On the reduction of Krasnoselskii’s theorem to Schauder’s theorem

On the reduction of Krasnoselskii’s theorem to Schauder’s theorem

Applied Mathematics and Computation 250 (2015) 339–351 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepag...

400KB Sizes 0 Downloads 0 Views

Applied Mathematics and Computation 250 (2015) 339–351

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

On the reduction of Krasnoselskii’s theorem to Schauder’s theorem T.A. Burton a,⇑, Bo Zhang b a b

Northwest Research Institute, 732 Caroline St., Port Angeles, WA, United States Department of Mathematics and Computer Science, Fayetteville State University, Fayetteville, NC 28301, United States

a r t i c l e

i n f o

Keywords: Fixed points Krasnoselskii’s theorem on sum of two maps Nonlinear integral equations

a b s t r a c t Krasnoselskii noted that many problems in analysis can be formulated as a mapping which is the sum of a contraction and compact map. He proved a theorem covering such cases which is the union of the contraction mapping principle and Schauder’s second fixed point theorem. In putting the two results together he found it necessary to add a condition which has been difficult to fulfill, although a great many problems have been solved using his result and there have been many generalizations and simplifications of his result. In this paper we point out that when the mapping is defined by an integral plus a contraction term, the integral can generate an equicontinuous map which is independent of the smoothness of the functions. Because of that, it is possible to set up that mapping, not as a sum of contraction and compact map, but as a continuous map on a compact convex subset of a normed space. An application of Schauder’s first fixed point theorem will then yield a fixed point without any reference to that difficult condition of Krasnoselskii. Finite and infinite intervals are handled separately. For the class of problems considered, application is parallel to the much simpler Brouwer fixed point theorem. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Throughout applied mathematics we see real world problems modeled by various differential equations which are often inverted as integral equations defining natural mappings of certain sets in a Banach space into themselves. In order to get a fixed point solving the differential equation we are frequently faced with severe compactness problems, particularly when we need a solution on an entire interval ½0; 1Þ. Indeed, there is a myriad of real world problems modeled by fractional differential equations which naturally invert as integral equations with singular kernels. These integral equations define a natural mapping which invites either Schauder’s or Krasnoselskii’s fixed point theorem. Frequently the mapping is continuous and we can locate a convex set mapped into itself. Our task is only beginning as we consider compactness questions and the mixing of contraction and compact maps. Here we arrive at the objective of this project. We show that if the kernel and its coefficient function satisfy reasonable conditions, then there is a natural equicontinuity condition on the part of the mapping generated by the integral. We then restrict our mapping to a convex set in the Banach space for which that equicontinuity holds even for the contraction part of the mapping. This allows us to use Schauder’s first fixed point theorem to get a fixed point.

⇑ Corresponding author. E-mail addresses: [email protected] (T.A. Burton), [email protected] (B. Zhang). http://dx.doi.org/10.1016/j.amc.2014.10.093 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

340

T.A. Burton, B. Zhang / Applied Mathematics and Computation 250 (2015) 339–351

Here is the advantage. In Krasnoselskii’s theorem there is a complicated condition (item (i) in the theorem below) which ties the contraction mapping to the compact mapping. By the above process we avoid that complication and get the fixed point directly from Schauder’s theorem. We now look at the details. Krasnoselskii [12] studied an old paper of Schauder on elliptic partial differential equations and deduced a working principle which we formalize as follows: The inversion of a perturbed differential operator yields the sum of a contraction and compact map. Accordingly, he offered the following result to facilitate treatment of that sum. Theorem 1.1 (Krasnoselskii). Let ðS; k  kÞ be a Banach space, M a closed, convex, nonempty subset of S. Suppose that A; B : M ! S such that

ðiÞ x; y 2 M ) Ax þ By 2 M: A is continuous and

ðiiÞ AM resides in a compact set; ðiiiÞ B is a contraction with constant a < 1. Then 9y 2 M with Ay þ By ¼ y. It is clear from (ii) and (iii) that it is intended to be a combination of Schauder’s second fixed point theorem and the contraction mapping principle. As such it would seem to be exactly what is needed in so many problems in differential and integral equations. But the marriage of the two principles takes place in (i) and that has been very challenging in so many standard problems from applied mathematics. We addressed item (i) in [3]. A very nice summary of selected results on Krasnoselskii’s theorem up through 2007 is found in [16]. Other recent results are found in [1,10,11,5]. It is very convenient to find Krasnoselskii’s result and proof in [17], as well as two forms of Schauder’s fixed point theorem used here. To focus on the need for such a result we consider a neutral functional differential equation. Example. Consider the scalar equation

x0 ðtÞ ¼ xðtÞ þ a

d xðt  hÞ þ uðt; xðtÞÞ dt

with a continuous initial function w : ½h; 0 ! R in which, for simplicity in this presentation, we ask that wð0Þ ¼ awðhÞ. By grouping terms and integrating we obtain

xðtÞ ¼ axðt  hÞ þ

Z

t

eðtsÞ ½axðs  hÞ þ uðs; xðsÞÞds:

0

A full treatment using Krasnoselskii’s fixed point theorem is found in [2, pp. 180–184]. The first term, axðt  hÞ, does not smooth but the integral smooths in a most remarkable way. When x is restricted to any given bounded set in BC with a bound of a fixed number K, then the integral maps that set into an equicontinuous set where the equicontinuity is completely independent of the behavior of x. This allows us to place an equicontinuity condition on the mapping set so that the integral equation maps that set into itself. The fact that the contraction term does not smooth causes no trouble at all. We then apply Schauder’s first fixed point theorem and obtain a bounded and continuous solution on any interval ½0; T. The victory is that in applying the result condition (i) of Krasnoselskii’s theorem is completely avoided. If the mapping set is essentially a ball then the work holds for 0 6 t < 1 in a weighted space. An integral equation with a mild singularity has a natural induced equicontinuity which can be of prime importance in fixed point theory. We will consider two essentially different forms:

xðtÞ ¼ Vðt; xðtÞÞ þ

Z

t

Rðt  sÞuðt; s; xðsÞÞds

ð1aÞ

Rðt  sÞuðt; s; xðÞÞds;

ð1bÞ

0

and

xðtÞ ¼ f ðt; xðÞÞ þ

Z

t

0

where V and f are of a nature to generate a contraction while R will generate a compact map. For example, we may find a closed convex nonempty set M in the Banach space ðBC; k  kÞ of bounded continuous functions / : ½0; 1Þ ! R with the supremum norm with the property that / 2 M and for P defined by either

ðP/ÞðtÞ ¼ Vðt; /ðtÞÞ þ

Z

t

Rðt  sÞuðt; s; /ðsÞÞds

ð2aÞ

0

or

ðP/ÞðtÞ ¼ f ðt; /ðÞÞ þ

Z

t

Rðt  sÞuðt; s; /ðÞÞds

0

with a given initial function w : ½h; 0 ! R;; we have P : M ! M.

