Fixed point theory for Mönch-type maps defined on closed subsets of Fréchet spaces: the projective limit approach

International Journal of Mathematics and Mathematical Sciences, Aug 2018

New Leray-Schauder alternatives are presented for Mönch-type maps defined between Fréchet spaces. The proof relies on viewing a Fréchet space as the projective limit of a sequence of Banach spaces.

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Fixed point theory for Mönch-type maps defined on closed subsets of Fréchet spaces: the projective limit approach

Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences RAVI P. AGARWAL JEWGENI H. DSHALALOW DONAL O'REGAN New Leray-Schauder alternatives are presented for Mo¨nch-type maps defined between Fre´chet spaces. The proof relies on viewing a Fre´chet space as the projective limit of a sequence of Banach spaces. This paper presents new Leray-Schauder alternatives for Mo¨nch-type maps defined between Fre´chet spaces. Two approaches [1, 2, 3, 6, 7] have recently been presented in the literature both of which are based on the fact that a Fre´chet space can be viewed as a projective limit of a sequence of Banach spaces {En}n∈N (here N = {1, 2, . . .}). Both approaches are based on constructing maps Fn defined on subsets of En whose fixed points converge to a fixed point of the original operator F. Both approaches have advantages and disadvantages over the other [1] and in this paper, we combine the advantages of both approaches to present very general fixed point results. Our theory in particular extends and improves the theory in [3] (in [3], the single-valued case was discussed). Finally in this section, we gather together some definitions and a fixed point result which will be needed in Section 2. Now, let I be a directed set with order ≤ and let {Eα}α∈I be a family of locally convex spaces. For each α ∈ I, β ∈ I for which α ≤ β, let πα,β : Eβ → Eα be a continuous map. Then the set 1. Introduction (1.1) α∈I is a closed subset of α∈I Eα and is called the projective limit of {Eα}α∈I and is denoted by lim← Eα (or lim←{Eα, πα,β} or the generalized intersection [4, page 439] α∈I Eα). Next, we recall a fixed point result from the literature [9] which we will use in Section 2. Theorem 1.1. Let K be a closed convex subset of a Banach space X, U a relatively open subset of K , x0 ∈ U, and suppose that F : U → CK (K ) is an upper semicontinuous map (here CK (K ) denotes the family of nonempty convex compact subsets of K ). Also assume that the following conditions hold: M ⊆ U, C ⊆ M M ⊆ co x0 ∪ F(M) with M = C, countable, implies M is compact, x ∈/ (1 − λ) x0 + λFx for x ∈ U \ U, λ ∈ (0, 1). Then there exist a compact set of U and an x ∈ with x ∈ Fx. Remark 1.2. In [9], we see that we could take to be y ∈ U : y ∈ (1 − λ) x0 + λF y for some λ ∈ [ 0, 1 ] . We did not show that is compact in [9] but this is easy to see as we will now show. First, notice that is closed since F is upper semicontinuous. Now let {yn}1∞ be a sequence in . Then there exists {tn}1∞ in [ 0, 1 ] with yn ∈ (1 − tn){x0} + tnF yn for n ∈ N = {1, 2, . . .}. Without loss of generality, assume that tn → t ∈ [ 0, 1 ]. Let C = {yn}1∞. Notice that C is countable and C ⊆ co({x0} ∪ F(C)). Now (1.2) with M = C guarantees that C is compact (so sequentially compact). Thus there exist a subsequence N1 of N and a y ∈ C with yn → y as n → ∞ in N1. This together with yn ∈ (1 − tn){x0} + tnF yn and the upper semicontinuity of F guarantees that y ∈ (1 − t){x0} + tF y, so y ∈ = . Consequently, is sequentially compact and hence compact. In fact, one could also of course take to be {y ∈ U : y ∈ F y} for the compact set in Theorem 1.1. 