Nonlinear Anal Research Articles

Applications of fixed point theory to extended Nash equilibriums of nonmonetized noncooperative games on posets

Abstract We say that a noncooperative game is nonmonetized if the ranges of the utilities of the players are posets. In this paper, we examine some nonmonetized noncooperative games of which both the collection of strategies and the ranges of the utilities for the players are posets. Then we carry the concept of generalized Nash equilibriums of noncooperative games defined in (Li in J. Nonlinear Anal. Forum 18:1-11, 2013; Li and Park in Br. J. Econ. Manag. Trade 4(1), 2014) to extended Nash equilibriums of nonmonetized noncooperative games. By applying some fixed point theorems in posets and by using the order-preserving property of mappings, we prove an existence theorem of extended Nash equilibriums for nonmonetized noncooperative games. MSC:46B42, 47H10, 58J20, 91A06, 91A10.

LOWER BOUND FOR THE RATE OF BLOW-UP OF SINGULAR SOLUTIONS OF THE ZAKHAROV SYSTEM IN ℝ3

We consider the scalar Zakharov system in ℝ3 for initial conditions (ψ(0), n(0), nt(0)) ∈ Hℓ+1/2 × Hℓ × Hℓ-1, 0 ≤ ℓ ≤ 1. Assuming that the solution blows up in a finite time t* < ∞, we establish a lower bound for the rate of blow-up of the corresponding Sobolev norms in the form [Formula: see text] with [Formula: see text]. The analysis is a reappraisal of the local well-posedness theory of Ginibre, Tsutsumi and Velo [On the Cauchy problem for the Zakharov system, J. Funct. Anal.151 (1997) 384–436] combined with an argument developed by Cazenave and Weissler [The Cauchy problem for the critical nonlinear Schrödinger equation in Hs, Nonlinear Anal.14 (1990) 807–836] in the context of nonlinear Schrödinger equations.

On an open question of Takahashi for nonspreading mappings in Banach spaces

In this paper, we first introduce a new class of mappings called asymptotically nonspreading mappings and establish weak and strong convergence theorems of the iterative sequences generated by these mappings in a real Banach space. We modify Halpern’s iterations for finding a fixed point of an asymptotically nonspreading mapping and provide an affirmative answer to an open problem posed by Kurokawa and Takahashi in their final remark of (Kurokawa and Takahashi in Nonlinear Anal. 73:1562-1568, 2010) for nonspreading mappings. Furthermore, we investigate the approximation of common fixed points of asymptotically nonspreading mappings and nonexpansive mappings and derive a strong convergence theorem by a new hybrid method for these mappings. Our results improve and generalize many known results in the current literature. MSC:47H10, 37C25.

Existence Results for Variational Inequalities with Surjectivity Consequences Related to Generalized Monotone Operators

We present existence results for variational inequalities given by generalized monotone operators. As a consequence, we deduce the existence of zeros, or even more, the surjectivity of some classes of set-valued operators. We show that by strengthening the continuity assumptions, similar surjectivity results can be obtained without any monotonicity assumption. In the framework of reflexive Banach spaces, we extend a related result due to Inoan and Kolumban (Nonlinear Anal. 68:47–53, 2008).

Expanding the applicability of Lavrentiev regularization methods for ill-posed problems

AbstractIn this paper, we are concerned with the problem of approximating a solution of an ill-posed problem in a Hilbert space setting using the Lavrentiev regularization method and, in particular, expanding the applicability of this method by weakening the popular Lipschitz-type hypotheses considered in earlier studies such as (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009). Numerical examples are given to show that our convergence criteria are weaker and our error analysis tighter under less computational cost than the corresponding works given in (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009).MSC:65F22, 65J15, 65J22, 65M30, 47A52.

The convergence of implicit Mann and Ishikawa iterations for weak generalized φ-hemicontractive mappings in real Banach spaces

Let E be a real Banach space and let D be a nonempty closed convex subset of E ,l et T : D → D be a continuous weak generalized ϕ-hemicontractive mapping. The existence theorem of a fixed point of a weak generalized ϕ-pseudocontractive mapping is obtained. And we also prove that implicit Mann and Ishikawa iterations converge strongly to the unique fixed point of T. Our results extend the corresponding results of Xiang (Nonlinear Anal. 70(6): 2277-2279, 2009). MSC: Primary 47H09; 47J25

On the Second-order Directional Derivatives of Singular Values of Matrices and Symmetric Matrix-valued Functions

The (parabolic) second-order directional derivatives of singular values of matrices and symmetric matrix-valued functions induced by real-valued functions play important roles in studying second-order optimality conditions for different types of matrix cone optimization problems. We propose a direct way to derive the formula for the second-order directional derivative of any eigenvalue of a symmetric matrix in Torki (Nonlinear Anal 46:1133–1150 2001), from which a formula for the second-order directional derivative of any singular value of a matrix is established. We demonstrate a formula for the second-order directional derivative of the symmetric matrix-valued function. As applications, the second-order derivative for the projection operator over the SDP cone is derived and used to get the second-order tangent set of the SDP cone in Bonnans and Shapiro (2000), and the tangent cone and the second-order tangent set of the epigraph of the nuclear norm are given as well.

