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: Assoc Prof Dr Tran Van An 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: Nguyen Thuc Hao Library and Information Center Vietnam National Library PREFACE 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 contractive mappings T : X → X in complete metric space (X, d) with the contractive condition (B) d(T x, T y) ≤ kd(x, y), for all x, y ∈ X where ≤ k < 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(T x, T y) ≤ 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(T x, T y) ≤ ϕ d(x, y) , for all x, y ∈ X, where ϕ : R+ → R+ is a semi right upper continuous function and satisfies ≤ ϕ(t) < t for all t ∈ R+ In 2001, B E Roades, while improving and extending a result of Y I Alber and S Guerre-Delabriere, gave a contractive condition of the form (R1) d(T x, T y) ≤ 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) = if and only if t = 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(T x, T y) ≤ 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) = = ϕ(t) if and only if t = In 2009, R K Bose and M K Roychowdhury introduced the notion of new generalized weak contractive mappings with the following contractive condition in order to study common fixed points of mappings (BR) ψ d(T x, 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) > for all t > and ψ(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(T x, T y) ≤ 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) < +∞ for all t > 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 k (BL) There exists k ∈ [0, 1) such that d F (x, y), F (u, v) ≤ 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) , for 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 < 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 T F (x, y, z), T F (u, v, w) ≤ φ max d(T x, T u), d(T y, T v), d(T z, T w) 1.4 The development of fixed point theory is motivated from its popular applications, 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 the above work well However, for boundary problems with unbounded domain, fixed point theorems in metric spaces are not enough to 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) = and < φα (t) < t for all t > Then he introduced the notion of Φ-contractive mappings, which are mappings T : M → X satisfying (A) dα (T x, T y) ≤ 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- ducing 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) ||T x − T y|| ≤ φ ||x − y|| for all x, y ∈ X, where φ : R+ → R+ is a nondecreasing continuous function, φ(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 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 Subject of the research Study objects of this thesis are uniform spaces, generalized contractive mappings 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 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 5 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 studying the topic 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 Overview and Organization of the research The content of this thesis is presented in 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 Continuously, 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 Theorems, 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 solutions 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 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 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 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 CHAPTER 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 U 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 ⊂ 