The author proposes an alternative way of using fixed point theory to get the existence for semilinear equations.As an example,a nonlocal ordinary differential equation is considered.The idea is to solve homogeneous e...The author proposes an alternative way of using fixed point theory to get the existence for semilinear equations.As an example,a nonlocal ordinary differential equation is considered.The idea is to solve homogeneous equations in the linearization.One feature of this method is that it does not need the equation to have special structures,for instance,variational structures,maximum principle,etc.Roughly speaking,the existence comes from good properties of the suitably linearized equation.The idea may have wider application.展开更多
In this paper we define a fixed point index theory for locally Lip., completely continuous and weakly inward mappings defined on closed convex sets in general Banach spaces where no other artificial conditions are imp...In this paper we define a fixed point index theory for locally Lip., completely continuous and weakly inward mappings defined on closed convex sets in general Banach spaces where no other artificial conditions are imposed. This makes ns to deal with these kinds of mappings more easily. As obvious applications, some results in [3],[5],[7],[9],[10] are deepened and extended.展开更多
The Department of Economic and Social Affairs of the United Nations has released seventeen goals for sustainable development and SDG No.3 is“Good Health and Well-being”,which mainly emphasizes the strategies to be a...The Department of Economic and Social Affairs of the United Nations has released seventeen goals for sustainable development and SDG No.3 is“Good Health and Well-being”,which mainly emphasizes the strategies to be adopted for maintaining a healthy life.The Monkeypox Virus disease was first reported in 1970.Since then,various health initiatives have been taken,including by the WHO.In the present work,we attempt a fractional model of Monkeypox virus disease,which we feel is crucial for a better understanding of this disease.We use the recently introduced ABC fractional derivative to closely examine the Monkeypox virus disease model.The evaluation of this model determines the existence of two equilibrium states.These two stable points exist within the model and include a disease-free equilibrium and endemic equilibrium.The disease-free equilibrium has undergone proof to demonstrate its stability properties.The system remains stable locally and globally whenever the effective reproduction number remains below one.The effective reproduction number becoming greater than unity makes the endemic equilibrium more stable both globally and locally than unity.To comprehensively study the model’s solutions,we employ the Picard-Lindelof approach to investigate their existence and uniqueness.We investigate the Ulam-Hyers and UlamHyers Rassias stability of the fractional order nonlinear framework for the Monkeypox virus disease model.Furthermore,the approximate solutions of the ABC fractional order Monkeypox virus disease model are obtained with the help of a numerical technique combining the Lagrange polynomial interpolation and fundamental theorem of fractional calculus with the ABC fractional derivative.展开更多
In this paper,we discuss the generalized Abelian differential equation.By using the fixed point theorem,we obtain sufficient conditions for the existence of two nonzero periodic solutions of the equation.We also discu...In this paper,we discuss the generalized Abelian differential equation.By using the fixed point theorem,we obtain sufficient conditions for the existence of two nonzero periodic solutions of the equation.We also discuss the case that there is no nonzero periodic solution and there is a unique nonzero periodic solution.展开更多
In this paper, we study the anti-periodic solutions for 2n-th order differential equations. By using the Schauder＇s fixed point theorem, we present some new results about the existence and uniqueness of anti-periodic...In this paper, we study the anti-periodic solutions for 2n-th order differential equations. By using the Schauder＇s fixed point theorem, we present some new results about the existence and uniqueness of anti-periodic solutions for 2n-th order differential equations.展开更多
This paper deals with a class of n-degree polynomial differential equations. By the fixed point theorem and mathematical analysis techniques, the existence of one (n is an odd number) or two (n is an even number) peri...This paper deals with a class of n-degree polynomial differential equations. By the fixed point theorem and mathematical analysis techniques, the existence of one (n is an odd number) or two (n is an even number) periodic solutions of the equation is obtained. These conclusions have certain application value for judging the existence of periodic solutions of polynomial differential equations with only one higher-order term.展开更多
In this paper. we discuss the existence and stability of solution for two semi-homogeneous boundary value problems. The relative theorems in [1.2] are extended. Meanwhile. we obtain some new results.
