Tóm tắt Luận án On the existence of fixed point for some mapping classes in spaces with uniform structure and applications

The first result on fixed points of mappings was obtained in 1911. At that time, L. Brouwer proved that: Every continuous mapping from a compact convex set in a finite-dimensional space into itself has at least one fixed point. In 1922, S. Banach introduced a class of contractive mappings in metric spaces and proved the famous contraction mapping principle: Each contractive mapping from a complete metric space (X, d) into itself has a unique fixed point. The birth of the Banach contraction mapping principle and its application to study the existence of solutions of differential equations marks a new development of the study of fixed point theory. After that, many mathematicians have studied to extend the Banach contraction mapping principle for classes of maps and different spaces. Expanding only contractive mappings, till 1977, was summarized and compared with 25 typical formats by B.E.Rhoades.

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MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY LE KHANH HUNG ON THE EXISTENCE OF FIXED POINT FOR SOME MAPPING CLASSES IN SPACES WITH UNIFORM STRUCTURE AND APPLICATIONS Speciality: Mathematical Analysis Code: 62 46 01 02 A SUMMARY OF MATHEMATICS DOCTORAL THESIS NGHE AN - 2015 Work is completed at Vinh University Supervisors: 1. Assoc. Prof. Dr. Tran Van An 2. Dr. Kieu Phuong Chi Reviewer 1: Reviewer 2: Reviewer 3: Thesis will be presented and defended at school - level thesis evaluating Council at Vinh University at ...... h ...... date ...... month ...... year ...... Thesis can be found at: 1. Nguyen Thuc Hao Library and Information Center 2. Vietnam National Library 1PREFACE 1 Rationale 1.1. The first result on fixed points of mappings was obtained in 1911. At that time, L. Brouwer proved that: Every continuous mapping from a compact convex set in a finite-dimensional space into itself has at least one fixed point. In 1922, S. Banach introduced a class of contractive mappings in metric spaces and proved the famous contraction mapping principle: Each contractive mapping from a complete metric space (X, d) into itself has a unique fixed point. The birth of the Banach contraction mapping principle and its application to study the existence of solutions of differential equations marks a new development of the study of fixed point theory. After that, many mathematicians have studied to extend the Banach contraction mapping principle for classes of maps and different spaces. Expanding only contractive mappings, till 1977, was summarized and compared with 25 typical formats by B.E. Rhoades. 1.2. The Banach contraction mapping principle associates with the class of con- tractive mappings T : X → X in complete metric space (X, d) with the contractive condition (B) d(Tx, Ty) ≤ kd(x, y), for all x, y ∈ X where 0 ≤ k < 1. There have been many mathematicians seeking to extend the Banach contraction mapping principle for classes of mappings and different spaces. The first extending was obtained by E. Rakotch by mitigating a contractive condition of the form (R) d(Tx, Ty) ≤ θ(d(x, y))d(x, y), for all x, y ∈ X, where θ : R+ → [0, 1) is a monotone decreasing function. In 1969, D. W. Boyd and S. W. Wong introduced an extended form of the above result by considering a contractive condition of the form (BW) d(Tx, Ty) ≤ ϕ(d(x, y)), for all x, y ∈ X, where ϕ : R+ → R+ is a semi right upper continuous function and satisfies 0 ≤ ϕ(t) < t for all t ∈ R+. In 2001, B. E. Roades, while improving and extending a result of Y. I. Alber and 2S. Guerre-Delabriere, gave a contractive condition of the form (R1) d(Tx, Ty) ≤ d(x, y) − ϕ(d(x, y)), for all x, y ∈ X, where ϕ : R+ → R+ is a continuous, monotone increasing function such that ϕ(t) = 0 if and only if t = 0. Following the way of reducing contractive conditions, in 2008, P. N. Dutta and B. S. Choudhury introduced a contractive condition of the form (DC) ψ ( d(Tx, Ty) ) ≤ ψ(d(x, y))−ϕ(d(x, y)), for all x, y ∈ X, where ψ, ϕ : R+ → R+ is a continuous, monotone non-decreasing