ð2bÞ

T.A. Burton, B. Zhang / Applied Mathematics and Computation 250 (2015) 339–351

341

Immediately we think of Krasnoselskii’s fixed point theorem with all its benefits, together with real challenges. But there is a more direct way if R has mild singularities of the following form. Assume that for the fixed set M there is a positive constant, q, with 0 < q < 1 so that for / 2 M then

0 6 Rðt  sÞ 6 ðt  sÞq1 :

ð3Þ

There are many sources for such problems. We can then show (see [6] and Theorem 6.1) that, independently of the particular / 2 M, the integral

ðL/ÞðtÞ :¼

Z

t

Rðt  sÞuðt; s; /ðsÞÞds

ð4Þ

0

satisfies

jL/ðtÞ  L/ðsÞj 6 HðtÞjt  sjq ;

ð5Þ

where HðtÞ is an increasing function. That is, LM is an equicontinuous set. In fact, let t P 0 be fixed. For a given  > 0, we find d > 0 corresponding to the  and t in the definition of equicontinuity by HðtÞdq <  or 1=q

d < ð=HðtÞÞ

ð6Þ

:

Next, suppose there is a b < 1 so that x; y 2 R implies that

jVðt; xÞ  Vðt; yÞj 6 bjx  yj:

ð7aÞ

Or, suppose there is an a < 1 and h > 0 so that for t P h,

jf ðt; /ðÞÞ  f ðt; wðÞÞj 6 aj/t  wt j½h;0

ð7bÞ

½h;0

for /; w 2 M, where j/t  wt j ¼ suph6h60 j/ðt þ hÞ  wðt þ hÞj. The process is now clear. We ask that we add to M the property that all functions satisfy the equicontinuity condition (6) in a certain way so that / 2 M implies P/ 2 M. 2. The integral equation without a delay We are going to continue to use the kernel in the stated form so that the reader can see clearly the exactness of the equicontinuity on which the entire process depends. However, with care one can do the same for a more general equation. For example, Garcia–Falset [10, p. 1746, Item 4] considers the integral equation

xðtÞ ¼ gðt; xÞ þ

Z

t

Fðt  s; s; uðsÞÞds

0

and obtains a bounded mapping set M in which there is a contraction condition on g and a relation

kFðt; s; xÞ  Fðh; s; xÞk 6 SGðjt  hjÞ where G is a continuous function with Gð0Þ ¼ 0 and S a constant depending on a bound on the functions in M. The reader is then left to carry out computations parallel to those which we provide below as a template for such work. Let T > 0 and ðBC; k  kÞ be the Banach space of bounded continuous functions / : ½0; T ! R with the supremum norm. We consider an integral equation of the form (1a)

xðtÞ ¼ Vðt; xðtÞÞ þ

Z

t

Rðt  sÞuðt; s; xðsÞÞds:

0

The following assumptions will be used. (i) R : ð0; 1Þ ! ½0; 1Þ is continuous, decreasing, and Rðt  sÞ 6 ðt  sÞq1 ; 0 < q < 1. (ii) u : ½0; T  ½0; T  R ! R is continuous. (iii) There is a closed, bounded, convex, nonempty set M  BC with the following properties: (a) There are positive J; S such that / 2 M and 0 6 s 6 t 6 T implies that juðt; s; /ðsÞÞj 6 S and

juðt; s; /ðsÞÞ  uðs; s; /ðsÞÞj 6 Jjt  sjq : (b) V : ½0; T  R ! R is continuous and there is a positive b with b < 1 such that / 2 M and t; s 2 ½0; T implies that

jVðt; /ðtÞÞ  Vðt; /ðsÞÞj 6 bj/ðtÞ  /ðsÞj: We will proceed with a view of constructing a nonempty compact convex subset M  of M which is mapped into itself by P defined in (2a) in such a way that Schauder’s first fixed point theorem can be applied to the restriction mapping P : M  ! M  . The resulting fixed point is then trivially a fixed point of the original mapping P : M ! M.

342

T.A. Burton, B. Zhang / Applied Mathematics and Computation 250 (2015) 339–351

Theorem 2.1. Let (i)–(iii) hold. Suppose that the mapping P defined by / 2 M implies that

ðP/ÞðtÞ ¼ Vðt; /ðtÞÞ þ

Z

t

Rðt  sÞuðt; s; /ðsÞÞds;

06t6T

0

maps M ! M. Then P has a fixed point. Proof. Since M is nonempty, we choose a fixed / 2 M and define

M ¼ f/ 2 M : j/ðtÞ  /ðsÞj 6 xðt; sÞ; 8t; s 2 ½0; Tg; where

xðt; sÞ ¼

 1   j/ ðtÞ  / ðsÞj þ HðtÞjt  sjq þ V  ðt; sÞ 1b

Rt with V  ðt; sÞ ¼ supjxj6D jVðt; xÞ  Vðs; xÞj; HðtÞ ¼ J 0 RðsÞds þ ð2S=qÞ and D the bound of M. We now see that M  is nonempty    since / 2 M . We shall show that M is a compact convex subset of M. First, we note that M is convex so if /; g 2 M and 0 6 k 6 1 then k/ þ ð1  kÞg 2 M. Thus, if /; g 2 M  then

jk½/ðt 1 Þ  /ðt 2 Þ þ ð1  kÞ½gðt1 Þ  gðt 2 Þj 6 xðt 1 ; t2 Þ; 

so M is also convex. In fact, M  is a compact subset of M. To see this, let f/n g denote a sequence in M  . As the sequence is uniformly bounded and equicontinuous on ½0; T, by the Ascoli–Arzela theorem there is a subsequence f/nk g with a limit / residing in the closed set M. If t1 ; t2 2 ½0; T then

j/ðt 1 Þ  /ðt 2 Þj 6 j/ðt1 Þ  /nk ðt 1 Þj þ j/nk ðt1 Þ  /nk ðt 2 Þj þ j/nk ðt2 Þ  /ðt2 Þj: The first and last terms on the right-hand-side tend to zero as k ! 1, while the middle term is less than xðt1 ; t2 Þ so that for large k the left-hand term is less than xðt 1 ; t 2 Þ. Thus, / 2 M , and so M  is a compact subset of M. Next, we show that / 2 M implies that P/ 2 M . Certainly, P/ 2 M. By Theorem 6.1, we see (5) holds on ½0; T. Now, for t; s 2 ½0; T, by (iii) and Theorem 6.1 we obtain

jðP/ÞðtÞ  ðP/ÞðsÞj 6 jVðt; /ðtÞÞ  Vðs; /ðsÞÞj þ jL/ðtÞ  L/ðsÞj 6 jVðt; /ðtÞÞ  Vðt; /ðsÞÞj þ jVðt; /ðsÞÞ  Vðs; /ðsÞÞj þ jL/ðtÞ  L/ðsÞj 6 bj/ðtÞ  /ðsÞj þ V  ðt; sÞ þ HðtÞjt  sjq 6 bxðt; sÞ þ ð1  bÞxðt; sÞ ¼ xðt; sÞ; so P : M ! M  . It is rather routine to show that P is continuous. Let

l

Z

 > 0 be given. First, for =2, find l > 0 so that

T

RðsÞds < =2:

