Bounded and periodic solutions of infinite delay evolution equations

The weak solution of a functional differential equation in a general Banach space

This chapter deals with a kind of semilinear differential inclusions in general Banach spaces . Firstly, we study different types of generalized solutions including limit and weak solutions . Under appropriate assumptions, we show that the set of the limit solutions is a compact \(R_\delta \) -set. When the right-hand side satisfies the one-sided Perron condition , a variant of the well-known lemma of Filippov-Pliś , as well as a relaxation theorem, are proved. Secondly, we study a kind of semilinear evolution inclusions . If the nonlinearity is one-sided Perron with sublinear growth, then we establish the relation between the solutions of the considered differential inclusion and the solutions of the relaxed one. A variant of the well known Filippov-Pliś lemma is also proved. Finally, we analyze the existence of pullback attractor for non-autonomous differential inclusions with infinite delays by using measures of noncompactness. As samples of applications, we apply the abstract results to control systems driven by semilinear partial differential equations and multivalued feedbacks.

The advantage of fractional-order derivative has attracted extensive attention in the field of dynamics. In this paper, we investigated the stability of the fractional-order Mathieu equation under forced excitation, which is based on a model of the pantograph–catenary system. First, we obtained the approximate analytical expressions and periodic solutions of the stability boundaries by the multi-scale method and the perturbation method, and the correctness of these results were verified through numerical analysis by Matlab. In addition, by analyzing the stability of the k’T-periodic solutions in the system, we verified the existence of the unstable k’T-resonance lines through numerical simulation, and visually investigated the effect of the system parameters. The results show that forced excitation with a finite period does not change the position of the stability boundaries, but it can affect the expressions of the periodic solutions. Moreover, by analyzing the properties of the resonant lines, we found that when the points with k’T-periodic solutions were perturbed by the same frequency of forced excitation, these points became unstable due to resonance. Finally, we found that both the damping coefficient and the fractional-order parameters in the system have important influences on the stability boundaries and the resonance lines.

Linear modulational and subharmonic dynamics of spectrally stable Lugiato-Lefever periodic waves

Periodic solutions for a class of neutral functional differential equations with infinite delay

Characterizations of the metric and generalized metric projections on subspaces of Banach spaces

By using Fourier multiplier theorems we characterize the existence and uniqueness of periodic solutions for a class of second-order differential equations with infinite delay. We also establish maximal regularity results for the equations in various space

In this paper we establish the existence of a unique classical solution to a class of quasilinear integrodifferential equations in a general Banach space.

In this paper we consider periodic functional differential and integral equations. Massera proved that the existence of a bounded solution implies the existence of a periodic solution for linear (inhomogeneous) ordinary differential equations. Chow proved a similar result for linear functional differential equations with finite delay. We give a new proof for Chow's result that works for linear functional differential equations with infinite delay and also for integral equations. We also give examples that do not satisfy uniform boundedness or uniform ultimate boundedness, the conditions frequently used to prove that there is a periodic solution.

In this paper, impulsive Lasota‐Wazewska model with infinite delay is studied. By using fixed point theorem of decreasing operator, we obtain sufficient conditions for the existence of unique almost periodic positive solution. Particularly, we give iterative sequence, which converges to the almost periodic positive solution. Moreover, we investigate exponential stability of the almost periodic positive solution by Liapunov functional. Copyright © 2014 John Wiley & Sons, Ltd.

Decades ago, fractional calculus arose to generalize ordinary derivation and integration, and then became a means of modeling and interpreting many phenomena in various fields such as engineering, physics, chemistry, biology and signal processing. The definition of the fractional derivative began with a derivative with a singular kernel, such as the Riemann-Liouville and Caputo derivative. The study of existence and uniqueness of solution of fractional partial differential equations are then proved by mani methods as: the well-known Lax–Milgram theorem, Energy estimate and fixed point theorem... Among them, In this work we investigate the existence and uniqueness of solutions for a new class of tripled system, which containing of mixed fractional derivatives between the Riemann-Liouville and the Caputo fractional derivatives of different orders with Laplacian operator, by using Leray-Schauder alternative fixed point theorem type in generalized Banach space and Banach contraction principle. The stability of the Ulam type of the proposed coupled system is also studied. Due to the importance of variable-order fractional calculus, both from a theoretical and an applied perspective, we think that the results obtained can be of interest to the research pursuits of many readers. At the end, to conclude this paper we provide an example illustrating our findings and approach.

In this paper, we prove some fixed point results dealing with rational type contractions on generalized Banach Space. Our results in the new cotext extend and completement fixed point theorems of contractive mappings. Also we present certain common point result for such contraction.

The solvability behavior on the real line of linear integrodifferential equations in a general Banach space is considered and several applications to integral partial differential equations are given.

This paper discusses a class of semilinear fractional evolution equations with infinite delay and almost sectorial operator on infinite interval in Banach space. By using the properties of analytic semigroups and Schauder’s fixed-point theorem, this paper obtains the existence of mild solutions of the fractional evolution equation. Moreover, this paper also discusses the existence of mild solution when the analytic semigroup lacks compactness by Kuratowski measures of noncompactness and Darbo–Sadovskii fixed-point theorem.

On a nonlocal problem for semilinear differential equations with upper semicontinuous nonlinearities in general Banach spaces