functions such that ψ(t) = 0 = ϕ(t) if and only if t = 0. In 2009, R. K. Bose and M. K. Roychowdhury introduced the notion of new gen- eralized weak contractive mappings with the following contractive condition in order to study common fixed points of mappings (BR) ψ ( d(Tx, Sy ) ≤ ψ(d(x, y))− ϕ(d(x, y)), for all x, y ∈ X, where ψ, ϕ : R+ → R+ are continuous functions such that ψ(t) > 0, ϕ(t) > 0 for all t > 0 and ψ(0) = 0 = ϕ(0), moreover, ϕ is a monotone non-decreasing function and ψ is a monotone increasing function. In 2012, B. Samet, C. Vetro and P. Vetro introduced the notion of α-ψ-contractive type mappings in complete metric spaces, with a contractive condition of the form (SVV) α(x, y)d(Tx, Ty) ≤ ψ(d(x, y)), for all x, y ∈ X where ψ : R+ → R+ is a monotone non-decreasing function satisfying ∑+∞ n=1 ψ n(t) 0 and α : X ×X → R+. 1.3. In recent years, many mathematicians have continued the trend of generalizing contractive conditions for mappings in partially ordered metric spaces. Following this trend, in 2006, T. G. Bhaskar and V. Lakshmikantham introduced the notion of coupled fixed points of mappings F : X ×X → X with the mixed monotone property and obtained some results for the class of those mappings in partially ordered metric spaces satisfying the contractive condition (BL) There exists k ∈ [0, 1) such that d(F (x, y), F (u, v)) ≤ k 2 ( d(x, u) + d(y, v) ) , for all x, y, u, v ∈ X such that x ≥ u, y ≤ v. In 2009, by continuing extending coupled fixed point theorems, V. Lakshmikantham and L. Ciric obtained some results for the class of mappings F : X × X → X with g-mixed monotone property, where g : X → X from a partially ordered metric space into itself and F satisfies the following contractive condition (LC) d ( F (x, y), F (u, v) ) ≤ ϕ(d(g(x), g(u))+ d(g(y), g(v)) 2 ) , 3for all x, y, u, v ∈ X with g(x) ≥ g(u), g(y) ≤ g(v) and F (X ×X) ⊂ g(X). In 2011, V. Berinde and M. Borcut introduced the notion of triple fixed points for the class of mappings F : X × X × X → X and obtained some triple fixed point theorems for mappings with mixed monotone property in partially ordered metric spaces satisfying the contractive condition (BB) There exists constants j, k, l ∈ [0, 1) such that j + k + l < 1 satisfy d ( F (x, y, z), F (u, v, w) ) ≤ jd(x, u) + kd(y, v) + ld(z, w), for all x, y, z, u, v, w ∈ X with x ≥ u, y ≤ v, z ≥ w. After that, in 2012, H. Aydi and E. Karapinar extended the above result and obtained some triple fixed point theorems for the class of mapping F : X×X×X → X with mixed monotone property in partially ordered metric spaces and satisfying the following weak contractive condition (AK) There exists a function φ such that for all x ≤ u, y ≥ v, z ≤ w we have d ( TF (x, y, z), TF (u, v, w) ) ≤ φ(max{d(Tx, Tu), d(Ty, Tv), d(Tz, Tw)}). 1.4. The development of fixed point theory is motivated from its popular ap- plications, especially in theory of differential and integral equations, where the first impression is the application of the Banach contraction mapping principle to study the existence of solutions of differential equations. In the modern theory of differential and integral equations, proving the existence of solutions or approximating the solutions are always reduced to applying appropriately certain fixed point theorems. For boundary problems with bounded domain, fixed point theorems in metric spaces are enough to do the above work well. However, for boundary problems with unbounded domain, fixed point theorems in metric spaces are not enough to do that work. So, in the 70s of last century, along with seeking to extend to mapping classes, one was looking to extend to classes of wider spaces. One of typical directions of this expansion is seeking to extend results on fixed points of mappings in metric spaces to the class of local convex spaces, more broadly, uniform spaces which has attracted the attention of many mathematical, notably V. G. Angelov. In 1987, V. G. Angelov considered the family of real functions Φ = {φα : α ∈ I} such that for each α ∈ I, φα : R+ → R+ is a monotone increasing, continuous, φα(0) = 0 and 0 0. Then he introduced the notion