire, J. Math. Pure Appl. 58 (1979) 309–337 ; D. Gourdin, M. Mechab, Solution globale d'un probleme de Cauchy lineaire, J. Funct. Anal. 202 (2003) 123–146], qui eux meme bases sur les normes formelles de Leray et Waelbroeck [J. Leray, L. Waelbroeck, Norme formelle d'une fonction composee (Preliminaire a l'etude des systemes non lineaires, hyperboliques non stricts), in : Colloque de Liege, CBRM, 1964, pp. 145–152. [23] ].

In this paper, we introduce a new concept of mild solution of some class of semilinear fractional dierential inclusions of order 0 < < 1. Also we establish an existence result when the multivalued function has convex values. The result is obtained upon the nonlinear alternative of Leray-Schauder type.