On Mild Solutions of the Mittag-Leffler Fractional-Order Equation Subjected to Nonlocal Delay and Impulsive Conditions

In this paper, by means of the technique of measures of noncompactness, we establish a generalized form of the fixed point theorem for the sum of $T+S$, where $S$ is noncompact, $I-T$ may not be injective, and $T$ is not necessarily continuous. The obtained results unify and significantly extend a number of previously known generalizations of the Krasnosel’skii fixed point theorem. The analysis presented here reveals the essential characteristics of the Krasnosel’skii type fixed point theorem in strong topology setups. Further, the results are used to prove the existence of periodic solutions of a nonlinear neutral differential equation with delay in the critical case.

Among time scales, periodic ones deserve a special interest since they enable researchers to develop a theory for the existence of periodic solutions of dynamic equations on time scales (see for example Bi et al. (Nonlinear Anal 68(5):1226–1245, 2008), Bohner and Warth (Appl Anal 86(8):1007–1015, 2007), Kaufmann and Raffoul (J Math Anal Appl 319(1):315–325, 2006; Electron J Differ Equ 2007(27):12, 2007)). This chapter is devoted to the study of periodic solutions based on the definition of periodicity given by Kaufmann and Raffoul (J Math Anal Appl 319(1):315–325, 2006) and Atici et al. (Dynamic Systems and Applications, vol. 3, pp. 43–48, 2001). In this section, we cover natural setup of periodicity. For the conventional definition of periodic time scale and periodic functions on periodic time scales we refer to Definitions 1.5.1 and 1.5.2, respectively. Based on the Definitions 1.5.1 and 1.5.2, periodicity and existence of periodic solutions of dynamic equations on time scales were studied by various researchers and for first papers on the subject we refer to Adivar and Raffoul (Ann Mat Pura Appl 188(4):543–559, 2009; Comput Math Appl 58(2):264–272, 2009), Bi et al. (Nonlinear Anal 68(5):1226–1245, 2008), Kaufmann and Raffoul (J Math Anal Appl 319(1):315–325, 2006; Electron J Differ Equ 2007(27):12, 2007), and Liu and Li (Nonlinear Anal 67(5):1457–1463, 2007). We begin by introducing the concept of periodicity and then apply the concept to finite delay neutral nonlinear dynamical equations and to infinite delay Volterra integro-dynamic equations of variant forms by appealing to Schaefer fixed point theorem. The a priori bound will be obtained by the aid of Lyapunov-like functionals. Effort will be made to expose the improvement when we set the time scales to be the set of reals or integers. Then we proceed to study the existence of periodic solutions of almost linear Volterra integro-dynamic equations by utilizing Krasnosel’skii fixed point theorem (see Theorem 1.1.17). We will be examining the relationship between boundedness and the existence of periodic solutions. We will also be giving existence results for almost automorphic solutions of the delayed neutral dynamic system. We will end this chapter by proving existence results by using a fixed point theorem based on the large contraction.

The main purpose of this paper is to present the existence results of solutions and positive solutions of nonlinear high-order fractional boundary value problems with integral boundary condition. By using the Banach fixed point theorem and the Krasnosel’skii fixed point theorem, we obtain the existence and uniqueness of real solution. By the Guo–Krasnosel’skii fixed point theorem on the cone, we obtain a desired result for guaranteeing the existence of positive solution. Several interesting examples relevant to the main results are also considered.

In this paper we study the existence of periodic solutions of a Volterra type integral equation with infinite heredity. Banach fixed point theorem, Krasnosel'skii's fixed point theorem, and a combination of Krasnosel'skii's and Schaefer's fixed point theorems are employed in the analysis. The combination theorem of Krasnosel'skii and Schaefer requires an a priori bound on all solutions. We employ Liapunov's direct method to obtain such an a priori bound. In the process, we compare these theorems in terms of assumptions and outcomes.

At the time of this book, there is considerable research devoted to positive solutions for boundary value problems in each of the areas of partial differential equations, ordinary differential equations, and finite difference equations. In view of the unification theory brought forth by dynamic equations on time scales, it is natural that some research has emerged in terms of positive solutions for boundary value problems for dynamic equations on time scales. In this chapter, we present a synopsis of some of this recent work in a number of venues, including positive solutions that arise as eigenfunctions in a positive cone associated with eigenvalue comparison results, existence of at least one positive solution for a nonlinear dynamic equation as an application of the Guo-Krasnosel'skii fixed point theorem, existence of at least two positive solutions as dual applications of the Guo-Krasnosel'ski fixed point theorem or in some cases of the Avery-Henderson fixed point theorem. Finally, we conclude with results concerning the existence of at least three positive solutions via applications of either the Leggett-Williams fixed point theorem or a generalization of this theorem due to Avery.