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 structure, uniform space with uniform structure generated by a family of pseudometrics, Cauchy 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 j (α) = α, j k (α) = j j k−1 (α) , k = 1, 2, 1.2 Fixed points of weak contractive mappings In the next presentations, (X, P) or X 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) = 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 dj n (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) = 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) = if only if t = 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 (T x, T y) ≤ 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 11 1.4 Applications to nonlinear integral equations In this section, we wish to investigate the existence of a unique solution to nonlinear integral equations, as an application to the fixed point theorems proved in the Section 1.3 Let us consider the following integral equations ∆(t) x(t) = (1.27) G(t, s)f s, x(s) ds, where the functions f : R+ × R → R and G : R+ × R+ → R+ continuous The deviation ∆ : R+ → R+ is a continuous function, in general case, unbounded Note that, since deviation ∆ : R+ → R+ is unbounded, we can not apply the known fixed point theorems in metric space for the above integral equations Assumption 1.4.1 A1) There exists a function u : R2 → R such that for each compact subset K ⊂ R+ , there exist a positive number λ and ψK ∈ Ψ1 such that for all t ∈ R+ , for all a, b ∈ R with u(a, b) ≥ 0, we have ∆(t) f (t, a) − f (t, b) ≤ λψK |a − b| and λ sup t∈K G(t, s)ds ≤ A2) There exists x0 ∈ C(R+ , R) such that for all t ∈ R+ , we have ∆(t) u x0 (t), G(t, s)f s, x0 (s) ds ≥ 0 A3) For all t ∈ R+ , x, y ∈ C(R+ , R), if u x(t), y(t) ≥ 0, then ∆(t) u ∆(t) G(t, s)f s, x(s) ds, G(t, s)f s, y(s) ds ≥ 0 A4) If {xn } is a sequence in C(R+ , R) such that xn → x ∈ C(R+ , R) and u(xn , xn+1 ) ≥ for all n ∈ N∗ , then u(xn , x) ≥ for all n ∈ N∗ A5) For each compact subset K ⊂ R+ , there exists a compact set K ⊂ R+ such that for all n ∈ N∗ , we have ∆n (K) ⊂ K Theorem 1.4.3 Suppose that Assumption 1.4 are fulfilled Then, equation (1.27) has at least one solution in C R+ , R Corollary 1.4.4 Suppose that 1) f : R+ × R → R+ is continuous and non-decreasing according to the second variable 2) For each compact subset K ⊂ R+ there exist the positive number λ and ψK ∈ Ψ1 such that for all t ∈ R+ , for all a, b ∈ R with a ≤ b, we have ∆(t) f (t, a) − f (t, b) ≤ λψK |a − b| and λ sup t∈K G(t, s)ds ≤ 12 3) For each compact subset K ⊂ R+ , there exists a compact set K ⊂ R+ such that for all n ∈ N∗ , ∆n (K) ⊂ K Then, the equation (1.27) has a unique solution in C R+ , R Example 1.4.5 Consider nonlinear functional integral equation t x(t) = G(t, s)f s, x(s) ds, (1.28) where G : R+ × R+ → R+ is given by G(t, s) = s−t 4e t−s 4e if t ≥ s ≥ if s ≥ t ≥ and f : R+ × R → R+ is defined by f (t, x) = √ x + + x2 √ + x − + x2 if x < if x ≥ 0, for every t ∈ R+ Applying Corollary 1.4.4 we get that the equation (1.28) has a unique solution Conclusions of Chapter In this chapter, we obtained the following main results • Give and prove theorems which confirm the existence and unique existence of a fixed point for the class of (Ψ, Π)-contractive maps in uniform space (Theorem 1.2.6, 1.2.9) These results are written in the article: Tran Van An, Kieu Phuong Chi and Le Khanh Hung (2014), Some fixed point theorems in uniform spaces, submitted to Filomat • Give and prove a theorem which confirm the existence and unique existence of a fixed point for the class of (β, Ψ1 )-contractive mappings in uniform spaces (Theorem 