Abstract
This paper investigates a class of resonance boundary value problems for Hadamard-type fractional differential equations on an infinite interval. Utilizing the Leggett-Williams norm-type theorem proposed by O’Regan and Zima, the existence of positive solutions is established. The main conclusions are illustrated with an example.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The study of boundary value problems (BVPs) for differential equations on infinite intervals originated from the discussion of radially symmetric solutions for nonlinear elliptic equations.
For example, when analyzing the radially symmetric solutions of the following elliptic equation:
(1) can be equivalently transformed into the following BVP for second-order differential equations defined on an infinite interval:
where \(\Im\) represents the radial coordinate [1].
BVPs for differential equations on infinite intervals have a widespread use in real world problems, such as solid phase transition problems, plasma physics research, colloid theory, nonlinear mechanics, laminar flow theory, and non-Newtonian fluid problems [1]. Due to the practical application background of BVPs for differential equations on infinite intervals, researchers have conducted in-depth studies on the existence of solutions of fractional differential equations (FDEs) on infinite intervals and obtained a series of interesting results [2,3,4,5,6,7,8]. Alongside Riemann-Liouville and Caputo derivatives in scholarly literature, another variant of fractional derivatives is the Hadamard fractional derivative, first introduced in 1892 [9]. This derivative is characterized by the inclusion of a logarithmic function with an arbitrary exponent within the kernel of the integral in its definition.
Ioakimidis [10] highlighted the potential application of singular integrals in solving crack problems in elastic materials, which has sparked interest among scholars in Hadamard fractional calculus [11,12,13]. There are at least two distinctions between Hadamard-type fractional calculus and Riemann-Liouville fractional calculus. Firstly, regarding their definitions: the kernel of the integral in the Hadamard derivative is the power of \((\ln ({t / s}))\), whereas in the Riemann-Liouville derivative, the kernel is the power of \(t-s\). Additionally, the Hadamard derivative is viewed as a generalization of the operator \({\left( {t\frac{d}{{dt}}} \right) ^n}\), while the Riemann-Liouville derivative is considered an extension of the classical operator \({\left( {\frac{d}{{dt}}} \right) ^n}\). Secondly, in terms of application: due to the logarithmic kernel of Hadamard calculus, the decay rate of solutions to Hadamard fractional differential equations is slower than that of Riemann-Liouville fractional differential equations, making them more suitable for describing ultra slow kinetics processes. For instance, Garra et al. [14] utilized Hadamard fractional calculus to study the Lomnitz logarithmic creep law of igneous rocks. The authors pointed out that although the mathematical model described by Hadamard fractional calculus appears complex, it is indeed effective in characterizing creep behavior.
Given the particularity of Hadamard fractional calculus, extensive research has been conducted in recent years on the existence of solutions to Hadamard fractional BVPs. Notably, there has been a growing interest among researchers in exploring the existence of solutions to Hadamard fractional BVPs on infinite intervals [15,16,17,18,19,20,21,22,23,24].
In [22], Wang et al employed the monotone iterative method to investigate the existence of positive solutions for the nonlinear Hadamard FDE subject to nonlocal Hadamard integral and discrete boundary conditions on an infinite interval:
where \({}^H{D^\nu }\) is Hadamard fractional derivative, \(2 < \nu \le 3\), \(1< \xi< {\eta _1}< {\eta _2}< \cdots< {\eta _{j - 2}} < + \infty .\)
In [23], Deren and Cerdik used the monotone iterative method to explore the existence of positive solutions for the nonlinear Hadamard fractional differential systems supplemented with multipoint boundary conditions on an infinite interval:
where \(\varsigma ,\zeta \in \mathbb {N},~\varsigma ,\zeta \ge 3,\) \({}^HD_{1+}^\vartheta\) are Hadamard fractional derivatives of order \(\vartheta \in \{ \kappa ,\epsilon ,{\tau _1},{\tau _2}\}\), \({\tau _1} \in [0,\kappa - 1],~{\tau _2} \in [0,\epsilon - 1],~{c_i} \ge 0\;(i = 1,2, \cdots , {k_1}),~{d_j} \ge 0\;(j = 1,2, \cdots , {k_2}),~1< {\varsigma _1}< {\varsigma _2}< \cdots< {\varsigma _{{k_1}}} < + \infty ,\) and \(1< {\zeta _1}< {\zeta _2}< \cdots< {\zeta _{{k_2}}} < + \infty .\)
Note that the existing literature on BVPs of Hadamard FDEs on infinite intervals mainly focuses on the non-resonant case [15,16,17,18,19,20,21,22,23]. In [24], Zhang and Liu proposed to study the resonant BVP of Hadamard FDEs, and used Mawhin’s continuation theorem to explore the existence of solutions for the following Hadamard FDE with integral boundary conditions at resonance on an infinite interval:
where \(2<\varpi \le 3\), \({}^HD_{1 + }^\varpi\) is Hadamard fractional derivative, \(\ell (\varrho ) \ge 0\) and \(\big (1/a(\varrho )\big ) > 0\) on \([1, + \infty )\), \(\hbar :[1, + \infty ) \times {\mathbb {R}^3} \rightarrow \mathbb {R}\) satisfies \(a-\)Carathéodory condition.
