Xed Point Theory Research Articles

Existence theory for implicit fractional q-difference equations in Banach spaces

"This paper deals with some existence results for a class of implicit fractional q-difference equations. The results are based on the fi xed point theory in Banach spaces and the concept of measure of noncompactness. An illustrative example is given in the last section."

Existence of positive solutions for a class of BVPs in Banach spaces

In this work, we use index xed point theory for perturbation of expan- sive mappings by l-set contractions to study the existence of bounded positive solutions for a class of two-point boundary value problem (BVP) associated to second-order nonlinear di erential equation on the positive half-line. The nonlin- earity, which may exhibit a singularity at the origin, is written as a sum of two functions which behave di erently. These functions, depend on the solution and its derivative, take values in a general Banach space and have at most polynomial growth. An example to illustrate the main results is given.

Variational inequalities on Hilbert C -modules

We introduce variational inequality problems on Hilbert C -modules and we prove several existence results for variational inequalities dened on closed convex sets. Then relation between variational inequalities, C -valued metric projection and xed point theory on Hilbert C -modules is studied.

FIXED FUZZY POINTS OF GENERALIZED GERAGHTY TYPE FUZZY MAPPINGS ON COMPLETE METRIC SPACES

Generalized Geraghty type fuzzy mappings on complete metric spaces are introduced and a xed point theorem that generalizes some recent comparable results for fuzzy mappings in contemporary literature is obtained. Example is provided to show the validity of obtained results over comparable classical results for fuzzy mappings in xed point theory. As an application, existence of coincidence fuzzy points and common xed fuzzy points for hybrid pair of single valued self mapping and a fuzzy mapping is also established.

A Fixed Point Approach to the Stability of a Generalized Quadratic and Additive Functional Equation

In this paper, we investigate the stability of the functional equation f(x + 2y) 2f(x + y) + 2f(x y) f(x 2y) = 0 by using the xed point theory in the sense of L. Cadariu and V. Radu.

Cauchy problems for functional evolution inclusions involving accretive operators

We study the existence and stability of solutions for a class of nonlinear functional evolution inclusions involving accretive operators. Our approach is employing the xed point theory for multivalued maps and using estimates via the Hausdor measure of noncompactness.

Coupled Coincidence and Common Fixed Point Theorems in Menger PM Spaces

Recently Many results on coupled xed point theory exist in the literature, for more de-tails, one can see in [2,11,22]. We established coupled coincidence and common xed point theoremsfor two self mixed g-monotone mappings in the settings of Menger PM-spaces. Our result is alsosubstantiated with the aid of an appropriate example and some open problems are also suggestedfor further studies

A FIXED POINT APPROACH TO THE STABILITY OF THE MIXED TYPE FUNCTIONAL EQUATION

Abstract. In this paper, we investigate the stability of a functionalequationf(x+y+z) f(x+y) f(y+z) f(x+z)+f(x)+f(y)+f(z) = 0by using the xed point theory in the sense of L. Cadariu and V.Radu. 1. IntroductionIn 1940, S. M. Ulam [19] raised a question concerning the stability ofhomomorphisms: Given a group G 1 a metric group G 2 with the metricd(;), and a positive number , does there exist a >0 such that if amapping f: G 1 !G 2 satis es the inequalityd(f(xy);f(x)f(y)) <for all x;y2G 1 then there exists a homomorphism F: G 1 !G 2 withd(f(x);F(x)) <for all x2G 1 ? When this problem has a solution, we say that the ho-momorphisms from G 1 to G 2 are stable. In the next year, D. H. Hyers[6] gave a partial solution of Ulam’s problem for the case of approximateadditive mappings under the assumption that G 1 and G 2 are Banachspaces. Hyers’ result was generalized by T. Aoki [1] for additive map-pings and by Th. M. Rassias [17] for linear mappings by considering thestability problem with unbounded Cauchy di erences. The paper of Th.M. Rassias had much inuence in the development of stability problems.The terminology Hyers-Ulam-Rassias stability originated from this his-torical background. During the last decades, the stability problems of