2. Projective limit approach Let E = (E, {| · |n}n∈N) be a Fre´chet space with the topology generated by a family of seminorms {| · |n : n ∈ N}. We assume that the family of seminorms satisfies |x|1 ≤ |x|2 ≤ |x|3 ≤ · · · for every x ∈ E. To E, we associate a sequence of Banach spaces {(En, | · |n)} described as follows. For every n ∈ N, we consider the equivalence relation ∼n defined by x ∼n y iff |x − y|n = 0. We denote by En = (E/∼n, | · |n) the quotient space, and by (En, | · |n) the completion of En with respect to | · |n (the norm on En induced by | · |n and its extension to En are still denoted by | · |n). This construction defines a continuous map µn : E → En. Now since (2.1) is satisfied, the seminorm | · |n induces a seminorm on Em for every m ≥ n (again this seminorm is denoted by | · |n). Also (2.2) defines an equivalence relation on Em from which we obtain a continuous map µn,m : Em → En since Em/ ∼n can be regarded as (1.2) (1.3) (1.4) (1.5) (2.1) (2.2) a subset of En. We now assume that the following condition holds: for each n ∈ N, there exist a Banach space En, | · |n and an isomorphism (between normed spaces) jn : En −→ En. (2.3) Remark 2.1. (i) For convenience, the norm on En is denoted by | · |n. (ii) In our applications, En = En for each n ∈ N. (iii) Note that if x ∈ En (or En), then x ∈ E. However if x ∈ En, then x is not necessarily in E and in fact En is easier to use in applications as we will see in Theorem 2.3 (even though En is isomorphic to En). Finally, we assume that E1 ⊇ E2 ⊇ · · · and for each n ∈ N, |x|n ≤ |x|n+1 ∀x ∈ En+1. (2.4) Let lim← En (or 1∞ En, where 1∞ is the generalized intersection [ 4 ]) denote the projective limit of {En}n∈N (note that πn,m = jnµn,m jm−1 : Em → En for m ≥ n) and note that lim← En ∼= E, so for convenience we write E = lim← En. For each X ⊆ E and each n ∈ N, we set Xn = jnµn(X) and we let Xn and ∂Xn denote, respectively, the closure and the boundary of Xn with respect to | · |n in En. Also the pseudointerior of X is defined by [ 2 ] pseudo − int(X) = x ∈ X : jnµn(x) ∈ Xn \ ∂Xn for every n ∈ N . (2.5) Our main result in this paper is the extension of Theorem 1.1 to an applicable result in the Fre´chet space setting (we refer the reader to [ 1 ]; in applications, usually the set U is bounded and as a result has empty interior in the nonnormable situation). Theorem 2.2. Let E and En be as described above and let F : X → 2E, where X ⊆ E (here 2E denotes the family of nonempty subsets of E). Suppose that the following conditions are satisfied: x0 ∈ pseudo − int(X), for each n ∈ N, F : Xn −→ CK En is an upper semicontinuous map, for each n ∈ N, M ⊆ Xn with M ⊆ co with M = C and C ⊆ M countable, implies that M is compact Proof. Fix n ∈ N. Let n = {x ∈ Xn : x ∈ Fx in En}. Now Theorem 1.1 (note that (2.6) implies that jnµn(x0) ∈ Xn \ ∂Xn) guarantees that there exists yn ∈ n with yn ∈ F yn. We look at {yn}n∈N. Now y1 ∈ 1. Also yk ∈ 1 for k ∈ N \ {1} since yk ∈ X1 from (2.10) (see also (2.4)). As a result, yn ∈ 1 for n ∈ N and since 1 is compact (see Remark 1.2), there exist a subsequence N1 of N and a z1 ∈ 1 with yn → z1 in E1 as n → ∞ in N1 . Let N1 = N1 \ {1}. Now yn ∈ 2 for n ∈ N1 so there exist a subsequence N2 of N1 and a z2 ∈ 2 with yn → z2 in E2 as n → ∞ in N2 . Note from (2.4) that z2 = z1 in E1 since N2 ⊆ N1. Let N2 = N2 \ {2}. Proceed inductively to obtain subsequences of integers N1 ⊇ N2 ⊇ · · · , and zk ∈ k with yn → zk in Ek as n → ∞ in Nk . Note that zk+1 = zk in Ek for k ∈ {1, 2, . . .