Inspired by the mentioned articles, this paper employs the Leggett-Williams norm-type theorem to discuss the existence of positive solutions for Hadamard fractional BVP at resonance on an infinite interval as follows:
where \({}^HD_{1 + }^\varpi\) is Hadamard fractional derivative, \(3< \varpi < 4,\) and the function \(\hbar :[1, + \infty ) \times \mathbb {R} \rightarrow \mathbb {R}\) satisfies the following condition:
-
(H)
The function \(\hbar :[1, + \infty ) \times \mathbb {R} \rightarrow \mathbb {R}\) is continuous, and for each constant \(\jmath > 1,\) there exists a nonnegative function \({\phi _\jmath } \in C[1, + \infty )\) satisfy the conditions \(\mathop {\sup }\limits _{\varrho \ge 1} |{\phi _\jmath }(\varrho )| < + \infty\) and \(\displaystyle \int _1^{+ \infty } {{\phi _\jmath }} (\varrho )\frac{{d\varrho }}{\varrho } < + \infty\), such that
$$\begin{aligned} |\Im | < \jmath \Rightarrow \big |\hbar \big (\varrho ,(1 + {(\ln \varrho )^{\varpi - 1}})\Im \big )\big | \le {\phi _\jmath }(\varrho ),\;\;\;\;a.e.\;\; \varrho \ge 1. \end{aligned}$$
The main challenges and innovations of this paper can be summarized as follows:
-
This paper studies the fractional BVP on an infinite interval. Owing to the non-compactness of the infinite domain, the classical Arzelá-Ascoli theorem cannot be directly applied to determine the compactness of the corresponding operator, which brings direct difficulties to the study of problem (6).
-
For the resonance BVPs of ordinary differential equations, a common research method is to use the continuation theorem. However, the integration on the infinite interval requires the convergence of the improper integral, which brings additional difficulties to the construction of the projection operator in the process of using the continuation theorem to deal with the resonance BVPs on the infinite intervals. In particular, the Hadamard fractional integral kernel function is a logarithmic function, which further increases the complexity of the problem.
-
There are few literature on the resonance BVPs of Hadamard FDEs on the infinite interval, especially on the existence of positive solutions for this problem. We have not found any related research work. In this paper, we use the Leggett-Williams norm-type theorem to give a adequate condition for the problem (6) exists a positive solutions. Therefore, our results are new and contribute significantly to the existing literature on the topic.
The structure of this paper is organized as follows. In Sect. 2, we introduce the fundamental definitions and properties of Hadamard fractional calculus, along with the Leggett-Williams norm-type theorem and criteria for operator compactness on infinite intervals. In Sect. 3, we apply the Leggett-Williams norm-type theorem to demonstrate the existence of positive solutions for problem (6). In Sect. 4, we validate the applicability of our theoretical results by presenting a specific example.
2 Preliminaries
In this part, we present a number of definitions, lemmas of Hadamard fractional calculus, the Leggett-Williams norm-type theorem and the lemma of compactness determination on infinite interval, all of which will be used in our forthcoming discussions.
Definition 2.1
[13, 25] The Hadamard fractional integral of order \(\varpi (\varpi > 0)\) of a function \(\Im :[1, + \infty ) \rightarrow \mathbb {R}\) is given by
provided the integral exists.
Definition 2.2
[13, 25] The Hadamard fractional derivative of order \(\varpi (\varpi > 0)\) of a function \(\Im :[1, + \infty ) \rightarrow \mathbb {R}\) is given by
where \(\mathfrak {n}-1<\varpi <\mathfrak {n}\), \(\mathfrak {n} = [\varpi ] + 1\), \([\varpi ]\) denotes the integer part of the real number \(\varpi\).