1.3.11) And apply Theorem 1.3.11 to prove the unique existence of solution of a class of integral equations with unbounded deviations These results are written in the article: Kieu Phuong Chi, Tran Van An, Le Khanh Hung (2014), Fixed point theorems for (α-Ψ)-contractive type mappings in uniform spaces and applications, Filomat (to appear) 13 CHAPTER FIXED POINTS OF SOME MAPPING CLASSES IN PARTIALLY ORDERED UNIFORM SPACES AND APPLICATIONS In this chapter, we prove some fixed point theorems for generalized contractive mappings in uniform spaces and apply them to study the existence problem of solutions for a class of nonlinear integral equations with unbounded deviations We also give some examples to show that our results are effective 2.1 Coupled fixed points in partially ordered uniform spaces In 2006, T G Bhaskar and V Lakshmikantham introduced the notion of coupled fixed points of mappings F : X ×X → X with mixed monotone property and obtained some results for the class of those mappings in partially ordered metric spaces Definition 2.1.1 Let (X, ≤) be a partially ordered set and F : X × X → X The mapping F is said to have the mixed monotone property if F is monotone nondecreasing in its first argument and is monotone non-increasing in its second argument, that is, for any x, y ∈ X if x1 , x2 ∈ X, x1 ≤ x2 then F (x1 , y) ≤ F (x2 , y) and if y1 , y2 ∈ X, y1 ≤ y2 then F (x, y1 ) ≥ F (x, y2 ) In this section, we prove some coupled fixed point theorems for generalized contractive mappings in partially ordered uniform spaces Let Φ1 = {φα : R+ → R+ ; α ∈ I} be a family of functions with the properties: i) φα is monotone non-decreasing ii) < φα (t) < t for all t > and φα (0) = 14 Let (X, ≤) be a partially ordered set Then, we consider the partial order on X ×X that defined by for (x, y), (u, v) ∈ X × X, (x, y) ≤ (u, v) if only if x ≤ u, y ≥ v Theorem 2.1.5 Let (X, ≤) be a partially ordered set and P = {dα (x, y) : α ∈ I} be a family of pseudometrics on X such that (X, P) is a Hausdorff sequentially complete uniform space Let F : X ×X → X be a mapping having the mixed monotone property on X Suppose that 1) For every α ∈ I, there exists φα ∈ Φ1 such that dα F (x, y), F (u, v) ≤ φα dj(α) (x, u) + dj(α) (y, v) , for all x ≤ u, y ≥ v 2) For each α ∈ I, there exists φα ∈ Φ1 such that sup{φj n (α) (t) : n = 0, 1, } ≤ φ (t) φα (t) and α is non-decreasing t 3) There are x0 , y0 ∈ X such that x0 ≤ F (x0 , y0 ), y0 ≥ F (y0 , x0 ) and dj n (α) x0 , F (x0 , y0 ) + dj n (α) y0 , F (y0 , x0 ) < 2p(α) < ∞ for all α ∈ I, n ∈ N Also, assume either a) F is continuous; or, b) X has the property i) If a non-decreasing sequence {xn } in X converges to x then xn ≤ x for all n ii) If a non-increasing sequence {yn } in X converges to y then yn ≥ y for all n Then, F has a coupled fixed point Moreover, if X is j-bounded and for every (x, y), (z, t) ∈ X ×X there exists (u, v) ∈ X × X which is comparable to them, then F has a unique coupled fixed point Corollary 2.1.6 In addition to hypotheses of Theorem 2.1.5, if x0 and y0 are comparable then F has a unique fixed point, that is, there exists x ∈ X such that F (x, x) = x 2.2 Triple fixed points in partially ordered uniform spaces In 2011, V Berinde and M Borcut introduced the notion of triple fixed points for a class of mapping F : X × X × X → X and obtained some triple fixed point theorems for mappings with mixed monotone property in partially ordered metric spaces After that, in 2012, H Aydi and E Karapinar extended the above result and 15 obtained some triple fixed point theorems for a class of mapping F : X × X × X → X with mixed monotone