0

Now M is bounded by a number D and uðt; s; xÞ is uniformly continuous for 0 6 s 6 t 6 T and jxj 6 D so for the l > 0 there is the d1 > 0 of uniform continuity so that k/  gk < d1 implies that juðt; s; /ðsÞÞ  uðt; s; gðsÞÞj < l. Also, since Vðt; xÞ is uniformly continuous for 0 6 t 6 T and jxj 6 D, there exists d2 > 0 such that k/  gk < d2 implies that

jVðt; /ðtÞÞ  Vðt; gðtÞÞj < =2 for all t 2 ½0; T: Take d ¼ min½d1 ; d2 . Then k/  gk < d implies that

e ð/Þ  V e ðgÞk þ l kðP/Þ  ðPgÞk 6 k V

Z

T

RðsÞds 6 ð=2Þ þ ð=2Þ;

0

e ð/ÞðtÞ ¼ Vðt; /ðtÞÞ, as required. where V Thus, P is a continuous map of a compact convex nonempty set M  into itself so, by Schauder’s first fixed point theorem, P has a fixed point in M   M. h 2.1. The case for T ¼ 1 For the case of 0 6 t 6 T just covered we allowed M to be any closed bounded convex nonempty set in BC. When we pass to ½0; 1Þ there is a large change since M  may no longer be a compact subset of M in BC even if it satisfies the equicontinuity condition. In this case M must be essentially a ball in order for M to be closed and M  compact in a weighted space being considered here. Theorem 2.2. Let (i)–(iii) hold with T ¼ 1. Suppose there are continuous functions v ; w : ½0; 1Þ ! R with v ðtÞ < wðtÞ for t P 0. Let

M ¼ f/ 2 BCjv ðtÞ 6 /ðtÞ 6 wðtÞg: If the mapping P of Theorem 2.1 maps M into M, then P has a fixed point.

T.A. Burton, B. Zhang / Applied Mathematics and Computation 250 (2015) 339–351

343

Proof. Let g : ½0; 1Þ ! ½1; 1Þ be continuous with g 2" 1. Then ðW; j  jg Þ is the Banach space of continuous functions / : ½0; 1Þ ! R for which

j/jg ¼: sup

06t<1

j/ðtÞj < 1: gðtÞ

Since g is an arbitrary continuous strictly increasing function with gðtÞ ! 1 as t ! 1, we may choose g so that

lim

t!1

v ðtÞ gðtÞ

¼ 0 and lim

t!1

wðtÞ ¼ 0: gðtÞ

ð8Þ

We see that v and w may be unbounded, but the functions in M restricted to any compact interval ½0; T are uniformly bounded. We define M  as in Theorem 2.1 with

xðt; sÞ ¼

 1   j/ ðtÞ  / ðsÞj þ HðtÞjt  sjq þ V  ðt; sÞ 1b

for t; s P 0. Note that M  is a closed convex nonempty subset of ðW; j  jg Þ and the work in the proof of Theorem 2.1 shows that P : M  ! M  . Let f/n g be a sequence in M  and use Ascoli’s theorem and the diagonalization process to show that there is a subsequence f/nk g converging to some / 2 BC uniformly on compact subsets of ½0; 1Þ and / 2 M. Moreover, j/nk  /jg ! 0 as k ! 1 since (8) holds. The proof of Theorem 2.1 shows that / satisfies the equicontinuity property so / 2 M . Therefore M  is a compact subset of ðW; j  jg Þ. An argument similar to that in [5] shows that P is continuous in the g-norm on M. Applying Schauder’s first fixed point theorem to P : M  ! M  in ðW; j  jg Þ, we obtain that there exists a point / 2 M  with P/ ¼ /. This completes the proof. h Remark 1. If b ¼ 1 in (iii)-(b), then the argument in the proof of theorems above fails. However, we may still be able to establish the existence of a fixed point for P under additional assumptions on V. The process goes as follows. We first prove the existence of an e-fixed point of P; that is, for each e > 0, there exists xe 2 M such that

kPxe  xe k < e: Next, we apply the approximation method to obtain a fixed point of P. This will be demonstrated in Theorem 2.3. Let I denote the identity map and ðI  PÞðMÞ denote the range of I  P on M. Theorem 2.3. Let (i)–(iii) hold with b ¼ 1 and T ¼ 1, and let M be defined in Theorem 2.2. Suppose that (iv) the set ðI  PÞðMÞ is closed in BC. If the mapping P of Theorem 2.1 maps M into M, then P has a fixed point. ~ 2 M and define, for any positive integer n, a mapping Pn by / 2 M implies Proof. Since M is nonempty, we choose a fixed /

ðP n /ÞðtÞ ¼

      Z t 1 1~ 1 1~ 1 ¼ 1 ðP/ÞðtÞ þ /ðtÞ Vðt; /ðtÞÞ þ /ðtÞ þ 1 1 Rðt  sÞuðt; s; /ðsÞÞds: n n n n n 0

~ for Since P/ 2 M and M is convex, we have Pn / 2 M and thus, Pn ðMÞ  M. Letting V n ðt; xÞ ¼ ð1  1=nÞVðt; xÞ þ ð1=nÞ/ðtÞ n ¼ 2; 3; . . ., we see for / 2 M that

jV n ðt; /ðtÞÞ  V n ðt; /ðsÞÞj ¼



1

   1 1 jVðt; /ðtÞÞ  Vðt; /ðsÞÞj 6 1  j/ðtÞ  /ðsÞj ¼: bj/ðtÞ  /ðsÞj n n

for t; s P 0. Thus, conditions (i)–(iii) are satisfied with Vðt; xÞ; uðt; s; xÞ: replaced by V n ðt; xÞ; ð1  1=nÞuðt; s; xÞ, respectively. By Theorem 2.2, there is a point /n 2 M such that P n /n ¼ /n ; that is,

 Z t 1 /n ðtÞ ¼ V n ðt; /n ðtÞÞ þ 1  Rðt  sÞuðt; s; /n ðsÞÞds: n 0 We also see from

ðP/n ÞðtÞ  /n ðtÞ ¼

  Z t 1 1~ Vðt; /n ðtÞÞ þ Rðt  sÞuðt; s; /n ðsÞÞds  /ðtÞ n n 0

that P has an e-fixed point in M for each

e > 0 since

344

T.A. Burton, B. Zhang / Applied Mathematics and Computation 250 (2015) 339–351

kP/n  /n k 6

l n

;

~ Let G ¼ I  P. Then l ¼ supfkPuk : u 2 Mg þ k/k.

where

kGð/n Þk ¼ kP/n  /n k ! 0 as n ! 1 and so 0 2 GðMÞ ¼ GðMÞ since GðMÞ is closed in BC. Thus, 9/ 2 M such that P/  / ¼ 0. This completes the proof.

h

e : M ! BC by V e ð/ÞðtÞ ¼ Vðt; /ðtÞÞ for all t P 0 and / 2 M. We now define a mapping V Corollary 1. Let (i)–(iii) hold with b ¼ 1 and T ¼ 1, and let

M ¼ f/ 2 BC : k/k 6 Kg for a constant K > 0. Suppose that (v) suptP0 jVðt; xÞ  Vðt; yÞj < jx  yj for all x; y 2 ½K; K with x – y. If the mapping P of Theorem 2.1 maps M into M, then P has a fixed point.