of Φ-contractive mappings, which are mappings T :M → X satisfying (A) dα(Tx, Ty) ≤ φα ( dj(α)(x, y) ) for all x, y ∈M and for all α ∈ I, where M ⊂ X and obtained some results on fixed points of the class of those mappings. By intro- 4ducing the notion of spaces with j-bounded property, V. G. Angelov obtained some results on the unique existence of a fixed point of the above mapping class. Following the direction of extending results on fixed points to the class of local convex spaces, in 2005, B. C. Dhage obtained some fixed point theorems in Banach algebras by studying solutions of operator equations x = AxBx where A : X → X, B : S → X are two operators satisfying that A is D-Lipschitz, B is completely continuous and x = AxBy implies x ∈ S for all y ∈ S, where S is a closed, convex and bounded subset of the Banach algebra X, such that it satisfies the contractive condition (Dh) ||Tx − Ty|| ≤ φ(||x − y||) for all x, y ∈ X, where φ : R+ → R+ is a non- decreasing continuous function, φ(0) = 0. 1.5. Recently, together with the appearance of classes of new contractive mappings, and new types of fixed points in metric spaces, the study trend on the fixed point theory has advanced steps of strong development. With above reasons, in order to extend results in the fixed point theory for classes of spaces with uniform structure, we chose the topic ‘‘On the existence of fixed points for some mapping classes in spaces with uniform structure and applications” for our doctoral thesis. 2 Objective of the research The purpose of this thesis is to extend results on the existence of fixed points in metric spaces to some classes of mappings in spaces with uniform structure and apply to prove the existence of solutions of some classes of integral equations with unbounded deviation. 3 Subject of the research Study objects of this thesis are uniform spaces, generalized contractive map- pings in uniform spaces, fixed points, coupled fixed points, triple fixed points of some mapping classes in spaces with uniform structure, some classes of integral equations. 4 Scope of the research The thesis is concerned with study fixed point theorems in uniform spaces and apply to the problem of the solution existence of integral equations with unbounded deviational function. 55 Methodology of the research We use the theoretical study method of functional analysis, the method of the differential and integral equation theory and the fixed point theory in process of study- ing the topic. 6 Contribution of the thesis The thesis is devoted to extend some results on the existence of fixed points in metric spaces to spaces with uniform structure. We also considered the existence of solutions of some classes of integral equations with unbounded deviation, which we can not apply fixed point theorems in metric spaces. The thesis can be a reference for under graduated students, master students and Ph.D students in analysis major in general, and the fixed point theory and applications in particular. 7 Overview and Organization of the research The content of this thesis is presented in 3 chapters. In addition, the thesis also consists Protestation, Acknowledgements, Table of Contents, Preface, Conclusions and Recommendations, List of scientific publications of the Ph.D. student related to the thesis, and References. In chapter 1, at first we recall some notions and known results about uniform spaces which are needed for later contents. Then we introduce the notion of (Ψ,Π)- contractive mapping, which is an extension of the notion of (ψ, ϕ)-contraction of P. N. Dutta and B. S. Choudhury in uniform spaces, and obtained a result on the existence of fixed points of the (Ψ,Π)-contractive mapping in uniform spaces. By introducing the notion of uniform spaces with j-monotone decreasing property, we get a result on the existence and uniqueness of a fixed point of (Ψ,Π)-contractive mapping. Con- tinuously, by extending the notion of α-ψ-contractive mapping in metric spaces to uniform spaces, we introduce the notion of (β,Ψ1)-contractive mappings in uniform spaces and obtain some fixed point theorems for the class of those