Computation of fixed point index and applications to superlinear periodic problem

Transcription

1 Wang and Li Fixed Point Theory and Applications 5 5:57 DOI.86/s R E S E A R C H Open Access Computation of fixed point index and applications to superlinear periodic problem Feng Wang,* and Shengjun Li 3 * Correspondence: fengwang88@63.com School of Mathematics and Physics, Changzhou University, Changzhou, 364, China College of Science, Hohai University, Nanjing, 98, China Full list of author information is available at the end of the article Abstract In this paper we compute the fixed point index for A-proper semilinear operators under certain boundary conditions. The proof is based on the partial order method combined with the properties of the fixed point index. As an application, we use the abstract results presented above to study the existence conditions of positive solutions for superlinear first-order and second-order periodic problems. MSC: 34B; 34B5 Keywords: computation; fixed point index; partial order method; superlinear periodic problem Introduction The topological degree and fixed point index are important theories in nonlinear functional analysis as they have had significant applications as regards obtaining results on theexistenceandthenumberofsolutionsfordifferential equations [], differential inclusions [] and dynamical systems [3, 4]. In recent years, many authors have focused on the computation ofthe topologicaldegree and fixed point index see [5 ] and the references therein. It is noticed that the results cited above apply to an operator equation of the form x Ax, which is closely connected to fixed point theory. However, in this paper we will be mainly interested in studying a more general operator equation of the form Lx Nx,. where L is not necessarily invertible. During the last three decades, the existence problem for. has been an interesting topic and has attracted the attention of many researchers [, 3 6] because it has especially broad applications in the existence of periodic solutions of nonlinear differential equations. Based on the approach of Fitzpatrick and Petryshyn [7], the conceptof a fixed point index for A-propermapsrelatedto.hasbeen introducedby Cremins [8, 9]. Since then, some existence results for the semilinear equation. in cones have been established in [, ] using the properties of the fixed point index and partial order method. Recently, the computation of the fixed point index for A-proper semilinear operators was proved in [] under the sublinear case, but to the best of the knowledge of the authors nothing has been published concerning the computation results available in the literature for the 5 Wang and Li. This article is distributed under the terms of the Creative Commons Attribution 4. International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Wang and Li Fixed Point Theory and Applications 5 5:57 Page of 4 superlinear case. In this paper, we will continue this study and focus on the computation of the fixed point index with certain boundary conditions in the superlinear case. The remaining part of the paper is organized as follows. Some preliminaries and a number of lemmas useful to the derivation of the main results are given in Section. InSection 3, we obtain some sufficient conditions that the fixed point index equals {} or {}, values more easily applicable. As an application, in Section 4, we use the new results presented in Section 3 to study the existence conditions of positive solutions for first-order and second-order superlinear periodic boundary value problems. Preliminaries In this section, we recall some standard facts on A-proper mappings and Fredholm operators. Let X and Y be Banach spaces with the zero element θ, D alinearsubspaceofx, {X n } D, and{y n } Y sequences of oriented finite dimensional subspaces such that Q n y y in Y for every y and distx, X n foreveryx D where Q n : Y Y n and P n : X X n are sequences of continuous linear projections. The projection scheme Ɣ {X n, Y n, P n, Q n } is then said to be admissible for maps from D X to Y.AmapT : D X Y is called approximation-proper abbreviated A-proper at a point y Y with respect to Ɣ if T n Q n T D Xn is continuous for each n N and whenever {x nj x nj D X nj } is bounded with T nj x nj y, then there exists a subsequence {x njk } such that x njk x D, and Tx y. T is said to be A-proper on a set D if it is A-proper at all points of Y. L : dom L X Y is a Fredholm operator of index zero if Im L is closed and dim Ker L codim Im L <.ThenX and Y may be expressed as direct sums X X X, Y Y Y with continuous linear projections P : X Ker L X and Q : Y Y.Therestriction of L to dom L X,denotedL,isabijectionontoIm L Y with continuous inverse L : Y dom L X.SinceX and Y have the same finite dimension, there exists a continuous bijection J : Y X. Cremins [8, 9] defined a fixed point index [L, N], for A-proper maps acting on cones, which has the usual properties of the classical fixed point index, that is, existence, normalization, additivity, and homotopy invariance. Let K be a cone in the Banach space X, thenx becomes a partial ordered Banach space under the partial ordering which is induced by K. K is said to be normal if there exists a positive constant Ñ such that θ x y implies x Ñ y. For the concepts and the properties as regards the cone we refer to [3 5]. Here we remark that the results in [8, 9, ] hold in partial ordered Banach spaces. Let X be open and bounded such that K K.Ifwe let K L + J PK dom L, then K is a cone in Y and the linear operator L + J P is inversely positive by using [8], Proposition. Throughout this paper we assume that the following conditions are satisfied: A L : dom L Y is Fredholm of index zero. A L λn is A-proper for λ [, ]. A 3 N is bounded and P + JQN + L I QN maps K to K. Lemma. [6], Lemma a If L is compact, then L λn isa-properforeachλ [, ]. To obtain some new methods of computing the fixed point index for the A-proper semilinear operator., we need the following two lemmas.