Lemma 2.1
[13, 25] Let \(\varpi > 0\) and \(\Im \in C[1, + \infty ) \cap {L^1}[1, + \infty )\). Then the solution of Hadamard fractional differential equation \({}^HD_{1 + }^\varpi \Im (\varrho ) =0\) is given by
and the following formula holds:
where \({c_i} \in \mathbb {R},~i = 1,2, \cdots ,\mathfrak {n}\), and \(\mathfrak {n} - 1< \varpi < \mathfrak {n}.\)
Lemma 2.2
[13, 25] Let \(\varpi> 0,~\epsilon > 0,\) then
in particular, \({}^HD_{1 + }^\varpi {(\ln \varrho )^{\varpi - i}} = 0,~i = 1,2, \cdots , [\varpi ] + 1.\)
Let \((X, {\left\| \cdot \right\| _X})\) and \((Y, {\left\| \cdot \right\| _Y})\) are two Banach spaces. Let \(N:X\rightarrow Y\) be a nonlinear operator, \(L:\text{ dom }L \subset X \rightarrow Y\) be a Fredholm operator with index zero, i.e., \({\text {Im}} L\) is closed and \(\dim {\text {Ker}}L = {\text {co}}\dim {\text {Im}} L < \infty\), which is also implies that there exist continuous projections \(P:X \rightarrow X\) and \(Q:Y \rightarrow Y\) such that
and \(L\left| {_{{\text {dom}}L \cap {\text {Ker}}P}} \right. :\text{ dom }L \rightarrow {\text {Im}} L\) is invertible. We denote by \({K_P}{\text { = }}{(L\left| {_{{\text {dom}}L \cap {\text {Ker}}P}} \right. )^{ - 1}}\). Let \(\Omega\) be an open bounded subset of X and \(\text{ dom }L \cap \bar{\Omega }\ne \emptyset\). The operator \(N:X \rightarrow Y\) is called L-compact on \(\bar{\Omega },\) if \(QN\left( {\bar{\Omega }} \right)\) is bounded and \(K_P (I - Q)N:\bar{\Omega }\rightarrow X\) is compact. On account of \(\dim {\mathop \textrm{Im}\nolimits } Q = \text {co}\dim {\mathop \textrm{Im}\nolimits } L\), there exists an isomorphism \(J:{\text {Im}} Q \rightarrow {\text {Ker}}L.\) Then the equation \(L\Im = \lambda N\Im\) is equivalent to
for all \(\lambda \in (0,1]\) ([26]).
Definition 2.3
[26] A nonempty convex closed set \(\mathcal {C}\subset X\) is named as a cone if
- \({\text {(i)} }\):
-
\(\lambda \Im \in \mathcal {C}\) for all \(\Im \in \mathcal {C}\) and \(\lambda \ge 0\);
- \({\text {(ii)} }\):
-
\(\Im ,-\Im \in \mathcal {C}\) implies \(\Im =\theta\).
Remark 2.1
[26] The cone \(\mathcal {C}\) induces a partial order in X by
Lemma 2.3
[26] Let \(\mathcal {C}\) be a cone in X. Then for every \(\zeta \in \mathcal {C}\backslash \{ \theta \}\) there exists a positive number \(\tau (\zeta )\) such that
for all \(\Im \in \mathcal {C}.\)
Let \(\gamma :X \rightarrow \mathcal {C}\) be a retraction, i.e., a continuous mapping such that \(\gamma (\Im ) = \Im\) for all \(\Im \in \mathcal {C}\). Define
and
Theorem 2.1
[26] Let \(\mathcal {C}\) be a cone in X and let \({\Omega _1}\), \({\Omega _2}\) be open bounded subsets of X with \({\bar{\Omega }_1} \subset {\Omega _2}\) and \(\mathcal {C} \cap ({\bar{\Omega }_2}\backslash {\Omega _1}) \ne \emptyset\). Suppose that