Proof. We only need to show that (iv) holds. To this end, let wn 2 ðI  PÞðMÞ with kwn  wk ! 0 as n ! 1 for some w 2 BC. We shall show that w 2 ðI  PÞðMÞ. Let /n 2 M with wn ¼ ðI  PÞ/n for n ¼ 1; 2; . . .. We write ðI  PÞ/n ðtÞ ¼ wn ðtÞ as

/n ðtÞ  Vðt; /n ðtÞÞ ¼ yn ðtÞ þ wn ðtÞ;

ð9Þ

where

yn ðtÞ ¼

Z

t

Rðt  sÞuðt; s; /n ðsÞÞds:

0

Since V is continuous on ½0; 1Þ  ½K; K and PðMÞ  M, we see that the sequence fyn g is uniformly bounded and equicontinuous on any compact subset of ½0; 1Þ by (iii)-(a) and Theorem 6.1. Thus, by the Ascoli–Arzela theorem, there is a subsequence fynk g converging to some y 2 BC uniformly on any closed bounded interval ½0; T. Since (v) holds, we have by e Þ1 is continuous on ðI  V e ÞðMÞ. It now follows from (9) that Theorem 6.2 that ðI  V

h i e Þ1 y þ w ðtÞ: /nk ðtÞ ¼ ðI  V nk nk

ð10Þ

eÞ This implies that f/nk g converges to a function / 2 M uniformly on ½0; T. Since ðI  V ting k ! 1 in (10) we obtain

1

e ÞðMÞ, by letis continuous on ðI  V

1

e Þ ½y þ wðtÞ for t 2 ½0; T: /ðtÞ ¼ ðI  V

ð11Þ

Taking the limit in

Z

ynk ðtÞ ¼

t

0

Rðt  sÞuðt; s; /nk ðsÞÞds;

we also obtain

yðtÞ ¼

Z

t

Rðt  sÞuðt; s; /ðsÞÞds:

ð12Þ

0

Combining (11) and (12), we see that wðtÞ ¼ ðI  PÞ/ðtÞ for all t P 0. Thus, w 2 ðI  PÞðMÞ, and the proof is complete. h Example. Consider the fractional differential equation of Caputo type c

Dq ðx  jðxÞÞ ¼ aðtÞx3 ðtÞ þ Gðt; xðtÞÞ;

with a : ½0; 1Þ ! R;

xð0Þ ¼ x0 ;

0 < q < 1;

j : R ! R; G : ½0; 1Þ  R ! R; continuous. See [6] for background and definitions. Suppose that

(~i) aðtÞ is bounded on ½0; 1Þ, e jGðt; xÞj 6 bðtÞjxj3 for jxj 6 1 and t P 0, ( ii) e aðtÞ  bðtÞ P d for all t P 0 and a constant d > 0. ( iii) f ( iv ) 9c > 0 such that j : ½c; c ! R is nondecreasing, odd with x  jðxÞ  x3 increasing on ½0; c, Then the zero solution of (13) is stable.

ð13Þ

345

T.A. Burton, B. Zhang / Applied Mathematics and Computation 250 (2015) 339–351

Proof. Choose a constant g > 0 with suptP0 aðtÞ < g and define

CðtÞ ¼

g CðqÞ

tq1 :

Then the resolvent R satisfies

RðtÞ ¼ CðtÞ 

Z

t

Cðt  sÞRðsÞds:

0

This resolvent R is completely monotone on ð0; 1Þ. Moreover,

0 6 RðtÞ 6 CðtÞ;

tRðtÞ ! 0 as t ! 1;

and

Z

1

RðsÞds ¼ 1:

0

If we write (13) as c

Dq ðx  jðxÞÞ ¼ aðtÞx3 ðtÞ þ Gðt; xðtÞÞ ¼ g½x  jðxÞ þ g½x  jðxÞ  x3  þ ½g  aðtÞx3 þ Gðt; xðtÞÞ;

then the solution xðtÞ satisfies

xðtÞ ¼ zðtÞ þ jðxÞ þ

Z

t

Rðt  sÞ½xðsÞ  jðxÞ  x3 ðsÞds þ

0

¼: jðxÞ þ zðtÞ þ

Z

Z

t

  Z t aðsÞ 3 1 x ðsÞds þ Rðt  sÞ 1  Rðt  sÞ Gðs; xðsÞÞds

g

0

0

g

t

Rðt  sÞuðs; xðsÞÞds ¼: ðPxÞðtÞ;

0

Rt where zðtÞ ¼ ðx0  jðx0 ÞÞð1  0 RðsÞdsÞ. pffiffiffi Let 0 < e < c. We may assume that c < 3=3; g P 1, and 0 < d < 1 so that de3 =g < e. Now let jx0 j < de3 =g and define

M ¼ f/ 2 BC : k/k 6 eg: For the mapping defined above, we can show that P : M ! M. To see this, we observe that r  jðrÞ  r3 is odd and increase Apply ( ii) e and (f iii) to obtain ing on ½0; e by ( iv).

jðPxÞðtÞj 6 jzðtÞj þ jðeÞ þ ðe  jðeÞ  e3 Þ þ e3

Z

t

  aðsÞ jbðsÞj ds 6 jx0 j þ ðe  e3 Þ þ e3 ð1  d=gÞ < e; Rðt  sÞ 1  þ

0

g

g

3

if jx0 j < de =g. Since x  jðxÞ  x3 is increasing on ½0; c, we see that x  jðxÞ is strictly increasing on ½c; c with

jjðxÞ  jðyÞj < jx  yj for all x; y 2 ½c; c with x – y. We now readily verify that all conditions of Corollary 1 are satisfied with Vðt; xÞ ¼ jðxÞ þ zðtÞ, and so, P has a fixed point x 2 M which is the solution of (13). Since e > 0 is arbitrary, this proves that the solution x ¼ 0 of (13) is stable. h Critique Notice that conditions (i)–(iii) are simply defining the sets and functions. Theorem 2.1 asks only that the investigator show that P : M ! M and P is continuous. Krasnoselskii’s theorem has been replaced by Schauder’s theorem for certain Banach spaces. 3. The integral equation with a delay Let T > 0 and ðBC; k  kÞ be the Banach space of bounded continuous functions / : ½0; T ! R with the supremum norm. We consider an integral equation with a given continuous initial function w : ½h; 0 ! R of the form

xðtÞ ¼ f ðxðt  hÞÞ þ

Z

t

Rðt  sÞuðs; xðsÞ; xðs  hÞÞds þ FðtÞ;

tP0

0

with xðtÞ ¼ wðtÞ for h 6 t 6 0. The following assumptions will be used. (i) R : ð0; 1Þ ! ½0; 1Þ is continuous, decreasing, and Rðt  sÞ 6 ðt  sÞq1 ; 0 < q < 1. (ii) u : ½0; T  R2 ! R and F : ½0; T ! R are continuous. (iii) There is a closed, bounded, convex, nonempty set M  BC with the following properties: (a) / 2 M implies that /ð0Þ ¼ wð0Þ. (b) There is a positive S such that / 2 M and 0 6 t 6 T implies that juðt; /ðtÞ; /ðt  hÞÞj 6 S. (c) f : R ! R is continuous and there is an a < 1 such that / 2 M and t; s 2 ½0; T implies