3 Wang and Li Fixed Point Theory and Applications 5 5:57 Page 3 of 4 Lemma. [9] Let θ X. If Lx μnx μj Px for all x K and μ [, ], then [L, N], {}. Lemma.3 [] If there exists e K \{θ} such that Lx Nx μe, for all x K and all μ, then [L, N], {}. 3 Main results Theorem 3. Let θ XandLx Nx for all x K. Assume that the following hypotheses hold: i there exists a positive bounded linear operator B : X X, such that N + J P x L + J P Bx, for all x K, where this partial order is induced by the cone K in Y. ii rb, where rb is the spectral radius of B. Then the fixed point index [L, N], K {}. Proof We show that Lx μnx μj Px, x K, μ [, ]. 3. Suppose the assertion of 3. is false. Then there exist x K and μ [, ] such that Lx μ Nx μ J Px.SinceLx Nx x K,weseethatμ [,. This and condition i imply that L + J P x μ N + J P x μ L + J P Bx. Applying L + J P to the above inequality, we obtain x μ Bx.Continuingthisprocess, we get x μ n Bn x. 3. Let {y y x }.Equation3. yields{μ n Bn x n,,...}. x K and θ K imply d dθ, >.Consequently,wegetby3. B n B n x d x μ n x, forn,,...

4 Wang and Li Fixed Point Theory and Applications 5 5:57 Page 4 of 4 This shows n rb lim Bn >. n μ This contradicts the condition ii. Hence 3.is trueand we see fromlemma. that the conclusion is true. Theorem 3. Let K be a normal cone in X. If there exist a constant C >and u K \{θ}, such that N + J P x C L + J P x u, for all x K, 3.3 where this partial order is induced by the cone K in Y, then there exists R >,and for R > R, the fixed point index [L, N], BR K {}, where B R {x X : x X < R}. Proof Setting W {x K Lx Nx λu,foreachλ }, we claim that W is bounded. For x W,thereexistsλ suchthatlx Nx λu.this and assumption 3.3implythat L + J P x N + J P x + λu N + J P x C L + J P x u. Operating on both sides of the latter inequality by L + J P,weobtain x Cx L + J P u. This shows that x C L + J P u. In view of the normality of K,thereexistsanÑ > such that x Ñ L + J P u, x W. C This shows that W is bounded. Let R sup x W x.forr > R,wehave Lx Nx λu, x B R K, λ. 3.4 Using Lemma.3,weinferby3.4 that the conclusion is true. Now the following two corollaries are immediate consequences of Theorems 3. and 3. and the facts that L + J P N + J PP + JQN + L I QN see [8], Lemma and we have linearity and positivity of the operator L + J P see [8], Proposition.

5 Wang and Li Fixed Point Theory and Applications 5 5:57 Page 5 of 4 Corollary 3.3 Let θ K and Lx Nx for all x K. If there exists a positive bounded linear operator B : X XwithrB, such that P + JQN + L I QN x Bx, for any x K, where this partial order is induced by the cone K in X, then the fixed point index [L, N], K {}. Corollary 3.4 Let K be a normal cone in X. If there exist a constant C >and u K\{θ}, such that P + JQN + L I QN x Cx u, for any x K, where this partial order is induced by the cone K in X, then there exists R >,and for R > R, the fixed point index [L, N], BR K {}, where B R {x X : x X < R}. Using Corollaries 3.3 and 3.4, we complete this section with a proof of the following importantresulttobeusedlater. Theorem 3.5 Let K be a normal cone in X, u K\{θ} and let there be a constant C >. If there exists a positive bounded linear operator B : X XwithrB, such that the following hypotheses hold: i P + JQN + L I QNx Cx u, x K, ii P + JQN + L I QNx Bx, x B r K, where B r {x X : x < r}, then there exists x dom L K \{θ} such that Lx Nx. Proof It follows from Corollary 3.4 and condition i that there exists R >withr > r such that [L, N], BR {}. 3.5 We assume Lx Nx on B r K dom L, otherwise the conclusion follows. Using Corollary 3.3 we get from condition ii [L, N], Br {}. 3.6 In view of 3.5, 3.6, and the additivity property, we obtain [L, N], BR \B r indk [L, N], BR indk [L, N], Br {} {} { } {}, which completes the proof from the existence property.