S-iteration process for quasi-contractive mappings

Abstract In this note, we show that the S-iteration process due to Sahu and Petrusel (Nonlinear Anal. TMA 74(17):6012-6023, 2011) is faster than the Picard, Mann, Ishikawa and Noor iteration processes for Zamfirescu operators. Also, using computer programs in C + + , we present some examples to compare the convergence rate of iterative processes due to Picard, Mann, Ishikawa, Noor, Agarwal et al. and Sahu and Petrusel.

General decay of solutions for a weak viscoelastic equation with acoustic boundary conditions

In this paper, we consider the weak viscoelastic equation $$u_{tt} - \Delta u + \alpha(t) \int\limits_{0}^{t} g(t-s)\Delta u(s)\, {\rm d}s=0$$ with a homogeneous Dirichlet condition on a portion of the boundary and acoustic boundary conditions on the rest of the boundary. We establish a general decay result, which depends on the behavior of both α and g, by using the perturbed energy functional technique. This is an extension and improvement of the previous result from Park and Park (Nonlinear Anal 74(3):993–998, 2011) (i.e., the similar problem with \({\alpha(t) \equiv 1}\)) to the time-dependent viscoelastic case.

Pth Moment Exponential Stability of Stochastic Recurrent Neural Networks with Markovian Switching

This paper investigates the problem of the pth moment exponential stability for a class of stochastic recurrent neural networks with Markovian jump parameters. With the help of Lyapunov function, stochastic analysis technique, generalized Halanay inequality and Hardy inequality, some novel sufficient conditions on the pth moment exponential stability of the considered system are derived. The results obtained in this paper are completely new and complement and improve some of the previously known results (Liao and Mao, Stoch Anal Appl, 14:165---185, 1996; Wan and Sun, Phys Lett A, 343:306---318, 2005; Hu et al., Chao Solitions Fractals, 27:1006---1010, 2006; Sun and Cao, Nonlinear Anal Real, 8:1171---1185, 2007; Huang et al., Inf Sci, 178:2194---2203, 2008; Wang et al., Phys Lett A, 356:346---352, 2006; Peng and Liu, Neural Comput Appl, 20:543---547, 2011). Moreover, a numerical example is also provided to demonstrate the effectiveness and applicability of the theoretical results.

Strong convergence theorems for two total asymptotically nonexpansive nonself mappings in Banach spaces

In this paper, we define and study the convergence theorems of a new two-steps iterative scheme for two total asymptotically nonexpansive nonself-mappings in Banach spaces. The results of this paper can be viewed as an improvement and extension of the corresponding results of (Shahzad in Nonlinear Anal. 61:1031-1039, 2005; Thianwan in Thai J. Math. 6:27-38, 2008; Ozdemir et al. in Discrete Dyn. Nat. Soc. 2010:307245, 2010) and all the others. MSC: 47H09; 47H10; 46B20

On coupled common fixed points for mixed weakly monotone maps in partially ordered S-metric spaces

In this paper, we use the notion of a mixed weakly monotone pair of maps of Gordji et al. (Fixed Point Theory Appl. 2012:95, 2012) to state a coupled common fixed point theorem for maps on partially ordered S-metric spaces. This result generalizes the main results of Gordji et al. (Fixed Point Theory Appl. 2012:95, 2012), Bhaskar, Lakshmikantham (Nonlinear Anal. 65(7):1379-1393, 2006), Kadelburg et al. (Comput. Math. Appl. 59:3148-3159, 2010) into the structure of S-metric spaces.

Common fixed point of single valued and multivalued mappings

In this paper, we prove the existence of common fixed point of commuting and weakly commuting pair of single valued and multivalued mapping in some classes of Banach spaces. Our results extend previously known ones, we cites Theorem 2, Theorem 3 and Theorem 4 by Itoh and Takahashi (J Math Anal Appl 59(3):514–521, 1977), Theorem 4.2 by Kaewcharoen (Nonlinear Anal Hybrid Syst 4(3):389–394, 2010), Theorem 4.2 by Dhompongsa et al. (Nonlinear Anal 64(5):958–970, 2006) and Theorem 3.1 by Nanan and Dhompongsa (Fixed Point Theory Appl 54:10, 2011).

Approximation of fixed points for nonexpansive semigroups in Hilbert spaces

Abstract In this paper, we propose two new algorithms for finding a common fixed point of a nonexpansive semigroup in Hilbert spaces and prove some strong convergence theorems for nonexpansive semigroups. Our results improve and generalize the corresponding results given by Shimizu and Takahashi (J. Math. Anal. Appl. 211:71-83, 1997), Shioji and Takahashi (Nonlinear Anal. TMA 34:87-99, 1998), Lau et al. (Nonlinear Anal. TMA 67:1211-1225, 2007) and many others. MSC:47H05, 47H10, 47H17.

