Noncompact Intervals Research Articles

Volterra nonautonomous evolution inclusions: topological structure of solution sets and applications

This paper concerns with a class of Volterra nonautonomous evolution inclusions with time delay effect defined on a noncompact interval. It involves a family of (possibly unbounded) operators, which generates a noncompact evolution family. In the framework of Fréchet spaces, the topological structure of the solution set for evolution inclusion is considered. Moreover, we obtain geometric features of the corresponding solution map. Our results extend essentially the existing ones which do not involve Volterra integral in nonlinearity. We finally present a concrete example to illustrate the abstract results.

Topological Structure of the Solution Sets for a Nonlinear Delay Evolution

AbstractWe consider a nonlinear delay evolution equation with multivalued perturbation on a noncompact interval. The nonlinearity, having convex and closed values, is upper hemicontinuous with respect to the solution variable. A basic question on whether there exists a solution set carrying $R_{\delta }$-structure remains unsolved when the operator families generated by the principal part lack compactness. One of our main goals is to settle this question in the affirmative. Moreover, we prove that the solution map, having compact values, is an $R_{\delta }$-map, which maps any connected set into a connected set. It is then exploited to deal with the existence in the large for a nonlocal problem. Finally, several examples are worked out in detail, illustrating the applicability of our general results.

Nonlinear Volterra delay evolution inclusions subjected to nonlocal initial conditions

This paper deals with a nonlinear Volterra delay evolution inclusion subjected to a nonlocal implicit initial condition. The evolution inclusion involves an $m$-dissipative operator (possibly multivalued and/or nonlinear) and a noncompact interval. We first consider the evolution inclusion subjected to a local initial condition and prove an existence result for bounded $C^0$-solutions. Then, using a fixed point theorem for upper semicontinuous multifunctions with contractible values, we obtain a global solvability result for the original problem. Finally, we present an example to illustrate the abstract result.

A Fixed-Point Approach to the Hyers–Ulam Stability of Caputo–Fabrizio Fractional Differential Equations

In this paper, we study Hyers–Ulam and Hyers–Ulam–Rassias stability of nonlinear Caputo–Fabrizio fractional differential equations on a noncompact interval. We extend the corresponding uniqueness and stability results on a compact interval. Two examples are given to illustrate our main results.

On the integral modulus of continuity of infinitely divisible distributions, especially of stochastic integrals

We derive estimates for the integral modulus of continuity of probability densities of infinitely divisible distributions. The paper is split into two parts. The first part deals with general infinitely divisible distributions. The second part is concerned with densities of random integrals with respect to a Lévy process. We will see major differences between integrals over compact and non-compact intervals.

Tail asymptotic behavior of the supremum of a class of chi-square processes

We analyze in this paper the supremum of a class of chi-square processes over non-compact intervals, which can be seen as a multivariate counterpart of the generalized weighted Kolmogorov–Smirnov statistic. The boundedness and the exact tail asymptotic behavior of the supremum are derived. As examples, the chi-square process generated from the Brownian bridge and the fractional Brownian motion are discussed.

Euler type linear and half-linear differential equations and their non-oscillation in the critical oscillation case

This paper is devoted to the analysis of the oscillatory behavior of Euler type linear and half-linear differential equations. We focus on the so-called conditional oscillation, where there exists a borderline between oscillatory and non-oscillatory equations. The most complicated problem involved in the theory of conditionally oscillatory equations is to decide whether the equations from the given class are oscillatory or non-oscillatory in the threshold case. In this paper, we answer this question via a combination of the Riccati and Prüfer technique. Note that the obtained non-oscillation of the studied equations is important in solving boundary value problems on non-compact intervals and that the obtained results are new even in the linear case.

Ground state solutions to nonlinear equations with p-Laplacian

We investigate the existence of positive radial solutions for a nonlinear elliptic equation with p-Laplace operator and sign-changing weight, both in superlinear and sublinear case. We prove the existence of solutions u, which are globally defined and positive outside a ball of radius R, satisfy fixed initial conditions u(R)=c>0, u′(R)=0 and tend to zero at infinity. Our method is based on a fixed point result for boundary value problems on noncompact intervals and on asymptotic properties of suitable auxiliary half-linear differential equations. The results are new also for the classical Laplace operator and may be used for proving the existence of ground state solutions and decaying solutions with exactly k-zeros which are defined in the whole space. Some examples illustrate our results.

Topological properties of C^{0}-solution set for impulsive evolution inclusions

In this paper, we study the topological properties to a C^{0}-solution set of impulsive evolution inclusions. The definition of C^{0}-solutions for impulsive functional evolution inclusions is introduced. The R_{delta}-property of C^{0}-solution set is studied for compact as well as noncompact semigroups on compact intervals. Applying the inverse limit method, the R_{delta}-structure on noncompact intervals is obtained.

Topological Structure of Solutions Sets for Semilinear Evolution Inclusions

This paper deals with a semilinear evolution inclusion involving a nondensely defined closed linear operator satisfying the Hille–Yosida condition and source term of multivalued type in Banach spaces. The topological structure of the set of solutions is investigated in the case that semigroup is noncompact. It is shown that the solution set is nonempty, compact and an R_\delta -set. It is proved on compact intervals and then, using the inverse limit method, obtained on non-compact intervals. As a sample of application, we consider a parabolic partial differential inclusion at end of the paper.