- \({{1^ \circ } }\):
-
L is a Fredholm operator of index zero;
- \({ {2^ \circ }}\):
-
\(QN:X \rightarrow Y\) is continuous and bounded and \({K_P}(I - Q)N:X \rightarrow X\) is compact on every bounded subset of X;
- \({ {3^ \circ }}\):
-
\(||N\Im || \le ||L\Im ||\) for \(\Im \in \mathcal {C} \cap \partial {\Omega _2} \cap {\text {dom}}L\) (or \(L\Im \ne \lambda N\Im ,\) for any \(\Im \in \mathcal {C} \cap \partial {\Omega _2} \cap {\text {dom}}L\) and \(\lambda \in (0,1)\));
- \({ {4^ \circ }}\):
-
\(\gamma\) maps subsets of \({\bar{\Omega }_2}\) into bounded subsets of \(\mathcal {C}\);
- \({ {5^ \circ }}\):
-
\({d_B}\big (\big [I - (P + JQN)\gamma \big ]{|_{{\text {Ker}}L}},{\text {Ker}}L \cap {\Omega _2},0\big ) \ne 0,\) where \({d_B}\) represent the Brouwer degree;
- \({ {6^ \circ }}\):
-
there exists \({\Im _0} \in \mathcal {C}\backslash \{ 0\}\) such that \(||\Im || \le \tau ({\Im _0})||\Xi \Im ||\) for \(\Im \in \mathcal {C}({\Im _0}) \cap \partial {\Omega _1}\), where \(\mathcal {C}({\Im _0}) = \{ \Im \in \mathcal {C}:\lambda {\Im _0}\underset{{\smash {\scriptscriptstyle -}}}{ \prec } \Im ~\text {for some}~ \lambda > 0\}\) and \(\tau ({\Im _0})\) is such that \(||\Im + {\Im _0}|| \ge \tau ({\Im _0})||\Im ||\) for every \(\Im \in \mathcal {C}\);
- \({ {7^ \circ }}\):
-
\((P + JQN)\gamma (\partial {\Omega _2}) \subset \mathcal {C}\);
- \({ {8^ \circ }}\):
-
\({\Xi _\gamma }({\bar{\Omega }_2}\backslash {\Omega _1}) \subset \mathcal {C}\).
Then the equation \(L\Im = N\Im\) has a solution in the set \(\mathcal {C}\cap ({\bar{\Omega }_2}\backslash {\Omega _1})\).
Lemma 2.4
[27] Let \(\aleph \subset X\) be a bounded set. Then \(\aleph\) is relatively compact in X if the following conditions are satisfied:
- \({\text {(i)} }\):
-
For any \(\Im (\varrho )\in \aleph ,\;\frac{{\Im (\varrho )}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}\) is equicontinuous on any compact interval of \([1,+\infty )\);
- \({\text {(ii)} }\):
-
For any \(\varepsilon >0,\) there exists a constant \(\mathcal {T}=\mathcal {T}(\varepsilon )>0\) such that, for any \({\varrho _1},{\varrho _2} \ge \mathcal {T}\) and \(\Im \in \aleph\), it holds
$$\begin{aligned} \left| {\frac{{\Im ({\varrho _2})}}{{1 + {{(\ln {\varrho _2})}^{\varpi - 1}}}} - \frac{{\Im ({\varrho _1})}}{{1 + {{(\ln {\varrho _1})}^{\varpi - 1}}}}} \right| <\varepsilon . \end{aligned}$$
3 Main Results
Firstly, we define two Banach spaces, and within the framework of these spaces, we use the Leggett-Williams norm-type theorem to successfully establish the results of the existence of positive solutions for problem (6).
Define two spaces
equipped with the norm
and
equipped with the norm
It is easily to verify that \((X,{\left\| \cdot \right\| _X})\) and \((Y,{\left\| \cdot \right\| _Y})\) are two Banach spaces.