ð14Þ

346

T.A. Burton, B. Zhang / Applied Mathematics and Computation 250 (2015) 339–351

jf ð/ðtÞÞ  f ð/ðsÞÞj 6 aj/ðtÞ  /ðsÞj: It is understood that /ðsÞ ¼ wðsÞ; 8s 2 ½h; 0, for all / 2 M. We also extend the domain of any function g : ½a; b ! R: to R: by assigning gðsÞ ¼ gðaÞ for s 6 a and gðsÞ ¼ gðbÞ for s P b. For a real-valued function u : R ! R, we set

jut  us j½a;b ¼ sup juðt þ sÞ  uðs þ sÞj a6s6b

for all t; s 2 R. Finally, we point out that functions f ; F in (14) may depend on w so that wð0Þ ¼ f ðwðhÞÞ þ Fð0Þ. Theorem 3.1. Let (i)–(iii) hold. Suppose that the mapping P defined by / 2 M implies that

ðP/ÞðtÞ ¼ f ð/ðt  hÞÞ þ

Z

t

Rðt  sÞuðs; /ðsÞ; /ðs  hÞÞds þ FðtÞ;

06t6T

0

maps M ! M. Then P has a fixed point. Proof. Let / 2 M be fixed and define

M ¼ f/ 2 M : j/ðtÞ  /ðsÞj 6 hðt; sÞ; 8t; s 2 Xg; where

X ¼ fðt; sÞ : t; s 2 ½0; h or t; s 2 ½h; Tg and

hðt; sÞ ¼

  1 2S j/t  /s jð1;0 þ jt  sjq þ jF t  F s jð1;0 þ supjf ðwðt þ sÞÞ  f ðwðs þ sÞÞj; 1a q s60

for all t; s 2 ½0; T. It is clear that hðt; sÞ ! 0 as jt  sj ! 0. This implies that M  is uniformly bounded and equicontinuous on ½0; T. Since / 2 M  , we see that M  is a compact convex nonempty subset of M. We now claim that hðt  h; s  hÞ 6 hðt; sÞ: for all t; s 2 ½h; T. In fact, we have

  1 2S j/th  /sh jð1;0 þ jt  sjq þ jF th  F sh jð1;0 þ supjf ðwðt  h þ sÞÞ  f ðwðs  h þ sÞÞj 1a q s60   1 2S q ð1;h   ð1;h þ sup jf ðwðt þ rÞÞ  f ðwðs þ rÞÞj j/t  /s j þ jt  sj þ jF t  F s j ¼ 1a q r6h   1 2S j/t  /s jð1;0 þ jt  sjq þ jF t  F s jð1;0 þ supjf ðwðt þ sÞÞ  f ðwðs þ sÞÞj ¼ hðt; sÞ: 6 1a q s60

hðt  h; s  hÞ ¼

Next, we show that / 2 M  implies that P/ 2 M  . Certainly, P/ 2 M. We may assume h < T and still denote the integral term in (14) by L/ðtÞ so that (5) holds with HðtÞ ¼ 2S=q. Now, for t; s 2 ½h; T, we have

jðP/ÞðtÞ  ðP/ÞðsÞj 6 jf ð/ðt  hÞÞ  f ð/ðs  hÞÞj þ jL/ðtÞ  L/ðsÞj þ jFðtÞ  FðsÞj 6 aj/ðt  hÞ  /ðs  hÞj þ

2S 2S jt  sjq þ jFðtÞ  FðsÞj 6 ahðt  h; s  hÞ þ jt  sjq þ jF t  F s jð1;0 q q

6 ahðt; sÞ þ ð1  aÞhðt; sÞ ¼ hðt; sÞ: For t; s 2 ½0; h, we observe that

jf ð/ðt  hÞÞ  f ð/ðs  hÞÞj ¼ jf ðwðt  hÞÞ  f ðwðs  hÞÞj and so

jðP/ÞðtÞ  ðP/ÞðsÞj 6 jf ðwðt  hÞÞ  f ðwðs  hÞÞj þ jL/ðtÞ  L/ðsÞj þ jFðtÞ  FðsÞj 6 jf ðwðt  hÞÞ  f ðwðs  hÞÞj þ

2S jt  sjq þ jFðtÞ  FðsÞj 6 hðt; sÞ: q

This implies that P : M  ! M  . The rest of the proof is exactly as in the proof of Theorem 2.1. h The extension to the interval ½0; 1Þ is exactly as before. 4. A fractional integral equation There is an interesting and important paper [9] dealing with the scalar integral equation

xðtÞ ¼ gðt; xðtÞÞ þ

f ðt; xðtÞÞ CðqÞ

Z 0

t

v ðt; s; xðsÞÞ ðt  sÞ1q

ds;

0 < q < 1;

ð15Þ

T.A. Burton, B. Zhang / Applied Mathematics and Computation 250 (2015) 339–351

347

which is studied by means of measures of noncompactness and a fixed point theorem of Darbo. It is important because it models a number of real-world problems with a view to showing that at least one solution exists and that any other solution converges to it at infinity. All of the conditions are introduced in Lipschitz form and from these are obtained rapid decay of the functions involved. It is interesting from our point of view because there are no sign conditions on any of the functions. Everything is controlled by the Lipschitz conditions and the decay of functions. We wish to show that we can avoid the measures of noncompactness and one of the Lipschitz conditions (but not the implied decay), and use Theorem 2.2 to obtain some elementary solutions. Indeed, the point of our results is that they are elementary and, in fact, fairly close to Brouwer’s theorem. The idea is to state the conditions in [9], denoted by ðhi Þ, and then state the consequences which we will use and denote these as ðci Þ. ðh1 Þ g : Rþ  R ! R is continuous, gðt; 0Þ is bounded, and g  ¼ suptP0 jgðt; 0Þj. Also, there is a continuous function ‘ðtÞ with

jgðt; xÞ  gðt; yÞj 6 ‘ðtÞjx  yj for x; y 2 R and t P 0. ðc1 Þ We will use ðh1 Þ, but will not ask for g  (see ðc5 Þ). ðh2 Þ f : Rþ  R ! R is continuous and there is a continuous function m : Rþ ! Rþ such that

jf ðt; xÞ  f ðt; yÞj 6 mðtÞjx  yj for x; y 2 R and t P 0. ðc2 Þ We will use ðh2 Þ, but will not ask for mðtÞnðtÞtq ! 0 as t ! 1. ðh3 Þ v : Rþ  Rþ  R ! R is continuous. There exists a continuous functions n : Rþ ! Rþ and a continuous nondecreasing function U : Rþ ! Rþ with Uð0Þ ¼ 0 so that for all t; s 2 Rþ with t P s we have

jv ðt; s; xÞ  v ðt; s; yÞj 6 nðtÞUðjx  yjÞ: Also, there is a function ðc3 Þ We will use

v  ðtÞ ¼ maxfjv ðt; s; 0Þj : 0 6 s 6 tg.

