(Co)bordism groups in quantum super PDEs. II: Quantum super PDEs

Following our previous works on the integral (co)bordism groups of quantum PDEs [A. Prástaro, Geometry of PDEs and Mechanics, World Scientific Publishing Co., Singapore, 1996, 760pp; A. Prástaro, (Co)bordisms in PDE's and quantum PDE's, Rep. Math. Phys. 38(3) (1996) 443–455; A. Prástaro, Quantum and integral (co)bordism groups in partial differential equations, Acta Appl. Math. 51(3) (1998) 243–302; A. Prástaro, (Co)bordism groups in PDE's, Acta Appl. Math. 59(2) (1999) 111–202; A. Prástaro, (Co)bordism groups in quantum PDE's, Acta Appl. Math. 64(2/3) (2000) 111–217; A. Prástaro, Quantum manifolds and integral (co)bordism groups in quantum partial differential equations, Nonlinear Anal. 47/4 (2001) 2609–2620; A. Prástaro, Quantized Partial Differential Equations, World Scientific Publishing Co., Singapore, 2004, 500pp; A. Prástaro, Quantum super Yang-Mills equations: global existence and mass-gap, Dynamic Syst. Appl. 4 (2004) 227–234; A. Prástaro, Th.M. Rassias, A geometric approach to a noncommutative generalized d’Alembert equation, C. R. Acad. Sci. Paris 330(I-7) (2000) 545–550; A. Prástaro, Th.M. Rassias, Results on the J.D’Alembert equation, Ann Acad. Paed. Crac. Stud. Math. 1 (2001) 117–128.], we specialize, now, on quantum super partial differential equations, i.e., partial differential equations built in the category of quantum supermanifolds. These are manifolds modeled on locally convex topological vector spaces built starting from quantum algebras endowed also with a ${\mathbb{Z}}_2$-gradiation, and a ${\mathbb{Z}}_2$-graded Lie algebra structure, (quantum superalgebra). Then, we extend to these manifolds, with such richer structure, our previous results, and build a geometric theory of quantum super PDEs, that allows us to obtain theorems of existence of (smooth) local and global solutions in the category of quantum supermanifolds. Some quantum (super) PDEs, arising from the Dirac quantization of some classical (super) PDEs, are considered in some details.     

(Co)bordism groups in quantum super PDE's. III: Quantum super Yang–Mills equations

In this third part of a series of three papers devoted to the study of geometry of quantum super PDE's [A. Prástaro, (Co)bordism groups in quantum super PDE's. I: quantum supermanifolds, Nonlinear Anal. Real World Appl., in press, doi:10.1016/j.nonrwa.2005.12.007; (Co)bordism groups in quantum super PDE's. II: quantum super PDE's, Nonlinear Anal. Real World Appl., in press, doi:10.1016/j.nonrwa.2005.12.008], we apply our theory, developed in the first two parts, to quantum super Yang–Mills equations and quantum supergravity equations. For such equations we determine their integral bordism groups, and by using some surgery techniques, we obtain theorems of existence of global solutions, also with nontrivial topology, for Cauchy problems and boundary value problems. Quantum tunnelling effects are described in this context. Furthermore, for quantum supergravity equations we prove existence of solutions of the type quantum black holes evaporation processes just by using an extension to quantum super PDEs of our theory of integral (co)bordism groups. Our proof is constructive, i.e., we give geometric methods to build such solutions. In particular a criterion to recognize quantum global (smooth) solutions with mass-gap, for the quantum super Yang–Mills equation, is given. Finally it is proved that quantum super PDE's contain also solutions that come from Dirac quantization of their superclassical counterparts. This proves that quantum super PDE's are (nonlinear) generalizations of Dirac quantized superclassical PDE's. Applications of this result to free quantum super Yang–Mills equations are given.     