property in partially ordered metric spaces and satisfying the following weak contractive condition Definition 2.2.1 Let (X, ≤) be a partially ordered set and F : X × X × X → X The mapping F is said to have the mixed monotone property if for any x, y, z ∈ X x1 , x2 ∈ X, x1 ≤ x2 ⇒ F (x1 , y, z) ≤ F (x2 , y, z), y1 , y2 ∈ X, y1 ≤ y2 ⇒ F (x, y1 , z) ≥ F (x, y2 , z) and z1 , z2 ∈ X, z1 ≤ z2 ⇒ F (x, y, z1 ) ≤ F (x, y, z2 ) Definition 2.2.2 Let F : X → X An element (x, y, z) is called a triple fixed point of F if F (x, y, z) = x, F (y, x, y) = y and F (z, y, x) = z In this section, with Φ1 is the function family defined in Section 2.2, we prove some tripled fixed point theorems for generalized contractive mappings in uniform spaces We also give some examples to show that our results are effective Let (X, ≤) be a partially ordered set Then, we define a partial order on X in the following way: For (x, y, z), (u, v, w) ∈ X then (x, y, z) ≤ (u, v, w) if and only if x ≤ u, y ≥ v and z ≤ w We say that (x, y, z) and (u, v, w) are comparable if (x, y, z) ≤ (u, v, w) or (u, v, w) ≤ (x, y, z) Also, we say that (x, y, z) is equal to (u, v, w) if and only if x = u, y = v and z = w Definition 2.2.4 Let X be a uniform space A mapping T : X → X is said to be ICS if T is injective, continuous and has the property: for every sequence {xn } in X, if sequence {T xn } is convergent then {xn } is also convergent Theorem 2.2.5 Let (X, ≤) be a partially ordered set and P = {dα (x, y) : α ∈ I} be a family of pseudometrics on X such that (X, P) is a Hausdorff sequentially complete uniform space Let T : X → X is an ICS mapping and F : X → X be a mapping having the mixed monotone property on X Suppose that 16 1) For every α ∈ I there exists φα ∈ Φ1 such that dα T F (x, y, z), T F (u, v, w) ≤ φα max dj(α) (T x, T u), dj(α) (T y, T v), dj(α) (T z, T w) , for all x ≤ u, y ≥ v and z ≤ w 2) For each α ∈ I, there exists φα ∈ Φ1 such that sup{φj n (α) (t) : n = 0, 1, } ≤ φ (t) is non-decreasing on (0, +∞) φα (t) for all t > and α t 3) There are x0 , y0 , z0 ∈ X such that x0 ≤ F (x0 , y0 , z0 ), y0 ≥ F (y0 , x0 , y0 ), z0 ≤ F (z0 , y0 , x0 ) and max dj n (α) T x0 , T F (x0 , y0 , z0 ) , dj n (α) T y0 , T F (y0 , x0 , y0 ) , dj n (α) T z0 , T F (z0 , y0 , x0 ) < p(α) < ∞, for every α ∈ I, n ∈ N Also, assume either a) F is continuous; or b) X has the property i) If a non-decreasing sequence {xn } in X converges to x then xn ≤ x for all n ii) If a non-increasing sequence {yn } in X converges to y then yn ≥ y for all n Then, F has a triple fixed point Moreover, if X is j-bounded and for every (x, y, z), (u, v, w) ∈ X there exists (a, b, c) ∈ X which is comparable to them, then F has a unique triple fixed point Corollary 2.2.6 In addition to hypotheses of Theorem 2.2.5, if x0 ≤ y0 and z0 ≤ y0 then F has a unique fixed point, that is, there exists x ∈ X such that F (x, x, x) = x We also gave some examples to illustrate theorems in Sections 2.1, 2.2 2.3 Applications to nonlinear integral equations The first, we apply the coupled fixed point theorems proved in the Section 2.1 to investigate the existence of a unique solution to nonlinear integral equations Let us consider the following integral equations ∆(t) x(t) = h(t) + K1 (t, s) + K2 (t, s) f (s, x(s)) + g(s, x(s)) ds, (2.49) where K1 , K2 ∈ C R+ × R+ , R , f, g ∈ C R+ × R, R and the unknown functions x ∈ C R+ , R) The deviation ∆ : R+ → R+ is a continuous function, in general case, 17 unbounded Note that, since deviation ∆ is unbounded, we can not apply the known coupled fixed point theorems in metric space for the above integral equations We shall