6 Wang and Li Fixed Point Theory and Applications 5 5:57 Page 6 of 4 4 Applications to superlinear periodic boundary value problem 4. First-order periodic boundary value problem We shall apply Theorem 3.5 to obtain positive solutions to the following first-order periodic boundary value problem PBVP: { x tft, xt, t,, 4. x x, where f :[,] R + R is a continuous function. Let X Y C[, ] with the usual norm x max t [,] xt. Define the linear operator L : dom L X Y,Lxtx t, t [, ], where dom L { x X : x C[, ], x x } and N : X Y with Nxtf t, xt, t [, ]. It is easy to check that Ker L { x dom L : xt c on [, ], c R }, { } Im L y Y : ys ds, dim Ker L codim Im L, so that L isafredholmoperatorofindexzero. Next, define the projections P : X X by Px xs ds, and Q : Y Y by Qy ys ds. Furthermore, we define the isomorphism J : Im Q Im P as Jy βy with β.wemay easy verify that the inverse operator L : Im L dom L Ker P of L dom L Ker P : dom L Ker P Im L is L yt Kt, sys ds,where { s +, s < t, Kt, s s, t s. For notational convenience, we set Gt, s+kt, s Kt, s ds.wecanverifythat Gt, s { 3 t s, s < t, +s t, t s, and Gt, s 3, t, s [, ].

7 Wang and Li Fixed Point Theory and Applications 5 5:57 Page 7 of 4 Define the cone K in X by K {x X : xt, xt 3 } x, t [, ], then K is a normal cone of X see [5]. Lemma 4. If H f t, x x, for all t [, ], x, 3 then P + JQN + L I QN is a positive operator, that is, P + JQN + L I QN K K. Proof It follows from condition H that Px + JQNx + L + I QNx xs ds + f s, xs ds Kt, s f s, xs xs ds + Gt, sf s, xs ds 3 Gt, s xs ds, f s, xs ds ds for all x K.ThusP + JQN + L I QNx. Now we are ready to prove P + JQNxt+L I QNxt P + JQNx + L 3 I QNx. Using condition H, we obtain P + JQNx + L [ max t [,] [ max t [,] 5 6 I QNx xs ds + Gt, sf s, xs ] ds 3 Gt, s xs ds + xs ds + 3 f s, xs + 3 xs ds. From the last inequality, we have from condition H Gt, s f s, xs + 3 ] xs ds min t [,] [ P + JQNxt+L I QNxt] min t [,] [ xs ds + Gt, sf s, xs ] ds

8 Wang and Li Fixed Point Theory and Applications 5 5:57 Page 8 of 4 min t [,] [ 3 Gt, s xs ds + xs ds + f s, xs + 3 xs ds [ 3 xs ds + 3 f s, xs + 3 ] 3 xs ds [ 5 xs ds + 3 f s, xs + 3 ] 3 6 xs ds P + JQNx + L 3 I QNx. Gt, s f s, xs + 3 ] xs ds Therefore, P + JQNx + L I QNx K. We can now state and prove our result on the existence of a positive solution for the PBVP 4.. Theorem 4. Suppose that condition H is satisfied. If f t,x H lim inf x + min t [,] >4, x f t,x H 3 lim sup x + max t [,] <, x then the PBVP 4. has at least one positive solution. Proof First, we note that L, as defined, is Fredholm of index zero, L is compact by Arzela- Ascoli theorem and thus L λn is A-proper for λ [, ] by Lemma.. Condition H guarantees that there exist ɛ >andτ >suchthat f t, x 4 + ɛx, for all t [, ], x τ. Hence, f t, x 4 + ɛx M, for all t [, ], x, 4. where M max t, x τ f t, x 4+ɛx +. Set C + ε 6, u 3M.ThenC >,u K\{θ}.Using4., we obtain Px + JQNx + L xs ds + I QNx xs ds +4+ɛ Gt, sf s, xs ds Gt, sxs ds M xs ds ɛ xs ds 3M 3+ ɛ xs ds 3M 3+ ɛ 3 x 3M Gt, s ds

