CAPITOLO 13 Ratio Mathematica Volume 37, 2019, pp. 25-38 25 Some results for Volterra integro- differential equations depending on derivative in unbounded domains Giuseppe Anichini1 Giuseppe Conti2 Alberto Trotta3 Abstract In this paper we study the existence of continuous solutions of an integro- differential equation in unbounded interval depending on derivative. This paper extends some results obtained by the authors using the technique developed in their previous paper. This technique consists in introducing, in the given problems, a function q, belonging to a suitable space, instead of the state variable x. The fixed points of this function are the solutions of the original problem. In this investigation we use a fixed point theorem in Fréchet spaces. Keywords: Fréchet spaces, semi-norms, acyclic sets, Ascoli-Arzelà theorem. 2010 AMS subject classification: 45G10, 47H09, 47H30* 1 Università degli Studi di Firenze, Department of Mathematics DIMAI, Viale Morgagni 67/A, 50134 Firenze. giuseppe.anichini@unifi.it 2 Università degli Studi di Firenze, Department of Mathematics DIMAI, Viale Morgagni 67/A, 50134 Firenze. gconti@unifi.it 3 IISS Santa Caterina-Amendola, Via L. Lazzarelli 12, 84132 Salerno. albertotrotta@virgilio.it * Received on November 15th, 2019. Accepted on December 30rd, 2019. Published on December 31st, 2019. doi:10.23755/rm.v36i1.471. ISSN: 1592-7415. eISSN: 2282-8214. doi: 10.23755/rm.v37i0.486. ©Anichini et al. Anichini, Conti, and Trotta 26 1 Introduction In this paper we study, in abstract setting, the solvability of a nonlinear integro- differential equation of Volterra type with implicit derivative, defined in unbounded interval, like (1) 𝑥′(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑓(𝑠, 𝑥(𝑠), 𝑥′(𝑠))𝑑𝑠 𝑥(0) = 0, 𝑡 ∈ 𝐽 = [0, +∞) 𝑡 0 We will look for solutions of this equation in the Fréchet space of all real C1 functions defined in the real unbounded interval 𝐽 = [0, +∞). Equation (1) is a special case of integro-differential