On the Existence of Bounded Solutions to a Class of Nonlinear Initial Value Problems with Delay

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1 Filomat : (27, Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: On the Existence of Bounded Solutions to a Class of Nonlinear Initial Value Problems with Delay Muhammad Usman Ali a, Fairouz Tchier b, Calogero Vetro c a Department of Mathematics, COMSATS Institute of Information Technology, Attock Pakistan b Mathematics Department College of Science (Malaz, King Saud University, PO Box Riyadh, King Saudi Arabia c Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 4, 92, Palermo, Italy Abstract. We consider a class of nonlinear initial value problems with delay. Using an abstract fixed point theorem, we prove an existence result producing a unique bounded solution. To the memory of Professor Lj. Ćirić ( Introduction In this paper, we study the existence and uniqueness of a bounded solution for the following nonlinear initial value problem with delay: u(t = (s, u(s, u (sds, t [, t ], t >, ( φ(t, t [ τ, ], φ C [ τ, ], under assumption φ( = τ (s, φ(s, φ (sds, φ ( = (, φ(, φ ( ( τ, φ( τ, φ ( τ. In this problem, u(t is the proportion of infectious individuals (in population at time t, τ > is the length of time for which an individual remains infectious; u (t is the speed of infectivity and (t, u(t, u (t is the proportion of new infectious individuals per unit time. For a comprehensive study of integral equations with delay, the reader is referred to Precup [7]. We denote by X the product space X = C [ τ, t ] C[ τ, t ]. Then, we consider the Bielecki metric d B : X X R 2, where R 2 is the set of all 2 matrices, given as d B ((u, v, (u 2, v 2 = ( u u 2 B, v v 2 B T, for all (u, v, (u 2, v 2 X, 2 Mathematics Subject Classification. Primary 4A2; Secondary 5F25, 47H Keywords. Nonlinear initial value problem with delay, Perov s fixed point theorem, Λ-admissible mapping Received: 2 November 26; Accepted: 5 April 27 Communicated by Vladimir Rakočević The authors extend their appreciation to the International Scientific Partnership Program ISPP at King Saud University for funding this research work through ISPP#68. addresses: muh_usman_ali@yahoo.com (Muhammad Usman Ali, ftchier@ksu.edu.sa (Fairouz Tchier, calogero.vetro@unipa.it (Calogero Vetro

2 M. U. Ali et al. / Filomat : (27, where z B = max{ z(t e ϑ(t+τ : t [ τ, t ]}, for a chosen ϑ > and any z C[ τ, t ]. We need the following functional space X + = {(u, v X : u(t, t [ τ, t ]}. Of course, X is a complete metric space, X + X is closed in X and so X + is a complete metric space too. Clearly, from ( we have u (t, u(t, u (t (t τ, u(t τ, u (t τ, t [, t ], (t = φ (t, t [ τ, ]. This problem is largely investigated by Bica-Muresan [2], where some existence and uniqueness results of solution for problem ( are obtained by using classical tools of fixed point theory (see Banach [] and Perov [6]. In this paper, by using the same approach in Bica-Muresan [2] and a concept of admissibility for mappings (based on an idea of Samet-Vetro-Vetro [9], we obtain the existence and uniqueness of a bounded solution of problem (. First we prove an abstract result which is a generalization of Perov s fixed point theorem [6], then we work with a suitable integral operator associated to a large class of nonlinear initial value problems. 2. Mathematical Background and Preliminaries We fix notation as follows. Let X be a non-empty set. By R + we denote the set of all non-negative numbers and by R m the set of all m real matrices. Let α, β R m, that is α = (α, α 2,..., α m T and β = (β, β 2,..., β m T, then by α β (resp., α < β we mean α i β i (resp., α i < β i for each i {, 2,..., m}. Also, we denote the set of all m m matrices with non-negative elements by M m,m (R +, the zero m m matrix by and the identity m m matrix by I. Let A M m,m (R +, then A is said to be convergent to zero if and only if A n as n (see Varga []. Also note that A = I. From Filip-Petruşel [4], we have: Theorem 2.. Let A M m,m (R +. The following conditions are equivalent: (i A is convergent to zero; (ii the eigenvalues of A are in the open unit disc, that is, λ < for every λ C with det(a λi = ; (iii the matrix I A is nonsingular (that is, its determinant is nonzero and (I A = I + A + + A n +. Thus, it is easy to give some examples of matrices convergent to zero, from the literature (see Filip-Petruşel [4]. For example, we consider the following: ( a a A :=, where a, b R b b + and a + b < ; ( a b B :=, where a, b, c R c + and max{a, c} <. Now, we work in the setting of generalized metric spaces. Precisely, a mapping d: X X R m is called a vector-valued metric on X if the following properties are satisfied: (d d(x, y for all x, y X; if d(x, y = then x = y, and viceversa; (d 2 d(x, y = d(y, x for all x, y X; (d d(x, y d(x, z + d(z, y for all x, y, z X.