Define the linear operator \(L:{\text {dom}}L \rightarrow Y\) by
where
Define the nonlinear operator \(N:X \rightarrow Y\) by
then the BVP (6) is equivalent to the operator equation
Lemma 3.1
Let \(L:{\text {dom}}L \subset X \rightarrow Y\) is defined by (7), then
Proof
By Lemma 2.1, we know that \({}^HD_{1 + }^\varpi \Im (\varrho ) = 0\) has following solution
From boundary conditions \(\Im (1) = \Im '(1) = \Im ''(1) = 0,\) we have \({c_2} = {c_3} = {c_4} = 0\), that is, \(\Im (\varrho ) = {c_1}{(\ln \varrho )^{\varpi - 1}}\). Conversely, for \(\Im \in \text {dom}L\) and \(\Im (\varrho )=c(\ln \varrho )^{\varpi -1}\), it follows from Lemma 2.2 that \({}^HD_{1 + }^\varpi \Im (\varrho ) = 0\). Hence, (8) holds. If \(\zeta \in {\text {Im}} L,\) there exists a function \(\Im \in {\text {dom}}L\) such that \(\zeta (\varrho ) = {}^HD_{1 + }^\varpi \Im (\varrho ).\) By Lemma 2.1, Lemma 2.2 and boundary condition \({}^HD_{1 + }^{\varpi - 1}\Im (1)= \mathop {\lim }\limits_{\varrho\to + \infty }{}^HD_{1 + }^{\varpi - 1}\Im (\varrho )\), we know that
On the other hand, if \(\zeta \in Y\) and satisfies (10). Let
then \(\Im (\varrho ) \in {\text {dom}}L\) and \(L\Im (\varrho ) = \zeta (\varrho )\). Therefore, (9) holds. The proof is completed. \(\square\)
Lemma 3.2
Assume that L is given by (7), then L is a Fredholm operator of index zero. Define the linear operators \(P:X \rightarrow X\) and \(Q:Y \rightarrow Y\) as follows
Proof
By the definition of the operator P, we can get \({\text {Im}} P = {\text {Ker}}L\) and \(P\Im (\varrho ) = {P^2}\Im (\varrho )\). For any \(\Im \in X\), then \(\Im = (\Im - P\Im ) + P\Im\), it follows
Furthermore, it is easily to verify that \({\text {Ker}}P \cap {\text {Ker}}L = \left\{ \theta \right\}\). Then, we obtain
Therefore, the operator \(P:X \rightarrow X\) is a projection operator. By the definition of operator Q, we have
For any \(\zeta \in Y\), then \(\zeta = (\zeta - Q\zeta ) + Q\zeta\). This implies that
It follows from \({\text {Ker}}Q = {\text {Im}} L\) and \({Q^2}\zeta = Q\zeta\) that \({\text {Im}} Q \cap {\text {Im}} L = \left\{ \theta \right\} .\) We also get
Hence, the operator \(Q:Y \rightarrow Y\) is a projection operator. Moreover,
that is, L is a Fredholm operator of index zero. This ends the proof. \(\square\)
Lemma 3.3
Define the operator \({K_P}:{\text {Im}} L \rightarrow {\text {dom}}L \cap {\text {Ker}}P\) by
Then \({K_P} = {(L{|_{{\text {dom}}L \cap {\text {Ker}}P}})^{ - 1}}\) and
Proof
Firstly, we will show that the definition of the \({K_P}\) is resonbale. For \(\zeta \in {\text {Im}} L,\) we have
and
From (11) and (12), we obtain \({K_P}\) is well defined. Next, we will prove that \({K_P} = {(L{|_{{\text {dom}}L \cap {\text {Ker}}P}})^{ - 1}}\). Actually, if \(\zeta \in {\text {Im}} L\), by Lemma 2.2, we have
On the other hand, for any \(\Im (\varrho )\in {{\text {dom}}L \cap {\text {Ker}}P}\). Since \({K_P}L\Im (\varrho ) \in {\text {Ker}}P\) and \(\Im (\varrho ) \in {\text {Ker}}P,\) it follows that
This combined with Lemma 2.1 and \(\Im (\varrho )\in {\text {dom}}L\), we deduce that
Form (13) and (14), we obtain \({K_P} = {(L{|_{{\text {dom}}L \cap {\text {Ker}}P}})^{ - 1}}\). Moreover,
The proof is complete.
Lemma 3.4
Suppose that the condition (H) holds, let \(\Omega \subset X\) be an open bounded subset and satisfy \(\text {dom}L \cap \bar{\Omega }\ne \emptyset\), then N is L-compact on \({\bar{\Omega }}\).
