ON THE EXISTENCE AND UNIQUENESS THEOREMS OF SOLUTIONS FOR A CLASS OF THE SYSTEMS OF MIXED MONOTONE OPERATOR EQUATIONS WITH APPLICATION

ＯＮＴＨＥＥＸＩＳＴＥＮＣＥＡＮＤＵＮＩＱＵＥＮＥＳＳＴＨＥＯＲＥＭＳＯＦＳＯＬＵＴＩＯＮＳＦＯＲＡＣＬＡＳＳＯＦＴＨＥＳＹＳＴＥＭＳＯＦＭＩＸＥＤＭＯＮＯＴＯＮＥＯＰＥＲＡＴＯＲＥＱＵＡＴＩＯＮＳＷＩＴＨＡＰＰＬＩＣＡＴＩＯＮＳＨＥＮＰＥＩＬＯＮＧ...     

THE RELATION BETWEEN SIGN-CHANGING SOLUTION AND POSITIVE-NEGATIVE SOLUTIONS FOR NONLINEAR OPERATOR EQUATIONS AND ITS APPLICATIONS

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.     

ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR 2nTH-ORDER BOUNDARY VALUE PROBLEMS

In this paper, by using the Leray-Schauder continuation theorem, we establish the existence and uniqueness theorems of solutions of two-point boundary value problems for 2nth-order nonlinear differential equations with nonlinear growth.     

ON THE EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS FOR A CLASSOF DEGENERATE ELLIPTIC SYSTEMS

ＯＮＴＨＥＥＸＩＳＴＥＮＣＥＡＮＤＵＮＩＱＵＥＮＥＳＳＯＦＰＯＳＩＴＩＶＥＳＯＬＵＴＩＯＮＳＦＯＲＡＣＬＡＳＳＯＦＤＥＧＥＮＥＲＡＴＥＥＬＬＩＰＴＩＣＳＹＳＴＥＭＳ￥ＣＨＥＮＺＨＥＮＴＡＯ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，Ｘｉａｎｇｔ...     

EXISTENCE AND UNIQUENESS OF WEAK SOLUTIONS FOR TWO-DIMENSIONAL MODIFIED NAVIER-STOKES EQUATIONS

This paper studies a two-dimensional modified Navier-stokes equations. The author shows the existence and uniqueness of weak solutions for this equation by Galerkin method in bounded domains. The result is further extended to the case of unbounded channel-like domains.     

ON THE EXISTENCE AND THE UNIQUENESS OF SOLUTIONS TO STOCHASTIC INTEGRAL EQUATIONS WITH TWODIMENSIONAL TIME PARAMETER

In this paper the following stochastic integral equation is considered where A is a continuous increasing process, M is a continuous martingale. Mappings φ, F, G map continuous adapted processes into continuous adapted processes. We will prove equation (*) has a unique continuous solution under some conditions.     

ON THE EXISTENCE OF POSITIVE ENTIRE SOLUTIONS FOR SINGULAR p-LAPLACE EQUATIONS

In this paper N-dimensional singular, p-Laplace equations of the following form are considered, where p ≥N, β > 0, and f : [0,∞)×(0,∞)×[0, ∞)→[0,∞) is a continuous function. Some sufficient conditions are obtained for the existence of infinitely many radially positive entire solutions of the equation which are asymptotic to positive constant multiples of |x|(p-N)/(p-1) for p > N or log|x| for N = p as |x|→∞.     

ON THE REGULARITY AND EXISTENCE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC EQUATIONS OF SECOND ORDER

Let Ω(?)R~n be a bounded domain with a smooth boundary (?)Ω L a strictly elliptic operator and c(x)≥0 in Ω. In this paper we are concerned with the following Dirichlet problem with the growth condition (P_1): a<2, for n=2. It is proved that if p(x, t) has all derivatives up to order l which are locally Hlder continuous in (?)×R. and if a_(ij)(x) ∈C_(l 1,α)(Ω) and c(x)∈C_(l,α)(Ω), then any weak solution in W_0~(1,2) of (1) lies in C_(l 2,α)(Ω). Moreover, under the growth condition (P_1) and some additional assumptions, the existence of nontrivial solution of (1) is proved. The main difficulity here is that the simple bootstrapping procedure fails to apply directly to the case of the growth condition (P_1).     

EXISTENCE OF SOLUTIONS OF STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS FOR NONLINEAR SECOND ORDER IMPULSIVE DIFFERENTIAL EQUATIONS IN BANACH SPACES

In this paper, the fixed point theorem is applied to investigate the existence of solutions of Sturm-Liouville boundary value problems for nonlinear second order impulsive differential equations in Banaeh spaces.     

New fixed point theorems and applications of mixed monotone operator

In this paper, we study the uniqueness and existence of fixed point of mixed monotone operators in the partially ordered Banach space. The results extend and improve recent related results.     

New existence and uniqueness theorems of positive fixed points for mixed monotone operators with perturbation

In this paper, we obtain some new and general existence and uniqueness theorems of positive fixed points for mixed monotone operators with perturbation, which extend the corresponding results in [Z.T. Zhang, New fixed point theorems of mixed monotone operators and applications, J. Math. Anal. Appl. 204 (1996) 307-319, Theorem 1, Corollaries 1 and 2]. Moreover, some applications to nonlinear integral equations on unbounded region are given.     

Minimax theorems and applications to the existence and uniqueness of solutions of some differential equations

In this paper, minimax theorems due to Stepan A. Tersian were generalized, the existence and uniqueness of solutions for several semilinear equations were proved by employing our generalized theorems, and the existence and uniqueness results of nonresonance problem for these semilinear equations under the asymptotic un-uniformity conditions were presented.     

An operator formula for the number of halved monotone triangles with prescribed bottom row

Monotone triangles are certain triangular arrays of integers, which correspond to n×n alternating sign matrices when prescribing (1,2,…,n) as bottom row of the monotone triangle. In this article we define halved monotone triangles, a specialization of which correspond to vertically symmetric alternating sign matrices. We derive an operator formula for the number of halved monotone triangles with prescribed bottom row which is analogous to our operator formula for the number of ordinary monotone triangles [I. Fischer, The number of monotone triangles with prescribed bottom row, Adv. in Appl. Math. 37 (2) (2006) 249-267].     

On some existence theorems for the solutions of a class of nonlinear operator equations

Theorems of M. A. Krasnosel'skii are generalized which concern fixed points of completely continuous operators which compress or expand a cone, and a topological principle for fixed points in the case of P-compact operators.Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 229–237, February, 1970.The author expresses warm thanks to his advisor I. A. Bakhtin for constant help in this work.     

Fixed points of mixed monotone operators for existence and uniqueness of nonlinear fractional differential equations

In this paper we study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems by using new fixed point results of mixed monotone operators on cones.     

On fixed point theorems of mixed monotone operators and applications

In this paper, we obtain some new existence and uniqueness theorems of positive fixed point of mixed monotone operators in Banach spaces partially ordered by a cone. Finally we apply the main theorem to a nonlinear parabolic partial differential equation. Some results are new even for increasing or decreasing operators.     

On multistability of equations with a distributed delay,monotone production and the Allee effect

We consider the delay population dynamics model dN/dt=f(_Nτ(t))−d(N(t))$dN/dt=f(N_{\tau }(t))-d(N(t))$ with an increasing fecundity function f and any mortality function d where the Allee effect or multistability can be observed. The study includes non-autonomous equations with a distributed delay. Attractivity of certain positive solutions, persistence, extinction of solutions with initial values below a threshold, boundedness and oscillation are discussed.