9 Wang and Li Fixed Point Theory and Applications 5 5:57 Page 9 of 4 + ɛ x 3M 6 Cx u. This implies that condition i of Theorem 3.5 is satisfied. It follows from condition H 3 that there exist σ, and r, τsuchthat f t, x σ x, for all t [, ], x r. 4.3 Take Bx σ xs ds. OnecanseethatB : X X is a positive bounded linear operator. It is clear that rb σ <.Thus,by4.3, we have Px + JQNx + L xs ds + I QNx xs ds σ xs ds σ σ Gt, sf s, xs ds Gt, sxs ds xs ds xs ds Bx, for all x K with x r. 4.4 This means that condition ii of Theorem 3.5 is verified. We see that all assumptions of Theorem 3.5 are satisfied. The proof is finished. 4. Second-order periodic boundary value problem We will discuss the existence of positive solutions of the second-order periodic boundary value problem PBVP { x tft, xt, t,, 4.5 x x, x x, where f :[,] R + R is a continuous function. Since some parts of the proof are in the same line as that of Theorem 4.,wewilloutline the proof with the emphasis on the difference. Let Banach spaces X, Y be as in Section 4.. In this case, we may define dom L { x X : x C[, ], x x, x x }, and let the linear operator L : dom L X Y be defined by Lxt x t, for x dom L, t [, ]. Then L is Fredholm of index zero, Ker L { x dom L : xt c on [, ], c R },

10 Wang and Li Fixed Point Theory and Applications 5 5:57 Page of 4 and { Im L y Y : } ys ds. Define N : X Y by Nxtf t, xt, t [, ]. Thus it is clear that the PBVP 4.5 is equivalent to the operator equation.. We use the same projections P, Q as in Section 4. and define the isomorphism J : Im Q Im P as Jy βy with β.itiseasytoverifythattheinverseoperatorl : Im L dom L Ker P of L dom L Ker P : dom L Ker P Im L is where L y t t, sys ds, Set t, s { s t + s, s < t, st s, t s. Ht, s 6 + t, s t, s ds. We can verify that and Ht, s { 4 + s t + s+ t t, s < t, 4 + st s+ t + t, t s, 8 Ht, s, t, s [, ]. 4 Define a normal cone K in X by K {x X : xt, xt 5 } x, t [, ]. Lemma 4.3 If H 4 f t, x 3x, for all t [, ], x, then P + JQN + L I QN is a positive operator, that is, P + JQN + L I QN K K.

11 Wang and Li Fixed Point Theory and Applications 5 5:57 Page of 4 Proof For each x K, by condition H 4 P + JQNx + L + xs ds + 6 I QNx f s, xs ds t, s f s, xs xs ds + Ht, sf s, xs ds 3Ht, s xs ds. f s, xs ds ds Thus P + JQN + L I QNx. We now show that P + JQNxt+L I QNxt P + JQNx + L 5 I QNx. In fact, we get from condition H 4 P + JQNx + L I QNx [ max xs ds + t [,] max t [,] 5 8 Ht, sf s, xs ] ds [ 3Ht, s xs ds + xs ds + 4 f s, xs +3xs ds. By the above inequality, we have from condition H 4 Ht, s f s, xs +3xs ] ds min t [,] [ P + JQNxt+L I QNxt] min t [,] min t [,] [ xs ds + [ 3Ht, s xs ds + Ht, sf s, xs ] ds Ht, s f s, xs +3xs ] ds xs ds + f s, xs +3xs ds 4 8 [ 5 xs ds + 5 ] f s, xs +3xs ds [ 5 xs ds + ] f s, xs +3xs ds P + JQNx + L 5 I QNx. Thus P + JQNx + L I QNx K.