Proof
It follows from \(\Omega \subset X\) is an open bounded subset that there exists a constant \(\jmath > 0\), such that \(||\Im |{|_X}\le \jmath\), for any \(\Im \in \Omega\). By (H), we obtain
Then
By Lemma 3.3, we also have
Form (15) and (16), we get \(QN(\bar{\Omega })\) and \({K_P}(I - Q)N(\bar{\Omega })\) are uniformly bounded. For \(\Im (\varrho ) \in \bar{\Omega }\), let
then
We now show that for any \(\Im \in \bar{\Omega }\), \({K_P}(I - Q)N\Im\) is equicontinuous on any compact interval of \([1,+\infty )\). In fact, for any \(T>1\), \(\Im \in \bar{\Omega }\) and \(1 \le {\varrho _1}< {\varrho _2} < T.\) Since the uniform continuity of \(\frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}\) on \([{\varrho _1},{\varrho _2}]\), and \(\mu (\varrho ,\varsigma )=\frac{{{{(\ln \frac{\varrho }{\varsigma })}^{\varpi - 1}}}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}}\) is also uniform continuity on \([{\varrho _1},{\varrho _2}] \times [1,{\varrho _1}]\), then
We finally prove that \(\Im \in \bar{\Omega }\), \({K_P}(I - Q)N\Im\) is equiconvergent at infinity. Indeed, for any \(\varepsilon >0,\) there exists a positive constant \(\mathcal {T}>1\) such that
Note that
then for any \(\varepsilon >0,\) there exists a constant \(\mathcal {T}(\varepsilon )>\mathcal {T}\), such that \(\varrho >\mathcal {T}(\varepsilon )\),
Hence, for any \(\Im \in \bar{\Omega }\) and \({\varrho _2},{\varrho _1} > \mathcal {T}(\varepsilon )\), without loss of generality, we assume that \({\varrho _2} > {\varrho _1}\), it follows
Therefore, \({K_P}(I - Q)N:\bar{\Omega }\rightarrow X\) is compact. The proof is complete. \(\square\)
Define a homeomorphism operator \(J:{\text {Im}} Q \rightarrow {\text {Ker}}L\) by
Then, \(JQN + {K_P}(I - Q)N:X \rightarrow X\) can be written as
where
For any \(\varrho ,\varsigma \in (1,+\infty )\), we have
and
It follows that
Theorem 3.1
Assume that condition (H) holds and there exist non-negative functions \({\vartheta _i}(\varrho )(i = 1,2,3),~{\omega _j}(\varrho )(j = 1,2)\) and \(\eta (\varrho )\) such that
and
where \(0 \le \displaystyle \frac{\Im }{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} \le M,~M > {M_0}\), \({\vartheta _1}(\varrho )\) is bounded on \([1, + \infty )\), \({\omega _1}(\varrho ) > 0\), \(\varrho \ge 1\), \({\vartheta _2}(\varrho )\), \({\vartheta _3}(\varrho )\), \({\omega _1}(\varrho )\), \({\omega _2}(\varrho ) \in Y\),
and
Then BVP (6) has at least one positive solution.
Proof
Define
then from (23) that \({\xi _0}< 1\). Consider the cone
Let
where \({M_2} \in ({M_0},M),~{M_1} \in (0,{M_2}),~{\xi _1} \in ({\xi _0},1),\) and
Obviously, \({\Omega _1}\) and \({\Omega _2}\) are open bounded set of X.
Step 1. By Lemma 3.2, we obtain L is a Fredholm operator of index zero, then the proof of \({1^ \circ }\) in Theorem 2.1 is complete.
Step 2. In view of Lemma 3.4, we have \(QN:X \rightarrow Y\) is continuous and bounded and \({K_P}(I - Q)N:X \rightarrow X\) is compact on each bounded subset of X, that is, the condition \({2^ \circ }\) of Theorem 2.1 holds.
Step 3. We show that \({3^ \circ }\) in Theorem 2.1 holds. Using proof by contradiction, suppose that there exists \({\Im ^ * } \in \mathcal {C} \cap \partial {\Omega _2} \cap {\text {dom}}L\) and \({\lambda ^ * } \in (0,1)\) such that \(L{\Im ^ * } = {\lambda ^ * }N{\Im ^ * }.\) Note that
then
It follows from (19) and (20) that
and
In view of the fact that
which combined with (26) and (28), it follows
that is,
On the other hand, from (27) and (28), we also have
that is,
This combined with (19), (22), (24) and (25), we obtain
which is contradict to \({\Im ^ * } \in \mathcal {C} \cap \partial {\Omega _2} \cap {\text {dom}}L\), the proof of \({3^ \circ }\) in Theorem 2.1 is complete.
Step 4. Let \((\gamma \Im )(\varrho ) = |\Im (\varrho )|\), it is easy to demonstrate that \(\gamma :X \rightarrow \mathcal {C}\) is a retraction. The proof of \({4^ \circ }\) in Theorem 2.1 is now complete.
