Higher order evolution equations and dynamic boundary value problems

Im ersten Teil dieser Arbeit untersuchen wir Cauchyprobleme höherer Ordnung $(ACP_n)$. Dazu führen wir in Kapitel 1 Operatoren, sogenannte Existenzfamilien, ein, die einen weiteren Banachraum Y in X abbilden. Damit erhalten wir eine grosse Flexibilität und können Existenz und stetige Abhängigkeit der Lösungen von ($ACP_n$) und seiner inhomogenen Version beweisen. Analog werden Eindeutigkeitsfamilien definiert zur Charakterisierung der Eindeutigkeit der Lösungen. Die Verbindung dieser beiden Konzepte gestattet die Verallgemeinerung aller bisher bekannten Resultate zur Lösung von ($ACP_n$). In Kapitel 2 werden dann multiplikative und additive Störungsresultate vom Desch-Schappacher-Typ für ($ACP_n$) bewiesen und angewandt. Im zweiten Teil der Arbeit untersuchen wir dynamische Randbedingungen für Cauchyprobleme erster und zweiter Ordnung. Kapitel 3 presentiert eine Lösung für ein Problem, das A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli [34] gestellt haben bezüglich des gemischten Problems für Wellengleichungen mit verallgemeinerten Wentzell Randbedingungen. Im vierten Kapitel wird der zugehörige nichtautonome Fall betrachtet. Hier erhalten wir nicht nur Existenz- und Eindeutigkeitsresultate sondern auch präzise Aussagen zur Regularität der Lösungen. Schliesslich enthalten Kapitel 5 und 6 eine einheitliche Behandlung gemischter Probleme (Anfangs-Randwert Probleme) mit dynamischen Randbedingungen für parabolische und hyperbolische oder allgemeine Gleichungen zweiter Ordnung. Wir beschäftigen uns direkt mit Problemen zweiter Ordnung, ohne sie auf erste Ordnung zu reduzieren. Es stellt sich heraus, dass diese direkte Methoden starke Lösungen von erwünschter Regularität liefern und sogar allgemeine Theoreme ermöglichen. Eine Reihe von ganz neuen Resultaten werden bewiesen. Die Ergebnisse werden dann auf konkrete partielle Differentialgleichungen angewandt.In the first part of this work we are concerned with the Cauchy problem for higher order evolution equations $(ACP_n)$ in a Banach space X. In Chapter 1 we introduce a new operator family of bounded linear operators from another Banach space Y to X, called an existence family for $(ACP_n)$, to study the existence and continuous dependence on initial data of the solutions of $(ACP_n)$ and its inhomogeneous version $(IACP_n)$, and obtain some basic results in a quite general setting. Chapter 2 is intended to establish Desch-Schappacher type multiplicative and additive perturbation theorems for existence families for $(ACP_n)$ (with $A_1=...=A_{n-1}=0$). In the second part of the work, we investigate the dynamic boundary value problems of first or second order. Chapter 3 presents a solution to an open problem put forward by A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli [34], concerning the mixed problem for wave equations with generalized Wentzell boundary conditions. Chapter 4 concerns with the nonautonomous heat equation with generalized Wentzell boundary conditions. In Chapter 5 we exhibit a unified treatment of the mixed initial boundary value problem for second order (in time) parabolic linear differential equations in Banach spaces whose boundary conditions are of a dynamical nature. Results regarding existence, uniqueness, continuous dependence and regularity of classical and strict solutions are established. Moreover, two examples are given as samples for possible applications. In the final Chapter 6 we continue to deal with the mixed initial boundary value problem for complete second order (in time) linear differential equations in Banach spaces, in which time-derivatives occur in the boundary conditions. General wellposedness theorems are obtained (for the first time) which are used to solve the corresponding inhomogeneous problems. Examples of applications to PDEs are also presented

Multiple positive solutions for second order impulsive boundary value problems in Banach spaces

By means of the fixed point index theory of strict set contraction operators, we establish new existence theorems on multiple positive solutions to a boundary value problem for second-order impulsive integro-differential equations with integral boundary conditions in a Banach space. Moreover, an application is given to illustrate the main result