mappings. Theo- rems, which are obtained in uniform spaces, are considered as extensions of theorems in complete metric spaces. Finally, applying our theorems about fixed points of the class of (β,Ψ1)-contractive mapping in uniform spaces, we prove the existence of so- lutions of a class of nonlinear integral equations with unbounded deviations. Note that, when we consider a class of integral equations with unbounded deviations, we can not apply known fixed point theorems in metric spaces. Main results of Chapter 61 is Theorem 1.2.6, Theorem 1.2.9, Theorem 1.3.11 and Theorem 1.4.3. In Chapter 2, we consider extension problems in partially ordered uniform spaces. Firstly, in section 2.1, we obtain results on couple fixed points for a mapping class in partially ordered uniform spaces when we extend (LC)-contractive condition of V. Lakshmikantham and L. Ciric for mappings in uniform spaces. In section 2.2, by extending the contractive condition (AK) of H. Aydi and E.Karapinar for mappings in uniform spaces, we get results on triple fixed points of a class in partially ordered uniform spaces. In section 2.3, by introducing notions of upper (lower) couple, upper (lower) triple solution, and applying results in section 2.1, 2.2, we prove the unique existence of solution of some classes of non-linear integral equations with unbounded deviations. Main results of Chapter 2 are Theorem 2.1.5, Corollary 2.1.6, Theorem 2.2.5, Corollary 2.2.6, Theorem 2.3.3 and Theorem 2.3.6. In Chapter 3, at first we present systematically some basic notions about locally convex algebras needed for later sections. After that, in section 3.2, by extending the notion of D-Lipschitz maps for mappings in locally convex algebras and by basing on known results in Banach algebras, and uniform spaces, we prove a fixed point theorem in locally convex algebras which is an extension of an obtained result by B. C. Dhage. Finally, in section 3.3, applying obtained theorems, we prove the existence of solution of a class of integral equations in locally convex algebras with unbounded deviations. Main results of Chapter 3 are Theorem 3.2.5, Theorem 3.3.2. In this thesis, we also introduce many examples in order to illustrate our results and the meaning of given extension theorems. 7CHAPTER 1 UNIFORM SPACES AND FIXED POINT THEOREMS In this chapter, firstly we present some basic knowledge about uniform spaces and useful results for later parts. Then, we give some fixed point theorems for the class of (Ψ,Π)-contractive mappings in uniform spaces. In the last part of this chapter, we extend fixed point theorems for the class of α-ψ-contractive mappings in metric spaces to uniform spaces. After that, we apply these new results to show a class of integral equations with unbounded deviations having a unique solution. 1.1 Uniform spaces In this section, we recall some knowledge about uniform spaces needed for later presentations. Let X be a non-empty set, U, V ⊂ X ×X. We denote by 1) U−1 = {(x, y) ∈ X ×X : (y, x) ∈ U}. 2) U ◦ V = {(x, z) : ∃y ∈ X, (x, y) ∈ U, (y, z) ∈ V } and U ◦ U is replaced by U2. 3) ∆(X) = {(x, x) : x ∈ X} is said to be a diagonal of X. 4) U [A] = {y ∈ X : ∃x ∈ A such that (x, y) ∈ U}, where A ⊂ X and U [{x}] is replaced by U [x]. Definition 1.1.1. An uniformity or uniform structure on X is a non-empty family U consisting of subsets of X ×X which satisfy the following conditions 1) ∆(X) ⊂ U for all U ∈ U . 2) If U ∈ U then U−1 ∈ U . 3) If U ∈ U then there exists V ∈ U such that V 2 ⊂ U . 4) If U, V ∈ U then U ∩ V ∈ U . 5) If U ∈ U and U ⊂ V ⊂ X ×X then V ∈ U . The ordered pair (X,U) is called a uniform space. In this section, we also present the concept of topology generated by uniform struc- ture, uniform space with uniform structure generated by a family of pseudometrics, 8Cauchy sequence, sequentially complete uniform space and the relationship between them. Remark 1.1.8. 