Geometry of PDE's. IV: Navier-Stokes equation and integral bordism groups

Following our previous results on this subject [R.P. Agarwal, A. Prástaro, Geometry of PDE's. III(I): Webs on PDE's and integral bordism groups. The general theory, Adv. Math. Sci. Appl. 17 (2007) 239-266; R.P. Agarwal, A. Prástaro, Geometry of PDE's. III(II): Webs on PDE's and integral bordism groups. Applications to Riemannian geometry PDE's, Adv. Math. Sci. Appl. 17 (2007) 267-285; A. Prástaro, Geometry of PDE's and Mechanics, World Scientific, Singapore, 1996; A. Prástaro, Quantum and integral (co)bordism in partial differential equations, Acta Appl. Math. (5) (3) (1998) 243-302; A. Prástaro, (Co)bordism groups in PDE's, Acta Appl. Math. 59 (2) (1999) 111-201; A. Prástaro, Quantized Partial Differential Equations, World Scientific Publishing Co, Singapore, 2004, 500 pp.; A. Prástaro, Geometry of PDE's. I: Integral bordism groups in PDE's, J. Math. Anal. Appl. 319 (2006) 547-566; A. Prástaro, Geometry of PDE's. II: Variational PDE's and integral bordism groups, J. Math. Anal. Appl. 321 (2006) 930-948; A. Prástaro, Th.M. Rassias, Ulam stability in geometry of PDE's, Nonlinear Funct. Anal. Appl. 8 (2) (2003) 259-278; I. Stakgold, Boundary Value Problems of Mathematical Physics, I, The MacMillan Company, New York, 1967; I. Stakgold, Boundary Value Problems of Mathematical Physics, II, Collier-MacMillan, Canada, Ltd, Toronto, Ontario, 1968], integral bordism groups of the Navier-Stokes equation are calculated for smooth, singular and weak solutions, respectively. Then a characterization of global solutions is made on this ground. Enough conditions to assure existence of global smooth solutions are given and related to nullity of integral characteristic numbers of the boundaries. Stability of global solutions are related to some characteristic numbers of the space-like Cauchy data. Global solutions of variational problems constrained by (NS) are classified by means of suitable integral bordism groups too.     

Quantum exotic PDE’s

Following the previous works on the Prástaro’s formulation of algebraic topology of quantum (super) PDE’s, it is proved that a canonical Heyting algebra (integral Heyting algebra) can be associated to any quantum PDE. This is directly related to the structure of its global solutions. This allows us to recognize a new inside in the concept of quantum logic for microworlds. Furthermore, the Prástaro’s geometric theory of quantum PDE’s is applied to the new category of quantum hypercomplex manifolds, related to the well-known Cayley–Dickson construction for algebras. Theorems of existence for local and global solutions are obtained for (singular) PDE’s in this new category of noncommutative manifolds. Finally, the extension of the concept of exotic PDE’s, recently introduced by Prástaro, has been extended to quantum PDE’s. Then a smooth quantum version of the quantum (generalized) Poincaré conjecture is given too. These results extend ones for quantum (generalized) Poincaré conjecture, previously given by Prástaro.     

On the singular PDE's geometry and boundary value problems

A geometric formulation of singular partial differential equations (PDEs) is considered. Surgery techniques and integral bordism groups are utilized, following previous works by Prástaro on PDEs, in order to build global solutions crossing also singular points and to study their stability properties.     

Extended crystal PDE’S stability,II: The extended crystal MHD-PDE’S

This paper is the second part of a work devoted to the algebraic topological characterization of PDE’s stability, and its relationship with an important class of PDE’s called extended crystals PDE’s in the sense introduced in [A. Prástaro, Extended crystal PDE’s (submitted for publication)]. In fact, their integral bordism groups can be considered as extensions of subgroups of crystallographic groups. This allows us to identify a characteristic class that measures the obstruction to the existence of global solutions. In part I [A. Prástaro, Extended crystal PDE’s stability, I: The general theory, Math. Comput. Modelling, 49 (9–10) (2009) 1759–1780] we identified criteria to recognize PDE’s that are stable (in extended Ulam sense) and in their regular smooth solutions, finite time instabilities do not occur (stable extended crystal PDE’s). Here, we study in some detail, a new PDE encoding anisotropic incompressible magnetohydrodynamics. Stable extended crystal MHD-PDE’s are obtained, where in their smooth solutions, instabilities do not occur in finite time. These results are considered first for systems without a body energy source, and later, by also introducing a contribution from an energy source, in order to take into account nuclear energy production. A condition in order that solutions satisfy the second principle of thermodynamics is given.     