A note on the fractional Cauchy problems with nonlocal initial conditions

AbstractOf concern is the Cauchy problems for fractional integro-differential equations with nonlocal initial conditions. Using a new strategy in terms of the compactness of the semigroup generated by the operator in the linear part and approximating technique, a new existence theorem for mild solutions is established. An application to a fractional partial integro-differential equation with a nonlocal initial condition is also considered

Existence of positive solutions for singular boundary value problem on time scales

AbstractThis paper deals with a class of boundary value problem of singular differential equations on time scales. The conditions we used here differ from those in the majority of papers as we know. An existence theorem of positive solutions is established by using the Krasnosel'skii fixed point theorem and an example is given to illustrate it

Multiple Positive Periodic Solutions for Delay Differential System

We obtain some existence results for multiple positive periodic solutions of some delay differential systems. Examples are presented as applications. For a general positive integer m≥2, main results of this paper do not appear in former literatures as we know. Comparing with the existing results, our results are new also when m=1

Almost Automorphic Solutions to Abstract Fractional Differential Equations

A new and general existence and uniqueness theorem of almost automorphic solutions is obtained for the semilinear fractional differential equation Dtαu(t)=Au(t)+Dtα−1f(t,u(t))  (1<α<2), in complex Banach spaces, with Stepanov-like almost automorphic coefficients. Moreover, an application to a fractional relaxation-oscillation equation is given

Search for Quasi-Periodical Oscillations in Precursors of Short and Long Gamma Ray Bursts

The precursors of short and long Gamma Ray Bursts (SGRBs and LGRBs) can serve as probes of their progenitors, as well as shedding light on the physical processes of mergers or core-collapse supernovae. Some models predict the possible existence of Quasi-Periodically Oscillations (QPO) in the precursors of SGRBs. Although many previous studies have performed QPO search in the main emission of SGRBs and LGRBs, so far there was no systematic QPO search in their precursors. In this work, we perform a detailed QPO search in the precursors of SGRBs and LGRBs detected by Fermi/GBM from 2008 to 2019 using the power density spectrum (PDS) in frequency domain and Gaussian processes (GP) in time domain. We do not find any convinced QPO signal with significance above 3 $\sigma$, possibly due to the low fluxes of precursors. Finally, the PDS continuum properties of both the precursors and main emissions are also studied for the first time, and no significant difference is found in the distributions of the PDS slope for precursors and main emissions in both SGRBs and LGRBs.Comment: submitte

Reliable Detection of Paternal SNPs within Deletion Breakpoints for Non-Invasive Prenatal Exclusion of Homozygous α0-Thalassemia in Maternal Plasma

Reliable detection of large deletions from cell-free fetal DNA (cffDNA) in maternal plasma is challenging, especially when both parents have the same deletion owing to a lack of specific markers for fetal genotyping. In order to evaluate the efficacy of a non-invasive prenatal diagnosis (NIPD) test to exclude α-thalassemia major that uses SNPs linked to the normal paternal α-globin allele, we established a novel protocol to reliably detect paternal SNPs within the (−−SEA) breakpoints and performed evaluation of the diagnostic potential of the protocol in a total of 67 pregnancies, in whom plasma samples were collected prior to invasive obstetrics procedures in southern China. A group of nine SNPs identified within the deletion breakpoints were scanned to select the informative SNPs in each of the 67 couples DNA by multiplex PCR based mini-sequencing technique. The paternally inherited SNP allele from cffDNA was detected by allele specific real-time PCR. A protocol for reliable detection of paternal SNPs within the (−−SEA) breakpoints was established and evaluation of the diagnostic potential of the protocol was performed in a total of 67 pregnancies. In 97% of the couples one or more different SNPs within the deletion breakpoint occurred between paternal and maternal alleles. Homozygosity for the (−−SEA) deletion was accurately excluded in 33 out of 67 (49.3%, 95% CI, 25.4–78.6%) pregnancies through the implementation of the protocol. Protocol was completely concordant with the traditional reference methods, except for two cases that exhibited uncertain results due to sample hemolysis. This method could be used as a routine NIPD test to exclude gross fetal deletions in α-thalassemia major, and could further be employed to test for other diseases due to gene deletion