Compactness criteria and new impulsive functional dynamic equations on time scales

In this paper, we introduce the concept of Δ-sub-derivative on time scales to define e-equivalent impulsive functional dynamic equations on almost periodic time scales. To obtain the existence of solutions for this type of dynamic equation, we establish some new theorems to characterize the compact sets in regulated function space on noncompact intervals of time scales. Also, by introducing and studying a square bracket function $[x(\cdot),y(\cdot) ]:\mathbb{T}\rightarrow\mathbb{R}$ on time scales, we establish some new sufficient conditions for the existence of almost periodic solutions for e-equivalent impulsive functional dynamic equations on almost periodic time scales. The final section presents our conclusion and further discussion of this topic.

On the uniform convergence of a q-series

The paper deals with a class of linear positive operators expressed by q-series. By using modulus of smoothness an upper bound of approximation error is determined. We identify functions for which these operators provide uniform approximation over noncompact intervals. A particular case is delivered.

Boundary value problems on noncompact intervals for the n-th order vector differential inclusions

Boundary value problems on noncompact intervals for the n-th order vector differential inclusions

Fractional Sturm–Liouville boundary value problems in unbounded domains: Theory and applications

Recently, Zayernouri and Karniadakis in (2013) [78] investigated two classes of fractional Sturm–Liouville eigenvalue problems on compact interval [a,b] in more detail. They were the first authors who not only obtained some explicit forms for the eigensolutions of these problems but also derived some useful spectral properties of the obtained eigensolutions. Until now, to the best of our knowledge, fractional Sturm–Liouville eigenvalue problems on non-compact interval, such as [0,+∞) are not analyzed. So, our aim in this paper is to study these problems in detail. To do so, we study at first fractional Sturm–Liouville operators (FSLOs) of the confluent hypergeometric differential equations of the first kind and then two special cases of FSLOs: FSLOs-1 and FSLOs-2 are considered. After this, we obtain the analytical eigenfunctions for the cases and investigate the spectral properties of eigenfunctions and their corresponding eigenvalues. Also, we derive two fractional types of the associated Laguerre differential equations. Due to the non-polynomial nature of the eigenfunctions obtained from the two fractional associated Laguerre differential equations, they are defined as generalized associated Laguerre functions of the first and second kinds, GALFs-1 and GALFs-2. Furthermore, we prove that these fractional Sturm–Liouville operators are self-adjoint and the obtained eigenvalues are all real, the corresponding eigenfunctions are orthogonal with respect to the weight function associated to FSLOs-1 and FSLOs-2 and form two sets of non-polynomial bases. At the end, two new quadrature rules and L2-orthogonal projections with respect to and based on GALFs-1 and GALFs-2 are introduced. The upper bounds of the truncation errors of these new orthogonal projections according to some prescribed norm are proved and then verified numerically with some test examples. Finally, some fractional differential equations are provided and analyzed numerically.

Topological theory of non-autonomous parabolic evolution inclusions on a noncompact interval and applications

In this paper we consider the topological structure of the solution set of non-autonomous parabolic evolution inclusions with time delay, defined on non-compact intervals. The result restricted to compact intervals is then extended to non-autonomous parabolic control problems with time delay. Moreover, as the applications of the information about the structure, we establish the existence result of global integral solutions for non-autonomous Cauchy problems subject to nonlocal condition, and prove the invariance of a reachability set for non-autonomous control problems under single-valued nonlinear perturbations. Finally, some illustrating examples are supplied.

Nonlinear evolution inclusions: Topological characterizations of solution sets and applications

This paper deals with a nonlinear delay differential inclusion of evolution type involving m-dissipative operator and source term of multi-valued type in a Banach space. Under rather mild conditions, the Rδ-structure of C0-solution set is studied on compact intervals, which is then used to obtain the Rδ-property on non-compact intervals. Secondly, the result about the structure is furthermore employed to show the existence of C0-solutions for the inclusion (mentioned above) subject to nonlocal condition defined on right half-line. No nonexpansive condition on nonlocal function is needed. As samples of applications, we consider a partial differential inclusion with time delay and then with nonlocal condition at the end of the paper.

On the Ulam–Hyers stability of first order differential systems with nonlocal initial conditions

In this paper we investigate the Ulam–Hyers stability of first order differential systems with nonlocal initial conditions on compact intervals (in W1,1(0,k)) and on noncompact intervals (in Ww1,2(0,∞), with suitable weight function w). The novelty consists in the simultaneous use of vectorial norms, convergent to zero matrices, Sobolev spaces for studying the Ulam–Hyers stability of equations with nonlocal conditions on noncompact intervals.

On Some Fixed Point Theorems for 1-Set Weakly Contractive Multi-Valued Mappings with Weakly Sequentially Closed Graph

In this paper we prove Leray-Schauder and Furi-Pera types fixed point theorems for a class of multi-valued mappings with weakly sequentially closed graph. Our results improve and extend previous results for weakly sequentially closed maps and are very important in applications, mainly for the investigating of boundary value problems on noncompact intervals.

Topological structure of solution sets for impulsive differential inclusions in Fréchet spaces

In this paper, we consider the existence of solutions as well as the topological and geometric structure of solution sets for first-order impulsive differential inclusions in some Fréchet spaces. Both the initial and terminal problems are considered. Using ingredients from topology and homology, the topological structures of solution sets (closedness and compactness) as well as some geometric properties (contractibility, acyclicity, A R and R δ ) are investigated. Some of our existence results are obtained via the method of taking the inverse system limit on noncompact intervals.

Topological structure of solution sets to asymptotic boundary value problems

Topological structure is investigated for second-order vector asymptotic boundary value problems. Because of indicated obstructions, the R δ -structure is firstly studied for problems on compact intervals and then, by means of the inverse limit method, on non-compact intervals. The information about the structure is furthermore employed, by virtue of a fixed-point index technique in Fréchet spaces developed by ourselves earlier, for obtaining an existence result for nonlinear asymptotic problems. Some illustrating examples are supplied.