jv ðt; s; xÞj 6 v  ðtÞ þ nðtÞUðjxjÞ: Now, everything is gathered. ðh4 Þ The functions /; w; n; g : Rþ ! Rþ defined by

/ðtÞ ¼ mðtÞnðtÞtq wðtÞ ¼ mðtÞv  ðtÞtq nðtÞ ¼ nðtÞjf ðt; 0Þjt q

gðtÞ ¼ v  ðtÞjf ðt; 0Þjtq are all bounded on Rþ and limt!1 /ðtÞ ¼ limt!1 nðtÞ ¼ 0. ðc4 Þ We will use the notation of ðh4 Þ, but will not ask boundedness or limit conditions of these functions. ðh5 Þ There exists a positive solution r0 of the inequality

ð‘ r þ g  ÞCðq þ 1Þ þ ½/ rUðrÞ þ w r þ n UðrÞ þ g  6 r Cðq þ 1Þ and ‘ Cðq þ 1Þ þ / Uðr 0 Þ þ w < Cðq þ 1Þ, where ‘ ¼ supf‘ðtÞ : t P 0g; / ¼ supf/ðtÞ : t P 0g; w ¼ supfwðtÞ : t P 0g; n ¼ supfnðtÞ : t P 0g and g ¼ supfgðtÞ : t P 0g. ðc5 Þ We will improve ðh5 Þ by asking that there exists a continuous function r : ½0; 1Þ ! ð0; 1Þ and a b < 1 such that for all tP0

‘ðtÞ þ ½wðtÞ þ /ðtÞUðr  ðtÞÞ

1

Cðq þ 1Þ

6b

and

cðtÞ þ ‘ðtÞr þ ½wðtÞr þ /ðtÞrUðr Þ þ nðtÞUðr Þ

1

Cðq þ 1Þ

6r

where r ¼ rðtÞ; r  ðtÞ ¼ sup06s6t rðsÞ; cðtÞ ¼ jgðt; 0Þj þ gðtÞ=Cðq þ 1Þ. Under the assumptions ðh1 Þ–ðh5 Þ it is shown in [9, p. 77] that there is at least one solution of the integral equation; moreover, if there are other solutions, then they converge to the given solution. This is proved using the full conditions ðh1 Þ—ðh5 Þ and methods of measures of noncompactness. Our purpose here is to show that with our theorems there is an elementary proof using only ðc1 Þ—ðc5 Þ to show that there is a solution x of (15) with jxðtÞj 6 rðtÞ without the Lipschitz condition on v or the limit conditions in ðh4 Þ.

348

T.A. Burton, B. Zhang / Applied Mathematics and Computation 250 (2015) 339–351

Theorem 4.1. Let ðc1 Þ—ðc5 Þ hold. Then the integral equation (15) has a solution x with jxðtÞj 6 rðtÞ for all t P 0. Proof. As in [9, p. 79], the stated conditions are sufficient to show that if

M ¼ f/ 2 BC : j/ðtÞj 6 rðtÞg; then the natural mapping defined by the integral equation maps M into itself. The proof of Theorems 2.1 and 2.2 will establish continuity of the map. The appendix will show the required equicontinuity and other technical details. h

5. A neutral delay equation 5.1. Neutral equations A large body of literature can be found concerning applications of neutral differential equations by simply putting ‘‘epidemics and neutral differential equations’’ into a search engine. The basic and heuristic idea of a neutral equation is that the rate of change, x0 ðtÞ, is influenced not only by a ‘‘position’’ of x in space and time, together with forces acting on x as we would deduce from Newton’s Second Law of Motion, but it is also influenced by the recent rate of change of x. Epidemics and other problems in mathematical biology are widely studied by neutral differential equations. See, for example, Gopalsamy [7], Gopalsamy and Zhang [8], Kuang [13–15]. Investigators have given heuristic arguments to support their use in describing biological phenomena and much of this is formalized in the final chapter of each of the books by Gopalsamy [7] and Kuang [13]. There are many other problems treated by neutral differential equations and our main contribution here is a quick sketch of the way to express a problem of the form

x0 ðtÞ ¼

d f ðxðt  hÞÞ  uðt; xðtÞ; xðt  hÞÞ; dt

xðtÞ ¼ wðtÞ;

h 6 t 6 0;

as a problem readily attacked by the extension of Theorem 3.1 to ½0; 1Þ. Here, h is a positive constant, u is continuous and bounded for x bounded, and f satisfies a contraction condition. The function w is a given continuous initial function and we want a solution for 0 6 t < 1. If we were to simply integrate that equation to obtain an integral equation defining a mapping, then almost everything in our theory would fail. Instead we employ a form of a ‘‘linearization trick’’. Let J be a positive constant to be determined, subtract and add JxðtÞ to obtain

x0 ðtÞ ¼ JxðtÞ þ JxðtÞ þ

d f ðxðt  hÞÞ  uðt; xðtÞ; xðt  hÞÞ: dt

Take all the terms except x0 ðtÞ ¼ JxðtÞ as an inhomogeneous term and use the variation of parameters formula to write

xðtÞ ¼ wð0ÞeJt þ

Z

t

eJðtsÞ ½JxðsÞ þ

0

d f ðxðs  hÞÞ  uðs; xðsÞ; xðs  hÞÞds: ds

Integration by parts of the derivative term in the integral yields

xðtÞ ¼ wð0ÞeJt þ f ðxðt  hÞÞ  eJt f ðwðhÞÞ þ

Z

t

eJðtsÞ ½JxðsÞ  Jf ðxðs  hÞÞ  uðs; xðsÞ; xðs  hÞds:

0

The positive constant, J, can be chosen at will to facilitate the construction of a self-mapping set M of the type required for our theory. This equation defines a mapping which is well-suited to Theorem 3.1 and its extension to ½0; 1Þ. We do not need to satisfy the difficult condition (i) of Krasnoselskii’s theorem. Details for a self-mapping set can be found in [4]. The details are quite lengthy and will not be repeated here. Appendix A Theorem 6.1. Let u : ½0; 1Þ  ½0; 1Þ  R ! R be continuous, and let R : ð0; 1Þ ! ½0; 1Þ be continuous, decreasing, and Rðt  sÞ 6 Dðt  sÞq1 with 0 < q < 1 and D > 0. Then there is a continuous increasing function H so that if t; s P 0, if x 2 BC with juðt; s; xðsÞÞj 6 K and

juðt; s; xðsÞÞ  uðs; s; xðsÞÞj 6 Jjt  sjq ; then

Z t Z s L :¼ Rðt  sÞuðt; s; xðsÞÞds  Rðs  sÞuðs; s; xðsÞÞds 6 HðtÞjt  sjq ; 0

where HðtÞ ¼ 2KD=q þ J

0

Rt 0

RðsÞds.