Tripled fixed point theorem in fuzzy metric spaces and applications

Abstract In this paper we prove an existence and uniqueness theorem for contractive type mappings in fuzzy metric spaces. In order to do that, we consider a slight modification of the concept of a tripled fixed point introduced by Berinde et al. (Nonlinear Anal. TMA 74:4889-4897, 2011) for nonlinear mappings. Additionally, we obtain some fixed point theorems for metric spaces. These results generalize, extend and unify several classical and very recent related results in literature. For instance, we obtain an extension of Theorem 4.1 in (Zhu and Xiao in Nonlinear Anal. TMA 74:5475-5479, 2011) and a version in non-partially ordered sets of Theorem 2.2 in (Bhaskar and Lakshmikantham in Nonlinear Anal. TMA 65:1379-1393, 2006). As application, we solve a kind of Lipschitzian systems in three variables and an integral system. Finally, examples to support our results are also given.

One-sided Perron Differential Inclusions

The main qualitative properties of the solution set of almost lower (upper) semicontinuous one-sided Perron differential inclusion with state constraints in finite dimensional spaces are studied. Using the technique introduced by Veliov (Nonlinear Anal 23:1027–1038, 1994) we give sufficient conditions for the solution map of the above state constrained differential inclusion to be continuous in the sense of Hausdorff metric. An application on the propagation of the continuity of the state constrained minimum time function associated with the nonautonomous differential inclusion and the target zero is given. Some relaxation theorems are proved, which are used afterward to derive necessary and sufficient conditions for invariance.

Strong convergence theorems for Bregman quasi-asymptotically nonexpansive mappings and equilibrium problem in reflexive Banach spaces

The purpose of this article is to propose an iteration algorithm for Bergman quasiasymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of reflexive Banach spaces. As applications, we apply our results to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Reich and Sabach [Nonlinear Anal. 73 (2010), 122–135], Suantai et al. [Comput. Math. Appl. 64 (2012), 489–499], Nilsrakoo and Saejung [Appl. Math. Comput. 217:14 (2011), 6577–6586], Qin et al. [Applied Math Letters, 22 (2009), 1051–1055], Wang et al. [J. Comput. Appl. Math., 235 (2011), 2364–2371], Su et al. [Nonlinear Anal. 73 (2010), 390–3906], Nartinez-Yanes et al. [Nonlinear Anal., 64 (2006), 2400–2411] and others. Mathematics subject classification (2010): 47J05, 47H09, 49J25.

Oscillation of p(x)-Laplacian elliptic inequalities with mixed variable exponents

Oscillation criteria are established for p(x) -Laplacian elliptic inequalities with mixed variable nonlinearities of the form u [ ∇ · (A(x)|∇u|p(x)−2∇u)+ 〈b(x), |∇u|p(x)−2∇u〉−h(x,u)+g(x,u) 0, x ∈Ω, where β(x) > p(x) > γ(x) > 1 , Ω is an exterior domain in RN , and h(x,u) = ln |u| |∇u|p(x)−2 (A(x)∇u) ·∇p(x), g(x,u) =c(x)|u|p(x)−2u+ c1(x)|u|β(x)−2u+ c2(x)|u|γ(x)−2u+ f (x). The function h(x,u) recently introduced in [N. Yoshida, Nonlinear Anal. 74 (2011) 2563–2575] allows employing the Riccati transformation technique commonly used in the oscillation theory of ordinary differential equations. It should be noted that the results obtained are new for one dimensional case as well. Examples are given to illustrate the results. Mathematics subject classification (2010): 35B05,34C10.

On the integral characterization of principal solutions for half-linear ODE

We discuss a new integral characterization of principal solutions for half-linear differential equations, introduced in the recent paper of S. Fisnarova and R. Marik, Nonlinear Anal. 74 (2011), 6427-6433. We study this characterization in the framework of the existing results and we show when this new integral characterization with a parameter $\alpha$ is equivalent with two extremal cases of the integral characterization used in the literature. We illustrate our results on the Euler and Riemann-Weber differential equations.

Multiple positive periodic solutions for an n-species competition predator-prey system on time scales

In this paper, we consider an n-species competition predator-prey system on time scales with Holling-type II functional response and multiple exploited (or harvesting) terms, which contains n−1 competing preys and one predator. By using the continuation theorem based on Gaines and Mawhin’s coincidence degree theory, easily verifiable criteria are established for global existence of multiple positive periodic solutions to the above system. In addition, our results generalize the corresponding results of Zhang and Hou (Nonlinear Anal. (RWA) 11:1560–1571, 2010), Fan and Wang (J. Math. Anal. Appl. 262:179–190, 2001), Ding and Lu (Appl. Math. Model. 33:2748–2756, 2009).