1) Let X be a uniform space. Then, uniform topology on X is generated by the family of uniform continuous pseudometrics on X 2) If E is locally convex space with a saturated family of seminorms {pα}α∈I , then we can define a family of associate pseudometrics ρα(x, y) = pα(x− y) for every x, y ∈ E. The uniform topology generated the family of associate pseudometrics coincides with the original topology of the space E. Therefore, as a corollary of our results, we obtain fixed point theorems in the locally convex space. 3) Let j : I → I be an arbitrary mapping of the index I into itself. The iterations of j can be defined inductively j0(α) = α, jk(α) = j ( jk−1(α) ) , k = 1, 2, . . . 1.2 Fixed points of weak contractive mappings In the next presentations, (X,P) orX we mean a Hausdorff uniform space whose uniformity is generated by a saturated family of pseudometrics P = {dα(x, y) : α ∈ I}, where I is an index set. Note that, (X,P) is Hausdorff if only if dα(x, y) = 0 for all α ∈ I implies x = y. Definition 1.2.2. A uniform space (X,P) is said to be j-bounded if for every α ∈ I and x, y ∈ X there exists q = q(x, y, α) such that djn(α)(x, y) ≤ q(x, y, α) < ∞, for all n ∈ N. Let Ψ = {ψα : α ∈ I} be a family of functions ψα : R+ → R+ which is monotone non-decreasing and continuous, ψα(t) = 0 if only if t = 0, for all α ∈ I. Denote Π = {ϕα : α ∈ I} be a family of functions ϕα : R+ → R+, α ∈ I such that ϕα is continuous, ϕα(t) = 0 if only if t = 0. Definition 1.2.4. Let X be a uniform space. A map T : X → X is called a (Ψ,Π)-contractive on X if ψα ( dα(Tx, Ty) ) ≤ ψα(dj(α)(x, y))− ϕα(dj(α)(x, y)), for all x, y ∈ X and for all ψα ∈ Ψ, ϕα ∈ Π, α ∈ I. Definition 1.2.5. A uniform space (X,P) is called to have the j-monotone decreasing property iff dα(x, y) ≥ dj(α)(x, y) for all x, y ∈ X and all α ∈ I. Theorem 1.2.6. Let X is a Hausdorff sequentially complete uniform space and T : X → X. Suppose that 91) T is a (Ψ,Π)-contractive map on X. 2) A map j : I → I is surjective and there exists x0 ∈ X such that the sequence {xn} with xn = Txn−1, n = 1, 2, . . . satisfying dα(xm, xm+n) ≥ dj(α)(xm, xm+n) for all m,n ≥ 0, all α ∈ I. Then, T has at least one fixed point. X. Moreover, if X has j-monotone decreasing property, then T has a unique fixed point. Example 1.2.7. Let X = R∞ = { x = {xn} : xn ∈ R, n = 1, 2, . . . } . For every n = 1, 2, . . . we denote by Pn : X → R a map is defined by Pn(x) = xn for all x = {xn} ∈ X. Denote I = N∗ × R+. For every (n, r) ∈ I we define a pseudometrics d(n,r) : X ×X → R, which is given by d(n,r)(x, y) = r ∣∣Pn(x)− Pn(y)∣∣, for every (x, y) ∈ X. Then, the collection of pseudometrics {d(n,r) : (n, r) ∈ I} generated a uniformity on X. Now for every (n, r) ∈ I we consider the functions, which is given by ψ(n,r)(t) = 2(n− 1) 2n− 1 t, for all t ≥ 0, and put Ψ = Φ = {ψ(n,r) : (n, r) ∈ I}. Denote by j : I → I a map is defined by j(n, r) = ( n, r ( 1− 12n )) , for all (n, r) ∈ I and define a mapping T : X → X which is defined by Tx = { 1− (1− 2 3 ) (1− x1), 1− ( 1− 2 3.2 ) (1− x2), . . . , 1− ( 1− 2 3n ) (1− xn), . . . } , for every x = {xn} ∈ X. Applying Theorem 1.2.6, T has a unique fixed point, that is x = {1, 1, . . .}. Theorem 1.2.9. Let X be a Hausdorff sequentially complete uniform space and T, S : X → X be mappings satisfying ψα ( dα(Tx, Sy) ) ≤ ψα(dj(α)(x, y))− ϕα(dj(α)(x, y)), for all x, y ∈ X, where ψα ∈ Ψ, ϕα ∈ Π for all α ∈ I. Suppose j : I → I be a surjective map and for some ti x0 ∈ X such that the sequence {xn} with x2k+1 = Tx2k, x2k+2 = Sx2k+1, k ≥ 0 satisfies dα(xm+n, xm) ≥ dj(α)(xm+n, xm) for all m,n ≥ 0, α ∈ I. Then, there exists u ∈ X such that u = Tu = Su. Moreover, if X has the j-monotone decreasing property, then there exists a unique point u ∈ X such that u = Tu = Su. 10 1.3 Fixed points of (β,Ψ1)-contractive type mappings Denote Ψ1 = {ψα : α ∈ I} be a family of functions with the properties (i) ψα : R+ → R+ is monotone non-decreasing and ψα(0) = 0. (ii) for each α ∈ I, there exists ψα ∈ Ψ1 such that sup { ψjn(α)(t) : n = 0, 1, . . . } ≤ ψα(t) and +∞∑ n=1 ψ n α(t) 0. Denote by β a family of functions β = {βα : X ×X → R+, α ∈ I}. Definition 1.3.7. Let (X,P) be a uniform space with P = {dα(x, y) : α ∈ I} and T : X → X be a given mapping. We say that T is an (β,Ψ1)-contractive if for every function βα ∈ β and ψα ∈ Ψ1 we have
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