adopt the following assumptions Assumption 2.3.1 B1) K1 (t, s) ≥ and K2 (t, s) ≤ for all t, s ≥ B2) For each compact subset K ⊂ R+ , there exist λ, µ ≥ and φK ∈ Φ1 such that for all x, y ∈ R, x ≥ y and for all t ∈ K, we have x−y x−y , −µφK 2 ≤ f (t, x) − f (t, y) ≤ λφK and ∆(t) max(λ, µ) sup t∈K ≤ g(t, x) − g(t, y) ≤ K1 (t, s) − K2 (t, s) ds ≤ B3) For each compact subset K ⊂ R+ , there exists a compact set K ⊂ R+ such that ∆n (K) ⊂ K, for all n ≥ φ (t) is B4) For each compact subset K ⊂ R+ , there exists φK ∈ Φ1 such that K t non-decreasing and φ∆n (K) (t) ≤ φK (t) for all n and for all t ≥ Definition 2.3.2 An element (α, β) ∈ C R+ , R × C R+ , R is a coupled lower and upper solution of the integral equation (2.49) if α(t) ≤ β(t) and ∆(t) α(t) ≤ h(t) + K1 (t, s) f (s, α(s)) + g(s, β(s)) ds ∆(t) + K2 (t, s) f (s, β(s)) + g(s, α(s)) ds and ∆(t) β(t) ≥ h(t) + K1 (t, s) f (s, β(s)) + g(s, α(s)) ds ∆(t) + K2 (t, s) f (s, α(s)) + g(s, β(s)) ds Theorem 2.3.3 Consider the integral equation (2.49) with K1 , K2 ∈ C R+ ×R+ , R , f, g ∈ C R+ ×R, R , h ∈ C R+ , R and suppose that Assumption 2.3 is fulfilled Then the existence of a coupled lower and upper solution for (2.49) provides the existence of a unique solution of (2.49) in C R+ , R The next, we wish to investigate the existence of a unique solution to a class of nonlinear integral equations, as an application of the tripled fixed point theorems 18 proved in the previous section Let us consider the following integral equations ∆(t) x(t) = k(t) + K1 (t, s) + K2 (t, s) + K3 (t, s) (2.50) × f s, x(s) + g s, x(s) + h s, x(s) ds, where K1 , K2 , K3 ∈ C R+ × R+ , R , f, g, h ∈ C R+ × R, R and an unknown function x ∈ C R+ , R), the deviation ∆ : R+ → R+ is a continuous function, We assume that the functions K1 , K2 , K3 , f, g, h fulfill the following conditions Assumption 2.3.4 C1) K1 (t, s) ≥ 0, K2 (t, s) ≤ and K3 (t, s) ≥ for all t, s ≥ C2) For each compact subset K ⊂ R+ , there exist the positive numbers λ, µ, η ≥ and φK ∈ Φ1 such that for all x, y ∈ R, x ≥ y and for all t ∈ K, we have ≤ f (t, x) − f (t, y) ≤ λφK x − y , −µφK x − y ≤ g(t, x) − g(t, y) ≤ 0, ≤ h(t, x) − h(t, y) ≤ ∆(t) K1 (t, s) − K2 (t, s) + K3 (t, s) ds ≤ ηφK x − y and max(λ, µ, η) supt∈K C3) For each compact subset K ⊂ R+ , there exists a compact set K ⊂ R+ such that for all n ∈ N, ∆n (K) ⊂ K φ (t) C4) For each compact subset K ⊂ R+ , there exists φK ∈ Φ1 such that K is t non-decreasing and φ∆n (K) (t) ≤ φK (t) for all n ∈ N and for all t ≥ Definition 2.3.5 An element (α, β, γ) ∈ C R+ , R × C R+ , R × C R+ , R is called a tripled lower and upper solution of the integral equation (2.50) if for every t ∈ R+ we have α(t) ≤ β(t), γ(t) ≤ β(t) and ∆(t) α(t) ≤ k(t)+ K1 (t, s) f s, α(s) + g s, β(s) + h s, γ(s) ds K2 (t, s) f s, β(s) + g s, α(s) + h s, β(s) ds K3 (t, s) f s, γ(s) + g s, β(s) + h s, α(s) ds, K1 (t, s) f s, β(s) + g s, α(s) + h s, β(s) ds K2 (t, s) f s, α(s) + g s, β(s) + h s, α(s) ds K3 (t, s) f s, β(s) + g s, α(s) + h s, β(s) ds, ∆(t) + ∆(t) + ∆(t) β(t) ≥ k(t)+ ∆(t) + ∆(t) + 19 ∆(t) γ(t) ≤ k(t)+ K1 (t, s) f s, γ(s) + g s, β(s) + h s, α(s) ds K2 (t, s) f s, β(s) + g s, γ(s) + h s, β(s) ds K3 (t, s) f s, α(s) + g s, β(s) + h s, γ(s) ds ∆(t) + ∆(t) + Theorem 2.3.6 Consider the integral equation (2.50) with K1 , K2 , K3 ∈ C R+ × R+ , R , f, g, h ∈ C R+ × R, R , k ∈ C R+ , R and suppose that Assumption 2.3 is fulfilled Then the existence of a tripled lower and upper solution for (2.50) provides the existence of a unique solution of (2.50) in C R+ , R Conclusions of Chapter In this chapter, we obtained the following main results • Give and prove results