Abstract: In this paper, we study the existence and uniqueness of solutions to the nonlocal problems for the fractional differential equation in Banach spaces. New sufficient conditions for the existence and uniqueness of solutions are established by means of fractional calculus and fixed point method under some suitable conditions. Two examples are given to illustrate the results.

Abstract: Comparison results of the nonlinear scalar Riemann-Liouville fractional dier- ential equation of order q, 0 < q 1, are presented without requiring Holder continuity assumption. Monotone method is developed for finite systems of fractional dierential equa- tions of order q, using coupled upper and lower solutions. Existence of minimal and maximal solutions of the nonlinear fractional dierential system is proved.

Abstract: In this paper, we investigate the existence and uniqueness of positive solutions to arbitrary order nonlinear fractional differential equations with advanced arguments. By applying some known fixed point theorems, sufficient conditions for the existence and uniqueness of positive solutions are established.

Abstract: Some conditions under which any subadditive function is periodic are presented. It is shown that the boundedness from below in a neighborhood of a point of a subadditive periodic (s.p.) function implies its nonnegativity, and the boundedness from above in a neighborhood of a point implies it nonnegativity and global boundedness from above. A necessary and sufficient condition for existence of a subadditive periodic extension of a function \(f_{0}:[0,1)\rightarrow \mathbb{R}\) is given. The continuity, differentiability of a s.p. function is discussed, and an example of a continuous nowhere differentiable s.p. function is presented. The functions which are the sums of linear functions and s.p. functions are characterized. The refinements of some known results on the continuity of subadditive functions are presented.

Abstract: Let G = (V;E) be a graph without isolated vertices. A dominating set S of G is called a neighbourhood total dominating set (ntd-set) if the induced subgraphhN(S)i has no isolated vertices. The minimum cardinality of a ntd-set of G is called the neighbourhood total domination number of G and is denoted by nt(G). The maximum order of a partition of V into ntd-sets is called the neighbourhood total domatic number of G and is denoted by dnt(G): In this paper we initiate a study of these parameters.

Abstract: We prove two sampling theorems for infinite (countable discrete) weighted graphs G; one example being "large grids of resistors" i.e., networks and systems of resistors. We show that there is natural ambient continuum X containing G, and there are Hilbert spaces of functions on X that allow interpolation by sampling values of the functions restricted only on the vertices in G. We sample functions on X from their discrete values picked in the vertex-subset G. We prove two theorems that allow for such realistic ambient spaces X for a fixed graph G, and for interpolation kernels in function Hilbert spaces on X, sampling only from points in the subset of vertices in G. A continuum is often not apparent at the outset from the given graph G. We will solve this problem with the use of ideas from stochastic integration.

Abstract: A graph G = (V,E) is arbitrarily vertex decomposable if for any sequence tau of positive integers adding up to |V |, there is a sequence of vertex-disjoint subsets of V whose orders are given by tau, and which induce connected graphs. The main aim of this paper is to study the recursive version of this problem. We present a solution for trees, suns, and partially for a class of 2-connected graphs called balloons.

Abstract: For a densely defined nonnegative symmetric operator \(\mathcal{A} = L_2^*L_1 \) in a Hilbert space, constructed from a pair \(L_1 \subset L_2\) of closed operators, we give expressions for the Friedrichs and Kreĭn nonnegative selfadjoint extensions. Some conditions for the equality \((L_2^* L_1)^* = L_1^* L_2\) are obtained. Applications to 1D nonnegative Hamiltonians, corresponding to point interactions, are given.

Abstract: The paper is devoted to investigation of operators of transition and the corre- sponding decompositions of Krein spaces. The obtained results are applied to the study of relationship between solutions of operator Riccati equations and properties of the associated operator matrix L. In this way, we complete the known result (see Theorem 5.2 in the paper of S. Albeverio, A. Motovilov, A. Skhalikov, Integral Equ. Oper. Theory 64 (2004), 455-486) and show the equivalence between the existence of a strong solution K (kKk < 1) of the Riccati equation and similarity of the J-self-adjoint operator L to a self-adjoint one.

Abstract: In this paper, we prove a general common fixed point theorem for two pairs of weakly compatible self-mappings of a metric space satisfying a weak Meir-Keeler type contractive condition by using a class of implicit relations. In particular, our result general- izes and improves a result of K. Jha, R.P. Pant, S.L. Singh, by removing the assumption of continuity, relaxing compatibility to weakly compatibility property and replacing the com- pleteness of the space with a set of four alternative conditions for maps satisfying an implicit relation. Also, our result improves the main result of H. Bouhadjera, A. Djoudi.