3 M. U. Ali et al. / Filomat : (27, Thus, a non-empty set X equipped with a vector-valued metric d is called a generalized metric space, say (X, d. Notice that the convergence and Cauchyness of a sequence in generalized metric spaces are defined in a similar manner as in usual metric spaces. So Perov [6] proved the following interesting generalization of Banach contraction principle in []. Theorem 2.2. Let (X, d be a complete generalized metric space and f : X X be a mapping for which there exists a matrix A M m,m (R + such that d( f x, f y Ad(x, y for all x, y X. If A is a matrix convergent to zero, then (i Fix( f = {x }, where Fix( f = {x X : x = f x}; (ii the sequence of successive approximations {x n } such that x n = f n x is convergent and admits the limit x, for all x X. Some interesting contributions to the development of fixed point theory and its applications in this context are obtained by Bica-Muresan [2], Bucur-Guran-Petruşel [], Filip-Petruşel [4], O Regan-Shahzad-Agarwal [5], Rus [8], Turinici [].. Fixed Point Theorem In this section we prove a fixed point theorem useful to obtain the existence and uniqueness of solution of problem (. The crucial key to establish our generalization of Theorem 2.2 (Perov [6] is the following notion of admissibility (inspired by Samet-Vetro-Vetro [9]. Definition.. Let X be a non-empty set, Λ : X X M m,m (R + and f : X X be a mapping. The function f is said to be Λ-admissible if x, y X, Λ(x, y I = Λ( f x, f y I, where I is the m m identity matrix and the inequality between matrices means entrywise inequality. Let Λ, A, A 2, A, A 4, B M m,m (R + such that (I A A 4 exists. Let f : X X. The hypotheses are the following: (i the matrix A = (I A A 4 (A + A 2 + A 4 converges to zero; (ii there exists x X such that Λ(x, f x I; (iii f is Λ-admissible; (iv a. for each sequence {x n } X such that lim n x n = x and Λ(x n, x n+ I for all n N, we have Λ(x n, x I for all n N; or b. f is continuous. Now we can have the first theorem producing existence and uniqueness of fixed point for a given mapping f. Theorem.2. Let (X, d be a complete generalized metric space and f : X X be a mapping such that, for all x, y X, we have Λ(x, yd( f x, f y A d(x, y + A 2 d(x, f x + A d(y, f y + A 4 d(x, f y + Bd(y, f x (2 with Λ, A, A 2, A, A 4, B M m,m (R + satisfying hypotheses (i-(iv. Then f has a fixed point. Moreover, if for all x, x Fix( f we have Λ(x, x I and A + A 4 + B converges to zero then the fixed point is unique.