}. Also let Nk = Nk \ {k}. Fix k ∈ N. Let y = zk in Ek. Notice that y is well defined and y ∈ lim← En = E. Now yn ∈ F yn in En for n ∈ Nk and yn → y in Ek as n → ∞ in Nk (since y = zk in Ek) together with the fact that F : Xk → CK (Ek) is upper semicontinuous (note that yn ∈ k for n ∈ Nk) imply that y ∈ F y in Ek. We can do this for each k ∈ N so as a result, we have y ∈ F y in E. Next, we present an application of Theorem 2.2. We discuss the differential equation y (t) = f t, y(t) a.e. t ∈ [0, T), y(0) = y0 ∈ R, ρn(u) = sup t∈[0,tn] u(t) , (2.12) (2.13) where 0 < T ≤ ∞ is fixed. First we introduce some notation. If u ∈ C[0, T), then for every n ∈ N, we define the seminorms ρn(u) by where tn ↑ T. Note that C[0, T) is a locally convex linear topological space. The topology on C[0, T), induced by the seminorms {ρn}n∈N, is the topology of uniform convergence on every compact interval of [0, T). Recall that a function g : [a, b] × R → R is an L1-Carathe´odory function if (a) the map t → g(t, y) is measurable for all y ∈ R, (b) the map y → g(t, y) is continuous for a.e. t ∈ [a, b]. Now, g : [a, b] × R → R is said to be an Lp-Carathe´odory function (1 ≤ p ≤ ∞) if g is a Carathe´odory function and (c) for any r > 0, there exists µr ∈ Lp[a, b] such that |y| ≤ r implies that |g(t, y)| ≤ µr (t) for a.e. t ∈ [a, b]. Finally, a function g : [0, T) × R → R is an Llpoc-Carathe´odory function if (a), (b), and (c) above hold when g is restricted to [0, tn] × R for any n ∈ N. Theorem 2.3. Suppose that the following conditions are satisfied: for each n ∈ N, the problem v (t) = g t, v(t) , a.e. t ∈ 0, tn , v(0) = y0 has a maximal solution rn(t) on 0, tn here rn ∈ C 0, tn . Then (2.12) has at least one solution y ∈ C[0, T). Remark 2.4. One could also obtain a multivalued version of Theorem 2.3 (with (2.12) replaced by a differential inclusion) by using the ideas in the proof below with the ideas in [ 6 ]. Proof. Here E = C[0, T), Ek consists of the class of functions in E which coincide on the interval [0, tk], Ek = C[0, tk] with of course πn,m = jnµn,m jm−1 : Em → En for m ≥ n defined by πn,m(x) = x|[0,tn]. We will apply Theorem 2.2 with X = u ∈ C[0, T) : |u|n ≤ wn for each n ∈ N ; here |u|n = supt∈In |u(t)|, where In = [0, tn] and wn = supt∈In rn(t) + 1. On any interval In = [0, tn] (n ∈ N), we let F on C(In) be defined by Fix n ∈ N. Notice that F y(t) = y0 + f s, y(s) ds. t 0 Xn = u ∈ C 0, tn : |u|n ≤ wn . Clearly, (2.6) holds with x0 = 0 and a standard argument from the literature [8] guarantees that F : Xn −→ En is continuous and compact, so (2.7) and (2.8) hold. To show that (2.9), fix n ∈ N and let y ∈ C(In) be such that y = λF y for λ ∈ (0, 1). We claim |y|n < wn and if this is true, then y ∈/ ∂Xn and hence (2.9) is true. Let t ∈ In and we now show that |y(t)| < wn. If |y(t)| ≤ |y0|, we are finished so it remains to discuss the (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) case when |y(t)| > |y0|. In this case, there exists a ∈ [0, t) with Also so for s ∈ (a, t], y(a) = y0 . y(s) ≤ y (s) ≤ g s, y(s) a.e. on (a, t), y(s) ≤ g s, y(s) , a.e. on (a, t), y(a) = y0 . (2.21) (2.22) (2.23) (2.24) (2.25) (2.26) (2.27) (2.28) (2.29) (2.30) Now a standard comparison theorem for ordinary differential equations in the real case [5, Theorem 1.10.2] guarantees that |y(s)| ≤ rn(s) for s ∈ [a, t], so in particular |y(t)| ≤ rn(t) < wn, so (2.9) is true. It remains to show that (2.10). To see this, fix n ∈ {2, 3, . . .} and suppose that y ∈ Xn solves Next, fix k ∈ {1, . . . , n − 1}. We must show that y ∈ Xk. Now since tn ↑ T, notice that [0, tk] ⊆ [0, tn] so as a result, y (t) = f t, y(t) , a.e. on 0, tn , y(0) = y0. y (t) = f t, y(t) , a.e. on 0, tk , y(0) = y0. Let t ∈ [0, tk] and essentially the same argument as above guarantees that |y(t)| < wk so |y|k < wk. Thus y ∈ Xk and (2.10) holds. The result now follows immediately from Theorem 2.2. Our final result was motivated by Urysohn-type operators. Theorem 2.5. Let E and En be as described in the beginning of Section 2 and let F : X → 2E, where X ⊆ E. Suppose that the following conditions are satisfied: x0 ∈ pseudo − int(X), X1 ⊇ X2 ⊇ · · · , for each n ∈ N, Fn : Xn −→ CK En is upper semicontinuous, for each n ∈ N, M ⊆ Xn with M ⊆ co y ∈/ (1 − λ) jnµn x0 + λFn y in En ∀λ ∈ (0, 1), y ∈ ∂Xn, for each n ∈ N, the map n : Xn −→ 2En , given by n(y) = ∞ m=n Fm(y) (see Remark 2.6), satisfies that if C ⊆ Xn is countable with C ⊆ n(C), then C is compact, if there exist a w ∈ X and a sequence yn n∈N with yn ∈ Xn and yn ∈ Fn yn in En such that for every k ∈ N there exists a subsequence S ⊆ {k + 1, k + 2, . . .