equations. These equations have been seen as an important tool in the study of many boundary problems that we can encounter in various applications, like, for exemple, heat flow in material, kinetic theory, electrical ingeneering, vehicular traffic theory, biology, population dynamics, control theory, mechanics, mathematical economics. The integro-differential equations have been studied in various papers with the help of several tools of functional analysis, topology and fixed point theory. For istance we can refer to [1], [2], [3], [4], [5], [11], [12] and the references therein. In [8] an Hammerstein equation, similar to (1), is consideren in the multivalued setting and bounded intervals. Our paper extend some results obtained by the authors Anichini and Conti, using the techniques developed in previous paper (see to [1], [2], [3], [4], [5]). The crucial key of our approach, in order to find solutions of equation (1), consists in the use of a very useful fixed point theorem for multivalued, compact, uppersemicontinuous maps with acyclic values in a Fréchet space. 2 Preliminaries and Notations Let C1(J, ℝ) be the Fréchet the of all real C1 functions defined in the real unbounded interval 𝐽 = [0, +∞)  ℝ, equipped with the following family of semi-norms ‖𝑥‖1,𝑛 = max{‖𝑥‖𝑛 , ‖𝑥′‖𝑛} where ‖𝑥‖𝑛 = sup{|𝑥(𝑡)|, 𝑡 ∈ [0, 𝑛]} and ‖𝑥′‖𝑛 = sup{|𝑥′(𝑡)|, 𝑡 ∈ [0, 𝑛]}. We recall that the topology of C1(J, ℝ) coincides with the topology of a complete metric space {𝐹, 𝑑} where Some results for Volterra integro-differential equations 27 𝑑(𝑥, 𝑦) = ∑ 2−𝑛‖𝑥 − 𝑦‖1,𝑛 1 + ‖𝑥 − 𝑦‖1,𝑛 +∞ 𝑛=1 A subset A  C1(J, ℝ) is said to be bounded if, for every natural number n, there exists Mn > 0 such that ‖𝑥‖1,𝑛 ≤ 𝑀𝑛 ∀ 𝑥 ∈ 𝐶 1(𝐽, ℝ). A subset A  C1(J, ℝ) is relatively compact set if and only if the functions of the set A are equicontinuous and uniformly bounded (with their derivatives) in any interval [0, n]. We will denote by C(F) the family of all nonempty and compact subset of a Fréchet space F. Let M