4 M. U. Ali et al. / Filomat : (27, Proof. Because of hypothesis (ii, we see that there exists x X such that Λ(x, f x I. By putting x = f x and x 2 = f x, from (2, we have d(x, x 2 = d( f x, f x = Id( f x, f x Λ(x, x d( f x, f x A d(x, x + A 2 d(x, f x + A d(x, f x + A 4 d(x, f x + Bd(x, f x = A d(x, x + A 2 d(x, x + A d(x, x 2 + A 4 d(x, x 2 + Bd(x, x A d(x, x + A 2 d(x, x + A d(x, x 2 + A 4 [d(x, x + d(x, x 2 ] + B. After routine calculations, we get d(x, x 2 (I A A 4 (A + A 2 + A 4 d(x, x = Ad(x, x. ( By putting x = f x 2, hypothesis (iii and (2 imply that This yields d(x 2, x = d( f x, f x 2 = Id( f x, f x 2 Λ(x, x 2 d( f x, f x 2 A d(x, x 2 + A 2 d(x, f x + A d(x 2, f x 2 + A 4 d(x, f x 2 + Bd(x 2, f x = A d(x, x 2 + A 2 d(x, x 2 + A d(x 2, x + A 4 d(x, x + Bd(x 2, x 2 A d(x, x 2 + A 2 d(x, x 2 + A d(x 2, x + A 4 [d(x, x 2 + d(x 2, x ] + B. d(x 2, x (I A A 4 (A + A 2 + A 4 d(x, x 2 = Ad(x, x 2. (4 Combining ( and (4, we deduce that d(x 2, x A 2 d(x, x. Iterating this process, we construct a sequence {x n } X such that x n = f x n, Λ(x n, x n I and d(x n, x n+ A n d(x, x, for all n N. Next we show that {x n } is a Cauchy sequence. Let n, m be arbitrary natural numbers. By using the triangular inequality (d, for all n, m N, we have d(x n, x n+m n+m i=n n+m i=n d(x i, x i+ A i d(x, x A n A i d(x, x i= = A n (I A d(x, x (by condition (iii of Theorem 2.. Letting n in the above inequality, we get lim d(x n, x n+m =, (see hypothesis (i, n {x n } is a Cauchy sequence. From completeness of (X, d, we deduce that there exists x X such that x n x. Next, we distinguish two cases. Case : If hypothesis (iv.a holds then we have Λ(x n, x I for all n N. Thus, from (2, we get d( f x n, f x = Id( f x n, f x Λ(x n, x d( f x n, f x A d(x n, x + A 2 d(x n, f x n + A d(x, f x + A 4 d(x n, f x + Bd(x, f x n = A d(x n, x + A 2 d(x n, x n+ + A d(x, f x + A 4 d(x n, f x + Bd(x, x n+.

5 M. U. Ali et al. / Filomat : (27, By passing to the limit as n in the above inequality, we obtain d(x, f x (A + A 4 d(x, f x, (I (A + A 4 d(x, f x. Since the matrix I (A + A 4 is nonsingular, we deduce that d(x, f x =, and hence x = f x. Case 2: If hypothesis (iv.b holds then, for n, we have f x n f x, that is x n+ f x and so f x = x. This concludes the existence part. The uniqueness part is obvious and is obtained by contradiction. Precisely, assume that there exist x, x Fix( f with x x. Clearly, we have Λ(x, x I and so (by (2 d( f x, f x = Id( f x, f x Λ(x, x d( f x, f x A d(x, x + A 2 d(x, f x + A d(x, f x + A 4 d(x, f x + Bd(x, f x = A d(x, x + A 2 d(x, x + A d(x, x + A 4 d(x, x + Bd(x, x = (A + A 4 + Bd(x, x. Consequently, by iterating this process, we obtain d(x, x (A + A 4 + B n d(x, x, for all n N, d(x, x = (letting n, x = x, a contradiction. Thus, the fixed point of f is unique. Example.. The following mappings and matrices satisfy the hypotheses of Theorem.2. Let X = R 2 be endowed with the generalized metric d defined by d(x, y = ( x y, x 2 y 2 T, for all x = (x, x 2, y = (y, y 2 X. Let f : X X be given by ( 2x f x = x 2 +, x 2 +, for all x = (x, x 2 X with x, ( x x 2 2 +, x 2 2 +, for all x = (x, x 2 X with x >. For the sake of simplicity, we put f x = f (x, x 2 = ( f (x, x 2, f 2 (x, x 2, where 2x f (x, x 2 = x 2 +, if x, x x 2 2 +, if x >, and f 2 (x, x 2 = x 2 + if x, x if x >. Consider Λ : X X M 2,2 (R + defined by, if x, x 2, y, y 2, 2 Λ(x, y = Λ((x, x 2, (y, y 2 = 2, if x, x 2, y, y 2 >,, otherwise.