} of N with yn −→ w in Ek as n −→ ∞ in S, then w ∈ Fw in E. (2.31) (2.32) Then F has a fixed point in X. Remark 2.6. The definition of n is as follows. If y ∈ Xn and y ∈/ Xn+1, then n(y) = Fn(y), whereas if y ∈ Xn+1 and y ∈/ Xn+2, then n(y) = Fn(y) ∪ Fn+1(y), and so on. Proof. Fix n ∈ N. Let n = {x ∈ Xn : x ∈ Fnx in En}. Now, Theorem 1.1 guarantees that there exists yn ∈ n with yn ∈ Fn yn in En. We look at {yn}n∈N. Note that yn ∈ X1 for n ∈ N from (2.27). In addition with C = {yn}1∞, we have from assumption (2.31) that C(⊆ E1) is compact; note that yn ∈ 1(yn) in E1 for each n ∈ N. Thus there exist a subsequence N1 of N and a z1 ∈ X1 with yn → z1 in E1 as n → ∞ in N1 . Let N1 = N1 \ {1}. Proceed inductively to obtain subsequences of integers and zk ∈ Xk with yn → zk in Ek as n → ∞ in Nk . Note that zk+1 = zk in Ek for k ∈ N. Also let Nk = Nk \ {k}. Fix k ∈ N. Let y = zk in Ek. Notice that y is well defined and y ∈ lim← En = E. Now yn ∈ Fn yn in En for n ∈ Nk and yn → y in Ek as n → ∞ in Nk (since y = zk in Ek) together with (2.32) imply that y ∈ F y in E. [7] Advances in ns Research Hindawi Publishing Corporation ht p:/ www.hindawi.com Advances in Journal of Algebra Hindawi Publishing Corporation ht p:/ www.hindawi.com Journal of Pro bability and Statistics Hindawi Publishing Corporation ht p:/ www.hindawi.com Hindawi Publishing Corporation ht p:/ www.hindawi.com Hindawi Publishing Corporation ht p:/ www.hindawi.com The Scientiifc World Journal Hindawi Publishing Corporation ht p:/ www.hindawi.com International Journal of Combinatorics Hindawi Publishing Corporation ht p:/ www.hindawi.com Submit your manuscr ipts Differential Equatio Journal of Mathematics Hindawi Publishing Corporation ht p:/ www.hindawi.com En gineering Hindawi Publishing Corporation ht p:/ www.hindawi.com International Journal of Mathematics and Mathematical Sciences Journal of Discrete Mathematics ht p:/ w w.hindawi.com Journal of [1] R. P. Agarwal , M. Frigon , and D. 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Vol. I: Ordinary Differential Equations, Mathematics in Science and Engineering , vol. 55 , Academic Press, New York, 1969 . [6] D. O'Regan , Maximal solutions and multivalued differential and integral inclusions on a noncompact interval , to appear in Nonlinear Funct. Anal. Appl. D. O'Regan and R. P. Agarwal , Fixed point theory for admissible multimaps defined on closed subsets of Fr´echet spaces , J. Math. Anal. Appl . 277 ( 2003 ), no. 2 , 438 - 445 . D. O'Regan and M. Meehan , Existence Theory for Nonlinear Integral and Integrodifferential Equations, Mathematics and Its Applications , vol. 445 , Kluwer Academic, Dordrecht, 1998 . D. O'Regan and R. Precup , Fixed point theorems for set-valued maps and existence principles for integral inclusions , J. Math. Anal. Appl . 245 ( 2000 ), no. 2 , 594 - 612 . Ravi P. Agarwal: Department of Mathematical Sciences, College of Science, Florida Institute of Technology, Melbourne, FL 32901 -6975, USA E-mail address: Jewgeni H. Dshalalow: Department of Mathematical Sciences, College of Science, Florida Institute of Technology, Melbourne, FL 32901 -6975, USA E-mail address: Donal O'Regan : Department of Mathematics, National University of Ireland, Galway, University Road, Galway, Ireland E-mail address: donal. Volume 2014 Journal of Hindawi Publishing Corporation ht p:/ www .hindawi.com


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Ravi P. Agarwal, Jewgeni H. Dshalalow, Donal O'Regan. Fixed point theory for Mönch-type maps defined on closed subsets of Fréchet spaces: the projective limit approach, International Journal of Mathematics and Mathematical Sciences, DOI: 10.1155/IJMMS.2005.2775