Step 5. In order to prove \({5^ \circ }\) in Theorem 2.1, let \(\Im \in {\text {Ker}}L \cap {\bar{\Omega }_2},\) then \(\Im (\varrho ) = c{(\ln \varrho )^{\varpi - 1}},~\varrho \ge 1,~c \in \mathbb {R}\). Define
where \(c \in [ - {M_2},{M_2}],~\upsilon \in [0,1].\) Define a homeomorphism operator \(\mathcal {J}:{\text {Ker}}L \cap {\bar{\Omega }_2} \rightarrow \mathbb {R}\) by
then
Using (20) and (23), we can show that \(\mathcal {J}H(\mathcal {J}^{ - 1}c,\upsilon ) = 0\) implies \(c \ge 0.\) Let \({a_0} \in \mathcal {J}({\text {Ker}}L \cap {\partial \Omega _2}),\) then \(|{a_0}| = {M_2}.\) Assume that \(\mathcal {J}H(\mathcal {J}^{ - 1}{a_0},\upsilon ) = 0,~\upsilon \in (0,1],\) then \({a_0} = {M_2}.\) From (20) and (22), we have
This leads to a contradiction. Moreover, if \(\upsilon = 0\), then \({M_2} = 0\), which is also a contradiction. Therefore, \(\mathcal {J}H(\mathcal {J}^{ - 1}c,\upsilon ) \ne 0\), for \(c \in \mathcal {J}({\text {Ker}}L \cap {\partial \Omega _2})\), \(\upsilon \in [0,1].\) Consequently,
The demonstration of \({5^ \circ }\) in Theorem 2.1 has been concluded.
Step 6. To prove \({6^ \circ }\) in Theorem 2.1, take \({\Im _0} = 1 + {(\ln \varrho )^{\varpi - 1}} \in \mathcal {C}\backslash \{ 0\}\) and \(\tau ({\Im _0}) = 1\), then
Since \(\mathop {\lim }\limits _{\varrho \rightarrow \infty } \frac{{{{(\ln \varrho )}^{\varpi - 1}}}}{{1 + {{(\ln \varrho )}^{\varpi - 1}}}} = 1,\) there exists a \({\varrho _0} > 0,\) such that
For \(\Im \in \mathcal {C}({\Im _0}) \cap \partial {\Omega _1}\), we have
Then, by (18), (20) and (23), we get
that is, \(||\Im |{|_X} \le \tau ({\Im _0})||\Xi \Im |{|_X},~\forall \Im \in \mathcal {C}({\Im _0}) \cap \partial {\Omega _1}\). Hence, the condition \({6^ \circ }\) of Theorem 2.1 is satisfied.
Step 7. To prove \({7^ \circ }\) in Theorem 2.1. For \(\Im \in \partial {\Omega _2}\), it follows from (20) and (23) that
So, \((P + JQN)\gamma (\partial {\Omega _2}) \subset \mathcal {C}.\) Subsequently, the condition \({7^ \circ }\) of Theorem 2.1 is met.
Step 8. To prove \({8^ \circ }\) in Theorem 2.1. For \(\Im \in {\bar{\Omega }_2}\backslash {\Omega _1},\) from (18), (20) and (23), we obtain
this implies, \({\Xi _\gamma }({\bar{\Omega }_2}\backslash {\Omega _1}) \subset \mathcal {C}.\) Hence, the condition \({8^ \circ }\) of Theorem 2.1 is fulfilled. Consequently, by Theorem 2.1, we deduce that BVP (6) has at least one positive solution in \(\mathcal {C} \cap ({\bar{\Omega }_2}\backslash {\Omega _1})\). This completes the proof. \(\square\)
4 Examples
Example 4.1
Consider the following BVP
where
Let \({\phi _\jmath }(\varrho ) = {\omega _1}(\varrho )\jmath + {\omega _2}(\varrho ),\) it is easy to verify that \(\hbar\) satisfies the condition (H). Take
then
and
Therefore, \(\hbar (\varrho ,\Im )\) satisfies (19) and (20). By calculation,
Hence, (21)–(23) hold. According to Theorem 3.1 that BVP (29) has at least one positive solution.
Data Availability
No datasets were generated or analysed during the current study.