(Co)bordism Groups in Quantum PDEs

In this paper we formulate a theory of noncommutative manifolds (quantum manifolds) and for such manifolds we develop a geometric theory of quantum PDEs (QPDEs). In particular, a criterion of formal integrability is given that extends to QPDEs previously obtained by D. C. Spencer and H. Goldschmidt for PDEs for commutative manifolds, and by Prástaro for super PDEs. Quantum manifolds are seen as locally convex manifolds where the model has the structure A ^m _{1 1}×···×A ^m _{s s} , with AA ₁×···×A _s a noncommutative algebra that satisfies some particular axioms (quantum algebra). A general theory of integral (co)bordism for QPDEs is developed that extends our previous for PDEs. Then, noncommutative Hopf algebras (full quantum p-Hopf algebras, 0pm–1) are canonically associated to any QPDE, Ê_k ^k _m(W) whose elements represent all the possible invariants that can be recognized for such a structure. Many examples of QPDEs are considered where we apply our theory. In particular, we carefully study QPDEs for quantum field theory and quantum supergravity. We show that the corresponding regular solutions, observed by means of quantum relativistic frames, give curvature, torsion, gravitino and electromagnetic fields as A-valued distributions on spacetime, where A is a quantum algebra. For such equations, canonical quantizations are obtained and the quantum and integral bordism groups and the full quantum p-Hopf algebras, 0p3, are explicitly calculated. Then, the existence of (quantum) tunnel effects for quantum superstrings in supergravity is proved.     

Permanence of nonautonomous Lotka–Volterra delay differential systems

In this work, applying the results offered by S. Ahmad and A.C. Lazer [On a property of nonautonomous Lotka–Volterra competition model, Nonlinear Anal. 37 (1999) 603–611] and the recent work of R. Redheffer [Mean values and the nonautonomous May–Leonald equations, Nonlinear Anal. Real World Appl. 4 (2003) 301–306] to an nonautonomous Lotka–Volterra differential system with finite delays, we establish sufficient conditions for the permanence of the system.     

Periodic solutions for predator–prey systems with Beddington–DeAngelis functional response on time scales

This paper deals with the question of existence of periodic solutions of nonautonomous predator–prey dynamical systems with Beddington–DeAngelis functional response. We explore the periodicity of this system on time scales. New sufficient conditions are derived for the existence of periodic solutions. These conditions extend previous results presented in [M. Bohner, M. Fan, J. Zhang, Existence of periodic solutions in predator–prey and competition dynamic systems, Nonlinear. Anal.: Real World Appl. 7 (2006) 1193–1204; M. Fan, Y. Kuang, Dynamics of a nonautonomous predator–prey system with the Beddington–DeAngelies functional response, J. Math. Anal. Appl. 295 (2004) 15–39; J. Zhang, J. Wang, Periodic solutions for discrete predator–prey systems with the Beddington–DeAngelis functional response, Appl. Math. Lett. 19 (2006) 1361–1366].     

Extended crystal PDE’s stability,I: The general theory

This work, divided in two parts, follows some our previous works devoted to the algebraic topological characterization of PDE’s. In this first part, the stability of PDE’s is studied in some details in the framework of the geometric theory of PDE’s, and bordism groups theory of PDE’s. In particular we identify criteria to recognize PDE’s that are stable (in extended Ulam sense) and in their regular smooth solutions do not occur finite time unstabilities, (stable extended crystal PDE’s). Applications to some important PDE’s are carefully considered. (In the second part a stable extended crystal PDE, encoding anisotropic incompressible magnetohydrodynamics is obtained Ref. [A. Prástaro, Extended crystal PDE’s (submitted for publication)].)     

The permanence and global attractivity of a Kolmogorov system with feedback controls

In this paper, we study the permanence and global asymptotic behavior for a Kolmogorov system with feedback controls. By means of lower and upper averages of a function, the average conditions for permanence, global attractivity and extinction of this system are established respectively. The corresponding results given by Chen in [F. Chen, The permanence and global attractivity of Lotka–Volterra competition system with feedback controls, Nonlinear Anal. 7 (2006) 133–143] and Zhao in [J.D. Zhao, J.F. Jiang, A.C. Lazer, The permanence and global attractivity in a nonautonomous Lotka–Volterra system, Nonlinear Anal. Real World Appl. 5 (2004) 265–276] are extended and improved.     