12 Wang and Li Fixed Point Theory and Applications 5 5:57 Page of 4 Theorem 4.4 Suppose that condition H 4 is satisfied. If f t,x H 5 lim inf x + min t [,] >, x f t,x H 6 lim sup x + max t [,] <, x then the PBVP 4.5 has at least one positive solution. Proof It is again easy to show that L λn is A-proper for λ [, ] by Lemma.. From condition H 5, we know that there exist ɛ >andτ > Thus f t, x + ɛx, for all t [, ], x τ. f t, x + ɛx M, for all t [, ], x, 4.6 where M max t, x τ f t, x +ɛx +. From 4.6, we have Px + JQNx + L xs ds + I QNx xs ds ++ɛ Ht, sf s, xs ds Ht, sxs ds M xs ds ɛ xs ds M ɛ xs ds M ɛ 8 + ɛ 5 x M 4 x M 4, Ht, s ds and condition i of Theorem 3.5 is satisfied with C + ε and u M 4. By condition H 6, we can find σ, 8 and r, τsuchthat f t, x σ x, for all t [, ], x r. 4.7 If we take Bx σ 8 xs ds,thenb : X X is a positive bounded linear operator and rb σ <. Finally, similar to the proof of 4.4, it follows from 4.7that 8 Px + JQNx + L I QNx Bx, for all x K, x r. Thus condition ii of Theorem 3.5is satisfied and theproofis complete. Example 4.5 Let the nonlinearity in 4.5be f t, xatx α kx,

13 Wang and Li Fixed Point Theory and Applications 5 5:57 Page 3 of 4 where α >,a C[, ] is positive -periodic function and k, 3. Then 4.5 has at least one positive -periodic solution. Proof We will apply Theorem 4.4 with f t, xatx α kx.sincek, 3, it is easy to see that H 4 holds. f t,x One may easily see that lim inf x + min t [,] lim inf x x + min t [,] atx α k + >,whichimpliesthath 5 holds. f t,x On the other hand, we have lim sup x + max t [,] lim sup x x + max t [,] atx α k k <, which implies that H 6 holds. Now we have the desired result. Remark 4.6 In [7, 8], some existence results for the second-order periodic problem were established by Graef, Kong, Wang and Torres, respectively. Their proofs are based on Krasnosel skii fixed point theorem in cones for completely continuous operators and the proofs are simpler and more clear than the proof presented in our paper. However, for us it seems difficult to obtain the same results in our paper using the fixed point theorem in cones. The main reason is that our condition H 5 is weaker than the classical superlinear condition near x + that is, lim inf x + f t, x/x +, yet being used in [7], Theorem. and [8], Corollary 4.. Therefore, in a sense, our results improve and generalize those in [7, 8]. Remark 4.7 The following natural question concerning the optimality of conditions H andh 5 remains open to the authors: Find an optimal constant λ such that if lim inf x + f t, x/x > λ then Theorems 4. and 4.4 remain valid. In other words, Are H andh 5 also necessary conditions in order to get at least one positive solution in Theorems 4.and4.4 respectively? Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript. Author details School of Mathematics and Physics, Changzhou University, Changzhou, 364, China. College of Science, Hohai University, Nanjing, 98, China. 3 College of Information Sciences and Technology, Hainan University, Haikou, 578, China. Acknowledgements The authors express their thanks to the referees and the editor for their valuable comments and suggestions. The research of Feng Wang is supported by the National Natural Science Foundation of China Grant No. 555 and No. 466, Natural Science Foundation of the Jiangsu Higher Education Institutions of China Grant No. 5KJB. The research of Shengjun Li is supported by the National Natural Science Foundation of China Grant No. 466 and Hainan Natural Science Foundation Grant No. 3. Received: 3 February 5 Accepted: 4 August 5 References. Mawhin, J: Topological Degree Methods in Nonlinear Boundary Value Problems. Regional Conf. in Math., vol. 4. Am. Math. Soc., Providence 979. Górniewicz, L: Topological approach to differential inclusions. In: Granas, A, Frigon, M eds. Topological Methods in Differential Equations and Inclusions. NATO ASI Series C, vol. 47, pp Kluwer Academic, Dordrecht Cid, JA, Infante, G, Tvrdy, M, Zima, M: A topological approach to periodic oscillations related to the Liebau phenomenon. J. Math. Anal. Appl. 43, Fonda, A, Toader, R: Periodic orbits of radially symmetric Keplerian-like systems: a topological degree approach. J. Differ. Equ. 44, Cui, Y: Computation of topological degree in ordered Banach spaces with lattice structure and applications. Appl. Math. 58,

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