References
Agarwal, R.P., O’Regan, D.: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht (2001)
Chen, Y., Tang, X.: Positive solutions of fractional differential equations at resonance on the half-line. Bound. Value Probl. 2012, 64 (2012)
Hao, X., Sun, H., Liu, L., Wang, D.B.: Positive solutions for semipositone fractional integral boundary value problem on the half-line. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(4), 3055–3067 (2019)
Liu, X., Jia, M.: A class of iterative functional fractional differential equation on infinite interval. Appl. Math. Lett. 136, 108473 (2023)
Zhai, C., Wang, W.: Properties of positive solutions for \(m\)-point fractional differential equations on an infinite interval. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(2), 1289–1298 (2019)
Wang, W., Liu, X.: Properties and unique positive solution for fractional boundary value problem with two parameters on the half-line. J. Appl. Anal. Comput. 11(5), 2491–2507 (2021)
Wang, F., Cui, Y.: Unbounded solutions to abstract boundary value problems of fractional differential equations on a half line. Math. Methods Appl. Sci. 44(10), 8166–8176 (2021)
Bouteraa, N., Inc, M., Hashemi, M.S., Benaicha, S.: Study on the existence and nonexistence of solutions for a class of nonlinear Erdélyi-Kober type fractional differential equation on unbounded domain. J. Geom. Phys. 178, 104546 (2022)
Hadamard, J.: Essai sur l’étude des fonctions données par leur développement de Taylor. J. Mat. Pure Appl. Ser. 8, 101–186 (1892)
Ioakimidis, N.I.: Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity. Acta Mech. 45(1–2), 31–47 (1982)
Ma, L., Li, C.: On Hadamard fractional calculus. Fractals 25(3), 1750033 (2017)
Kilbas, A.A.: Hadamard-type fractional calculus. J. Korean Math. Soc. 38(6), 1191–1204 (2001)
Ahmad, B., Alsaedi, A., Ntouyas, S.K., Tariboon, J.: Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities. Springer, Cham (2017)
Garra, R., Mainardi, F., Spada, G.: A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus. Chaos Solitons Fractals 102, 333–338 (2017)
Pei, K., Wang, G., Sun, Y.: Successive iterations and positive extremal solutions for a Hadamard type fractional integro-differential equations on infinite domain. Appl. Math. Comput. 312, 158–168 (2017)
Zhang, W., Ni, J.: New multiple positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite interval. Appl. Math. Lett. 118, 107165 (2021)
Cerdik, T.S., Deren, F.Y.: New results for higher-order Hadamard-type fractional differential equations on the half-line. Math. Methods Appl. Sci. 45(4), 2315–2330 (2022)
Li, Y., Bai, S., O’Regan, D.: Monotone iterative positive solutions for a fractional differential system with coupled Hadamard type fractional integral conditions. J. Appl. Anal. Comput. 13(3), 1556–1580 (2023)
Luca, R., Tudorache, A.: On a system of Hadamard fractional differential equations with nonlocal boundary conditions on an infinite interval. Fractal Fract. 7(6), 458 (2023)
Zhai, C., Liu, R.: Positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite interval. Nonlinear Anal. Model. Control 29(2), 224–243 (2024)
Nyamoradi, N., Ahmad, B.: Hadamard fractional differential equations on an unbounded domain with integro-initial conditions. Qual. Theory Dyn. Syst. 23(4), 183 (2024)
Wang, G., Pei, K., Agarwal, R.P., Zhang, L., Ahmad, B.: Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line. J. Comput. Appl. Math. 343, 230–239 (2018)
Deren, F.Y., Cerdik, T.S.: Extremal positive solutions for Hadamard fractional differential systems on an infinite interval. Mediterr. J. Math. 20(3), 158 (2023)
Zhang, W., Liu, W.: Existence of solutions for several higher-order Hadamard-type fractional differential equations with integral boundary conditions on infinite interval. Bound. Value Probl. 2018, 134 (2018)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Math. Stud., vol. 204. Elsevier Science B.V., Amsterdam (2006)
O’Regan, D., Zima, M.: Leggett-Williams norm-type theorems for coincidences. Arch. Math. 87(3), 233–244 (2006)
Thiramanus, P., Ntouyas, S.K., Tariboon, J.: Positive solutions for Hadamard fractional differential equations on infinite domain. Adv. Differ. Equ. 2016, 83 (2016)
Funding
This research is supported by the Anhui Provincial Natural Science Foundation (2208085QA05, 2208085MA04).
Author information
Authors and Affiliations
Contributions
WZ was a major contributor to writing the manuscript and funding acquisition. XF and JN made the formal analysis, writing-review, and editing. All authors have read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
About this article
Cite this article
Zhang, W., Fu, X. & Ni, J. Existence of Positive Solutions for Hadamard-Type Fractional Boundary Value Problems at Resonance on an Infinite Interval. J Nonlinear Math Phys 31, 60 (2024). https://doi.org/10.1007/s44198-024-00230-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s44198-024-00230-z
Keywords
- Hadamard fractional derivative
- Fractional differential equation
- Infinite interval
- Resonance
- Cone
- Positive solution