Solvability and Construction of a Solution to the Fredholm Integral Equation of the First Kind

The issues of solvability and construction of a solution of the Fredholm integral equation of the first kind are considered. It is done by immersing the original problem into solving an extremal problem in Hilbert space. Necessary and sufficient conditions for the existence of a solution are obtained. A method of constructing a solution of the Fredholm integral equation of the first kind is developed. A constructive theory of solvability and construction of a solution to a boundary value problem of a linear integrodifferential equation with a distributed delay in control, generated by the Fredholm integral equation of the first kind, has been created.

Exact solution of the one-dimensional Klein-Gordon equation with scalar and vector linear potentials in the presence of a minimal length

Using the momentum space representation, we solve the Klein--Gordon equation in one spatial dimension for the case of mixed scalar and vector linear potentials in the context of deformed quantum mechanics characterized by a finite minimal uncertainty in position. The expressions of bound state energies and the associated wave functions are exactly obtained.

An exact inflationary solution in the chaotic model with non-minimal coupling

This paper presents a new exact inflationary solution to the non-minimMly coupled scalar field. The inflation is driven by the evolution of a scalar field with inflation potential V（φ） = （λ/4）φ4 ＋ b1φ2 ＋ b2 ＋ b3φ-2 ＋ b4φ-4. The spectral index of the scalar density fluctuations ns is consistent with the result of WMAP3 （Wilkinson Microwave Anisotropy Probe 3） for ACDM （Lambda-Cold Dark Matter）. This model relaxes the constraint to the quartic coupling constant. And it can enter smoothly into a radiation-dominated stage when inflation ends.

New Approach to Find Initial Basic Feasible Solution (IBFS) for Optimal Solution in Transportation Problem

Minimizing transportation time and getting optimal solutions are always considered as important factors while solving transportation problem. This paper shows a new approach for finding initial basic solution for transportation problem which reduces cost of transportation more than any transportation method such as LCM, northwest, Vogel’s approximation and so on. This method has been illustrated by taking an example;afterwards, it compares basic initial feasible solution with other methods IBF and optimal dictate solutions such as MODI and Steppingstone method.

The Existence of Meromorphic Solutions to Non-Linear Delay Differential Equations

In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational coefficients, k is a positive integer. Under the assumption when above equations own transcendental meromorphic solutions with minimal hyper-type, we derive the concrete conditions on the degree of the right side of them. Specially, when w(z)=0 is a root of , its multiplicity is at most k. Some examples are given here to illustrate that our results are accurate.

Generalization of the Global Error Minimization for Constructing Analytical Solutions to Nonlinear Evolution Equations

The global error minimization is a variational method for obtaining approximate analytical solutions to nonlinear oscillator equations which works as follows. Given an ordinary differential equation, a trial solution containing unknowns is selected. The method then converts the problem to an equivalent minimization problem by averaging the squared residual of the differential equation for the selected trial solution. Clearly, the method fails if the integral which defines the average is undefined or infinite for the selected trial. This is precisely the case for such non-periodic solutions as heteroclinic (front or kink) and some homoclinic (dark-solitons) solutions. Based on the fact that these types of solutions have vanishing velocity at infinity, we propose to remedy to this shortcoming of the method by averaging the product of the residual and the derivative of the trial solution. In this way, the method can apply for the approximation of all relevant type of solutions of nonlinear evolution equations. The approach is simple, straightforward and accurate as its original formulation. Its effectiveness is demonstrated using a Helmholtz-Duffing oscillator.

New Procedure of Finding an Initial Basic Feasible Solution of the Time Minimizing Transportation Problems

Minimization of transportation time is a great concern of the transportation problems like the cost minimizing transportation problems. In this writing, a transportation algorithm is developed and applied to obtain an Initial Basic Feasible Solution (IBFS) of transportation problems in minimizing transportation time. The developed method has also been illustrated numerically to test the efficiency of the method where it is observed that the proposed method yields a better result.

EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR SEMILINEAR ELLIPTIC SYSTEMS INVOLVING m CRITICAL HARDY-SOBOLEV EXPONENTS AND m SIGN-CHANGING WEIGHT FUNCTION

In this article, we consider a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical Hardy-Sobolev exponent and one sign- changing function. The existence and multiplicity results of positive solutions are obtained by variational methods.

Extremal ranks of the solution to a system of real quaternion matrix equations

In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new result.

EXISTENCE OF SOLUTIONS FOR PERIODIC BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER INTEGRO-DIFFERENTIAL EQUATIONS

By establishing a comparison result and using monotone iterative methods, the theorem of existence for minimal and maximal solutions of periodic boundary value problems for second-order nonlinear integro-differential equations in Banach spaces is proved.