In this paper, the famous Amann three-solution theorem is generalized. Multiplicity question of fixed points for nonlinear operators via two coupled parallel sub-super solutions is studied. Under suitable conditions, ...In this paper, the famous Amann three-solution theorem is generalized. Multiplicity question of fixed points for nonlinear operators via two coupled parallel sub-super solutions is studied. Under suitable conditions, the existence of at least six distinct fixed points of nonlinear operators is proved. The theoretical results are then applied to nonlinear system of Hammerstein integral equations.展开更多
By fixed point index theory and a result obtained by Amann, existence of the solution for a class of nonlinear operator equations x = Ax is discussed. Under suitable conditions, a couple of positive and negative solut...By fixed point index theory and a result obtained by Amann, existence of the solution for a class of nonlinear operator equations x = Ax is discussed. Under suitable conditions, a couple of positive and negative solutions are obtained. Finally, the abstract result is applied to nonlinear Sturm-Liouville boundary value problem, and at least four distinct solutions are obtained.展开更多
In this paper, we consider stochastic Volterra-Levin equations. Based on semigroup of operators and fixed point method, under some suitable assumptions to ensure the existence and stability of pth-mean almost periodic...In this paper, we consider stochastic Volterra-Levin equations. Based on semigroup of operators and fixed point method, under some suitable assumptions to ensure the existence and stability of pth-mean almost periodic mild solutions to the system.展开更多
This paper discusses the existence and multiplicity of positive solutions for a class of singular boundary value problems of Hadamard fractional differential systems involving the p-Laplacian operator. First, for the ...This paper discusses the existence and multiplicity of positive solutions for a class of singular boundary value problems of Hadamard fractional differential systems involving the p-Laplacian operator. First, for the sake of overcoming the singularity, sequences of approximate solutions to the boundary value problem are obtained by applying the fixed point index theory on the cone. Next, it is demonstrated that these sequences of approximate solutions are uniformly bounded and equicontinuous. The main results are then established through the Ascoli-Arzelà theorem. Ultimately, an instance is worked out to test and verify the validity of the main results.展开更多
There are two algebraic lower bounds of the number of n-periodic points of a self-map f ： M → M of a compact smooth manifold of dimension at least 3： NFn（f） = min{#Fix（gn）;g - f; g is continuous} and NJDn（f） ...There are two algebraic lower bounds of the number of n-periodic points of a self-map f ： M → M of a compact smooth manifold of dimension at least 3： NFn（f） = min{#Fix（gn）;g - f; g is continuous} and NJDn（f） = min{#Fix（gn）; g - f; g is smooth}. In general, NJDn（f） may be much greater than NFn（f）. If M is a torus, then the invariants are equal. We show that for a self-map of a nonabelian compact Lie group, with free fundamental group, the equality holds 〈=〉 all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.展开更多
Hepatitis B virus (HBV) remains a persistent global health concern, with recent research advancing our understanding of its transmission dynamics and potential interventions.The present study proposes a mathematical m...Hepatitis B virus (HBV) remains a persistent global health concern, with recent research advancing our understanding of its transmission dynamics and potential interventions.The present study proposes a mathematical model of Hepatitis B Virus (HBV) epidemics using fractional calculus, with a special emphasis on the influence of spontaneous clearance across diverse population groups. Using the Atangana-Baleanu derivative, the model accounts for the complications of vertical and horizontal transmission, therapy, immunisation, and spontaneous clearance. Numerical simulations with different fractional orders demonstrate how spontaneous clearance affects the dynamics of susceptible, chronic, treated, and recovered populations. The findings indicate that in vulnerable populations, increasing spontaneous clearance reduces vulnerability because people either clear the illness naturally or gain resistance.However, in chronic populations, spontaneous clearance is insufficient for complete recovery without treatment. The combination of therapy and spontaneous clearance improves the treated population, demonstrating the beneficial effects of both medical intervention and natural immunity. Furthermore, increased spontaneous clearance boosts the restored population, demonstrating the immune system's ability to eliminate the virus over time. The fractional-order framework captures the memory effect of illness development, revealing how healing is time-dependent and how immune responses have a long-term impact. This study emphasises the need of combining spontaneous clearance with medical therapies to improve HBV management and public health consequences. Hepatitis B virus (HBV) remains a persistent global health concern, with recent research advancing our understanding of its transmission dynamics and potential interventions. This study presents a fractional mathematical model of HBV infection, employing the Atangana-Baleanu derivative with Mittag-Leffler kernels to capture memory-dependent and nonlocal transmission processes. The model integrates vertical and horizontal transmission pathways, treatment strategies, immunization efforts, and spontaneous clearance, providing a nuanced perspective compared to classical models. Stability conditions are analyzed through fixed-point theory, revealing the global stability of both disease-free and endemic states under specific values of the basic reproduction number R0. Numerical simulations demonstrate the model's effectiveness in capturing the complex dynamics of HBV, with fractional-order parameters enhancing prediction accuracy. This approach offers valuable insights into optimizing public health interventions and treatment strategies for managing HBV infections effectively.展开更多
