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In this paper, we consider the Cauchy problem for the 2D inviscid Boussinesq equations with N being the buoyancy frequency. It is proved that for general initial data u 0 ∈ H s with s > 3 , the life span of the classical solutions satisfies T > C ln N 3 / 4 .

In this paper, we consider the initial boundary value problem for a class of nonlinear fractional partial integro-differential equations of mixed type with non-instantaneous impulses in Banach spaces. Sufficient conditions of existence and uniqueness of PC-mild solutions for the equations are obtained via general Banach contraction mapping principle, Krasnoselskii’s fixed point theorem, and α-order solution operator.

In this paper, we study the mild solutions of a class of nonlinear fractional reaction–diffusion equations with delay and Caputo’s fractional derivatives. By the measure of noncompactness, the theory of resolvent operators, the fixed point theorem and the Banach contraction mapping principle, we develop various theorems for the existence and uniqueness of the global mild solutions for the problem.

By using the method of mixed monotone operator, a unique positive solution is obtained for a singular p-Laplacian boundary value system with infinite-point boundary conditions in this paper. Green’s function is derived and some useful properties of the Green’s function are obtained. Based upon these new properties and by using mixed monotone operator, the existence results of the positive solutions for the boundary value problem are established. Moreover, the unique positive solution that we obtained in this paper is dependent on $\lambda ,\mu $ , and an iterative sequence and convergence rate, which are important for practical application, are given. An example is given to demonstrate the application of our main results.

We investigate a class of semipositone fractional integral boundary value problem on the half-line. Using the fixed point index theorems in a cone, we obtain the existence result of positive solutions.

In this paper, we study the initial boundary value problem for a Petrovsky type equation with a memory term, nonlinear weak damping, and a superlinear source: $$ u_{tt}+\Delta ^{2} u- \int _{0}^{t} g(t-\tau )\Delta ^{2} u(\tau )\,\mathrm{d} \tau + \vert u_{t} \vert ^{m-2}u_{t}= \vert u \vert ^{p-2}u,\quad \text{in }\varOmega \times (0,T). $$ When the source is stronger than dissipations, we obtain the existence of certain weak solutions which blow up in finite time with initial energy $E(0)=R$ for any given $R\geq 0$ .

A Kirchhoff-type problem with concave-convex nonlinearities is studied. By constrained variational methods on a Nehari manifold, we prove that this problem has a sign-changing solution with least energy. Moreover, we show that the energy level of this sign-changing solution is strictly larger than the double energy level of the ground state solution.

The present paper deals with the Cauchy problem of a multi-dimensional non-conservative viscous compressible two-fluid system. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In the functional setting as close as possible to the physical energy spaces, we prove the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we establish the time decay rates for the constructed global solutions. The proof relies on an application of Fourier analysis to a complicated parabolic-hyperbolic system, and on a refined time-weighted inequality.

In this paper, we study the initial boundary value problem for a Petrovsky type equation with a memory term, a linear weak damping and superlinear source. Finite time blow-up results have been obtained for the case in which the initial energy $$E(0)\le M$$ , where M is a positive constant. By utilizing Levine’s classical concavity method, we give a new blow-up criterion which includes the case of $$E(0)>M$$ and derive an explicit upper bound for the blow-up time. By using the Fountain Theorem, we show that the problem with arbitrary positive initial energy always admits weak solutions blowing up in finite time.

In this paper, we establish the results of nonexistence and existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator. Under some suitable growth conditions for nonlinearity, the result of nonexistence of blow-up solutions is established, a sufficient and necessary condition on existence of blow-up solutions is given, and some further results are obtained.

In this paper, we are concerned with the existence of the maximum and minimum iterative solutions for a tempered fractional turbulent flow model in a porous medium with nonlocal boundary conditions. By introducing a new growth condition and developing an iterative technique, we establish new results on the existence of the maximum and minimum solutions for the considered equation; at the same time, the iterative sequences for approximating the extremal solutions are performed, and the asymptotic estimates of solutions are also derived.

In the paper, we prove that the N -Kirchhoff type problem in R N has at least three solutions by using the Ekeland variational principle and fibering map.

In this paper we investigate a system of impulsive integral boundary value problems with sign-changing nonlinearities. Using the fixed point theorem in double cones, we prove the existence of multiple positive solutions.

In this paper, we study a three-dimensional (3D) viscoelastic wave equation with nonlinear weak damping, supercritical sources and prescribed past history u0(x,t) , t≤0 : utt-k(0)Δu-∫0+∞k′(s)Δu(t-s...

In this paper, we consider the initial boundary value problem for a class of thin-film equations in R n with a p-Laplace term and a nonlocal source term | u | q − 2 u − 1 | Ω | ∫ Ω | u | q − 2 u dx . We prove that there exist weak solutions for the problem with arbitrarily initial energy that blow up in finite time. We also obtain the upper bounds for the blow-up time.