be a subset of a Fréchet space F; a multivalued map 𝑆: 𝑀 → 𝐶(𝐹) is said to be uppersemicontinuous (u.s.c.) if the graph is closed in 𝑀 × 𝐹, i.e. for any sequence {𝑥𝑛 }  𝑀, 𝑥𝑛→ 𝑥0 and 𝑦𝑛 ∈ 𝑆(𝑥𝑛), 𝑦𝑛→ 𝑦0, we have 𝑦0 ∈ 𝑆(𝑥0). A multivalued map 𝑆: 𝑀 → 𝐶(𝐹) is said to be compact if it sends bounded sets into relatively compact sets. We apply the same definition for singlevalued maps. A subset A of a metric space E is said to be an R - set if A is the intersection of a countable decreasing sequence of absolute retracts contained in E (see [10]). It is known that an R - set is an acyclic set, i.e. it is acyclic with respect to any cohomology theory (see [7]). Let M be a subset of the Fréchet space C1(J, ℝ) and consider an operator 𝑇: 𝑀 → 𝐶1(𝐽, ℝ) . Let {𝜖𝑛} be an infinitesimal sequence of real numbers. A sequence {𝑇𝑛} of maps 𝑇𝑛 : 𝑀 → 𝐶 1(𝐽, ℝ) is said to be an 𝜖𝑛-approximation of T on M if ‖𝑇𝑛(𝑥) − 𝑇(𝑥)‖1,𝑛 ≤ 𝜖𝑛 for every 𝑥 ∈ 𝑀 and for any natural number n. Define 𝑈𝑛 = {𝑥 ∈ 𝐹 ∶ ‖𝑥‖1,𝑛 < 1}. Let T be a compact map 𝑇: 𝑀 → C1(J, ℝ), where M is a closed set of the Fréchet space C1(J, ℝ), and let {𝑇𝑛} be a 𝜖𝑛-approximation of T on M, where 𝑇𝑛 : 𝑀 → 𝐶1(𝐽, ℝ) are compact maps; then the set of fixed point of T is a compact R - set if the equation 𝑥 − 𝑇𝑛 (𝑥) = 𝑦 has at most a solution for every 𝑦 ∈ 𝜀𝑛𝑈𝑛 for any natural number n (see [5]). In the sequel we will use the following result (see [9]). Proposition 1 (Kirszebraun’s Theorem) Let F : M → ℝ be a Lipschitz map defined on arbitrary subset M of ℝn. Then F admits a Lipschitz extension  : ℝn → ℝ with the same Lipschtiz constant. The well known Gronwall’s Lemma, from the standard theory of Ordinary Differential Equations, will be used. Anichini, Conti, and Trotta 28 Proposition 2 (Gronwall’s Lemma) Let g, h : 𝐽 → 𝐽 be continuous functions such that the following inequality: 𝑔(𝑡) ≤ 𝑢(𝑡) + ∫ ℎ(𝑠)𝑔(𝑠)𝑑𝑠 𝑡 ∈ 𝐽 𝑡 0 , holds, where u : 𝐽 → 𝐽 is a continuous nondecreasing function. Then we have: 𝑔(𝑡) ≤ 𝑢(𝑡)exp (∫ ℎ(𝑠)𝑑𝑠) 𝑡 ∈ 𝐽 𝑡 0 . In