For any pair of orientable closed hyperbolic 3-manifolds,this paper shows that any isomorphism between the profinite completions of their fundamental groups witnesses a bijective correspondence between the Zariski den...For any pair of orientable closed hyperbolic 3-manifolds,this paper shows that any isomorphism between the profinite completions of their fundamental groups witnesses a bijective correspondence between the Zariski dense PSL(2,Q^(ac))-representations of their fundamental groups,up to conjugacy;moreover,corresponding pairs of representations have identical invariant trace fields and isomorphic invariant quaternion algebras.(Here,Q^(ac)denotes an algebraic closure of Q.)Next,assuming the p-adic Borel regulator injectivity conjecture for number fields,this paper shows that uniform lattices in PSL(2,C)with isomorphic profinite completions have identical invariant trace fields,isomorphic invariant quaternion algebras,identical covolume,and identical arithmeticity.展开更多
This paper discusses a class of the bidirectional associative memories（BAM） type neural networks with impulse.By using the Banach fixed point theory and some analysis technology,we obtain the existence of almost per...This paper discusses a class of the bidirectional associative memories（BAM） type neural networks with impulse.By using the Banach fixed point theory and some analysis technology,we obtain the existence of almost periodic solution and stability under some sufficient conditions.展开更多
We study the existence of positive solutions to boundary value problems for one- dimensional p-Laplacian under some conditions about the first eigenvalues correspon- ding to the relevant operators by the fixed point t...We study the existence of positive solutions to boundary value problems for one- dimensional p-Laplacian under some conditions about the first eigenvalues correspon- ding to the relevant operators by the fixed point theory. The main difficulties are the computation of fixed point index and the subadditivity for positively 1-homogeneous operator.展开更多
In this paper, we consider a second-order periodic boundary value problem. By the topological degree theory and fixed point index theory, we prove the existence of positive solutions which gives the relationship betwe...In this paper, we consider a second-order periodic boundary value problem. By the topological degree theory and fixed point index theory, we prove the existence of positive solutions which gives the relationship between the first positive eigenvalue of the associated eigenvalue problem and the behavior of the nonlinear term of the system.展开更多
The primary varicella-zoster virus(VzV)infection that causes chickenpox(also known as varicella),spreads quickly among people and,in severe circumstances,can cause to fever and encephalitis.In this paper,the Mittag-Le...The primary varicella-zoster virus(VzV)infection that causes chickenpox(also known as varicella),spreads quickly among people and,in severe circumstances,can cause to fever and encephalitis.In this paper,the Mittag-Leffler fractional operator is used to examine the mathematical representation of the vzV.Five fractional-order differential equations are created in terms of the disease's dynamical analysis such as S:Susceptible,V:Vaccinated,E:Exposed,I:Infectious and R:Recovered.We derive the existence criterion,positive solution,Hyers-Ulam stability,and boundedness of results in order to examine the suggested fractional-order model's wellposedness.Finally,some numerical examples for the VzV model of various fractional orders are shown with the aid of the generalized Adams-Bashforth-Moulton approach to show the viability of the obtained results.展开更多
By using fixed point index theory, we consider the existence of positive solutions for singular nonlinear Neumann boundary value problems. Our main results extend and improve many known results even for non-singular c...By using fixed point index theory, we consider the existence of positive solutions for singular nonlinear Neumann boundary value problems. Our main results extend and improve many known results even for non-singular cases.展开更多
文摘The author proposes an alternative way of using fixed point theory to get the existence for semilinear equations.As an example,a nonlocal ordinary differential equation is considered.The idea is to solve homogeneous equations in the linearization.One feature of this method is that it does not need the equation to have special structures,for instance,variational structures,maximum principle,etc.Roughly speaking,the existence comes from good properties of the suitably linearized equation.The idea may have wider application.
文摘In this paper we define a fixed point index theory for locally Lip., completely continuous and weakly inward mappings defined on closed convex sets in general Banach spaces where no other artificial conditions are imposed. This makes ns to deal with these kinds of mappings more easily. As obvious applications, some results in [3],[5],[7],[9],[10] are deepened and extended.
基金sponsored by Prince Sattam Bin Abulaziz University(PSAU)as part of funding for its SDG Roadmap Research Funding Programme Project Number PSAU-2023-SDG-107.