6 M. U. Ali et al. / Filomat : (27, 25 5 We show only that the condition (2 holds for all x, y X, by distinguishing some cases; we leave to the reader to check the remaining hypotheses of Theorem.2. Let A = Case : If x, x 2, y, y 2, then we have ( f (x Λ(x, yd( f x, f y =, x 2 f (y, y 2 f 2 (x, x 2 f 2 (y, y 2 Case 2: If x, x 2, y, y 2 >, then we have ( 2 Λ(x, yd( f x, f y = f (x, x 2 f (y, y 2 2 f 2(x, x 2 f 2 (y, y 2 ( 2 ( 2 ( 2. ( x y x 2 y 2 Case : For other choices of x, x 2, y and y 2, we have ( ( 2 ( Λ(x, yd( f x, f y = x y = A x 2 y 2 d(x, y. Thus (2 holds for all x, y X with A = ( 2 (I A A 4 (A + A 2 + A 4 = A, which is convergent to zero. ( x y x 2 y 2 = A d(x, y. = A d(x, y. and A 2 = A = A 4 = B =. Of course, we have A = Using two generalized metrics, one can have the following variant of Theorem.2. Theorem.4. Let (X, d be a complete generalized metric space, ρ a second generalized metric and f : (X, ρ (X, ρ be a mapping such that, for all x, y X, we have Λ(x, yρ( f x, f y A ρ(x, y + A 2 ρ(x, f x + A ρ(y, f y + A 4 ρ(x, f y + Bρ(y, f x (5 with Λ, A, A 2, A, A 4, B M m,m (R + satisfying hypotheses (i-(iii. Further, assume that (v there exists C M m,m (R + such that d( f x, f y Cρ(x, y, whenever there exists a sequence {x i } n i= with Λ(x i, x i+ I, where x = x and x n = y; (vi f : (X, d (X, d is Λ-continuous, that is, if {x n } is a sequence in X such that lim n d(x n, x = and Λ(x n, x n+ I for all n N, then we have lim n d( f x n, f x =. Then f has a fixed point. Moreover, if for all x, y Fix( f we have Λ(x, y I and A + A 4 + B converges to zero then the fixed point is unique. Remark.5. The proof of Theorem.4 essentially follows step by step the proof of Theorem.2, by replacing the generalized metric d with ρ. The difference between the two proofs is relative to the fact that here we have to establish the Cauchyness of the sequence {x n } in respect both of (X, ρ and (X, d; because only (X, d is complete by hypothesis. To this aim, by construction of {x n }, for all n, m N, we have Λ(x i, x i+ I for each i {n, n +,, n + m }. So, by using hypothesis (v, we get d(x n+, x n+m+ = d( f x n, f x n+m Cρ(x n, x n+m C[A n (I A ρ(x, x ] (by triangular inequality for ρ and (iii of Theorem 2.. So, passing to the limit as n, we deduce easily that {x n } is Cauchy in (X, d. Finally, by using hypothesis (vi, we deduce that f has a fixed point. Example.6. The following mappings and matrices satisfy the hypotheses of Theorem.4. Let X = R + \ {} be endowed with the generalized metrics ρ and d defined by ρ(x, y = ( x y, x y T, for all x, y X,

7 M. U. Ali et al. / Filomat : (27, 25 5 and ( T x y +, x y +, if x (, or y (, or x, y (, with x y, d(x, y = (, T, if x = y (,, ( T x y, x y, otherwise. Let f : X X be given by x, if x (,, f x = x+2 5, otherwise. Consider Λ : X X M 2,2 (R + defined by Λ(x, y =, if x, y,, otherwise. In particular, the condition (5 holds true with A = ( 5 5 and A 2 = A = A 4 = B =. 4. Solution of an Initial Value Problem with Delay In this section we prove a theorem producing the existence of a unique bounded solution of problem (. We follow the presentation in Bica-Muresan [2] and in Samet-Vetro-Vetro [9]. First, we consider a more general integral operator than the one in Bica-Muresan [2]. Let f : X + X + be the integral operator defined for all (u, v X + by ( f (u, v(t = (s, u(s, v(sds, h(t, u(t, v(t, t [, t ], (φ(t, φ (t, t [ τ,. The hypotheses are the following: H : ζ : R 2 R 2 R is a function such that (i there exists (u, v X + such that ζ((u (t, v (t, f (u, v (t for all t [ τ, t ]; (ii for all t [ τ, t ], (u, v, (u 2, v 2 X +, we have ζ((u (t, v (t, (u 2 (t, v 2 (t = ζ( f (u, v (t, f (u 2, v 2 (t ; (6 (iii for each sequence {(u n, v n } X + such that (u n, v n (u, v as n and ζ((u n, v n, (u n+, v n+ for all n N, we have ζ((u n, v n, (u, v for all n N. H 2 :, h C([ τ, t ] R + R are functions such that (i there exist,, φ, φ R + such that (t, u, v, t [ τ, t ], u R +, v R, φ φ(t φ, t [ τ, ];