In this paper, inspired by the idea of Meir–Keeler contractive mappings, we introduce Meir–Keeler expansive mappings, say MKE, in order to obtain Krasnosel’skii-type fixed point theorems in Banach spaces. The idea of the paper is to combine the notion of Meir–Keeler mapping and expansive Krasnosel’skii fixed point theorem. We replace the expansion condition by the weakened MKE condition in some variants of Krasnosel’skii fixed point theorems that appear in the literature, e.g., in [T. Xiang, R. Yuan, A class of expansive-type Krasnosel’skii fixed point theorems, Nonlinear Anal., Theory Methods Appl., 71(7–8):3229–3239, 2009].

In this note, a couple of unclear and unnecessary points made in the two existing papers by Liu and Li (Proc. Am. Math. Soc. 136:1213-1220, 2008) and Xiang and Yuan (Proc. Am. Math. Soc. 139:1033-1044, 2011) are first pointed out and clarified. Second, a few additional remarks are observed. Upon these observations, three corresponding refined and unified Krasnosel’skii fixed point theorems in strong topology setup are formulated. As an illustration, several new classes of Krasnosel’skii fixed point theorems are obtained, which expand and complement some known related results by Agarwal, O’Regan and Taoudi (Fixed Point Theory Appl. 2010:243716, 2010) and Edmunds (Math. Ann. 174:233-239, 1967).

<abstract><p>We investigate the existence of solutions for a Caputo fractional differential equation with periodic boundary condition. Using the positivity of Green's function of the corresponding linear equation, we show the existence of positive solutions by using Krasnosel'skii fixed point theorem. Meanwhile, by using monotone iterative method and lower and upper solutions method, we also discuss the existence of extremal solutions for a special case.</p></abstract>

In Chap. 7, we had to require the time scale to be additive in order to show the existence of periodic solutions. We devote this chapter to the study of periodic solutions of functional delay dynamical systems. The periodic delays are expressed in terms of shift operators (see Definition 1.4.4) that were developed by Adivar (Electron J Qual Theory Differ Equ 2010(7):1–22, 2010). We begin the chapter with the discussion of periodicity in shift operators over different time scales and provide examples. We proceed to unifying Floquet theory for homogeneous and nonhomogeneous dynamical systems. Then we consider neutral dynamical systems using Floquet theory and Krasnosel’skii fixed point theorem. We move to the existence of almost automorphic solutions of delayed neutral dynamic system using exponential dichotomy and Krasnosel’skii fixed point theorem. This chapter should serve as the foundation and guidance for future research on periodicity using the well-defined shift operators, by the authors. Part of this chapter is new and the rest can be found in Adivar (Math Slovaca 63(4):817–828, 2013), Adivar and Koyuncuoglu (Appl Math Comput 273:1208–1233, 2016; Appl Math Comput 242:328–339, 2014), and Koyuncuoglu (q-Floquet Theory and Its Extensions to Time Scales Periodic in Shifts. Thesis (Ph.D.), 2016).

Existence results for time scale boundary value problem

It is well known that Krasnose'skii fixed point theorem is very important. It was extensively used for studying the boundary value problems. In this paper, Krasnose'skii fixed point theorem is extended. A new fixed point theorem is obtained. The second order quasilinear differential equation (Phi(y'))' + a(t)f(t,y,y') = 0, 0 < t < 1 subject to Dirichlet boundary condition is studied, where f is a non-negative continuous function, Phi(v) = |v| <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p-2</sup> v, p > 1. We show the existence of at least one positive solution by using the new fixed point theorem in cone.

This paper establishes new fixed-point theorems in the framework of complete p-normed spaces, where p∈(0,1]. By extending the classical Banach, Schauder, and Krasnosel’skii fixed-point theorems, we derive several results for the sum of contraction and compact operators acting on s-convex subsets. The analysis is further generalized to multivalued upper semi-continuous operators by employing Kuratowski and Hausdorff measures of noncompactness. These results lead to new Darbo–Sadovskii-type fixed-point theorems and global versions of Krasnosel’skii’s theorem for multifunctions in p-normed spaces. The theoretical findings are then applied to demonstrate the existence of solutions for nonlinear integral equations formulated in p-normed settings. A section on numerical applications is also provided to illustrate the effectiveness and applicability of the proposed results.

In this paper we will study the existence, uniqueness and positivity of solution of a boundary value problem for nonlinear fractional differential equation, by using Leray-Schauder nonlinear alternative, Banach contraction and Guo-Krasnosel’skii fixed point theorems.

The objective of this paper is to determine the eigenvalue intervals for which positive solutions are guaranteed for the iterative system of the 3rd -order impulsive boundary value problem. The existence of solutions is established by applying the well-known Guo-Krasnosel’skii fixed point theorem. An illustrative example is provided to demonstrate the applicability of the theoretical results.

On the existence of positive solutions of fourth-order difference equations