the sequel we will use the following proposition that can be deduced from Theorem 1 of [6]. Proposition 3 (a fixed poin theorem) Let F be a Fréchet space and M  𝑋 be a bounded, closed and convex subset; let 𝑆: 𝐹 → 𝑀 be a multivalued, uppersemicontinuous map with acyclic values. If S(F) is (relatively) compact, then S has a fixed point. 3 Main result The following result holds. Theorem Consider integral equation (1). Assume that i) k : 𝐽 × 𝐽 → ℝ is a C1 function; moreover we assume that there exists a continuous function h : 𝐽 → 𝐽 with |𝑘(𝑡, 𝑠)| ≤ ℎ(𝑠) and | 𝜕𝑘(𝑡,𝑠) 𝜕𝑡 | ≤ ℎ(𝑠). ii) f : 𝐽 × ℝ × ℝ → ℝ is a C1 function; moreover we assume that there exist continuous functions a, b : 𝐽 → 𝐽, with ∫ 𝑎(𝑠)𝑑𝑠 = 𝐴 < +∞ +∞ 0 and ∫ 𝑏(𝑠)𝑑𝑠 = 𝐵 < +∞ +∞ 0 , such that: |𝑓(𝑠, 𝑥, 𝑦)| ≤ 𝑎(𝑠) + 𝑏(𝑠)|𝑦|. iii) Assume that ∫ ℎ(𝑠)𝑏(𝑠)𝑑𝑠 =  < 1 +∞ 0 . Then equation (1) has at least one solution in the space C1(J, ℝ). Some results for Volterra integro-differential equations 29 Proof Let q be a function belonging to C1(J, ℝ) and consider the following integral equation: (2) 𝑦(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 𝑡 0 𝑡 ∈ 𝐽 = [0, +∞) Let S : C1(J, ℝ) → C1(J, ℝ) be the multivalued map which associates to every q ∈ C1(J, ℝ) the set of solutions of equation (2). Clearly, putting 𝑥(𝑡) = ∫ 𝑦(𝑠)𝑑𝑠 𝑡 0 (hence x’(t) = y(t) and x(0) = 0), we have that the fixed points of the map S are the solution of equation (1). In order to find the fixed points of multivalued map S, the following steps in the proof have to be established (Proposition 3): a) There exists a bounded, closed and convex set M  C1(J, ℝ) such that S(C1(J, ℝ))  M. b) The set S(C1(J, ℝ)) is relatively compact. c) The map S is uppersemicontinuous. d) The set S(q) is an acyclic set for every q ∈ C1(J, ℝ). a) Let q ∈ C1(J, ℝ) and consider equation (2); assume that t ∈ [0, n], from hypotheses we have, : |𝑦(𝑡)| = |∫ 𝑘(𝑡, 𝑠)𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 𝑡 0 | ≤ ≤ |∫ ℎ(𝑠)(𝑎(𝑠) + 𝑡 0 𝑏(𝑠)|𝑦(𝑠)|)𝑑𝑠| ≤ ≤ |∫ ℎ(𝑠)𝑎(𝑠)𝑑𝑠 𝑡 0 | + |∫ ℎ(𝑠)𝑏(𝑠)|𝑦(𝑠)|𝑑𝑠 𝑡 0 | ≤ ‖ℎ‖𝑛𝐴 +  ‖𝑦‖𝑛. So that, since  < 1, we have ‖𝑦‖𝑛 ≤ ‖ℎ‖𝑛𝐴 1− . Moreover, we have for t ∈ [0, n]: Anichini, Conti, and Trotta 30 𝑦′(𝑡) = ∫ 𝜕𝑘(𝑡, 𝑠) 𝜕𝑡 𝑡 0 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 + 𝑘(𝑡, 𝑡)𝑓 (𝑡, ∫ 𝑞(𝑠)𝑑𝑠 𝑡 0 , 𝑦(𝑡)) and we obtain: |𝑦′ (𝑡)| ≤ |∫ 𝜕𝑘(𝑡, 𝑠) 𝜕𝑡 𝑡 0 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠| + |𝑘(𝑡, 𝑡)𝑓 (𝑡, ∫ 𝑞(𝑠)𝑑𝑠 𝑡 0 , 𝑦(𝑡)) | ≤ ≤ ∫ ℎ(𝑠)𝑎(𝑠)𝑑𝑠 𝑡 0 + ∫ ℎ(𝑠)𝑏(𝑠)|𝑦(𝑠)|𝑑𝑠 𝑡 0 + ℎ(𝑡)(𝑎(𝑡) + 𝑏(𝑡)|𝑦(𝑡)|) ≤ ≤ ‖ℎ‖𝑛 𝐴 +  ‖𝑦‖𝑛 + ‖ℎ𝑎‖𝑛 + ‖ℎ𝑏‖𝑛‖𝑦‖𝑛 ≤ ≤ ‖ℎ‖𝑛 𝐴 + ‖ℎ𝑎‖𝑛 + ‖𝑦‖𝑛( + ‖ℎ𝑏‖𝑛) ≤ ‖ℎ‖𝑛𝐴 + ‖ℎ𝑎‖𝑛 + ‖ℎ‖𝑛𝐴 1− ( + ‖ℎ𝑏‖𝑛 ). So that there exists Mn > 0 such that ‖𝑦‖1,𝑛 ≤ 𝑀𝑛 . Then we have S(C1(J, ℝ))  M, where 𝑀 = {𝑦 ∈ 𝐶1(𝐽, ), ‖𝑦‖1,𝑛 ≤ 𝑀𝑛 }. b) Now, we want to prove that the set S(C1(J, ℝ)) is relatively compact. Let y ∈ S(C1(J, ℝ)) and fix 𝜀 > 0. For any u, w ∈ [0, n] we have: 𝑦′(𝑤) − 𝑦′(𝑢) = ∫ 𝜕𝑘(𝑤, 𝑠) 𝜕𝑡 𝑤 0 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 + 𝑘(𝑤, 𝑤)𝑓 (𝑤, ∫ 𝑞(𝑠)𝑑𝑠 𝑤 0 , 𝑦(𝑤)) − ∫ 𝜕𝑘(𝑢, 𝑠) 𝜕𝑡 𝑢 0 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 − 𝑘(𝑢, 𝑢)𝑓 (𝑢, ∫ 𝑞(𝑠)𝑑𝑠 𝑢 0 , 𝑦(𝑢)) Some results for Volterra integro-differential equations 31 = ∫ 𝜕𝑘(𝑤, 𝑠) 𝜕𝑡 𝑢 0 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 + 𝑘(𝑤, 𝑤)𝑓 (𝑤, ∫ 𝑞(𝑠)𝑑𝑠 𝑤 0 , 𝑦(𝑤)) − ∫ 𝜕𝑘(𝑢, 𝑠) 𝜕𝑡 𝑢 0 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 −𝑘(𝑢, 𝑢)𝑓 (𝑢, ∫ 𝑞(𝑠)𝑑𝑠 𝑢 0 , 𝑦(𝑢)) + ∫ 𝜕𝑘(𝑤, 𝑠) 𝜕𝑡 𝑤 𝑢 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠 It follows that |𝑦′(𝑤) − 𝑦′(𝑢)| ≤ ≤ ∫ | 𝜕𝑘(𝑤, 𝑠) 𝜕𝑡 − 𝜕𝑘(𝑢, 𝑠) 𝜕𝑡 | 𝑢 0 (𝑎(𝑠) + 𝑏(𝑠)|𝑦(𝑠)|)𝑑𝑠 + ‖ℎ‖𝑛 |𝑓 (𝑤, ∫ 𝑞()𝑑 𝑤 0 , 𝑦(𝑤)) − 𝑓 (𝑢, ∫ 𝑞()𝑑 𝑢 0 , 𝑦(𝑢))| + ‖ℎ‖𝑛 |∫ 𝜕𝑘(𝑤, 𝑠) 𝜕𝑡 𝑤 𝑢 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) 𝑑𝑠| By continuity of the functions q, h, f and 𝜕𝑘 𝜕𝑡 it follows that there exists  > 0 such that for |𝑤 − 𝑢| < , u, w ∈ [0, n], we have |𝑦′(𝑤) − 𝑦′(𝑢)| < 𝜀 Since |𝑦(𝑤) − 𝑦(𝑢)| ≤ 𝑀𝑛 |𝑤 − 𝑢|, we can conclude that the set S(C 1(J, ℝ)) is relatively compact. c) Let us now show that the map S is uppersemicontinuous. Let {𝑞𝑚} be a sequence, 𝑞𝑚 ∈ C 1(J, ℝ), with ‖𝑞𝑚 − 𝑞0‖1,𝑛 → 0 , 𝑦𝑚 ∈ 𝑆(𝑞𝑚), i.e. 