which confirm the existence and unique existence of coupled fixed points for a class of contractive mappings in partially ordered uniform spaces (Theorem 2.1.5, Corollary 2.1.6) • Give and prove results which confirm the existence and unique existence of triple fixed points for a class of contractive mappings in partially ordered uniform spaces (Theorem 2.2.5, Corollary 2.2.6) • Apply Theorem 2.1.5 to prove the unique existence of solution of a class of integral equations with unbounded deviation Apply Theorem 2.2.5 to prove the unique existence of solution of a class of integral equations with unbounded deviations These results were published in the articles • Tran Van An, Kieu Phuong Chi and Le Khanh Hung (2014), Coupled fixed point theorems in uniform spaces and application, Journal of Nonlinear Convex Analysis, Vol 15, No 5, 953-966 • Le Khanh Hung (2014), Triple fixed points in ordered uniform spaces, Bulletin of Mathematical Analysis and Applications, Vol 6, Issue 2, 1-22 20 CHAPTER FIXED POINT THEOREMS IN LOCALLY CONVEX ALGEBRAS AND APPLICATIONS In this chapter, we give a fixed point theorem in locally convex algebras and apply it to investigate the existence of solution for a class of nonlinear integral equations with unbounded deviations 3.1 Locally convex algebras In this section, we introduce some basic knowledge of locally convex algebra Throughout this chapter, we consider associative and commutative algebras over the field K of complex numbers or real numbers Definition 3.1.1 Let X be an algebra over K X is called a topological algebra if there is a topology τ on X such that 1) (X, τ ) is a topological vector space; 2) The multiplication in X is continuous Definition 3.1.2 Let X be a topological algebra Then 1) A seminorm p : X → R is called to be submultiplicative if p(xy) ≤ p(x)p(y) for all x, y ∈ X 2) A set U ⊂ X is called to be multiplicative if U.U ⊂ U Definition 3.1.3 The topological algebra X is called a locally multiplicatively convex algebra if X has a local basis consisting of multiplicative and convex sets Definition 3.1.6 Let X be a locally convex space and T : X → X Then, 2) T is totally bounded if for any bounded set S of X, T (S) is a totally bounded set of X 3) T is completely continuous if it is continuous and totally bounded 21 3.2 Fixed points in locally convex algebras Let Φ = {φα : R+ → R+ , α ∈ I} be a class of increasing and continuous functions satisfying < φα (t) < t for all t > and φα (0) = 0; Ψ = {ψα : R+ → R+ , α ∈ I} be a class of increasing and continuous functions and ψα (0) = Let X be a locally convex algebra with a saturated family of seminorms {pα }α∈I Definition 3.2.4 The mapping T : X → X is D-Lipschitz with the family of functions {ψα }α∈I if pα (T x − T y) ≤ ψα pj(α) (x − y) , for all x, y ∈ X and for all α ∈ I, where {ψα }α∈I is a subfamily of Ψ If ψα (t) = kα t for all t ≥ 0, where kα is a real number for all α ∈ I, then T is called Lipschitzian with the family of Lipschitz constants {kα }α∈I Extending the results in uniform spaces and Banach algebras, we obtained following theorem on local convex algebras Theorem 3.2.5 Let X be a locally convex algebra such that topology of X is Hausdorff sequentially complete Let S be a closed, convex and bounded subset of X and A : X → X, B : S → X be two operators such that 1) A is D-Lipschitzian with the family of functions {ψα }α∈I 2) B is completely continuous and for every y ∈ S, x = AxBy implies x ∈ S 3) pj(α) (x − y) ≤ pα (x − y) for every x, y ∈ S and α ∈ I 4) For every x ∈ X and for every α ∈ I, there exists q(α, x) such that pj k (α) (x) ≤ q(α, x) < ∞ for all