EXISTENCE OF SOLUTIONS FOR THE FRACTIONAL(p,q)-LAPLACIAN PROBLEMS INVOLVING A CRITICAL SOBOLEV EXPONENT

In this article,we study the following fractional(p,q)-Laplacian equations involving the critical Sobolev exponent:(Pμ,λ){(−Δ)s 1 p u+(−Δ)s 2 q u=μ|u|q−2 u+λ|u|p−2 u+|u|p∗s 1−2 u,u=0,inΩ,in R N∖Ω,whereΩ⊂R N is a smooth and bounded domain,λ,μ>0,0<s 2<s 1<1,1<q<p<Ns 1.We establish the existence of a non-negative nontrivial weak solution to(Pμ,λ)by using the Mountain Pass Theorem.The lack of compactness associated with problems involving critical Sobolev exponents is overcome by working with certain asymptotic estimates for minimizers.

Sparse Solutions of Mixed Complementarity Problems

In this paper, we consider an extragradient thresholding algorithm for finding the sparse solution of mixed complementarity problems (MCPs). We establish a relaxation l1 regularized projection minimization model for the original problem and design an extragradient thresholding algorithm (ETA) to solve the regularized model. Furthermore, we prove that any cluster point of the sequence generated by ETA is a solution of MCP. Finally, numerical experiments show that the ETA algorithm can effectively solve the l1 regularized projection minimization model and obtain the sparse solution of the mixed complementarity problem.

Improvement of the minimal residual method for solving nonsymmetric linear systems

In this paper, the minimal residual （MRES） method for solving nonsymmetric equation systems was improved, the recurrence relation was deduced between the approximate solutions of the linear equation system Ax = b, and a more effective method was presented, which can reduce the operational count and the storage.

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A COUPLED SYSTEM OF KIRCHHOFF TYPE EQUATIONS

In this paper,we study the coupled system of Kirchhoff type equations−(a+b∫R 3|∇u|2 dx)Δu+u=2αα+β|u|α−2 u|v|β,−(a+b∫R 3|∇v|2 dx)Δv+v=2βα+β|u|α|v|β−2 v,u,v∈H 1(R 3),x∈R 3,x∈R 3,where a,b>0,α,β>1 and 3<α+β<6.We prove the existence of a ground state solution for the above problem in which the nonlinearity is not 4-superlinear at infinity.Also,using a discreetness property of Palais-Smale sequences and the Krasnoselkii genus method,we obtain the existence of infinitely many geometrically distinct solutions in the case whenα,β≥2 and 4≤α+β<6.

Approximate Weak Minimal Solutions of Set-Valued Optimization Problems

This paper deals with approximate weak minimal solutions of set-valued optimization problems under vector and set optimality criteria.The relationships between various concepts of approximate weak minimal solutions are investigated.Some topological properties and existence theorems of these solutions are given.It is shown that for set-valued optimization problems with upper(outer)cone-semicontinuous objective values or closed objective maps the approximate weak minimal and strictly approximate lower weak minimal solution sets are closed.By using the polar cone and two scalarization processes,some necessary and sufficient optimality conditions in the sense of vector and set criteria are provided.

A QMR-Type Algorithm for Drazin-Inverse Solution of Singular Nonsymmetric Linear Systems

In this paper, we propose DQMR algorithm for the Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax=b where is a singular and in general non-hermitian matrix that has an arbitrary index. DQMR algorithm for singular systems is analogous to QMR algorithm for non-singular systems. We compare this algorithm with DGMRES by numerical experiments.

四元数矩阵方程(A_(1)XB_(1),…,A_(k)XB_(k))=(C_(1),…,C_(k))的极小范数最小二乘Toeplitz解

基于四元数矩阵实表示,结合矩阵H-表示和矩阵半张量积提出一种求解四元数矩阵方程(A_(1)XB_(1),…,A_(k)XB_(k))=(C_(1),…,C_(k))的极小范数最小二乘Toeplitz解的有效方法,给出该四元数矩阵方程存在Toeplitz解的充要条件及通解表达式.给出数值算法并通过算例分别从误差与计算时间两个方面验证该方法的有效性.

NON-NEGATIVE SOLUTIONS FOR A CLASS OF NON-POSITONE PROBLEMS

In the recent past maily results have been established on non-negative solu tions to boundry value problems of the form u(0) = 0= u(1) where λ > 0, f(0) > 0 (positone problem). In this paper we consider the impact on the non-negative solutions when f(0) <0. We find that we need f(u) to be convex to guarantee uniquenness of positive solutions and f(u) to be appropriately concave for multiple positive solutions.

一类耦合代数Riccati方程的解

耦合代数Riccati方程在控制论中具有广泛的应用,文章研究了一类耦合代数Riccati方程的解。基于M-矩阵和非负矩阵的理论,在一定假设条件下,证明了这类耦合代数Riccati方程存在最小非负解。最后通过数值算例,验证了所得结果的有效性。

一类退化椭圆方程解的存在性与爆破行为

研究了一类具有位势函数的锥形退化椭圆方程。通过探究约束极小化问题,建立了方程基态解的存在性定理,并分析了其爆破行为。证明了当参数满足一定条件时,约束极小化问题至少存在1个可达元,但当参数不满足此条件时,不存在可达元。详细分析了当参数趋近于临界值时,可达元的爆破行为。