𝑦𝑚(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑓 (𝑠, ∫ 𝑞𝑚()𝑑 𝑠 0 , 𝑦𝑚(𝑠)) 𝑑𝑠 𝑡 ∈ [0, 𝑛] 𝑡 0 Assume that ‖𝑦𝑚 − 𝑦0‖1,𝑛 → 0 . We need to show that 𝑦0 ∈ 𝑆(𝑞0). From the Dominated Lebesgue Convergence Theorem it follows: Anichini, Conti, and Trotta 32 𝑙𝑖𝑚𝑚→+∞ 𝑓 (𝑠, ∫ 𝑞𝑚()𝑑 𝑠 0 , 𝑦𝑚(𝑠)) = 𝑓 (𝑠, ∫ 𝑞0()𝑑 𝑠 0 , 𝑦0(𝑠)) and 𝑙𝑖𝑚𝑚→+∞ 𝑦𝑚(𝑡) = = 𝑙𝑖𝑚𝑚→+∞ ∫ 𝑘(𝑡, 𝑠)𝑓 (𝑠, ∫ 𝑞𝑚()𝑑 𝑠 0 , 𝑦𝑚(𝑠)) 𝑑𝑠 = 𝑡 0 = ∫ 𝑙𝑖𝑚𝑚→+∞ 𝑘(𝑡, 𝑠)𝑓 (𝑠, ∫ 𝑞𝑚()𝑑 𝑠 0 , 𝑦𝑚(𝑠)) 𝑑𝑠 = 𝑡 0 = ∫ 𝑘(𝑡, 𝑠)𝑓 (𝑠, ∫ 𝑞0()𝑑 𝑠 0 , 𝑦0(𝑠)) 𝑑𝑠 . 𝑡 0 Hence, we obtain 𝑦0(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑓 (𝑠, ∫ 𝑞0()𝑑 𝑠 0 , 𝑦0(𝑠)) 𝑑𝑠 𝑡 0 i. e. 𝑦0 ∈ 𝑆(𝑞0) d) Now we want to show that, for every fixed q ∈ C1(J, ℝ), the set S(q) is acyclic. Consider equation (2) (with q fixed). Put 𝑓 (𝑠, ∫ 𝑞()𝑑 𝑠 0 , 𝑦(𝑠)) = 𝑙(𝑠, 𝑦). Then equation (2) can be written in the following way: 𝑦(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑙(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 ∈ [0, +∞) 𝑡 0 We have: |𝑦(𝑡)| ≤ |∫ 𝑘(𝑡, 𝑠)𝑙(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 0 | ≤ ∫ ℎ(𝑠)𝑎(𝑠)𝑑𝑠 + ∫ ℎ(𝑠)𝑏(𝑠)|𝑦(𝑠)|𝑑𝑠. 𝑡 0 𝑡 0 Some results for Volterra integro-differential equations 33 From Gronwall’s Lemma it follows that: |𝑦(𝑡)| ≤ ∫ ℎ(𝑠)𝑎(𝑠)𝑑𝑠 exp(∫ ℎ(𝑠)𝑏(𝑠)𝑑𝑠) = 𝑚(𝑠) 𝑡 0 𝑡 0 where m is a continuous function. Let 𝑈: ℝ → [0, 1] the Uryshon (continuous) function defined by 𝑈(𝑧) = 1 if |𝑧| ≤ 1 and 𝑈(𝑧) = 0 if |𝑧| ≥ 2. Now we define the function 𝑔(𝑠, 𝑦) = 𝑈 ( 𝑦 𝑚(𝑠) + 1 ) 𝑙(𝑠, 𝑦). Clearly 𝑔(𝑠, 𝑦) = 𝑙(𝑠, 𝑦) when |𝑦| ≤ 𝑚(𝑠). Hence the set of solutions of the following equation 𝑦(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑔(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 ∈ [0, +∞) 𝑡 0 coincides with the set of solutions of equation (2) with q fixed. Consider now the integral operator 𝐻: 𝐶1(𝐽, ℝ)→ 𝐶1(𝐽, ℝ): (𝐻(𝑦))(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑔(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 