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 Chinese Journal of Mathematics , 2014, DOI: 10.1155/2014/609476 Abstract: Here we aim at establishing certain new fractional integral inequalities involving the Gauss hypergeometric function for synchronous functions which are related to the Chebyshev functional. Several special cases as fractional integral inequalities involving Saigo, Erdélyi-Kober, and Riemann-Liouville type fractional integral operators are presented in the concluding section. Further, we also consider their relevance with other related known results. 1. Introduction Fractional integral inequalities have many applications; the most useful ones are in establishing uniqueness of solutions in fractional boundary value problems and in fractional partial differential equations. Further, they also provide upper and lower bounds to the solutions of the above equations. These considerations have led various researchers in the field of integral inequalities to explore certain extensions and generalizations by involving fractional calculus operators. One may, for instance, refer to such type of works in the book  and the papers [2–11]. In a recent paper, Purohit and Raina  investigated certain Chebyshev type  integral inequalities involving the Saigo fractional integral operators and also established the -extensions of the main results. The aim of this paper is to establish certain generalized integral inequalities for synchronous functions that are related to the Chebyshev functional using the fractional hypergeometric operator, introduced by Curiel and Galu . Results due to Purohit and Raina  and Belarbi and Dahmani  follow as special cases of our results. In the sequel, we use the following definitions and related details. Definition 1. Two functions and are said to be synchronous on , if for any . Definition 2. A real-valued function is said to be in the space , if there exists a real number such that , where . Definition 3. Let , , and ; then a generalized fractional integral (in terms of the Gauss hypergeometric function) of order for a real-valued continuous function is defined by  (see also ): where the function appearing as a kernel for the operator (2) is the Gaussian hypergeometric function defined by and is the Pochhammer symbol: The object of the present investigation is to obtain certain Chebyshev type integral inequalities involving the generalized fractional integral operators  which involves in the kernel, the Gauss hypergeometric function (defined above). The concluding section gives some special cases of the main results. 2. Main Results Our results in this section are based on the preliminary assertions giving
 Jeffrey S. Case Mathematics , 2015, Abstract: Under a spectral assumption on the Laplacian of a Poincar\'e--Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order $2\gamma\in(0,2)$ or $2\gamma\in(2,4)$ and the energy of the weighted conformal Laplacian or weighted Paneitz operator, respectively. This spectral assumption is necessary and sufficient for such an inequality to hold. We prove the energy inequalities by introducing conformally covariant boundary operators associated to the weighted conformal Laplacian and weighted Paneitz operator which generalize the Robin operator. As an application, we establish a new sharp weighted Sobolev trace inequality on the upper hemisphere.
 Abstract and Applied Analysis , 2014, DOI: 10.1155/2014/143741 Abstract: We study existence of solutions for the fractional Laplacian equation in , , with critical exponent , , , where has a potential well and is a lower order perturbation of the critical power . By employing the variational method, we prove the existence of nontrivial solutions for the equation. 1. Introduction In the last 20 years, the classical nonlinear Schr？dinger equation has been extensively studied by many authors [1–10] and the references therein. We just mention some earlier work about it. Brézis and Nirenberg  proved that the critical problem with small linear perturbations can provide positive solutions. In , Rabinowitz proved the existence of standing wave solutions of nonlinear Schr？dinger equations. Making a standing wave ansatz reduces the problem to that of studying a class of semilinear elliptic equations. Floer and Weinstein  proved that Schr？dinger equation with potential and cubic nonlinearity has standing wave solutions concentrated near each nondegenerate critical point of . However, a great attention has been focused on the study of problems involving the fractional Laplacian recently. This type of operator seems to have a prevalent role in physical situations such as combustion and dislocations in mechanical systems or in crystals. In addition, these operators arise in modelling diffusion and transport in a highly heterogeneous medium. This type of problems has been studied by many authors [11–18] and the references therein. Servadei and Valdinoci [11–14] studied the problem where , is an open bounded set of , , with Lipschitz boundary, is a real parameter, and is a fractional critical Sobolev exponent. is defined as follows: Here is a function such that there exists such that and for any . They proved that problem (1) admits a nontrivial solution for any . They also studied the case and , respectively. Felmer et al.  studied the following nonlinear Schr？dinger equation with fractional Laplacian: where , , and is superlinear and has subcritical growth with respect to . The fractional Laplacian can be characterized as , where is the Fourier transform. They gave the proof of existence of positive solutions and further analyzed regularity, decay, and symmetry properties of these solutions. In this paper, we consider the following problem: with critical exponent , , , where has a potential well, where is the fractional Laplace operator, which may be defined as is the usual fractional Sobolev space. is a lower order perturbation of the critical power . Now we give our main assumptions. In order to find weak solutions of (5),
 Journal of Inequalities and Applications , 2009, Abstract: We give some results concerning stability in the Fredholm operators and Browder operators set, via the concept of measure of noncompactness. Moreover, we prove some localization results on the essential spectra of bounded operators on Banach space. As application, we describe the essential spectra of weighted shift operators. Finally, we describe the spectra of polynomially compact operators, and we use the obtained results to study the solvability for operator equations in Banach spaces.