ð16Þ

349

T.A. Burton, B. Zhang / Applied Mathematics and Computation 250 (2015) 339–351

Proof. Note that since RðtÞ is decreasing and there is a constant D with 0 6 RðtÞ 6 Dt q1 we have, for 0 6 s 6 t, that

Z t Z s jRðt  sÞ  Rðs  sÞjjuðt; s; xðsÞÞjds þ jRðt  sÞjjuðt; s; xðsÞÞjds þ jRðs  sÞjjuðt; s; xðsÞÞ  uðs; s; xðsÞÞjds 0 s 0 Z s Z t Z s 6 K½Rðs  sÞ  Rðt  sÞds þ K Rðt  sÞds þ jRðs  sÞjdsJjt  sjq 0 s 0 Z s Z s Z t Z s ¼K Rðs  sÞds  K Rðt  sÞds þ K Rðt  sÞds þ RðsÞdsJjt  sjq 0 0 s 0 Z s Z t Z t Z s ¼K RðsÞds  K RðsÞds þ K Rðt  sÞds þ RðsÞdsJjt  sjq 0 ts s 0 Z s Z t Z t Z s ¼K RðsÞds  K RðsÞds þ 2K Rðt  sÞds þ RðsÞdsJjt  sjq 0 0 s 0 Z t Z s the sum of the first two terms is negative 6 2DK ðt  sÞq1 ds þ RðsÞdsJjt  sjq s 0   Z s Z s ¼ 2ðKD=qÞðt  sÞq jts þ RðsÞdsJjt  sjq ¼ 2KD=q þ J RðsÞds jt  sjq 0 0   Z t 6 2KD=q þ J RðsÞds jt  sjq ¼: HðtÞjt  sjq :

L6

Z

s

0

The same equality holds if 0 6 t 6 s. This completes the proof.

h

Theorem 6.2. Let (i)–(iii) hold with b ¼ 1 and T ¼ 1, and let

M ¼ f/ 2 BC : k/k 6 Kg for a constant K > 0. Suppose that (v) suptP0 jVðt; xÞ  Vðt; yÞj < jx  yj for all x; y 2 ½K; K with x: ¼ y. e Þ1 is continuous on ðI  V e ÞðMÞ, where ð V e /ÞðtÞ ¼ Vðt; /ðtÞÞ for / 2 M. Then ðI  V

1

1

e Þ is one to one, and hence the inverse ðI  V e Þ exists. We now show that ðI  V e Þ is Proof. Since (v) holds, we see that ðI  V e ÞðMÞ. To this end, let fy g be a sequence in ðI  V e ÞðMÞ with ky  y k ! 0 as n ! 1 for a function continuous on ðI  V n n e ÞðMÞ. We need to show that y 2 ðI  V 1

1

e Þ y ! ðI  V e Þ y as n ! 1: ðI  V n 1

1

e Þ y and x ¼ ðI  V e Þ y . Then ðI  V e Þxn ¼ y and ðI  V e Þx ¼ y . Suppose that xn 9x . Then there exists an Set xn ¼ ðI  V n n e0 > 0 and a subsequence fxnk g of fxn g such that kxnk  x k P e0 for all k ¼ 1; 2; . . .. Now choose tk 2 ½0; 1Þ with

jxnk ðtk Þ  x ðt k Þj P e0 =2: Observe that the function suptP0 jVðt; xÞ  Vðt; yÞj is continuous on the compact set X ¼ ½K; K  ½K; K. By (v), we have



sup jxyjPe0 =2

suptP0 jVðt; xÞ  Vðt; yÞj : x; y 2 ½K; K ¼ d < 1: jx  yj

and therefore

jVðtk ; xnk ðt k ÞÞ  Vðtk ; x ðt k ÞÞj 6 djxnk ðtk Þ  x ðt k Þj: We now have

e xn Þðt k Þ  ðx  V e x Þðt k Þj P jxn ðtk Þ  x ðt k Þj  jð V e xn Þðt k Þ  ð V e x Þðtk Þj jynk ðt k Þ  y ðtk Þj ¼ jðxn  V k k P jxnk ðtk Þ  x ðt k Þj  djxnk ðt k Þ  x ðt k Þj ¼ ð1  dÞjxnk ðtk Þ  x ðt k Þj P ð1  dÞe0 =2 > 0: This yields

ð1  dÞe0 =2 6 jynk ðt k Þ  y ðt k Þj 6 kynk  y k ! 0 as n ! 1 e Þ1 is continuous on ðI  V e ÞðMÞ. This completes the proof. h a contradiction. So we obtain xn ! x as n ! 1, and thus ðI  V Proof of Theorem 4.1. Let BC be the Banach space of bounded continuous functions / : ½0; 1Þ ! R with the supremum norm k  k. To simplify notations, we set

350

T.A. Burton, B. Zhang / Applied Mathematics and Computation 250 (2015) 339–351

g  ðt; sÞ ¼ sup jgðt; yÞ  gðs; yÞj; jyj6rðsÞ 

f ðt; sÞ ¼ sup jf ðt; yÞ  f ðs; yÞj jyj6rðsÞ

and

Iðt; xðÞÞ ¼

1 CðqÞ

Z

t

v ðt; s; xðsÞÞ ðt  sÞ1q

0

ds:

We define

M ¼ fx 2 BC : jxðtÞj 6 rðtÞ; 8t P 0g where rðtÞ is given in (c5). Now let P be the natural mapping defined by the integral equation (15). We will show that P maps M into itself. To see this, letting x 2 M, we have

jðPxÞðtÞj ¼ jgðt; xðtÞÞ þ f ðt; xðtÞÞIðt; xðÞÞj 6 jgðt; xðtÞÞ  gðt; 0Þj þ jgðt; 0Þj þ jf ðt; xðtÞÞ  f ðt; 0ÞjjIðt; xðÞÞj þ jf ðt; 0ÞjjIðt; xðÞÞj 6 ‘ðtÞjxðtÞj þ jgðt; 0Þj þ mðtÞjxðtÞjjIðt; xðÞÞj þ jf ðt; 0ÞjjIðt; xðÞÞj ðuse ð16Þ belowÞ 6 ‘ðtÞrðtÞ þ jgðt; 0Þj þ mðtÞrðtÞ½v  ðtÞ þ nðtÞUðr  ðtÞÞt q