0 𝑡 ∈ [0, +∞) If z = H(y), we have 𝑧(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑈 ( 𝑦(𝑠) 𝑚(𝑠) + 1 ) 𝑙(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 0 . Notice that 𝑈 ( 𝑦(𝑠) 𝑚(𝑠)+1 ) 𝑙(𝑠, 𝑦(𝑠)) = 𝑙(𝑠, 𝑦(𝑠)) if 𝑦(𝑠) ≤ 𝑚(𝑠) + 1 Anichini, Conti, and Trotta 34 and 𝑈 ( 𝑦(𝑠) 𝑚(𝑠)+1 ) 𝑙(𝑠, 𝑦(𝑠)) = 0 if 𝑦(𝑠) ≥ 2𝑚(𝑠) + 2. So that: ‖𝑧‖𝑛 ≤ ‖ℎ‖𝑛𝐴 + 2(‖𝑚‖𝑛 + 1). Moreover we obtain: |𝑧′(𝑡)| ≤ ∫ ℎ(𝑠)𝑈 ( 𝑦(𝑠) 𝑚(𝑠) + 1 ) |𝑙(𝑠, 𝑦(𝑠))|𝑑𝑠 𝑡 0 + ℎ(𝑡)𝑈 ( 𝑦(𝑡) 𝑚(𝑡) + 1 ) |𝑙(𝑡, 𝑦(𝑡))| Hence ‖𝑧′‖𝑛 ≤ ‖ℎ‖𝑛𝐴 + 2(‖𝑚‖𝑛 + 1) + ‖ℎ𝑎‖𝑛 + 2‖ℎ𝑏‖𝑛(‖𝑚‖𝑛 + 1) = 𝐴𝑛 It follows that ‖𝑧‖1,𝑛 ≤ 𝐴𝑛, where z = H(y). So that the set of solutions of equation 𝑦(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑔(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 ∈ [0, +∞) 𝑡 0 coincides with the set of fixed points of operator H in the set 𝐴 = {𝑧 ∈ 𝐶1(𝐽, ℝ), ‖𝑧‖1,𝑛 ≤ 𝐴𝑛}. It is easy to see (again as consequence of the Ascoli- Arzelà Theorem) that the set H(A) is relatively compact set. Moreover H is a continuous operator; to show the last assertion, let us take 𝑦0 , 𝑦𝑚 ∈ 𝐴, ‖𝑦𝑚 − 𝑦0‖1,𝑛 → 0, 𝑧𝑚 ∈ 𝐻(𝑦𝑚), ‖𝑧𝑚 − 𝑧0‖1,𝑛 → 0; we are going to prove that 𝑧0 ∈ 𝐻(𝑦0). For every 𝑡 ∈ [0, 𝑛] we have: Some results for Volterra integro-differential equations 35 𝑙𝑖𝑚𝑚→+∞ |∫ 𝑘(𝑡, 𝑠)𝑔(𝑠, 𝑦𝑚(𝑠))𝑑𝑠 − ∫ 𝑘(𝑡, 𝑠)𝑔(𝑠, 𝑦0(𝑠))𝑑𝑠 𝑡 0 𝑡 0 | ≤ (from the Dominated Lebesgue Convergence Theorem and the continuity of function g) ≤ ∫ 𝑙𝑖𝑚𝑚→+∞ ℎ (𝑠)|𝑔(𝑠, 𝑦𝑚(𝑠)) − 𝑔(𝑠, 𝑦0(𝑠))|𝑑𝑠 . 𝑡 0 Hence 𝑧0(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝑔(𝑠, 𝑦0(𝑠))𝑑𝑠 = (𝐻(𝑦0))(𝑡). 𝑡 0 Fix now a natural number n. We know (Proposition 1) that there exists a Lipschitz function 𝑔𝑛: [0, 𝑛] × [−𝐴𝑛, 𝐴𝑛 ] → ℝ such that, for every (𝑠, 𝑦) ∈ [0, 𝑛] × [−𝐴𝑛 , 𝐴𝑛 ] , we have: |𝑔𝑛(𝑠, 𝑦) − 𝑔(𝑠, 𝑦)| ≤ 1 (𝑛 + 1)2‖ℎ‖𝑛 and |𝑔𝑛(𝑠, 𝑦) − 𝑔𝑛(𝑠, 𝑦1)| ≤ 𝐿𝑛|𝑦 − 𝑦1| for every (𝑠, 𝑦), (𝑠, 𝑦1) ∈ [0, 𝑛] × [−𝐴𝑛, 𝐴𝑛 ] , Let 𝐺𝑛 : 𝐽 × ℝ → ℝ be the Lipschitz extension of the function 𝑔𝑛; hence 𝐺𝑛(𝑠, 𝑦) = 𝑔𝑛(𝑠, 𝑦) for every (𝑠, 𝑦) ∈ [0, 𝑛] × [−𝐴𝑛 , 