 Journal of Inequalities and Applications , 2009, DOI: 10.1155/2009/284526 Abstract: We give some results concerning stability in the Fredholm operators and Browder operators set, via the concept of measure of noncompactness. Moreover, we prove some localization results on the essential spectra of bounded operators on Banach space. As application, we describe the essential spectra of weighted shift operators. Finally, we describe the spectra of polynomially compact operators, and we use the obtained results to study the solvability for operator equations in Banach spaces.
 Mathematics , 2013, Abstract: In this work, we consider boundary value problems involving Caputo and Riemann-Liouville fractional derivatives of order $\alpha\in(1,2)$ on the unit interval $(0,1)$. These fractional derivatives lead to non-symmetric boundary value problems, which are investigated from a variational point of view. The variational problem for the Riemann-Liouville case is coercive on the space $H_0^{\alpha/2}(0,1)$ but the solutions are less regular, whereas that for the Caputo case involves different test and trial spaces. The numerical analysis of these problems requires the so-called shift theorems which show that the solutions of the variational problem are more regular. The regularity pickup enables one to establish convergence rates of the finite element approximations. Finally, numerical results are presented to illustrate the error estimates.
 Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/682465 Abstract: A new class of generalized dynamical systems involving generalized f-projection operators is introduced and studied in Banach spaces. By using the fixed-point theorem due to Nadler, the equilibrium points set of this class of generalized global dynamical systems is proved to be nonempty and closed under some suitable conditions. Moreover, the solutions set of the systems with set-valued perturbation is showed to be continuous with respect to the initial value.
 Gerd Grubb Mathematics , 2014, DOI: 10.1016/j.jmaa.2014.07.081 Abstract: In the first part of the paper we show Weyl type spectral asymptotic formulas for pseudodifferential operators $P_a$ of order $2a$, with type and factorization index $a\in R_+$, restricted to compact sets with boundary; this includes fractional powers of the Laplace operator. The domain and the regularity of eigenfunctions is described. In the second part, we apply this in a study of realizations $A_{\chi ,\Sigma _+}$ in $L_2(\Omega )$ of mixed problems for a second-order strongly elliptic symmetric differential operator $A$ on a bounded smooth set $\Omega \subset R^n$; here the boundary $\partial\Omega =\Sigma$ is partioned smoothly into $\Sigma =\Sigma _-\cup \Sigma _+$, the Dirichlet condition $\gamma _0u=0$ is imposed on $\Sigma _-$, and a Neumann or Robin condition $\chi u=0$ is imposed on $\Sigma _+$. It is shown that the Dirichlet-to-Neumann operator $P_{\gamma ,\chi }$ is principally of type $\frac12$ with factorization index $\frac12$, relative to $\Sigma _+$. The above theory allows a detailed description of $D(A_{\chi ,\Sigma _+})$ with singular elements outside of $H^{\frac32}(\Omega )$, and leads to a spectral asymptotic formula for the Krein resolvent difference $A_{\chi ,\Sigma _+}^{-1}-A_\gamma ^{-1}$.