文摘The Department of Economic and Social Affairs of the United Nations has released seventeen goals for sustainable development and SDG No.3 is“Good Health and Well-being”,which mainly emphasizes the strategies to be adopted for maintaining a healthy life.The Monkeypox Virus disease was first reported in 1970.Since then,various health initiatives have been taken,including by the WHO.In the present work,we attempt a fractional model of Monkeypox virus disease,which we feel is crucial for a better understanding of this disease.We use the recently introduced ABC fractional derivative to closely examine the Monkeypox virus disease model.The evaluation of this model determines the existence of two equilibrium states.These two stable points exist within the model and include a disease-free equilibrium and endemic equilibrium.The disease-free equilibrium has undergone proof to demonstrate its stability properties.The system remains stable locally and globally whenever the effective reproduction number remains below one.The effective reproduction number becoming greater than unity makes the endemic equilibrium more stable both globally and locally than unity.To comprehensively study the model’s solutions,we employ the Picard-Lindelof approach to investigate their existence and uniqueness.We investigate the Ulam-Hyers and UlamHyers Rassias stability of the fractional order nonlinear framework for the Monkeypox virus disease model.Furthermore,the approximate solutions of the ABC fractional order Monkeypox virus disease model are obtained with the help of a numerical technique combining the Lagrange polynomial interpolation and fundamental theorem of fractional calculus with the ABC fractional derivative.
基金Supported by the Senior Personnel of Jiangsu University(Grant No.14JDG176)。
文摘In this paper,we discuss the generalized Abelian differential equation.By using the fixed point theorem,we obtain sufficient conditions for the existence of two nonzero periodic solutions of the equation.We also discuss the case that there is no nonzero periodic solution and there is a unique nonzero periodic solution.
文摘In this paper, we study the anti-periodic solutions for 2n-th order differential equations. By using the Schauder＇s fixed point theorem, we present some new results about the existence and uniqueness of anti-periodic solutions for 2n-th order differential equations.
文摘This paper deals with a class of n-degree polynomial differential equations. By the fixed point theorem and mathematical analysis techniques, the existence of one (n is an odd number) or two (n is an even number) periodic solutions of the equation is obtained. These conclusions have certain application value for judging the existence of periodic solutions of polynomial differential equations with only one higher-order term.
文摘In this paper. we discuss the existence and stability of solution for two semi-homogeneous boundary value problems. The relative theorems in [1.2] are extended. Meanwhile. we obtain some new results.
基金This research is supported by NSFC (10071042)NSFSP (Z2000A02).
文摘In this paper, the famous Amann three-solution theorem is generalized. Multiplicity question of fixed points for nonlinear operators via two coupled parallel sub-super solutions is studied. Under suitable conditions, the existence of at least six distinct fixed points of nonlinear operators is proved. The theoretical results are then applied to nonlinear system of Hammerstein integral equations.
基金. This work is supported by the WNSFC(60304003, 10371066) the NSF of Shandong Province(Z2003A01, Y02P01) and the doctoral Foundation of Shandong Province(03B5092)
文摘By fixed point index theory and a result obtained by Amann, existence of the solution for a class of nonlinear operator equations x = Ax is discussed. Under suitable conditions, a couple of positive and negative solutions are obtained. Finally, the abstract result is applied to nonlinear Sturm-Liouville boundary value problem, and at least four distinct solutions are obtained.
基金partially supported by the NNSF of China(Grant No.11271093)
文摘In this paper, we consider stochastic Volterra-Levin equations. Based on semigroup of operators and fixed point method, under some suitable assumptions to ensure the existence and stability of pth-mean almost periodic mild solutions to the system.
文摘This paper discusses the existence and multiplicity of positive solutions for a class of singular boundary value problems of Hadamard fractional differential systems involving the p-Laplacian operator. First, for the sake of overcoming the singularity, sequences of approximate solutions to the boundary value problem are obtained by applying the fixed point index theory on the cone. Next, it is demonstrated that these sequences of approximate solutions are uniformly bounded and equicontinuous. The main results are then established through the Ascoli-Arzelà theorem. Ultimately, an instance is worked out to test and verify the validity of the main results.
文摘There are two algebraic lower bounds of the number of n-periodic points of a self-map f ： M → M of a compact smooth manifold of dimension at least 3： NFn（f） = min{#Fix（gn）;g - f; g is continuous} and NJDn（f） = min{#Fix（gn）; g - f; g is smooth}. In general, NJDn（f） may be much greater than NFn（f）. If M is a torus, then the invariants are equal. We show that for a self-map of a nonabelian compact Lie group, with free fundamental group, the equality holds 〈=〉 all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.