k = 0, 1, 2, , in particular pj k (α) (x) ≤ q(α) < ∞ for every x ∈ S and for all k = 0, 1, 2, 5) For each α ∈ I, then Mα ψα (t) < t for all t > and there exists φα ∈ Φ such that φα (t) is non-decreasing and sup{Mj k (α) ψj k (α) (t) : k = 0, 1, 2, } ≤ φα (t) for every t t > 0, where Mα = sup pα B(x) : x ∈ S Then, the operator equation x = AxBx has a solution Corollary 3.2.7 Let X be a locally convex algebra such that topology of X is Hausdorff sequentially complete Let S be a closed, convex and bounded subset of X and A : X → X, B : S → X be two operators such that 1) A is Lipschitzian with the family of Lipschitz constants {kα }α∈I 2) B is completely continuous and for every y ∈ S, x = AxBy implies x ∈ S 22 3) pj(α) (x − y) ≤ pα (x − y) for every x, y ∈ S and α ∈ I 4) For every x ∈ X and for every α ∈ I, there exists q(α, x) such that pj k (α) (x) ≤ q(α, x) < ∞ for all k = 0, 1, 2, , in particular pj k (α) (x) ≤ q(α) < ∞ for every x ∈ S and for all k = 0, 1, 2, 5) For each α ∈ I, then Mα kα < and sup{Mj k (α) kj k (α) : k = 0, 1, 2, } ≤ rα < 1, where Mα = sup pα B(x) : x ∈ S Then, the operator equation x = AxBx has a solution 3.3 Applications to nonlinear integral equations In this section, we apply the previous result to investigate the existence of a solution to nonlinear integral equations with unbounded deviations Let us consider the following integral equation ∆1 (t) ∆m (t) x(s)ds, , x(t) = F t, x(s)ds, x τ1 (t) , , x τn (t) (3.3) t f s, x(s) ds , × q(t) + where an unknown function x ∈ C(R+ , R), ∆i , τj : R+ → R+ are continuous functions, in general case, unbounded and, q : R+ → R, f : R+ ×R → R are continuous functions, F : R+ × Rm+n → [0, 1] By a solution of the (3.3), we mean a continuous function x : R+ → R that satisfies (3.3) on R+ Let X = C(R+ , R) be the locally convex algebra (in fact Frechet algebra) of all continuous real valued functions on R+ with family of seminorms p[0,n] (x) = max |x(t)| : t ∈ [0, n] , n ∈ N∗ We shall adopt the following assumptions Assumption 3.3.1 D1) The functions ∆i (t) : R+ → R+ , i = 1, 2, , m; τl (t) : R+ → R+ , l = 1, 2, , n are continuous and ∆i (t) ≤ t, τl (t) ≤ t for every t > D2) The function F : (t, u1 , u2 , , um , v1 , , ) : R+ × Rm+n → [0, 1] is continuous and satisfies the conditions F (t, u1 , , um , v1 , , ) − F (t, u1 , , um , v , , v n ) ≤ Ω t, |u1 − u1 |, , |um − um |, |v1 − v |, , |vn − v n | , 23 where the function Ω(t, x1 , , xm , y1 , , yn ) : Rm+n+1 → R+ is continuous in t, + non-decreasing and continuous in each xi , yl , Ω(t, ay, , ay, y, , y) < y for every Ω(t, ay, , ay, y, , y) constant a > and is non-decreasing in y y D3) q : R+ → R is uniformly continuous on R+ , q +∞ f s, x(s) ds < − q ∞ ∞ = supt∈R+ |q(t)| < and for every x ∈ C(R+ , R) with |x(t)| ≤ for all t Theorem 3.3.2 Under assumptions D1), D2) and D3), then equation (3.3) has at least one solution x = x(t) which belongs to the space C(R+ , R) The following example is an illustration for the Theorem 3.3 Example 3.3.3 Consider the following nonlinear functional integral equation x(t) = + x τ (t) te −t2 t + se−s 1+x2 (s) ds , (3.9) where τ (t) is continuous function on R+ and τ (t) ≤ t for all t ∈ R+ Applying Theorem 3.3, we proved that the equation (3.9) has a solution in C(R+ , R) Conclusions of Chapter In this chapter, we obtained the following main results • Give and prove a fixed point theorem in locally convex algebras, basing on ideals of known results in Banach algebras and uniform spaces (Theorem 3.2.5) • Apply Theorem 3.2.5 to prove the existence of solution of a class of integral equations with unbounded deviations These