(Co)Bordism Groups in PDEs

We introduce a geometric theory of PDEs, by obtaining existence theorems of smooth and singular solutions. Within this framework, following our previous results on (co)bordisms in PDEs, we give characterizations of quantum and integral (co)bordism groups and relate them to the formal integrability of PDEs. An explicit proof that the usual Thom–Pontryagin construction in (co)bordism theory can be generalized also to a singular integral (co)bordism on the category of differential equations is given. In fact, we prove the existence of a spectrum that characterizes the singular integral (co)bordism groups in PDEs. Moreover, a general method that associates, in a natural way, Hopf algebras (full p-Hopf algebras, 0 p n – 1), to any PDE, recently introduced, is further studied. Applications to particular important classes of PDEs are considered. In particular, we carefully consider the Navier–Stokes equation (NS) and explicitly calculate their quantum and integral bordism groups. An existence theorem of solutions of (NS) with a change in sectional topology is obtained. Relations between integral bordism groups and causal integral manifolds, causal tunnel effects, and the full p-Hopf algebras, 0 p 3, for the Navier–Stokes equation are determined.     

Uniform in Time $L^{\infty }$ -Estimates for Nonlinear Aggregation-Diffusion Equations

In this paper, we consider the general reaction–diffusion system proposed in Abdelmalek and Bendoukha (Nonlinear Anal., Real World Appl. 35:397–413, 2017) as a generalization of the original Lengyel–Epstein model developed for the revolutionary Turing-type CIMA reaction. We establish sufficient conditions for the global existence of solutions. We also follow the footsteps of Lisena (Appl. Math. Comput. 249:67–75, 2014) and other similar studies to extend previous results regarding the local and global asymptotic stability of the system. In the local PDE sense, more relaxed conditions are achieved compared to Abdelmalek and Bendoukha (Nonlinear Anal., Real World Appl. 35:397–413, 2017). Also, new extended results are achieved for the global existence, which when applied to the Lengyel–Epstein system, provide weaker conditions than those of Lisena (Appl. Math. Comput. 249:67–75, 2014). Numerical examples are used to affirm the findings and benchmark them against previous results.

     

Multiple periodic solutions in generalized Gause-type predator–prey systems with non-monotonic numerical responses

By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of multiple positive periodic solutions in periodic Gause-type predator–prey systems with non-monotonic numerical responses and time delays. As corollaries, some applications are listed. In particular, our results improve and supplement those obtained by Chen [Y. Chen, Multiple periodic solutions of delayed predator–prey systems with type IV functional responses, Nonlinear Anal. Real World Appl. 5 (2004) 45–53].     

Convergence in a multi-layer population model with age-structure

In this paper we investigate the convergence of a multi-layer population model to a single-layer limit. In a previous paper [Cusulin, C., Iannelli, M., Marinoschi, G. Age-structured diffusion in a multi-layer environment, Nonlinear Anal. Real World Appl. 6(1) (2005) 207–223], we considered a Gurtin–MacCamy type model based on the fact that the population diffuses through a one dimensional habitat, partitioned into $n$ homogeneous layers. In each layer the population has its own age-dependent growth and diffusion parameters, so that within each layer the dynamics is not subject to environmental variations, while changes occur from a layer to another, according to different conditions. Such kind of a model may describe the growth of a population in a fragmented environment, but the same framework may be used to give an approximate view of the population growth and diffusion in a general spatially heterogeneous habitat, because the layer structure may arise by approximation of the original problem.In the present paper we show that this view is actually mathematically sound and justified. In fact, on the basis of the previous results (see [Cusulin, C., Iannelli, M., Marinoschi, G. Age-structured diffusion in a multi-layer environment, Nonlinear Anal. Real World Appl. 6(1) (2005) 207–223]) the approximating problem actually converges and the multi-layer solution may be considered a patch-wise picture of the original problem.     