FIXED POINT THEOREMS FOR GENERALIZED NONEXPANSIVE SET-VALUED MAPPINGS IN CONE METRIC SPACES

Abstract. In 2007, Huang and Zhang [1] introduced a cone metric spacewith a cone metric generalizing the usual metric space by replacing thereal numbers with Banach space ordered by the cone. They consideredsome xed point theorems for contractive mappings in cone metric spaces.Since then, the xed point theory for mappings in cone metric spaces hasbecome a subject of interest in [1-6] and references therein. In this paper,we consider some xed point theorems for generalized nonexpansive set-valued mappings under suitable conditions in sequentially compact conemetric spaces and complete cone metric spaces. 1. Introduction and PreliminariesIn 2007, Huang and Zhang [1] introduced a cone metric space with a conemetric generalizing the usual metric space by replacing the real numbers withBanach space ordered by the cone. They considered some xed point theoremsfor contractive mappings in cone metric spaces. Since then, the xed pointtheory for mappings in cone metric spaces has become a subject of interest in[1-6] and references therein. Especially, for single-valued mappings, Choudhuryand Metiya [6] considered some xed point theorems for weak contraction incone metric spaces in 2010. In 2011, Wardowski [2] introduced H-cone metricin the collection of subsets of a given cone metric space. And he consideredthe concept of set-valued contraction of Nadler-type and proved a xed pointtheorem for contractive set-valued mappings in H-cone metric spaces. Inspiredand encouraged by the previous works, in this paper we consider some xedpoint theorems for generalized nonexpansive set-valued mappings under suit-able conditions in sequentially compact cone metric spaces and complete conemetric spaces.Let Ebe a real Banach space and Pbe a subset of E.Pis called a cone if and only if

Boundedness and stability in nonlinear delay difference equations employing fixed point theory

In this paper we study stability and boundedness of the nonlinear dierence equation x(t + 1) = a(t)x(t) + c(t) x(t g(t)) + q(x(t); x(t g(t)) : In particular we study equi-boundedness of solutions and the stability of the zero solution of this equation. Fixed point theorems are used in the analysis. Liapunov's method is normally used to study the stability properties of the zero solution of dieren tial and dierence equations. Certain diculties arise when Liapunov's method is applied to equations with unbounded de- lay or equations containing unbounded terms (11), (20). It has been found that some of these diculties can be eliminated if xed point theory is used instead (3). In the present paper we study certain type of boundedness and stability properties of solutions of linear and nonlinear dierence equations with delay using xed point theory as the main mathematical tool. In particular we study equi-boundedness of solutions and stability of the zero solution of

Intuitionistic Fixed Point Theories for Strictly Positive Operators

In this paper it is shown that the intuitionistic xed point theory c i (strict) for times iterated xed points of strictly positive operator forms is conservative for negative arithmetic and 0 sentences over the theory ACA i for times iterated arithmetic comprehension without set parameters. This generalizes results previously due to Buchholz [5] and Arai [2].

The uniqueness of the periodic solution for a class of differential equations

In this paper we are concerned with a class of nonlinear dieren tial equations and obtaining the sucien t conditions for the uniqueness of the periodic solution by using Brouwer's xed point theory and the Sturm Theorem. Keywords. uniqueness, xed point, existence, control theory method, Sturm comparison theorem. AMS (MOS) subject classication: 34C25 where (t; x) 2 (0; 2 ) R and (t; x; z) 2 C 0 ((0; 2 ) R R), f 2 C 2 (0; 2 ) R , and (t; x; x 0 ); f(t; x) are 2 -periodic functions with respect to t.

Some random approximation theorems with applications

Fan [8] proved a very interesting theorem which provides a tool to study xed point theory in connection with best approximation. Various aspects of that theorem have been studied by Fan [9], Ha [10], Singh and Watson [22], Lin [14], Lin and Yen [16] and many others. Fan’s theorem has been of great importance in nonlinear analysis, approximation theory, game theory and minimax theorems. The study of random approximations and random xed points have been a very active area of research in probabilistic functional analysis in the last decade (see Sehgal and Waters [21], Sehgal and Singh [20], Papageorgiou [18], Lin [15], Xu [25], Tan and Yuan [23, 24] and Beg and Shahzad [1–4]). Most random xed point theorems in Banach spaces deal with condensing or nonexpansive random operators. What about the random operators which are neither of the above cases?. The interesting case would be a 1-set-contractive random operator. The class of 1-set-contractive random operators includes condensing and nonexpansive random operators. Besides, it also includes other important random operators such as semicontractive random operators and LANE [locally almost nonexpansive] random operators. Recently, Beg and Shahzad [3] studied these random operators in details and gave many results regarding random approximations and random

On the Dirichlet problem for the nonlinear wave equation in bounded domains with corner points

Using Mawhin’s coincidence topological degree arguments and xed point theory for non-expansive mappings results, we discuss the solvability of the Dirichlet problem for the semilinear equation of the vibrating string uxx uyy +f(x;y;u) = 0 in bounded domain with corner points. When the winding number associated to the domain is rational, we improve and extend some results of Lyashenko [8] and Lyashenko{Smiley [9]. The case where the winding number is irrational is also examined.