8 M. U. Ali et al. / Filomat : (27, (ii there exist a, a 2, ρ, ϑ > with a ϑ +ρa 2 < such that, for all u, u R +, v, v R with ζ((u, v, (u, v, we have and (t, u, v (t, u, v a u u + a 2 v v, for all t [ τ, t ], h(t, u, v h(t, u, v ρ(a u u + a 2 v v, for all t [ τ, t ]. (iii we have φ( = τ (s, φ(s, φ (sds, φ ( = (, φ(, φ ( ( τ, φ( τ, φ ( τ = h(, φ(, φ (. Now, we can have the theorem producing a unique fixed point of f. Theorem 4.. If hypotheses H and H 2 hold, then the integral operator (6 has a unique fixed point in X +. Proof. Note that hypothesis H 2 (i implies that f (X + X + and so f is well-defined. Since f (u, v (t = f (u 2, v 2 (t for all t [ τ, ], then we have ( d B ( f (u, v, f (u 2, v 2 = max t [,t ] (s, u (s, v (sds (s, u 2 (s, v 2 (sds e ϑ(t+τ, T max h(t, u (t, v (t h(t, u 2 (t, v 2 (t e ϑ(t+τ, t [,t ] for all (u, v, (u 2, v 2 X +. If ζ((u (t, v (t, (u 2 (t, v 2 (t for all t [ τ, t ], then we have (s, u (s, v (sds (s, u 2 (s, v 2 (sds (s, u (s, v (s (s, u 2 (s, v 2 (s ds = (a u (s u 2 (s + a 2 v (s v 2 (s ds ( a u (s u 2 (s e ϑ(s+τ + a 2 v (s v 2 (s e ϑ(s+τ e ϑ(s+τ ds (a u u 2 B + a 2 v v 2 B e ϑ(s+τ ds ( a = ϑ u u 2 B + a 2 (e ϑ v v 2 ϑ(t+τ B e ϑt. It follows that (s, u (s, v (sds (s, u 2 (s, v 2 (sds ( a ϑ u u 2 B + a 2 ϑ v v 2 B ( e ϑτ a ϑ u u 2 B + a 2 ϑ v v 2 B, t [, t ]. e ϑ(t+τ

9 Therefore max t [,t ] (s, u (s, v (sds a ϑ u u 2 B + a 2 ϑ v v 2 B, M. U. Ali et al. / Filomat : (27, 25 5 (s, u 2 (s, v 2 (sds e ϑ(t+τ for all (u, v, (u 2, v 2 X + such that ζ((u (t, v (t, (u 2 (t, v 2 (t for all t [ τ, t ]. Similarly, we get h(t, u (t, v (t h(t, u 2 (t, v 2 (t ρ(a u (t u 2 (t + a 2 v (t v 2 (t = ρ ( a u (t u 2 (t e ϑ(t+τ + a 2 v (t v 2 (t e ϑ(t+τ e ϑ(t+τ ρ (a u u 2 B + a 2 v v 2 B e ϑ(t+τ, for all (u, v, (u 2, v 2 X + such that ζ((u (t, v (t, (u 2 (t, v 2 (t for all t [ τ, t ]. Consequently, we obtain max h(t, u (t, v (t h(t, u 2 (t, v 2 (t e ϑ(t+τ t [,t ] ρ (a u u 2 B + a 2 v v 2 B, for all (u, v, (u 2, v 2 X + such that ζ((u (t, v (t, (u 2 (t, v 2 (t for all t [ τ, t ]. Then, for all (u, v, (u 2, v 2 X + such that ζ((u (t, v (t, (u 2 (t, v 2 (t for all t [ τ, t ], we have where d B ( f (u, v, f (u 2, v 2 Ad B ((u, v, (u 2, v 2, A = ( a ϑ a 2 ϑ ρa ρa 2. Consider Λ : X X M 2,2 (R + defined by, if ζ((u (t, v (t, (u 2 (t, v 2 (t for all t [ τ, t ], Λ((u, v, (u 2, v 2 =, otherwise. Finally, for all (u, v, (u 2, v 2 X +, we have Λ((u, v, (u 2, v 2 d B ( f (u, v, f (u 2, v 2 Ad B ((u, v, (u 2, v 2. Next, the eigenvalues of A are λ = and λ 2 = a ϑ + ρa 2. (8 Therefore, from (8 and hypothesis H 2 (ii (i.e., a ϑ + ρa 2 < we infer that λ, λ 2 are in the open unit disc and so A is convergent to zero (see conditions (i and (ii of Theorem 2.. Also, the above calculations and the relation between the matrix Λ and function ζ (by using hypotheses H show that the hypotheses of Theorem.2 hold with A = A given by (7 and A 2 = A = A 4 = B =. For instance, by H (ii, we have Λ((u, v, (u 2, v 2 I ζ((u (t, v (t, (u 2 (t, v 2 (t ζ( f (u, v (t, f (u 2, v 2 (t Λ( f (u, v (t, f (u 2, v 2 (t I, so f is Λ-admissible. Thus, the existence and uniqueness of a fixed point of f in X + is an immediate consequence of Theorem.2. (7