Abstract: In this paper, we study a class of integrodifferential evolution equations with nonlocal initial conditions in Banach spaces. Existence results of mild solutions are proved for a class of integrodifferential evolution equations with nonlocal initial conditions in Banach spaces. The main results are obtained by using the Schaefer fixed point theorem and semigroup theory. Finally, an example is given for demonstration.

Abstract: In this paper, some random fixed point theorems for continuous and condens- ing multi-valued random operators are proved and they are further applied to the random integral inclusions for proving the existence of the solutions via the priori bound method.

Abstract: In this paper, we discuss a class of integrodifferential equations with nonlocal conditions via a fractional derivative of the type: \[\begin{aligned}D_{t}^{q}x(t)=Ax(t)+\int\limits_{0}^{t}B(t-s)x(s)ds+t^{n}f\left(t,x(t)\right),&\;t\in [0,T],\;n\in Z^{+},\\qx(0)=g(x)+x_{0}.\end{aligned}\] Some sufficient conditions for the existence of mild solutions for the above system are given. The main tools are the resolvent operators and fixed point theorems due to Banach's fixed point theorem, Krasnoselskii's fixed point theorem and Schaefer's fixed point theorem. At last, an example is given for demonstration.

Abstract: For a graph \(G=(V,E)\), a set \(S\subseteq V\) is a dominating set if every vertex in \(V-S\) has at least a neighbor in \(S\). A dominating set \(S\) is a global offensive (respectively, defensive) alliance if for each vertex in \(V-S\) (respectively, in \(S\)) at least half the vertices from the closed neighborhood of \(v\) are in \(S\). The domination number \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\), and the global offensive alliance number \(\gamma_{o}(G)\) (respectively, global defensive alliance number \(\gamma_{a}(G)\)) is the minimum cardinality of a global offensive alliance (respectively, global deffensive alliance) of \(G\). We show that if \(T\) is a tree of order \(n\), then \(\gamma_{o}(T)\leq 2\gamma(T)-1\) and if \(n\geq 3\), then \(\gamma_{o}(T)\leq \frac{3}{2}\gamma_{a}(T)-1\). Moreover, all extremal trees attaining the first bound are characterized.

Abstract: We show that a class of countable weighted graphs arising in the study of electric resistance networks (ERNs) are naturally associated with groupoids. Starting with a fixed ERN, it is known that there is a canonical energy form and a derived energy Hilbert space HE. From HE, one then studies resistance metrics and boundaries of the ERNs. But in earlier research, there does not appear to be a natural algebra of bounded operators acting on HE. With the use of our ERN-groupoid, we show that HE may be derived as a representation Hilbert space of a universal representation of a groupoid algebra AG, and we display other representations. Among our applications, we identify a free structure of AG in terms of the energy form.

Abstract: We consider the stability problem for a class of functional equations related to the Popoviciu equation.

Abstract: Let \(p\in \mathbb{N}^*\) and \(\beta,\gamma\in \mathbb{C}\) with \(\beta eq 0\) and let \(\Sigma_p\) denote the class of meromorphic functions of the form \(g(z)=\frac{a_{-p}}{z^p}+a_0+a_1 z+\ldots,\,z\in \dot U\), \(a_{-p} eq 0\). We consider the integral operator \(J_{p,\beta,\gamma}:K_{p,\beta,\gamma}\subset\Sigma_p\to \Sigma_p\) defined by \[J_{p,\beta,\gamma}(g)(z)=\left[\frac{\gamma-p\beta}{z^\gamma }\int_0^zg^{\beta}(t) t^{\gamma-1}dt\right]^{\frac{1}{\beta}},\,g\in K_{p,\beta,\gamma},\,z\in \dot U.\] We introduce some new subclasses of the class \(\Sigma_p\), associated with subordination and superordination, such that, in some particular cases, these new subclasses are the well-known classes of meromorphic starlike functions and we study the properties of these subclasses with respect to the operator \(J_{p,\beta,\gamma}\).

Abstract: We investigate properties of a non symmetric Markov's chain on an infinite graph. We show the connection with matrix valued random walk polynomials which satisfy the orthogonality formula with respect to non a symmetric matrix valued measure.

Abstract: We establish the existence of principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We consider the Dirichlet and Neumann boundary conditions.

Abstract: This paper is concerned with ground state solutions for the Henon type equation u(x) =jyj u p 1 (x) in , where = B k (0; 1) B n k (0; 1) R n and x = (y;z)2 R k R n k . We study the existence of cylindrically symmetric and non-cylindrically symmetric ground state solutions for the problem. We also investigate asymptotic behavior of the ground state solution when p tends to the critical exponent 2 = 2n n 2 if n 3.