𝐴𝑛] and |𝐺𝑛 (𝑠, 𝑦) − 𝐺𝑛(𝑠, 𝑦1)| ≤ 𝐿𝑛|𝑦 − 𝑦1| for every (𝑠, 𝑦), (𝑠, 𝑦1) ∈ 𝐽 × ℝ. Let 𝐻𝑛 ∶ 𝐴 → 𝐶 1(𝐽, ℝ)) be the operator defined as follows: (𝐻𝑛(𝑦))(𝑡) = ∫ 𝑘(𝑡, 𝑠)𝐺𝑛(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 0 𝑡 ∈ [0, +∞) Anichini, Conti, and Trotta 36 Clearly this operator is compact for every natural number n. Moreover, for every t ∈ [0, n] and y ∈ A, we have: |(𝐻𝑛(𝑦))(𝑡) − (𝐻(𝑦))(𝑡)| ≤ ∫ 𝑘(𝑡, 𝑠)|𝐺𝑛(𝑠, 𝑦(𝑠)) − 𝑔(𝑠, 𝑦(𝑠))|𝑑𝑠 ≤ 𝑡 0 ≤ 𝑛‖ℎ‖𝑛 1 (𝑛+1)2‖ℎ‖𝑛 < 1 𝑛 . So that ‖𝐻𝑛(𝑦) − 𝐻(𝑦)‖𝑛 < 1 𝑛 . Moreover we have for every y ∈ A: |(𝐻′𝑛(𝑦))(𝑡) − (𝐻 ′(𝑦))(𝑡)| ≤ ≤ |∫ 𝜕𝑘(𝑡, 𝑠) 𝜕𝑡 𝐺𝑛(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 0 − 𝑘(𝑡, 𝑡)𝐺𝑛(𝑡, 𝑦(𝑡)) + ∫ 𝜕𝑘(𝑡, 𝑠) 𝜕𝑡 𝑔(𝑠, 𝑦(𝑠))𝑑𝑠 𝑡 0 − 𝑘(𝑡, 𝑡)𝑔(𝑡, 𝑦(𝑡))| ≤ ∫ ℎ(𝑠)|𝐺𝑛(𝑠, 𝑦(𝑠)) − 𝑔(𝑠, 𝑦(𝑠))|𝑑𝑠 𝑡 0 + ℎ(𝑡)|𝐺𝑛(𝑡, 𝑦(𝑡)) − 𝑔(𝑡, 𝑦(𝑡))| ≤ ≤ 𝑛‖ℎ‖𝑛 1 (𝑛+1)2‖ℎ‖𝑛 + ‖ℎ‖𝑛 1 (𝑛+1)2‖ℎ‖𝑛 = 𝑛+1 (𝑛+1)2 < 1 𝑛 . Hence ‖𝐻′𝑛(𝑦) − 𝐻′(𝑦)‖𝑛 < 1 𝑛 . Let now b ∈ A. We consider the equation 𝑦 − 𝐻𝑛(𝑦) = 𝑏. We want to prove that it has at most one solution. Consider the equation 𝑧 − 𝐻𝑛(𝑧) = 𝑏; then, for every t ∈ J and by Gronwall’s Lemma we have: |𝑦(𝑡) − 𝑧(𝑡)| ≤ ∫ ℎ(𝑠)|𝐺𝑛(𝑠, 𝑦(𝑠)) − 𝐺𝑛 (𝑠, 𝑧(𝑠))|𝑑𝑠 ≤ 𝑡 0 ≤ ∫ ℎ(𝑠)𝐿𝑛|𝑦(𝑠) − 𝑧(𝑠)|𝑑𝑠 ≤ 0 𝑡 0 . Some results for Volterra integro-differential equations 37 So that we can say that y(t) = z(t) for every t ∈ J. Finally, we are able to conclude that, for every q ∈ C1(J, ℝ), the set S(q) is acyclic and the theorem is proved. 4 An example Consider the following integro-differential equation: (3) 𝑥′(𝑡) = ∫ 3𝑡 𝑒−𝑠+2 1 + 𝑡3 ( 3𝑠2 𝑒−2𝑠 1 + (sin(𝑥(𝑠))) 2 + 𝑠 𝑒−𝑠−2𝑥′(𝑠)) 𝑑𝑠 𝑡 0 𝑥(0) = 0, 𝑡 ∈ 𝐽 = [0, +∞). We have 𝑘(𝑡, 𝑠) = 3𝑡 𝑒−𝑠+2 1 + 𝑡3 , 𝑓(𝑠, 𝑥(𝑠), 𝑥′(𝑠)) = 3𝑠2 𝑒−2𝑠 1 + (sin(𝑥(𝑠))) 2 + 𝑠 𝑒−𝑠−2𝑥′(𝑠) ℎ(𝑠) = 3 𝑒−𝑠+2, 𝑎(𝑠) = 3𝑠2 𝑒−2𝑠, 𝑏(𝑠) = 𝑠 𝑒−𝑠−2. Hence, we obtain: ∫ 𝑎(𝑠)𝑑𝑠 = 3 4 +∞ 0 ∫ 𝑏(𝑠)𝑑𝑠 = 𝑒−2 +∞ 0 ∫ ℎ(𝑠)𝑏(𝑠)𝑑𝑠 = ∫ 3𝑠𝑒−2𝑠𝑑𝑠 = 3 4 < 1 +∞ 0 +∞ 0 So that the assumptions of our theorem are satisfied and integro-differential equation (3) has solutions. 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