An improved local convergence analysis for a two-step Steffensen-type method

We study a class of Steffensen-type algorithm for solving nonsmooth variational inclusions in Banach spaces. We provide a local convergence analysis under ω-conditioned divided difference, and the Aubin continuity property. This work on the one hand extends the results on local convergence of Steffensen’s method related to the resolution of nonlinear equations (see Amat and Busquier in Comput. Math. Appl. 49:13–22, 2005; J. Math. Anal. Appl. 324:1084–1092, 2006; Argyros in Southwest J. Pure Appl. Math. 1:23–29, 1997; Nonlinear Anal. 62:179–194, 2005; J. Math. Anal. Appl. 322:146–157, 2006; Rev. Colomb. Math. 40:65–73, 2006; Computational Theory of Iterative Methods, 2007). On the other hand our approach improves the ratio of convergence and enlarges the convergence ball under weaker hypotheses than one given in Hilout (Commun. Appl. Nonlinear Anal. 14:27–34, 2007).     

On a non-homogeneous bi-layer shallow-water problem: smoothness and uniqueness results

In this paper, we present several results for a non-homogeneous bi-layer shallow-water model in depth-mean velocity formulation. The homogeneous case was studied in (Nonlinear Anal.: Real World Appl. 4(1) (2003) 139–171). In (On a non-homogeneous bi-layer shallow water problem: an existence theorem, Diff. and Integral Equations 17 (9–10) (2004) 1175–1200) we proved the existence of a solution for the non-homogeneous problem and now we give some smoothness and uniqueness results. The main difficulties in the non-homogeneous case arise from the treatment of the boundary terms.     

Multidimensional fixed point theorems in partially ordered complete metric spaces

In this paper we propose a notion of coincidence point between mappings in any number of variables and we prove some existence and uniqueness fixed point theorems for nonlinear mappings verifying different kinds of contractive conditions and defined on partially ordered metric spaces. These theorems extend and clarify very recent results that can be found in [T. Gnana-Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (7)(2006) 1379–1393], [V. Berinde, M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal. 74 (2011) 4889–4897] and [M. Berzig, B. Samet, An extension of coupled fixed point’s concept in higher dimension and applications, Comput. Math. Appl. 63 (8) (2012) 1319–1334].     

Approximation of fixed points of nonexpansive mapping in Banach spaces

In this paper, some iterative schemes are given to approximate a fixed point of the nonexpansive non-self-mapping and nonexpansive self-mapping. Furthermore, the strong convergence of the scheme to a fixed point is shown in a Banach space with uniformly Gâteaux differentiable norm. The theorems extend and improve some corresponding results of Matsushita and Takahashi [S. Matsushita, W. Takahashi, Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions, Nonlinear Anal. 68 (2008) 412–419], Chang et al. [S.S. Chang, H.W. Joseph Lee, C.K. Chan, On Reich’s strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces, Nonlinear Anal. 66 (2007) 2364–2374], Chidume and Chidume [C.E. Chidume, C.O. Chidume, Iterative approximation of fixed points of nonexpansive mappings, J. Math. Anal. Appl. 318 (2006) 288–295] and Suzuki [T. Suzuki, A sufficient and necessary condition for Halpern-type strong convergence to fixed point of nonexpansive mappings, Proc. Amer. Math. Society 135 (1) (2007) 99–106].     

A remark on weak solutions to the barotropic compressible quantum Navier–Stokes equations

In [A. Jüngel, Global weak solutions to compressible Navier–Stokes equations for quantum fluids, SIAM J. Math. Anal. 42 (2010) 1025–1045], Jüngel proved the global existence of the barotropic compressible quantum Navier–Stokes equations for when the viscosity constant is bigger than the scaled Planck constant. Recently, Dong [J. Dong, A note on barotropic compressible quantum Navier–Stokes equations, Nonlinear Anal. TMA 73 (2010) 854–856] extended Jüngel’s result to the case where the viscosity constant is equal to the scaled Planck constant by using a new estimate of the square root of the solutions. In this paper we show that Jüngel’s existence result still holds when the viscosity constant is bigger than the scaled Planck constant. Consequently, with our result, the existence for all physically interesting cases of the scaled Planck and viscosity constants is obtained.