In this paper, we study a class of nonlinear singular system with coupled integral boundary condition. Based on the Guo–Krasnosel'skii fixed point theorem, some new results on the existence of symmetric positive solutions for the coupled singular system are obtained. The impact of the two different parameters on the existence of symmetric positive solutions is also investigated. Finally, an example is then given to demonstrate the applicability of our results.

In this paper, we investigate the existence of positive solutions for a system of nonlinear fractional differential equations nonlocal boundary value problems with parameters and p-Laplacian operator. Under different combinations of superlinearity and sublinearity of the nonlinearities, various existence results for positive solutions are derived in terms of different values of parameters via the Guo-Krasnosel’skii fixed point theorem.

In this paper, we study the entire blow-up solutions for a quasilinear p -Laplacian Schrodinger elliptic equation with a non-square diffusion term. By using the dual approach and some new iterative techniques, the difficulty due to the non-square diffusion term and the p -Laplacian operator is overcome and the nonexistence and existence of entire blow-up solutions are established.

In this paper, we investigate the initial boundary value problem for a pseudo-parabolic equation under the influence of a linear memory term and a nonlinear source term ut-Δu-Δut+∫0tg(t-τ)Δu(τ)dτ=|...

By introducing a subspace of H 2 ( Ω ) with constraints ∂ u ∂ n | ∂ Ω = 0 and ∫ Ω u d x = 0 and using the Fountain Theorem, we obtain the existence of infinitely many sign-changing high energy solutions for a biharmonic equations with p -Laplacian and Neumann boundary condition.

In this paper, by constructing a new weighted norm method and analysis technique, we establish the conditions for the existence and uniqueness of positive monotone solutions to a class of nonlinear Schrdinger equations in planar exterior domains. Some examples are also given to demonstrate the application of our main results.

In this paper, we first discuss the geometric properties of the Lorentz cone and the extended Lorentz cone. The self-duality and orthogonality of the Lorentz cone are obtained in Hilbert spaces. These properties are fundamental for the isotonicity of the metric projection with respect to the order, induced by the Lorentz cone. According to the Lorentz cone, the quasi-sublattice and the extended Lorentz cone are defined. We also obtain the representation of the metric projection onto cones in Hilbert quasi-lattices. As an application, solutions of the classic variational inequality problem and the complementarity problem are found by the Picard iteration corresponding to the composition of the isotone metric projection onto the defining closed and convex set and the difference in the identity mapping and the defining mapping. Our results generalize and improve various recent results obtained by many others.

We give some background of fractional Cauchy problems and correct an existing result of Cauchy problem for the fractional differential inequality, and we also give an example to illustrate our statement.

In this paper, we investigate the existence of at least three positive solutions to a singular boundary value problem of Caputo's fractional differential equations with a boundary condition involving values at infinite number of points. Firstly, we establish Green's function and its properties. Then, the existence of multiple positive solutions is obtained by Avery–Peterson's fixed point theorem. Finally, an example is given to demonstrate the application of our main results.

In this paper, we study a higher-order fractional differential equation with integral boundary conditions and a parameter. Under different conditions of nonlinearity, existence and nonexistence results for positive solutions are derived in terms of different intervals of parameter. Our approach relies on the Guo–Krasnoselskii fixed point theorem on cones.

Based on the relation between Leray–Schauder degree and a pair of strict lower and upper solutions, we focus on the bifurcation analysis for a singular differential system with two parameters, explicit bifurcation points for relative parameters are obtained by using the property of solution for the akin systems and topological degree theory.

In this paper, we investigate the local and global existence of mild solutions to a class of nonlinear fractional reaction–diffusion equations with delay via the theory of β -resolvent family and fixed point theorems.

In this paper, we discuss isotonicity characterizations of the metric projection operator, including its necessary and sufficient conditions for isotonicity onto sublattices in Banach spaces. Then, we demonstrate their applications to variational inequalities and fixed point theory in Banach spaces. Our work generalizes many existing results obtained in earlier work.

In this paper, by constructing a cone K 1 × K 2 in the Cartesian product space C[0, 1] × C[0, 1], and using spectral analysis of the relevant linear operator for the corresponding differential system, some properties of the first eigenvalue corresponding to the relevant linear operator are obtained, and the fixed-point index of nonlinear operator in the K 1 × K 2 is calculated explicitly and the existence of at least one positive solution or two positive solutions of the singular differential system with integral boundary conditions is established.

In this paper, we consider a Modified Nonlinear Schrodinger equation Δ u − Δ ( | u | 2 γ ) | u | 2 γ − 2 u = q ( x ) F ( u ) on R n ( n ≥ 3 ) , where γ > 1 2 , q is a nonnegative continuous function on R n , F is a continuous positive and non-decreasing function on [ 0 , ∞ ) . We establish sufficient conditions for the existence and nonexistence of positive entire large solutions for the given equation by using the dual approach.