1

Cðq þ 1Þ

¼ cðtÞ þ ‘ðtÞrðtÞ þ ½wðtÞrðtÞ þ /ðtÞrðtÞUðr  ðtÞÞ þ nðtÞUðr  ðtÞÞ

þ jf ðt; 0Þj½v  ðtÞ þ nðtÞUðr ðtÞÞt q 1

Cðq þ 1Þ

1

Cðq þ 1Þ

6 rðtÞ

by applying ðc1 Þ—ðc3 Þ and ðc5 Þ. Next, we define M  by

M ¼ fx 2 M : jxðtÞ  xðsÞj 6 mðt; sÞ; 8t; s P 0g where mðt; sÞ is a continuous function to be defined below with mðt; sÞ ! 0 as jt  sj ! 0. We want to show that jðPxÞðtÞ  ðPxÞðsÞj 6 mðt; sÞ for / 2 M. To this end, we proceed to estimate the terms on the right-hand side of (15) for x 2 M. By (c3), we have

jIðt; xðÞÞj 6

1 CðqÞ

Z

0

t

jv ðt; s; xðsÞÞj ðt  sÞ

1q

6 ½v  ðtÞ þ nðtÞUðr  ðtÞÞ

ds 6 1

1 CðqÞ Z t

CðqÞ

0

Z

v  ðtÞ þ nðtÞUðjxðsÞjÞ

t

ðt  sÞ1q

0

1 1q

ðt  sÞ

ds

ds ¼ ½v  ðtÞ þ nðtÞUðr  ðtÞÞtq

1

Cðq þ 1Þ

¼: J  ðtÞ

ð16Þ

and

jf ðt; xðtÞÞj 6 jf ðt; xðtÞÞ  f ðt; 0Þj þ jf ðt; 0Þj 6 mðtÞjxðtÞj þ jf ðt; 0Þj 6 mðtÞrðtÞ þ jf ðt; 0Þj ¼: ^f ðtÞ:

ð17Þ

From (c1), we have

jgðt; xðtÞÞ  gðs; xðsÞÞj 6 jgðt; xðtÞÞ  gðt; xðsÞÞj þ jgðt; xðsÞÞ  gðs; xðsÞÞj 6 ‘ðtÞjxðtÞ  xðsÞj þ g  ðt; sÞ:

ð18Þ

Apply (c2) and the estimate in (16) to obtain

jf ðt; xðtÞÞ  f ðs; xðsÞÞj jIðt; xðÞÞj 6 jf ðt; xðtÞÞ  f ðt; xðsÞÞj jIðt; xðÞÞj þ jf ðt; xðsÞÞ  f ðs; xðsÞÞj jIðt; xðÞÞj 

6 mðtÞjxðtÞ  xðsÞjJ ðtÞ þ f ðt; sÞJ  ðtÞ ¼ ½wðtÞ þ /ðtÞUðr  ðtÞÞ

1



Cðq þ 1Þ

jxðtÞ  xðsÞj þ f ðt; sÞJ  ðtÞ:

ð19Þ

An argument similar to that in the proof of Theorem 6.1 yields, for 0 6 s 6 t, that

jIðt; xðÞÞ  Iðs; xðÞÞj 6 ½v  ðtÞ þ nðtÞUðr  ðtÞÞ

2

Cðq þ 1Þ

jt  sjq þ

1

Cðq þ 1Þ

tq v  ðt; sÞ ¼: I ðt; sÞ

where v  ðt; sÞ ¼ supjyj6rðsÞ jv ðt; s; yÞ  v ðs; s; yÞj for 0 6 s 6 s 6 t. Taking into account (17), we get

jIðt; xðÞÞ  Iðs; xðÞÞjjf ðs; xðsÞÞj 6 I ðt; sÞ^f ðsÞ:

ð20Þ

Combine (18)–(20) to obtain

jðPxÞðtÞ  ðPxÞðsÞj 6 ½‘ðtÞCðq þ 1Þ þ wðtÞ þ /ðtÞUðr  ðtÞÞ

1

Cðq þ 1Þ

 jxðtÞ  xðsÞj þ g  ðt; sÞ þ f ðt; sÞJ  ðtÞ þ I ðt; sÞ^f ðsÞ

ðassigning the last three terms as ð1  bÞmðt; sÞÞ 6 bjxðtÞ  xðsÞj þ ð1  bÞmðt; sÞ 6 bmðt; sÞ þ ð1  bÞmðt; sÞ ¼ mðt; sÞ;

T.A. Burton, B. Zhang / Applied Mathematics and Computation 250 (2015) 339–351

351

where b is given in (c5). Thus, P : M  ! M  . The rest of the proof follows that of Theorem 2.2 so P has a fixed point in M which is a solution of (15). The proof is complete. h References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

C.S. Barroso, Krasnoselskii’s fixed point theorem for weakly continuous maps, Nonlinear Anal.: TMA 55 (2003) 25–31. T.A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover, Mineola, NY, 2006. T.A. Burton, A fixed point theorem of Krasnoselskii, Appl. Math. Lett. 11 (1998) 85–88. T.A. Burton, I.K. Purnaras, A unification theory of Krasnoselskii for differential equations, Nonlinear Anal. 89 (2013) 121–133. T.A. Burton, Bo Zhang, Fractional equations and generalizations of Schaefer’s and Krasnoselskii’s fixed point theorems, Nonlinear Anal.: TMA 75 (2012) 6485–6495. T.A. Burton, Bo Zhang, Fixed points and fractional differential equations: examples, Fixed Point Theory 14 (2) (2013) 313–326. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer, Dordrecht, 1992. K. Gopalsamy, B.G. Zhang, On a neutral delay logistic equation, Dyn. Stab. Syst. 2 (1988) 183–195. Mohammed Abdalla Darwish, Johnny Henderson, Existence and asymptotic stability of solutions of a perturbed quadratic fractional integral equation, Fract. Calc. Appl. Anal. 12 (1) (2009) 71–86. Jesus Garcia-Falset, Existence of fixed points for the sum of two operators, Math. Nachr. 283 (2010) 1736–1757. J. Garcia-Falset, K. Latrach, Krasnoselskii-type fixed-point theorems for weakly sequentially continuous mappings, Bull. London Math. Soc. 44 (1) (2012) 25–38. M.A. Krasnoselskii, Am. Math. Soc. Transl. 10 (2) (1958) 345–409. Y. Kuang, Delay Differential Equations with Applications to Population Dynamics, Academic Press, Boston, 1993. Y. Kuang, Global stability in one or two species neutral delay population models, Can. Appl. Math. Q. 1 (1993) 23–45. Y. Kuang, On neutral delay logistic Gause-type predator-prey systems, Dyn. Stab. Syst. 6 (1991) 173–189. Sehie Park, Generalizations of the Krasnoselskii fixed point theorem, Nonlinear Anal.: TMA 67 (2007) 3401–3410. D.R. Smart, Fixed Point Theorems, Cambridge Univ, Press, 1980.