Ravi P. Agarwal

Fixed Point Theorems in Ordered Banach Spaces and Applications to Nonlinear Integral Equations

We present some new common fixed point theorems for a pair of nonlinear mappings defined on an ordered Banach space. Our results extend several earlier works. An application is given to show the usefulness and the applicability of the obtained results.

Multiplicity of positive periodic solutions to second order differential equations

In this paper, we study the existence of positive periodic solutions to the equation x ″ = f ( t, x ). It is proved that such a equation has more than one positive periodic solution when the nonlinearity changes sign. The proof relies on a fixed point theorem in cones.

Uniformly rd-piecewise almost periodic functions with applications to the analysis of impulsive Δ -dynamic system on time scales

We introduce some new concepts of uniformly rd-piecewise almost periodic functions on time scales and obtain some properties.Some new estimation inequalities and completely new theorems are established on time scales for impulsive dynamic systems.Several applications are given to illustrate our feasible results. In the present work, we introduce two equivalent concepts of uniformly rd-piecewise almost periodic functions on time scales and their equivalence is proved, based on this, some basic properties of them are obtained. Then, some new criteria of exponential dichotomy are established for homogeneous Δ -dynamic system on time scales. Also, some completely new theorems are established on time scales for impulsive almost periodic dynamic systems such as Favards theorem and exponential dichotomy theorem. As applications, we provide a method to obtain an almost periodic solution for a given nonhomogeneous impulsive dynamic system. Furthermore, we introduce an impulsive non-autonomous Nicholsons blowflies system model with patch structure and multiple nonlinear harvesting terms for which the existence and exponential stability of almost periodic solutions are studied, which shows that our results can be applied feasibly and effectively.

Disconjugacy via Lyapunov and Vallée-Poussin type inequalities for forced differential equations

In the case of oscillatory potentials, we present some new Lyapunov and Vallee-Poussin type inequalities for second order forced differential equations. No sign restriction is imposed on the forcing term. The obtained inequalities generalize and compliment the existing results in the literature.

Coincidence and fixed points for multi-valued mappings and its application to nonconvex integral inclusions

In this paper, we consider some problems on coincidence point and fixed point theorems for multi-valued mappings. Applying the characterizations of P -functions, we establish some new existence theorems for coincidence point and fixed point distinct from Nadlers fixed point theorem, Berinde-Berindes fixed point theorem, Mizoguchi-Takahashis fixed point theorem and Dus fixed point theorem for nonlinear multi-valued contractive mappings in complete metric spaces. Our results compliment and extend the main results given by some authors in the literature. In the sequel, we consider a nonconvex integral inclusion and prove the Filippov type existence theorem by using an appropriate norm on the space of selection of a multi-function and a multi-valued contraction for set-valued mappings.

On the Dimension of the Solution Set for Semilinear Fractional Differential Inclusions

We investigate the existence and dimension of the solution set for a nonlocal problem of semilinear fractional differential inclusions. The main tools of our study include some well-known results on multivalued maps.

Multiple Solutions for a System of

We consider the system of boundary value problems u (ni) i (t) + fi(t, u1(t), . . . , um(t)) = 0 u (j) i (0) = 0 u (pi) i (1) = 0 9 >= >; for t ∈ [0, 1], i = 1, . . . , m and 0 ≤ j ≤ ni−2 where ni ≥ 2 and 1 ≤ pi ≤ ni−1. Several criteria are offered for the existence of single and twin solutions of the system that are of fixed signs.

(n_i, p_i) Boundary Value Problems

In this paper, we first prove some perturbation results on the ranges of maximal monotone operators, one of which is then used to show that the non-linear elltic equation involving the generalized -Laplacian operator with Neumann boundary conditions has a unique solution in . This unique solution is shown to be the zero point of a suitably defined non-linear m-accretive mapping. Finally, two kinds of iterative sequences are constructed and proved to converge strongly and weakly to the unique solution, respectively. Some new techniques of constructing appropriate operators and decomposing the equations are employed, which extend and complement some of the previous work.

Non-linear boundary value problems with generalized p-Laplacian, ranges of m-accretive mappings and iterative schemes

We present an abstract result for the existence and uniqueness of the solution of nonlinear integro-differential systems involving the generalized ( p , q ) -Laplacian. The method used involves result on surjection of the sums of ranges of m -accretive mappings and strongly accretive mappings. The systems and technique discussed in this paper extend and complement some of the previous work.

Discussion on the existence and uniqueness of solution to nonlinear integro-differential systems

Abstract This paper is concerned with weak uniformly normal structure and the structure of the set of fixed points of Lipschitzian mappings. It is shown that in a Banach space X with weak uniformly normal structure, every asymptotically regular Lipschitzian semigroup of self-mappings defined on a weakly compact convex subset of X satisfies the (ω)-fixed point property. We show that if X has a uniformly Gâteaux differentiable norm, then the set of fixed points of every asymptotically nonexpansive mapping is nonempty and sunny nonexpansive retract of C. Our results improve several known fixed point theorems for the class of Lipschitzian mappings in a general Banach space.