文摘Hepatitis B virus (HBV) remains a persistent global health concern, with recent research advancing our understanding of its transmission dynamics and potential interventions.The present study proposes a mathematical model of Hepatitis B Virus (HBV) epidemics using fractional calculus, with a special emphasis on the influence of spontaneous clearance across diverse population groups. Using the Atangana-Baleanu derivative, the model accounts for the complications of vertical and horizontal transmission, therapy, immunisation, and spontaneous clearance. Numerical simulations with different fractional orders demonstrate how spontaneous clearance affects the dynamics of susceptible, chronic, treated, and recovered populations. The findings indicate that in vulnerable populations, increasing spontaneous clearance reduces vulnerability because people either clear the illness naturally or gain resistance.However, in chronic populations, spontaneous clearance is insufficient for complete recovery without treatment. The combination of therapy and spontaneous clearance improves the treated population, demonstrating the beneficial effects of both medical intervention and natural immunity. Furthermore, increased spontaneous clearance boosts the restored population, demonstrating the immune system's ability to eliminate the virus over time. The fractional-order framework captures the memory effect of illness development, revealing how healing is time-dependent and how immune responses have a long-term impact. This study emphasises the need of combining spontaneous clearance with medical therapies to improve HBV management and public health consequences. Hepatitis B virus (HBV) remains a persistent global health concern, with recent research advancing our understanding of its transmission dynamics and potential interventions. This study presents a fractional mathematical model of HBV infection, employing the Atangana-Baleanu derivative with Mittag-Leffler kernels to capture memory-dependent and nonlocal transmission processes. The model integrates vertical and horizontal transmission pathways, treatment strategies, immunization efforts, and spontaneous clearance, providing a nuanced perspective compared to classical models. Stability conditions are analyzed through fixed-point theory, revealing the global stability of both disease-free and endemic states under specific values of the basic reproduction number R0. Numerical simulations demonstrate the model's effectiveness in capturing the complex dynamics of HBV, with fractional-order parameters enhancing prediction accuracy. This approach offers valuable insights into optimizing public health interventions and treatment strategies for managing HBV infections effectively.
文摘For any pair of orientable closed hyperbolic 3-manifolds,this paper shows that any isomorphism between the profinite completions of their fundamental groups witnesses a bijective correspondence between the Zariski dense PSL(2,Q^(ac))-representations of their fundamental groups,up to conjugacy;moreover,corresponding pairs of representations have identical invariant trace fields and isomorphic invariant quaternion algebras.(Here,Q^(ac)denotes an algebraic closure of Q.)Next,assuming the p-adic Borel regulator injectivity conjecture for number fields,this paper shows that uniform lattices in PSL(2,C)with isomorphic profinite completions have identical invariant trace fields,isomorphic invariant quaternion algebras,identical covolume,and identical arithmeticity.
基金Supported by the National Natural Science Foundation of China (Grant No.11061031)
文摘This paper discusses a class of the bidirectional associative memories（BAM） type neural networks with impulse.By using the Banach fixed point theory and some analysis technology,we obtain the existence of almost periodic solution and stability under some sufficient conditions.
基金supported by the National Natural Science Foundation of China(11371221,11071141)the Specialized Research Foundation for the Doctoral Program of Higher Educationof China(20123705110001)the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province and Foundation of SDUST
文摘We study the existence of positive solutions to boundary value problems for one- dimensional p-Laplacian under some conditions about the first eigenvalues correspon- ding to the relevant operators by the fixed point theory. The main difficulties are the computation of fixed point index and the subadditivity for positively 1-homogeneous operator.
基金Supported by National Natural Science Foundation of China (11161022)Natural Science Foundation of Jiangxi Province (20114BAB211006 and 20122BAB201015)Educational Department of Jiangxi Province (GJJ12280)
文摘In this paper, we consider a second-order periodic boundary value problem. By the topological degree theory and fixed point index theory, we prove the existence of positive solutions which gives the relationship between the first positive eigenvalue of the associated eigenvalue problem and the behavior of the nonlinear term of the system.
文摘The primary varicella-zoster virus(VzV)infection that causes chickenpox(also known as varicella),spreads quickly among people and,in severe circumstances,can cause to fever and encephalitis.In this paper,the Mittag-Leffler fractional operator is used to examine the mathematical representation of the vzV.Five fractional-order differential equations are created in terms of the disease's dynamical analysis such as S:Susceptible,V:Vaccinated,E:Exposed,I:Infectious and R:Recovered.We derive the existence criterion,positive solution,Hyers-Ulam stability,and boundedness of results in order to examine the suggested fractional-order model's wellposedness.Finally,some numerical examples for the VzV model of various fractional orders are shown with the aid of the generalized Adams-Bashforth-Moulton approach to show the viability of the obtained results.
基金Project supported by NSFC(10471075) NSFSP(Y2003A01, J02P01, XJ03001).
文摘By using fixed point index theory, we consider the existence of positive solutions for singular nonlinear Neumann boundary value problems. Our main results extend and improve many known results even for non-singular cases.