results were published in the article Le Khanh Hung (2015), Fixed point theorems in locally convex algebras and applications to nonlinear integral equations, Fixed point theory and applications, DOI 10.1186/s13663-015-0310-9 24 GENERAL CONCLUSIONS AND RECOMMENDATIONS General conclusions The thesis is devoted to investigate the existence of fixed points of some mapping classes in spaces with uniform structure and apply them to consider the existence of solutions of some classes of integral equations with unbounded deviations Main results of the thesis are Give and prove theorems which confirm the existence and unique existence of a fixed point for the class of (Ψ, Π)-contractive mappings in uniform spaces Give and prove theorems which confirm the existence and unique existence of a fixed point for the class of (β, Ψ1 )-contractive mappings in uniform spaces Apply them to prove the unique existence of solution of a class of integral equations with unbounded deviations Give and prove a theorem which confirm the existence and unique existence of a coupled fixed point for a class of contractive mappings in partially ordered uniform spaces (Theorem 2.1.5, Corollary 2.1.6) Apply Theorem 2.1.5 to prove the unique existence of solution of a class of integral equations with unbounded deviation Give and prove a theorem which confirm the existence and unique existence of a triple fixed point for a class of contractive mappings in partially ordered uniform spaces (Theorem 2.2.5, Corollary 2.2.6) Apply Theorem 2.2.5 to prove the unique existence of solution of a class of integral equations with unbounded deviations Give and prove a fixed point theorem in locally convex algebras, basing on ideals of known results in Banach algebras and uniform spaces (Theorem 3.2.5) Apply Theorem 3.2.5 to prove the existence of solution of a class of integral equations with unbounded deviations Give some examples to illustrate our theorems and to show that our results are properly extensions of known results Some recommendations Next time, we will continue the study on the following problems Study the existence of fixed points of some classes of multi-valued mappings in spaces with uniform structure and apply them to investigate the existence of solution of integral equations, differential equations Study the existence of fixed points of contractive mappings in spaces with the weak topology LIST OF PUBLICATIONS RELATED TO THE THESIS Tran Van An, Kieu Phuong Chi and Le Khanh Hung (2014), Coupled fixed point theorems in uniform spaces and application, Journal of Nonlinear Convex Analysis, Volume 15, Number 5, 953-966 Le Khanh Hung (2014), Triple fixed points in ordered uniform spaces, Bulletin of Mathematical Analysis and Applications, Volume 6, Issue 2, 1-22 Kieu Phuong Chi, Tran Van An, Le Khanh Hung (2014), Fixed point theorems for (α-Ψ)-contractive type mappings in uniform spaces and applications, Filomat (to appear) Le Khanh Hung (2015), Fixed point theorems in locally convex algebras and applications to nonlinear integral equations, Fixed point theory and applications, DOI 10.1186/s13663-015-0310-9 Tran Van An, Kieu Phuong Chi and Le Khanh Hung (2014), Some fixed point theorems in uniform spaces, submited to Filomat THE RESULTS OF THE THESIS ARE REPORTED AT • Seminars of the Analysis section of Faculty of Math Vinh university; • Ph.D student conferences of Vinh University (2010 - 2014); • Associated mathematics conference Vietnam-France in Hue 20-24/8/2012; • Vietnamese national mathematics congress VIII in Nha Trang 10-14/8/2013 ... 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