 Advances in Mathematical Physics , 2013, DOI: 10.1155/2013/754248 Abstract: The main object of this paper is to investigate the Helmholtz and diffusion equations on the Cantor sets involving local fractional derivative operators. The Cantor-type cylindrical-coordinate method is applied to handle the corresponding local fractional differential equations. Two illustrative examples for the Helmholtz and diffusion equations on the Cantor sets are shown by making use of the Cantorian and Cantor-type cylindrical coordinates. 1. Introduction In the Euclidean space, we observe several interesting physical phenomena by using the differential equations in the different styles of planar, cylindrical, and spherical geometries. There are many models for the anisotropic perfectly matched layers , the plasma source ion implantation , fractional paradigm and intermediate zones in electromagnetism [3, 4], fusion , reflectionless sponge layers , time-fractional heat conduction , singular boundary value problems , and so on (see also the references cited in each of these works). The Helmholtz equation was applied to deal with problems in such fields as electromagnetic radiation, seismology, transmission, and acoustics. Kre？ and Roach  discussed the transmission problems for the Helmholtz equation. Kleinman and Roach  studied the boundary integral equations for the three-dimensional Helmholtz equation. Karageorghis  presented the eigenvalues of the Helmholtz equation. Heikkola et al.  considered the parallel fictitious domain method for the three-dimensional Helmholtz equation. Fu and Mura  suggested the volume integrals of the inhomogeneous Helmholtz equation. Samuel and Thomas  proposed the fractional Helmholtz equation. Diffusion theory has become increasingly interesting and potentially useful in solids [15, 16]. Some applications of physics, such as superconducting alloys , lattice theory , and light diffusion in turbid material , were considered. Fractional calculus theory (see [20–28]) was applied to model the diffusion problems in engineering, and fractional diffusion equation was discussed (see, e.g., [29–36]). Recently, the local fractional calculus theory was applied to process the nondifferentiable phenomena in fractal domain (see [37–48] and the references cited therein). There are some local fractional models, such as the local fractional Fokker-Planck equation , the local fractional stress-strain relations , the local fractional heat conduction equation , wave equations on the Cantor sets , and the local fractional Laplace equation . The main aim of this paper
 Mathematics , 2014, Abstract: In this paper we study the influence of the Hardy potential in the fractional heat equation. In particular, we consider the problem $$(P_\theta)\quad \left\{ \begin{array}{rcl} u_t+(-\Delta)^{s} u&=&\l\dfrac{\,u}{|x|^{2s}}+\theta u^p+ c f\mbox{ in } \Omega\times (0,T),\\ u(x,t)&>&0\inn \Omega\times (0,T),\\ u(x,t)&=&0\inn (\ren\setminus\Omega)\times[ 0,T),\\ u(x,0)&=&u_0(x) \mbox{ if }x\in\O, \end{array} \right.$$ where $N> 2s$, $01$, $c,\l>0$, $u_0\ge 0$, $f\ge 0$ are in a suitable class of functions and $\theta=\{0,1\}$. Notice that $(P_0)$ is a linear problem, while $(P_1)$ is a semilinear problem. The main features in the article are: \begin{enumerate} \item Optimal results about \emph{existence} and \emph{instantaneous and complete blow up} in the linear problem $(P_0)$, where the best constant $\Lambda_{N,s}$ in the fractional Hardy inequality provides the threshold between existence and nonexistence. Similar results in the local heat equation were obtained by Baras and Goldstein in \cite{BaGo}. However, in the fractional setting the arguments are much more involved and they require the proof of a weak Harnack inequality for a weighted operator that appear in a natural way. Once this Harnack inequality is obtained, the optimal results follow as a simpler consequence than in the classical case. \item The existence of a critical power $p_+(s,\lambda)$ in the semilinear problem $(P_1)$ such that: \begin{enumerate} \item If $p> p_+(s,\lambda)$, the problem has no weak positive supersolutions and a phenomenon of \emph{complete and instantaneous blow up} happens. \item If $p< p_+(s,\lambda)$, there exists a positive solution for a suitable class of nonnegative data. \end{enumerate} \end{enumerate}
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