In this paper, we are concerned with the multi-dimensional (N≥3) compressible viscoelastic flows in the whole space. We prove the optimal convergence rates of strong solutions to the system for the initial data close to a stable equilibrium state in critical Besov spaces. Our main ideas are based on the low-high frequency decomposition and the smooth effect of dissipative operator.

In this paper, we focus on the existence and asymptotic analysis of positive solutions for a class of singular fractional differential equations subject to nonlocal boundary conditions. By constructing suitable upper and lower solutions and employing Schauder’s fixed point theorem, the conditions for the existence of positive solutions are established and the asymptotic analysis for the obtained solution is carried out. In our work, the nonlinear function involved in the equation not only contains fractional derivatives of unknown functions but also has a stronger singularity at some points of the time and space variables.

In this paper, we study periodic boundary value problems of fractional semilinear integro-differential equations with non-instantaneous impulses in Banach spaces. By the measure of noncompactness, the theory of β-resolvent family, and the fixed point theorem, we obtain several sufficient conditions on the existence of mild solutions for such problems. Finally, an example is given to show the main results of this paper.

In this paper, we consider a class of fractional differential equations with infinite-point boundary value conditions. Under some conditions concerning the spectral radius with respect to the relevant linear operator, both the existence of uniqueness and the nonexistence of positive solution are obtained by means of the iterative technique.

In this paper, we focus on the convergence analysis and error estimation for the unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity. By introducing a double iterative technique, in the case of the nonlinearity with singularity at time and space variables, the unique positive solution to the problem is established. Then, from the developed iterative technique, the sequences converging uniformly to the unique solution are formulated, and the estimates of the error and the convergence rate are derived.

In this article, by using the spectral analysis of the relevant linear operator and Gelfand’s formula, some properties of the first eigenvalue of a fractional differential equation are obtained. Based on these properties and through the fixed point index theory, the singular nonlinear fractional differential equations with Riemann–Stieltjes integral boundary conditions involving fractional derivatives are considered under some appropriate conditions, and the nonlinearity is allowed to be singular in regard to not only time variable but also space variable and it includes fractional derivatives. The existence of positive solutions for boundary conditions involving fractional derivatives is established. Finally, an example is given to demonstrate the validity of our main results.

This paper is concerned with the uniqueness of positive solutions for a class of singular fractional differential equations with integral boundary conditions. The nonlinear term and boundary conditions of fractional differential equation contain the fractional order derivatives. The uniqueness of positive solutions is derived by the fixed point theorem of mixed monotone operator. An example is given to demonstrate the validity of our main results.

We discuss Ky Fan’s theorem and the variational inequality problem for discontinuous mappings f in a Banach space X. The main tools of analysis are the variational characterizations of the metric projection operator and the order-theoretic fixed point theory. Moreover, we derive some properties of the metric projection operator in Banach spaces. As applications of our best approximation theorems, three fixed point theorems for non-self maps are established and proved under some conditions. Our results are generalizations and improvements of various recent results obtained by many authors.

The abstract fractional dynamics model is based on a class of bioprocesses of HIV infection.The nonlinear terms and boundary conditions all depend on fractional derivatives of unknown functions.The system is singular and semipositone.The system involves some uncertain parametrical variations λ . Fractional order derivative is nonlocal which exhibits a long time memory behavior. With advantage of these, fractional order dynamic system models are more accurate than integer order ones in understanding the dynamic behavior of bioprocesses such as HIV infection. In this paper, we systematically study the existence of positive solutions of an abstract fractional semipositone differential system involving integral boundary conditions arising from the study of HIV infection models. By using the fixed point theorem in cone, some new results are established and an example is given to demonstrate the application of our main results.

The Cauchy problem for the compressible flow of nematic liquid crystals in dimension ( N ≧ 2) is considered. Employing the Littlewood-Paley theory and Schauder-Tychonoff fixed point theorem, we prove the local well-posedness of the system for large initial data in critical Besov spaces based on the L p framework under the sole natural assumption that the initial density is bounded away from zero.

In this paper, by using the spectral analysis of the relevant linear operator and Gelfand's formula, we obtain some properties of the first eigenvalue of a fractional differential equation. Based on these properties, the fixed point index of the nonlinear operator is calculated explicitly and some sufficient conditions for the existence of positive solutions are established.

In this paper, we study the nonexistence and existence of radial large positive solutions for a class of Schrodinger systems with a nonlinear random operator under certain growth conditions. We also establish the necessary and sufficient conditions for the existence of radial large positive solutions by a successive iteration technique. The results of the paper improve and extend many previous work in this field of research.

In this paper, we are concerned with the existence and uniqueness of solutions for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary condition. Our results are based on the Banach contraction mapping principle and the Krasnoselskii fixed point theorem. Some examples are also given to illustrate our results.