10 M. U. Ali et al. / Filomat : (27, By particularizing the choice of h C([ τ, t ] R + R, we can have the theorem producing a unique bounded solution of problem (. This theorem is more general than the analogous of Bica-Muresan ([2], Theorem 5. Let h(t, u(t, v(t = (t, u(t, v(t (t τ, u(t τ, v(t τ for all t [, t ] and consider the integral operator ( f (u, v(t = (s, u(s, v(sds, (t, u(t, v(t (t τ, u(t τ, v(t τ, t [, t ], (φ(t, φ (t, t [ τ, ]. Theorem 4.2. If hypotheses H and H 2 hold, then problem ( has a unique bounded solution in X +. Proof. The similar reasoning as in the proof of Theorem 4. shows that f has a unique fixed point in X +, say {(u, v } = Fix( f. To avoid repetition, we leave the details and point out just the difference. Precisely, here we obtain the matrix ( a A = ϑ a 2 ϑ a ( + e ϑτ a 2 ( + e ϑτ with eigenvalues λ = and λ 2 = a ϑ + a 2 ( + e ϑτ, that is, we have ρ = ( + e ϑτ. Next, we show that u is a unique bounded solution of (. In fact, from hypothesis H 2(i, we get τ u (t = (s, u (s, v (sds τ, t [, t ], φ u (t φ, t [ τ, ], and hence the boundedness is proved. It remain to prove that (u (t = v (t, t [ τ, t ], (see Bica-Muresan [2], p. 25. We distinguish the following two cases: Case : If t [, t ] then, from (6, we have So (u (t, v (t = f (u (t, v (t, t [, t ]. u (t = (s, u (t(s, v (sds, v (t = (t, u (t, v (t (t τ, u (t τ, v (t τ. It follows easily that (u (t = (t, u (t, v (t (t τ, u (t τ, v (t τ and so (u (t = v (t for all t [, t ]. Case 2: If t [ τ, ], again from (6, we have f (u (t, v (t = (φ(t, φ (t = (u (t, v (t and so u (t = φ(t and v (t = φ (t. Remark 4.. Every non-negative constant function ζ reduces Theorem 4.2 to Theorem 5 of Bica-Muresan [2], where (for the sake of exactness the authors assume a 2 (, 2. On the other hand, other choices of function ζ are possible. So Theorem 4. covers a large class of situations than those of the original version in [2]. For example, by assuming ζ((u (t, v (t, (u 2 (t, v 2 (t = u (t u 2 (t, for all t [ τ, t ], (u, v, (u 2, v 2 X +, the ordered approach to the study of initial value problems with delay arises naturally. Technically, this means to consider the product space X endowed with the partial order defined by (u, v, (u 2, v 2 X, (u, v (u 2, v 2 u (t u 2 (t, t [ τ, t ]. Thus, we have to check the contractive condition in Theorem 4. only for couples of points satisfying the partial order. Also, the hypothesis H (i reduces to the existence of an upper solution for problem ( and so on.

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