Abstract: This paper concerns the oscillation of solutions of the dierential eq. h r(t) (x(t))f ( x(t)) i +q (t)'(g (x(t));r(t) (x(t))f ( x(t))) = 0; where u'(u;v) > 0 for all u6 0; xg (x) > 0; xf (x) > 0 for all x6 0; (x) > 0 for all x2 R; r(t) > 0 for t t0 > 0 and q is of arbitrary sign. Our results complement the results in (A.G. Kartsatos, On oscillation of nonlinear equations of second order, J. Math. Anal. Appl. 24 (1968), 665-668), and improve a number of existing oscillation criteria. Our main results are illustrated with examples.

Abstract: The existence of a solution, as well as some properties of the obtained solution for a Hammerstein type nonlinear integral equation have been investigated. For a certain class of functions the uniqueness theorem has also been proved.

Abstract: This paper deals with some operator representations \(\Phi\) of a weak*-Dirichlet algebra \(A\), which can be extended to the Hardy spaces \(H^{p}(m)\), associated to \(A\) and to a representing measure \(m\) of \(A\), for \(1\leq p\leq\infty\) A characterization for the existence of an extension \(\Phi_p\) of \(\Phi\) to \(L^p(m)\) is given in the terms of a semispectral measure \(F_\Phi\) of \(\Phi\) For the case when the closure in \(L^p(m)\) of the kernel in \(A\) of \(m\) is a simply invariant subspace, it is proved that the map \(\Phi_p|H^p(m)\) can be reduced to a functional calculus, which is induced by an operator of class \(C_\rho\) in the Nagy-Foias sense A description of the Radon-Nikodym derivative of \(F_\Phi\) is obtained, and the log-integrability of this derivative is proved An application to the scalar case, shows that the homomorphisms of \(A\) which are bounded in \(L^p(m)\) norm, form the range of an embedding of the open unit disc into a Gleason part of \(A\)

Abstract: The aim of this paper is to introduce various convolution algebras associated with a topological groupoid with locally compact fibres. Instead of working with continuous functions on \(G\), we consider functions having a uniformly continuity property on fibres. We assume that the groupoid is endowed with a system of measures (supported on its fibres) subject to the "left invariance" condition in the groupoid sense.

Abstract: In this paper, we use the method of coincide degree theory to establish new results on the existence and uniqueness of anti-periodic solutions for a class of nonlinear \(n\)-th order functional differential equations of the form \[x^{(n)}(t)=F(t, x_t, x^{(n-1)}_t, x(t), x^{(n-1)}(t), x(t-\tau(t)), x^{(n-1)}(t-\sigma(t))).\]

Abstract: In this paper, we study the steady-state of a semilinear pulse-width sampler controlled system on infinite dimensional spaces. Firstly, by virtue of Schauder's fixed point theorem, the existence of periodic solutions is given. Secondly, utilizing a generalized Gronwall inequality given by us and the Banach fixed point theorem, the existence and stabilizability of a steady-state for the semilinear control system with pulse-width sampler is also obtained. At last, an example is given for demonstration.

Abstract: In this paper, by using Mawhin's continuation theorem of coincidence degree theory, we establish the existence of at least four positive periodic solutions for a discrete time Lotka-Volterra competitive system with harvesting terms. An example is given to illustrate the effectiveness of our results.

Abstract: Matrices of operators with respect to frames are sometimes more natural and easier to compute than the ones related to bases. The present work investigates such opera- tors on the Segal-Bargmann space, known also as the Fock space. We consider in particular some properties of matrices related to Toeplitz and Hankel operators. The underlying frame is provided by normalised reproducing kernel functions at some lattice points.

Abstract: Let P = (Pt)t 0 be a sub-Markovian semigroup on L 2 (m), let = ( t)t 0 be a Bochner subordinator and let P = (P t )t 0 be the subordinated semigroup of P by means of , i.e. P s := R 1 0 Pr s(dr). Let ' := ('t)t>0 be a P-exit law, i.e. Pt's = 's+t; s;t > 0 and let ' t := R 1 0 's t(ds). Then ' := (' t )t>0 is a P -exit law whenever it lies in L 2 (m). This paper is devoted to the converse problem when is without drift. We prove that a P -exit law := ( t)t>0 is subordinated to a (unique) P-exit law ' (i.e. = ' ) if and only if (Ptu)t>0 D(A ), where u = R 1 0 e s sds and A is the L 2 (m)-generator of P.